0 where there is only one stable equilibrium position but will keep the formulation general so that it also applies to negative detuning.\nA fully equivalent way of finding the semiclassical eigenstates and eigenenergies near the equilibrium position is to start from the Hamiltonian $\\tilde H\\_c$ (at Ω2 = 0) in terms of *c*, *c*†; see Eq. ([eq:rotatedHc]). One first makes a displacement transformation to eliminate the linear in *c* term:\n \n$$\\begin{aligned}\n\\label{eq:displacement\\_semiclassical}\nc \\rightarrow c + Q\\_0/\\sqrt{2\\lambda}.\\end{aligned}$$ \n*Q*0 is the value of *Q* at the equilibrium position of the oscillator and is given by one of the real solutions to the cubic equation:\n \n$$\\begin{aligned}\n\\label{eq:equation\\_for\\_Q0}\nQ\\_0(Q\\_0^2+\\sgn(\\delta\\_1)) = \\overline \\Omega\\_1. \\end{aligned}$$ \nNote that the displacement $Q\\_0/\\sqrt{2\\lambda}$ is denoted as $\\overline \\xi\\_1$ in the main text (neglecting the phase in the drive amplitude Ω1). In terms of the scaled drive strength *ξ*1 used in the main text, the above equation becomes $\\overline \\xi\\_1 (\\alpha|\\overline \\xi\\_1|^2/\\delta\\_1 + 1) = \\xi\\_1$. The dependence of *Q*02 on the scaled drive power $\\overline \\Omega\\_1^2$ is shown in Fig. [fig:squeezing] for *δ*1 > 0.\nFor negative detuning, there can be three real roots where two of them correspond to stable equilibrium positions. For the considered case of positive detuning, there is only one real root. The value of P at the equilibrium position is *P*0 = 0 since the Hamiltonian is even in *P*. Note that such a displacement differs from the displacement discussed in Sec. [sec:weakanharmonicity] as it takes into account the nonlinearity of the oscillator.\nAfter the displacement transformation, the Hamiltonian $\\tilde H\\_c$ becomes,\n \n$$\\begin{aligned}\n\\label{eq:displaced\\_Hamiltonian}\n\\tilde H\\_c/\\hbar = -\\delta\\_1c^\\dagger c -2\\delta\\_1 Q\\_0^2 c^\\dagger c - \\delta\\_1 Q\\_0^2 (c^{\\dagger2}+c^2)/2 +...\\end{aligned}$$ \nwhere... represent terms that are non-quadratic in *c* and *c*†. We note that at the quadratic level, the drive induces frequency shift and squeezing of the ancilla mode. This Hamiltonian has a similar form as the Hamiltonian in Eq. ([eq:HD]) of the main text but does not have linear in *c* or *c*† terms and has *ξ* replaced by $Q\\_0/\\sqrt{2\\lambda}$.\nNext, we diagonalize the quadratic part of the above Hamiltonian via a squeezing transformation\n \n$$\\begin{aligned}\n\\label{eq:squeezing}\nc = c\\_{\\rm aux} \\cosh\\phi - c\\_{\\rm aux}^+\\sinh\\phi,\\end{aligned}$$ \nwhere $c\\_{\\rm aux},c^\\dagger \\_{\\rm aux}$ can be thought of as the annihilation and creation operators of an auxiliary mode that corresponds to the small vibrations about the equilibrium position of the oscillator. The squeezing angle *ϕ* in Eq. ([eq:squeezing]) expressed in terms of the frequency $\\omega\\_{\\rm aux}$ (see below) of the auxiliary mode obeys\n \n$$\\begin{aligned}\n\\sinh \\phi = \\sgn(Q\\_0) \\left [ \\frac{(2Q\\_0^2+\\sgn(\\delta\\_1))\\sgn(Q\\_0)-\\omega\\_{\\rm aux}/|\\delta\\_1|}{2\\omega\\_{\\rm aux}/|\\delta\\_1|} \\right]^{1/2}\\end{aligned}$$ \nThe squeezing parameter sinh*ϕ* is only a function of *Q*0 which is controlled by the dimensionless drive amplitude $\\overline \\Omega\\_1$ through Eq. ([eq:equationforQ0]).\nThe dependence of the squeezing parameter sinh*ϕ* on the scaled drive power for positive detuning *δ*1 > 0 is shown in Fig. [fig:squeezing]. For small $\\overline \\Omega\\_1$, it is linear in $\\overline \\Omega\\_1^2$; for large $\\overline \\Omega\\_1$, it saturates to a value equal to $[(2-\\sqrt{3})/2\\sqrt{3}]^{1/2}$. This saturation can be understood as a result of the competition between the drive-induced frequency shift [the second term in Eq. ([eq:displacedHamiltonian])] and squeezing [the third term in Eq. ([eq:displacedHamiltonian])]. For positive detuning (*δ*1 > 0), the drive-induced frequency shift pushes the effective ancilla frequency further away from the drive frequency which in turn limits the amount of squeezing. Since both the frequency shift term and squeezing term in Eq. ([eq:displacedHamiltonian]) grow linearly in *Q*02, the amount of squeezing one can achieve saturates at large drive power.\n [fig:squeezing]\n \n$$\\begin{aligned}\n\\tanh \\phi = \\frac{Q\\_0^2}{2Q\\_0^2 + \\sgn(\\delta\\_1) + \\sgn(Q\\_0)\\omega\\_{\\rm aux}/|\\delta\\_1|}.\\end{aligned}$$ \nFor a large detuning over anharmonicity (∣*δ*1∣ ≫ *α*) or a relatively strong drive, one can neglect the nonlinear in $c\\_{\\rm aux},c\\_{\\rm aux}^+$ terms in $\\tilde H\\_c$. The resulting Hamiltonian in terms of $c\\_{\\rm aux}$ and $c^\\dagger \\_{\\rm aux}$ reads,\n \n$$\\begin{aligned}\n\\label{eq:Hamiltonian\\_auxiliary}\n\\tilde H\\_c/\\hbar \\approx -\\sgn(Q\\_0)\\omega\\_{\\rm aux} c\\_{\\rm aux}^+c\\_{\\rm aux},\\end{aligned}$$ \nwhere the frequency of the auxiliary mode is\n \n$$\\begin{aligned}\n\\label{eq:omega\\_aux}\n\\omega\\_{\\rm aux} &= |\\delta\\_1|\\sqrt{( Q\\_0^2 + \\sgn(\\delta\\_1))(3Q\\_0^2+\\sgn(\\delta\\_1))}.\\end{aligned}$$ \nWe note that eigenenergies of $\\tilde H\\_c$ can be negative, and excited states can have lower eigenenergies than the ground state, a consequence of being in the rotating frame of the drive. The frequency $\\omega\\_{\\rm aux}$ depends on the drive amplitude through the drive dependence in *Q*0. For the considered case of positive detuning, *Q*0 > 0; $\\omega\\_{\\rm aux}$ monotonically increases as the drive amplitude increases.\nEquation ([eq:omegaaux]) allows us to calculate the approximate AC Stark shift of the transition frequency *E*10 of the ancilla in the regime where the semiclassical analysis applies. $\\omega\\_{\\rm aux}$ relates to the AC Stark shift *δ**E*10 througth the relation\n \n$$\\begin{aligned}\n\\delta E\\_{10}/\\hbar \\approx \\delta\\_1 - \\omega\\_{\\rm aux}. \\end{aligned}$$ \nWe note that *δ**E*10 starts off being linear in the drive power [see also Eq. ([eq:Starkshiftweakdrive])], then becomes sublinear, and for a relatively strong drive, it becomes proportional to Ω12/3.\nIn the presence of two drives, in principle, one can follow the same procedure to find semiclassical quasienergy states and quasienergies of Hamiltonian $\\tilde H\\_c(t)$ or *g*(*Q*, *P*, *t*) in Eq. ([eq:g]) by expanding the Hamiltonian about the ``equilibrium position\" (now a periodic orbit in the phase space, or a point in the Poincaré section; see Fig. [fig:Poincaresection]). The resulting Hamiltonian does not have terms linear in *c*, *c*† but is still periodic in time with periodicity *τ*. Similar to the case of one drive, one can find the approximate quasienergies by diagonalizing the quadratic part of the resulting Hamiltonian. One can take the point of view that the quasiperiodic orbits surrounding such a equilibrium position now become quantized quasienergy levels just like the case of static Hamiltonian where periodic orbits surrounding a stable equilibrium position form quantized energy levels. Going away from the equilibrium position, there also exists a general semiclassical quantization scheme for Floquet systems ; such an analysis is beyond the scope of this paper.\n### Dissipation- and dephasing-induced heating in the semiclassical regime\nBecause of the squeezing transformation from *c* to $c\\_{\\rm aux}$, transition down in the ladder of Fock states of operator *c* (due to dissipation) can correspond to both transition up and down in the ladder of Fock states of operator $c\\_{\\rm aux}$. This leads to an effective “heating” of the auxiliary mode as discussed in Sec. [sec:dissipation] even when the bath that leads to the damping of the ancilla is at zero temperature. Importantly, the squeezing vanishes if *α* = 0 where the system is linear.\nLikewise, due to the displacement transformation in Eq. ([eq:displacementsemiclassical]), dephasing noise leads to transitions between different eigenstates of the auxiliary modes. To see the effects of ancilla dissipation and dephasing on the auxiliary mode, we substitute *c* in Eq. ([eq:masterequationMarkovian]) with $c\\_{\\rm aux} \\cosh\\phi - c\\_{\\rm aux}^+\\sinh\\phi+Q\\_0/\\sqrt{2\\lambda}$ and $\\tilde H\\_c$ with Eq. ([eq:Hamiltonianauxiliary]). Only keeping the terms in the Lindbladian that contain equal number of $c\\_{\\rm aux}$ and $c^\\dagger \\_{\\rm aux}$, we obtain the quantum master equation for the auxiliary mode to be ,\n \n$$\\begin{aligned}\n\\label{eq:master\\_equation\\_auxiliary}\n&\\dot \\rho = -i[\\tilde H\\_c,\\rho]/\\hbar + \\bigg\\{\\mathcal D[\\sqrt{\\gamma^\\downarrow\\_{\\rm aux}} c\\_{\\rm aux}] + \\mathcal D[\\sqrt{\\gamma^\\uparrow\\_{\\rm aux}}c^\\dagger \\_{\\rm aux}] \\nonumber \\\\\n&+\\mathcal D[\\sqrt{\\gamma\\_2}c^2\\_{\\rm aux}] + \\mathcal D[\\sqrt{\\gamma\\_2}c^{\\dagger 2}\\_{\\rm aux}]+\\mathcal D[\\sqrt{2\\tilde \\gamma\\_{\\rm ph}}c^\\dagger \\_{\\rm aux}c\\_{\\rm aux}]\\bigg\\}\\rho, \\nonumber \\\\\\end{aligned}$$ \nThe transition down and up rates between neighboring levels of the auxiliary mode read,\n \n$$\\begin{aligned}\n\\label{eq:semiclassical\\_gamma\\_up}\n\\gamma^{\\downarrow}\\_{\\rm aux} = (n\\_{\\rm th}+1)\\gamma \\cosh^2\\phi + n\\_{\\rm th}\\gamma \\sinh^2\\phi + \\gamma\\_{\\rm ph}^{\\rm (hf)} Q\\_0^2/\\lambda, \\nonumber \\\\\n\\gamma^{\\uparrow}\\_{\\rm aux} = (n\\_{\\rm th}+1)\\gamma \\sinh^2\\phi + n\\_{\\rm th}\\gamma \\cosh^2\\phi + \\gamma\\_{\\rm ph}^{\\rm (hf)} Q\\_0^2/\\lambda\\end{aligned}$$ \nWe emphasize that even at $n\\_{\\rm th}=0$, there is a finite transition up rate $\\gamma^\\uparrow\\_{\\rm aux}$. The transition up and down rates induced by the dephasing noise are the same as a result of the assumed symmetric noise spectrum in Eq. ([eq:masterequationMarkovian]). Note that we have replaced in Eq. ([eq:semiclassicalgammaup]) $\\gamma\\_{\\rm ph}$ with $\\gamma\\_{\\rm ph}^{\\rm (hf)}$ as introduced in Sec. [sec:heatingexp] of the main text to emphasize that the rates of transitions caused by dephasing is determined by the dephasing noise at the drive detuning frequency which as we found is much smaller than the dephasing rate obtained from the Ramsey or spin echo measurement.\nIn addition, the dephasing noise also induces two-photon transitions with a rate $\\gamma\\_2 = 2\\gamma\\_{\\rm ph}^{\\rm (hf)}\\sinh^2\\phi \\cosh^2\\phi$. However, this rate is small compared to the transition rates between neighboring levels in the considered semiclassical limit of *λ* ≪ 1. The dephasing rate of the auxiliary mode $\\tilde \\gamma\\_{\\rm ph}$ is proportional to the original ancilla dephasing rate $\\gamma\\_{\\rm ph}$ and becomes equal to $\\gamma\\_{\\rm ph}$ in the limit of weak drive.\nNeglecting the two-photon transitions, the steady-state distribution among the eigenstates of the auxiliary mode is of the Boltzmann form and can be written as follows\n \n$$\\begin{aligned}\nP\\_n^{\\rm st} = \\left(\\frac{\\tilde n\\_{\\rm th}}{\\tilde n\\_{\\rm th}+1}\\right)^n/(\\tilde n\\_{\\rm th}+1)\\end{aligned}$$ \nwhere the effective thermal population is given by\n \n$$\\begin{aligned}\n\\label{eq:tilde\\_n\\_th}\n\\tilde n\\_{\\rm th} = n\\_{\\rm th} + (2n\\_{\\rm th}+1)\\sinh^2\\phi + (Q\\_0^2/\\lambda)\\gamma\\_{\\rm ph}^{\\rm (hf)}/\\gamma\\end{aligned}$$ \nIn the absence of ancilla dephasing, the above equation reduces to that found in Ref. .\nThe total population in the excited eigenstates of the auxiliary mode $1-P^{\\rm st}\\_0 = \\tilde n\\_{\\rm th}/(\\tilde n\\_{\\rm th}+1)$ as a function of the scaled drive power is shown in Fig. [fig:heating] of the main text. For the case $\\gamma\\_{\\rm ph}^{\\rm (hf)}/\\gamma = 0$, the effective thermal population $\\tilde n\\_{\\rm th}$ saturates at strong drive due to the saturation of sinh2*ϕ* in Eq. ([eq:tildenth]) (see also Fig. [fig:squeezing]). For a non-zero $\\gamma\\_{\\rm ph}^{\\rm (hf)}/\\gamma$, the effective temperature $\\tilde n\\_{\\rm th}$ would rise indefinitely according to Eq. ([eq:tildenth]) but eventually will be constrained by the finite bandwidth of the noise that leads to ancilla dephasing. We note that for not very small *λ*, the semiclassical analysis already captures qualitatively the behavior of the results from the full Floquet analysis.\n### Decoherence rates of the driven ancilla\nWe discuss in this section the drive-dependence of the decoherence rates *V**m**n* of superpositions of Floquet states in Eq. ([eq:decoherencerate]) in the semiclassical approximation.\nWe approximate the Floquet states as the eigenstates of the auxiliary mode described by the Hamiltonian in Eq. ([eq:Hamiltonianauxiliary]). Using Eq. ([eq:Vgammasimplified]) and the squeezing transformation in Eq. ([eq:squeezing]), we find the dissipation-induced decoherence rate to be\n \n$$\\begin{aligned}\nV^{\\gamma}\\_{mn} \\approx \\gamma [ (\\sinh^2 \\phi +1/2)(m+n)+\\sinh^2 \\phi].\\end{aligned}$$ \nThe result above is shown in Fig. [fig:linewidth](a) and qualitatively matches the full Floquet results.\nSimilarly, using Eq. ([eq:Vgammaphsimplified]), we find that the dephasing-induced decoherence rate to be\n \n$$\\begin{aligned}\n\\label{eq:semiclassics\\_V\\_gamma\\_ph}\nV^{\\gamma\\_{\\rm ph}}\\_{mn} &\\approx \\gamma\\_{\\rm ph}^{\\rm (hf)}\\sinh^2\\phi\\cosh^2\\phi(m^2+n^2+m+n+2) \\nonumber \\\\\n&+\\gamma\\_{\\rm ph}(0)(2\\sinh^2 \\phi + 1)^2(m-n)^2.\\end{aligned}$$ \nIn the case $\\gamma\\_{\\rm ph}^{\\rm (hf)}\\ll \\gamma\\_{\\rm ph}(0)$, the above result shows that $V\\_{mn}^{\\gamma\\_{\\rm ph}}$ increases as a function of drive power and saturates to a value slightly larger than its value in the absence of the drive. However, it does not capture the sharp decrease of $V^{\\gamma\\_{\\rm ph}}\\_{mn}$ as the drive power increases found using the full Floquet analysis shown in Fig. [fig:linewidth](b). This indicates the significance of the quantum correction from the non-quadratic terms in Eq. ([eq:displacedHamiltonian]) which are neglected in obtaining Eq. ([eq:Hamiltonianauxiliary]).\n### Nonlinearities of the auxiliary mode\nWe discuss now the effects of the terms nonlinear in $c\\_{\\rm aux},c^\\dagger \\_{\\rm aux}$ that we have neglected in arriving at Eq. ([eq:Hamiltonianauxiliary]). Those nonlinear terms come from the original ancilla nonlinearity − *α**c*† 2*c*2/2. For clarification, we list those terms below:\n \n$$\\begin{aligned}\n\\label{eq:nonlinear\\_c\\_aux}\n-\\frac{\\alpha}{2}c^{\\dagger 2}c^2 \\rightarrow -\\frac{\\alpha}{2}(c^\\dagger \\_{\\rm aux}\\cosh\\phi - c\\_{\\rm aux}\\sinh \\phi + Q\\_0/\\sqrt{2\\lambda})^2 \\nonumber \\\\\n\\times(c\\_{\\rm aux}\\cosh\\phi - c^\\dagger \\_{\\rm aux}\\sinh \\phi + Q\\_0/\\sqrt{2\\lambda})^2\\end{aligned}$$ \nOn the one hand, nonlinear terms in $c\\_{\\rm aux},c^\\dagger \\_{\\rm aux}$ make the levels of the auxiliary mode slightly non-equidistant. Collecting all the resonant (i.e. non-rotating) nonlinear terms, we find that the effective self-Kerr of the auxiliary mode (corresponding to a term $-\\hbar\\alpha\\_{\\rm aux}c^{\\dagger 2}\\_{\\rm aux}c\\_{\\rm aux}^2/2$ in the Hamiltonian) has the following form\n \n$$\\begin{aligned}\n\\label{eq:alpha\\_aux\\_general}\n\\frac{\\alpha\\_{\\rm aux}}{\\alpha} &= C\\_4(\\phi) - C\\_3(\\phi)\\frac{Q\\_0^2}{(\\omega\\_{\\rm aux}/|\\delta\\_1|)}\\sgn(Q\\_0)\\end{aligned}$$ \nwhere *C*4, *C*3 are positive functions of the squeezing angle *ϕ*. The term proportional to *C*4 comes from the quartic terms in $c\\_{\\rm aux},c^\\dagger \\_{\\rm aux}$ in Eq. ([eq:nonlinearcaux]) whereas the term proportional to *C*3 comes from the cubic terms in $c\\_{\\rm aux},c^\\dagger \\_{\\rm aux}$ taken to second order. In general, both terms are of the same order of magnitude and can have different signs depending on the sign of *Q*0.\nFor the case *Q*0 > 0, we find after some algebra that\n \n$$\\begin{aligned}\n\\label{eq:alpha\\_aux}\n\\frac{\\alpha\\_{\\rm aux}}{\\alpha} &= \\frac{-3Q\\_0^4 + 2}{2[3Q\\_0^2+\\sgn(\\delta\\_1)]^2}.\\end{aligned}$$ \nThe above result applies to both positive detuning (*δ*1 > 0) and negative detuning (*δ*1 < 0) as long as *Q*0 > 0. Interestingly, for a strong drive-1 where $\\overline \\Omega\\_1 \\gg 1$ such that *Q*0 ≫ 1, the effective Kerr of the auxiliary mode changes from being positive to negative (with respect to the sign of *α*) and approaches − *α*/6 in the limit $\\overline\\Omega\\_1 \\rightarrow \\infty.$ This sign change in $\\alpha\\_{\\rm aux}$ is a consequence of the *C*3 term dominating over the *C*4 term in Eq. ([eq:alphaauxgeneral]).\nThe change in the sign of the effective anharmonicity $\\alpha\\_{\\rm aux}$ has an interesting consequence. If one now turns on the second drive on the ancilla, the AC Stark shift to the transition frequencies of the auxiliary mode due to this drive can also change sign depending on the sign of $\\alpha\\_{\\rm aux}$. Indeed, one finds that, to leading order in the drive amplitude Ω2, the AC Stark shift to the frequency *ε*10/ℏ of transition between the ground and excited state of the auxiliary mode reads\n \n$$\\begin{aligned}\n&\\delta\\epsilon\\_{10}/\\hbar \\nonumber \\\\\n&= -2\\alpha\\_{\\rm aux} |\\Omega\\_2|^2 \\frac{2\\sinh^2\\phi (\\omega\\_{21}^2+\\omega\\_{\\rm aux}^2)+(\\omega\\_{21}-\\omega\\_{\\rm aux})^2}{(\\omega\\_{21}^2-\\omega\\_{\\rm aux}^2)^2}.\\end{aligned}$$ \nClearly, *ε*10/ℏ changes sign when $\\alpha\\_{\\rm aux}$ changes sign. This sign change in the AC Stark shift due to drive-2 was also observed in the experiment where the strength of drive-1 has not yet reached the semiclassical regime yet; see Sec. [sec:Starkshift]. Quite interestingly, for stronger drive-2, the AC Stark shift becomes non-monotonic in its amplitude as shown in the main text.\nWe point out that the expression for the effective anharmonicity $\\alpha\\_{\\rm aux}$ in Eq. ([eq:alphaaux]) can also be found using the semiclassical quantization rule: *g**n* − *g**n* − 1 = − *λ**ν*(*g**n* − 1) where *g**n* is the eigenenergy of the *n*-th excited state of the Hamiltonian *g* about the equilibrium position and *ν*(*g**n* − 1) is the frequency of the orbit for the classical Hamiltonian at energy *g**n* − 1. Expanding *ν*(*g**n*) about the ground state energy *g*0 and keeping to leading order term in *λ*, one finds that \n$$\\frac{\\alpha\\_{\\rm aux}}{\\alpha} = -\\frac{1}{2}\\frac{d\\nu}{dg}\\bigg|\\_{g=g\\_0}\\nu(g\\_0).$$ \n The expression above is invariant with respect to the choice of coordinate system.\nIn addition to renormalizing the effective Kerr $\\alpha\\_{\\rm aux}$, the cubic terms in $c\\_{\\rm aux}, c^\\dagger \\_{\\rm aux}$ in Eq. ([eq:nonlinearcaux]) also provide a capability of three-wave mixing. If one now turns on the second drive on the ancilla and couples the ancilla to two off-resonant cavity modes, the three-wave mixing among the two cavities and drive-2 can lead to beam-splitter or two-mode squeezing interaction between the two cavities. We note that the strength of such three-wave mixing is proportional to *Q*0 which follows Eq. ([eq:equationforQ0]). When $\\overline \\Omega\\_1$ becomes of order one, *Q*0 becomes sublinear in $\\overline \\Omega\\_1$ as can be seen in Fig. [fig:beam-splitter] of the main text.\nQuasienergy level anti-crossing and experimental evidence\n=========================================================\nIn this section, we present experimental evidence of quasienergy level anti-crossing due to multi-photon resonance and show that the Floquet theory accurately predicts the locations of the level anti-crossings.\nAs we discussed in Sec. [sec:multi-photonresonance], if the ancilla comes close to, or passes through, a quasienergy level anti-crossing near the peak of the drive envelopes where the drive amplitudes change slowly, the probability of diabatic vs. adiabatic transition between the two levels can become comparable. In an AC Stark shift measurement of the ancilla transition frequency *E*10/ℏ as described in Sec. [sec:Starkshiftexp], if the above situation occurs, we would observe that the ancilla has a finite probablity of not remaining in the ground state regardless of the frequency of the spectrosopy tone.\nWe show in Fig. [fig:levelanti-crossingexp] an example of this situation where we vary the strength of drive-2 and keep the strength of drive-1 fixed. At a particular strength of drive-2, there is a sharp vertical line in the spectrum indicating a high probability of the ancilla not being in the ground state regardless of the frequency of the spectroscopy tone. A comparison with the theoretical quasienergy spectrum shows that there is an anti-crossing between quasienergy levels *ε*0 and *ε*2 at that particular drive strength; as a result, the ancilla has a significant probablity of transitioning from the state Ψ0 to Ψ2 during the ramping up and down of the drives. Near this resonance, we observed weak oscillations in the probability of the ancilla not being in the ground state, a typical situation in a Landau-Zener transition. Far away from the resonance, these oscillations damp out indicating a (mostly) diabatic transition when the ancilla passes through the resonance.\nLastly, we emphasize that the process of multi-photon resonance is strongly suppressed when the frequency difference of the two drive tones is much larger than their drive amplitudes. In particular, this is the case for the beam-splitter experiment presented in Fig. [fig:beam-splitter], where the frequency difference of the two drives *ω*21/*α* ≈ 15, was much larger than what we used here.\n![Quasienergy level anti-crossing due to multi-photon resonance. Top: Spectroscopy of the two-tone driven ancilla as a function of the scaled drive-2 power. The power of drive-1 is fixed at |\\xi_1|^2\\delta_1/(\\delta_1+\\alpha) = 0.22. The scaled drive detunings \\delta_1/\\alpha = 1,\\delta_2/\\alpha =4.5. The vertical axis is the frequency of the spectroscopy tone (a \\pi pulse) counted from the ancilla transition frequency \\omega_c. The color indicates the population of the ancilla not in the ground state. Bottom: the quasienergy spectrum of the driven ancilla for the same range of drive strengths as in the top panel. From top to down at \\xi_2 = 0, the blue, green, red and black lines refer to the quasienergy levels \\epsilon_1, \\epsilon_3, \\epsilon_2, \\epsilon_0, respectively, projected into the same Brillouin zone. The anti-crossing between \\epsilon_0 and \\epsilon_2 indicates a multi-photon resonance where the ancilla is excited from the ground to the second excited state by absorbing three drive-1 photons and emitting one drive-2 photon. The gap of the anti-crossing is too weak to be seen on the scale of the plot. In the top panel, the dip at around -30 MHz at the drive strength where the anti-crossing occurs is likely due to that the ancilla decays from the second to the first excited state and then be de-excited to the ground state by the spectroscopy tone.](./{level_anticrossing_theory_v2}.jpg \"fig:\") [fig:levelanti-crossingexp]\nTransient susceptibilities\n==========================\nAlso of interest to us are the transient susceptibilities where the ancilla is initialized in a given Floquet state. The analysis of the transient susceptibilities greatly simplifies in the limit of weak damping and dephasing where Eqs. ([eq:diagonal],[eq:offdiagonal]) hold and the probe detunings from the ancilla resonances are much larger than the corresponding linewidths. Assuming that the ancilla is initially in a pure Floquet state Ψ*j* at *t* = 0, we find that the ensemble-averaged susceptibilities read\n \n$$\\begin{aligned}\n\\label{eq:chi1\\_transient}\n&\\chi (\\omega,\\omega+K\\omega\\_{21},t) = \\sum\\_{m} P\\_m (t) \\chi \\_m(\\omega,\\omega+K\\omega\\_{21}) \\nonumber\\\\\n&+ i \\sum\\_{mnK'}[P\\_m(t) V\\_{mn} + \\dot P\\_m(t)]\\left( \\frac{\\mathcal F\\_{mnKK'}}{\\delta\\_{mnK'}^2(\\omega)} - \\frac{\\mathcal F\\_{nmKK'}}{\\delta\\_{nmK'}^2(\\omega)}\\right)\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\label{eq:X1\\_transient}\n&X (-\\omega,2\\omega\\_1+K\\omega\\_{21}-\\omega,t) \\nonumber \\\\\n&= \\sum\\_m P\\_m(t) X \\_m(-\\omega,2\\omega\\_1+K\\omega\\_{21}-\\omega)+ i \\sum\\_{mnK'}[P\\_m(t) V\\_{mn} \\nonumber \\\\\n& + \\dot P\\_m(t)]\\left( \\frac{\\tilde{\\mathcal F}\\_{mnKK'}}{\\delta\\_{mnK'}^2(-\\omega)} - \\frac{ \\tilde{\\mathcal F}\\_{nmKK'}}{\\delta\\_{nmK'}^2(-\\omega)}\\right),\\nonumber \\\\ \\end{aligned}$$ \nHere *P**m*(*t*) denotes the time-dependent population of the ancilla in the state Ψ*m* and it satisfies the rate equation ([eq:diagonal]) with *P**m*(*t*) ≡ *ρ**m**m*(*t*) and the initial condition *ρ**m**m*(0) = *δ**m**j*. The (unitary) partial susceptibilities *χ**m* and *X**m* are given in Eqs. ([eq:chi1], [eq:X1]). We have introduced a shorthand notation for the squared Floquet matrix elements and the detuning of the probe frequency from the corresponding Floquet resonances\n \n$$\\begin{aligned}\n&\\mathcal F\\_{mn KK'} = c\\_{mn,K'-K}(c^\\dagger )\\_{nm,-K'}, \\\\\n&\\tilde{\\mathcal F}\\_{mn KK'} = c\\_{mn,K'-K}c\\_{nm,-K'}, \\\\\n&\\delta\\_{mnK'}(\\omega) = (\\omega-\\omega\\_1) -\\epsilon\\_{nm}/\\hbar + K'\\omega\\_{21}.\\end{aligned}$$ \nEqs. ([eq:chi1transient],[eq:X1transient]) apply when the detunings *δ**m**n**K*ʹ are much larger than the linewidths *V**m**n* and the changing rate of *P**m*(*t*).\nAfter a time set by the inverse relaxation time of the ancilla, the transient susceptibilities become the steady-state susceptibilities. This can be seen by setting *P**m*(*t*) to the steady-state population $P\\_m^{\\rm st}$ and $\\dot P\\_m(0)$ to zero in Eqs. ([eq:chi1transient],[eq:X1transient]). A more general result for the steady-state susceptibilities beyond the regime ∣*δ**m**n**K*ʹ∣ ≫ *V**m**n* is given in Eq. ([eq:averagedsusceptibilities]) of the main text.\nOne can think of the ensemble-averaged susceptibilities in Eqs. ([eq:chi1transient],[eq:X1transient]) in a quantum-trajectory-like picture where the ancilla randomly jumps from one Floquet state to another. In between the jumps, the ancilla remains in a given Floquet state, say, Ψ*j*. Then the ``instantaneous\" susceptibilities of the ancilla are given by the short-time limit of Eqs. ([eq:chi1transient],[eq:X1transient]), where *t* is much shorter than the relaxation time of the ancilla. Specifically, we need to set all probabilities *P**m* ≠ *j*(*t*) in Eqs. ([eq:chi1transient],[eq:X1transient]) to zero, *P**j* to 1, and the time derivative of the probabilities to $\\dot P\\_{m\\neq j} = W\\_{jm}$, $\\dot P\\_j = - \\sum\\_{j'\\neq j} W\\_{jj'}$ based on Eq. ([eq:diagonal]). We denote the resulting susceptibility as $\\chi\\_j^{\\rm tr}$ which is given by,\n \n$$\\begin{aligned}\n\\label{eq:chi1\\_j}\n&\\chi\\_j^{\\rm tr}(\\omega,\\omega+K\\omega\\_{21}) =\\chi \\_j(\\omega,\\omega+K\\omega\\_{21}) \\nonumber\\\\\n&+ i \\sum\\_{nK'}[V\\_{jn} - \\sum\\_{j'\\neq j} W\\_{jj'}]\\left( \\frac{\\mathcal F\\_{jnKK'}}{\\delta\\_{jnK'}^2(\\omega)} - \\frac{\\mathcal F\\_{njKK'}}{\\delta\\_{njK'}^2(\\omega)}\\right) \\nonumber \\\\\n& + i \\sum\\_{nK',j'\\neq j} W\\_{jj'}\\left( \\frac{\\mathcal F\\_{j'nKK'}}{\\delta\\_{j'nK'}^2(\\omega)} - \\frac{\\mathcal F\\_{nj'KK'}}{\\delta\\_{nj'K'}^2(\\omega)}\\right) \\end{aligned}$$ \nNote that the expression above is different from the partial susceptibility $\\chi\\_j^{\\rm st}$ in Eq. ([eq:chi1decay]). One can show that summing the instantaneous susceptibility $\\chi\\_j^{\\rm tr}$ over the steady-state distribution $P\\_j^{\\rm st}$ recovers the ensemble-averaged steady-state susceptibility $\\chi^{\\rm st}$ in Eq. ([eq:averagedsusceptibilities]). The instantaneous susceptibility $X\\_j^{\\rm tr}$ can be found similarly.\nIncoherent hopping between the cavity and ancilla induced by ancilla dephasing\n==============================================================================\nIn this section, we study in detail the dephasing-induced incoherent hopping between the cavity and ancilla using a two-state approximation. To be concrete, we consider the situation described by the Hamiltonian in Eq. ([eq:inversePurcellRabi]) where cavity mode *a* is close to the resonance *ν*0*m**K* and other resonance processes can be neglected.\nImplicitly assumed in the formula for the inverse Purcell decay rate in Eq. ([eq:deltakappa]) is that the drives are turned on relatively slowly so that any initial state of the cavity mode stays in the adiabatic state of the coupled ancilla-cavity system described by Eq. ([eq:inversePurcellRabi]) as the drives are being turned on. Here, we make the same assumption and show that ancilla dissipation and dephasing lead to incoherent hopping between the adiabatic states of the Hamiltonian in Eq. ([eq:inversePurcellRabi]), and the hopping rate reduces to the inverse Purcell decay rate in the weak coupling limit.\n [fig:heatingratesscaling]\nTo relate to the experiment in Sec. [sec:inversePurcellexp], we consider that cavity mode *a* is initially in the Fock state ∣1*a*⟩ and the ancilla is in the vacuum state ∣0*c*⟩ before the drives are turned on. After the drives have been turned on adiabatically, the state ∣1*a*, 0*c*⟩ becomes an eigenstate of the the Hamiltonian in Eq. ([eq:inversePurcellRabi]): *ψ*1 = *α*1∣1*a*, *u*0⟩ + *β*1∣0*a*, *u**m*⟩. We denote another eigenstate of the Hamiltonian in Eq. ([eq:inversePurcellRabi]) in the same subspace as *ψ*2 = *α*2∣1*a*, *u*0⟩ + *β*2∣0*a*, *u**m*⟩. In the weak coupling limit, *ψ*1 will be mostly ∣1*a*, *u*0⟩ and *ψ*2 will be mostly ∣0*a*, *u**m*⟩.\nFinite ancilla dissipation and dephasing induce coupling between these two states and thus they acquire finite widths. In the limit where the decoherence-induced widths are much smaller than the energy splitting between these two states, there occurs incoherent hopping between these two states as well as pure loss from these states which can be described by the following rate equation,\n \n$$\\begin{aligned}\n\\label{eq:two\\_state\\_rate\\_equation}\n\\dot \\rho\\_{1} &= - (R\\_{11}+R\\_{12})\\rho\\_{1} + R\\_{21} \\rho\\_{2} \\nonumber \\\\\n\\dot \\rho\\_{2} &= - (R\\_{22}+R\\_{21})\\rho\\_{2} + R\\_{12} \\rho\\_{1},\\end{aligned}$$ \nwhere *ρ*1, 2 refer to the population in the states *ψ*1, 2, respectively. *R*12, *R*21 refer to the hopping rates between the two states; *R*11, *R*22 refer to the individual loss rates of the two states due to hopping to states outside the subspace. To simplify the analysis, we neglected the finite heating rate of the ancilla from lower to higher Floquet states which is small for not too strong drive. Therefore, we can restrict ourselves to the two state subspace spanned by the states Ψ1 and Ψ2 without considering the finite rates of hopping to higher Floquet states and then quickly hopping back.\nThe rates *R**i**j* can be calculated using Fermi’ s golden rule. The incoherent hopping between the two states is caused by the pure dephasing of the ancilla Floquet states and the hopping rates are found to be\n \n$$\\begin{aligned}\n\\label{eq:two\\_state\\_incoherent\\_hopping}\n&R\\_{12} = \\sum\\_K \\frac {1}{2}\\gamma |\\alpha\\_1\\alpha\\_2^\\*c\\_{00,-K}+\\beta\\_1\\beta\\_2^\\*c\\_{mm,-K}|^2 \\nonumber \\\\\n&+2 |\\alpha\\_1\\alpha\\_2^\\* (c^\\dagger c)\\_{00,K}+\\beta\\_1\\beta\\_2^\\*(c^\\dagger c)\\_{mm,K}|^2 \\gamma\\_{\\rm ph}(K\\omega\\_{21}+\\tilde \\Omega\\_R),\\end{aligned}$$ \nand *R*21 has the same form as *R*12 except that the argument of $\\gamma\\_{\\rm ph}$ changes sign. Expression ([eq:twostateincoherenthopping]) reduces to the pure dephasing rate of the Floquet states in Eq. ([eq"},"data_id":{"kind":"string","value":"arxiv_0002435"}}},{"rowIdx":589,"cells":{"content":{"kind":"string","value":"A Noncommutative Gauss Map\n==========================\nThe aim of this paper is to transfer the Gauss map, which is a Bernoulli shift for continued fractions, to the noncommutative setting. We feel that a natural place for such a map to act is on the AF algebra A considered separately by F. Boca and D. Mundici. The center of A is isomorphic to *C*[0, 1], so we first consider the action of the Gauss map on *C*[0, 1] and then extend the map to A and show that the extension inherits many desirable properties.\nIntroduction and Notation\n=========================\nFlorin Boca in and Daniele Mundici in separately considered an AF algebra A that is associated with the Farey tessellation. The algebra A exhibits many interesting properties, not the least of which is the connection between A and the unit interval [0, 1]. This connection is not merely topological, but also number theoretic. We briefly explain this connection\nLet *Z*(A) denote the center of A. As noted in, we have *C*[0, 1] ≅ *Z*(A). Moreover, the maximal ideal space of A is homeomorphic (when equipped with the topology induced by Prim(A)) to [0, 1] in a natural way. For each irrational 0 < *θ* < 1, let J*θ* denote the maximal ideal of A associated to *θ*. It was shown in that A/J*θ* ≅ F*θ*, the Effros-Shen algebra, defined in, associated with the continued fraction expansion of *θ*.\nIn other words, if we employ the topological decomposition theory of *C*\\*-algebras and visualize A as continuous, operator-valued functions on its maximal ideal space (a visualization which is usually “incorrect, but fruitful” ), then each function evaluated at *θ* takes values in the Effros-Shen algebra F*θ*. So it is not simply the topology around *θ* that determines this visualization, but also the continued fraction expansion of *θ*.\nGiven the close connection between A and the continued fraction expansions of numbers in [0, 1], it is natural to try and extend important functions from number theory (especially those related to continued fractions) to the *C*\\*-algebra A. The Gauss map might be the most fundamental such function, hence we take it as our starting point\nRecall the Gauss map *G* : [0, 1] → [0, 1] defined by *G*(0) = 0 and *G*(*x*) = 1/*x* − ⌊1/*x*⌋ if *x* ≠ 0, where ⌊ ⋅ ⌋ denotes the greatest integer function. One can think of *G* as the Bernoulli shift for continued fractions. Indeed, given *θ* ∈ [0, 1] with continued fraction expansion *θ* = [*a*1, *a*2, ...] then *G*([*a*1, *a*2, ...]) = [*a*2, *a*3, ...]. One can also recover the continued fraction expansion of *θ* by implementations of *G* and ⌊ ⋅ ⌋.\nWe first consider the induced action of *G* on *Z*(A) = *C*[0, 1]. First note that *f* ∘ *G* ∈ *C*[0, 1] if and only if *f* is a constant function. Therefore we will consider the “adjoint” action of the Gauss map on *C*[0, 1]. Let *μ* denote Gauss measure on [0, 1] defined by $d\\mu=\\frac{d\\theta}{\\ln 2 (\\theta+1)}$, where *d**θ* denotes Lebesgue measure. Then *G* is *μ*-invariant, i.e. *μ*(*G*− 1(*E*)) = *μ*(*E*) for every Borel set *E* ⊆ [0, 1] (see for details). From this it follows that the map\n \n*V**G*(*f*)(*θ*) = *f*(*G*(*θ*)) for *f* ∈ *L*2(*μ*), *θ* ∈ [0, 1] \nis an isometry. A standard calculation reveals that\n \n$$V\\_G^\\*(f)(\\theta)=\\sum\\_{s=1}^\\infty f\\Big( \\frac{1}{\\theta+s}\\Big) \\frac{1+\\theta}{(\\theta+s)(\\theta+s+1)} \\label{eq:V\\_G\\*definition}$$ \nand it is routine to verify that *V**G*\\*(*f*) ∈ *C*[0, 1] when *f* ∈ *C*[0, 1]. We mention that, symbolically, *V**G*\\* is the Perron-Frobenius operator of *G* under *μ* and refer the reader to for details about Perron-Frobenius operators and their connections to continued fractions.\nFurthermore, if we embed *C*[0, 1] into *B*(*L*2(*μ*)) as *f* ↦ M*f* where M*f*(*g*) = *f**g* for all *g* ∈ *L*2(*μ*), then\n \n*V**G*\\*M*f**V**G* = M*V**G*\\*(*f*). \nThis defines a unital completely positive map, which we will henceforth denote by G, on *C*[0, 1] that not only respects the action of *G* on its maximal ideal space, i.e. for each *E* ⊆ [0, 1] let *J**E* denote the ideal of *C*[0, 1] consisting of those functions that vanish on *E*, then\n \nG(*J**G*− 1(*E*)) ⊆ *J**E*, \nbut is also *μ* − invariant, i.e.\n \n∫*f**d**μ* = ∫G(*f*)*d**μ* for all *f* ∈ *C*[0, 1]. \nHence we are looking for an extension of G to A that satisfies the natural analogs of ([eq:conjugationproperty])-([eq:muinvariant]). In order to do this we must first consider what Gauss measure should mean on A. Our first step is proving that every state on *C*[0, 1] has a unique extension to a trace on A (Theorem [thm:statetrace]). D. Mundici showed that the state space of *C*[0, 1] and the space of tracial states on A are affinely, weak\\* homeomorphic. But for our purposes, we will need the extension property from Theorem [thm:statetrace].\nFor reasons that will become clear, we have to slightly modify the natural analogs of ([eq:conjugationproperty]) and ([eq:muinvariant]). In particular, we will use Theorem [thm:statetrace] to define two separate state extensions, *ϕ* and *τ*, of *μ* and intertwine between these two GNS representations to obtain analogs of ([eq:conjugationproperty]) and ([eq:muinvariant]). Let (*π**ϕ*, *L*2(A, *ϕ*)) and (*π**τ*, *L*2(A, *τ*)) be the GNS representations of A associated with *ϕ* and *τ*. Since *ϕ* and *τ* are extensions of *μ*, it follows that *L*2(*μ*) ⊆ *L*2(A, *ϕ*), *L*2(A, *τ*) and\n \n*π**ϕ*(*f*)∣*L*2(*μ*) = M*f* for every *f* ∈ *Z*(A) ≅ *C*[0, 1]. \nThis allows us to prove the main theorem:\n[thm:maintheorem] There is a unital completely positive map $\\widetilde{\\mathbb{G}}:{\\mathfrak{A}}\\rightarrow{\\mathfrak{A}}$ and an isometry $\\widetilde{V}\\_G: L^2({\\mathfrak{A}},\\tau)\\rightarrow L^2({\\mathfrak{A}},\\phi)$ such that\n1. $\\widetilde{\\mathbb{G}}|\\_{C[0,1]}=\\mathbb{G}.$\n2. $\\widetilde{\\mathbb{G}}(\\mathcal{J}(G^{-1}(E)))\\subseteq \\mathcal{J}(E),$ for each *E* ⊂ [0, 1] J*E* are the ideals of A defined in.\n3. $\\widetilde{V}\\_G|\\_{L^2(\\mu)}=V\\_G$ and $\\widetilde{V}\\_G^\\*|\\_{L^2(\\mu)}=V\\_G^\\*.$\n4. $\\widetilde{V}\\_G^\\*\\pi\\_\\phi(x)\\widetilde{V}\\_G=\\pi\\_\\tau(\\widetilde{\\mathbb{G}}(x))$ for *x* ∈ A. Hence $\\widetilde{V}\\_G^\\*\\pi\\_\\phi(f)\\widetilde{V}\\_G|\\_{L^2(\\mu)}=\\mathcal{M}\\_{\\mathbb{G}(f)}$ for *f* ∈ *C*[0, 1].\n5. $\\phi(x)=\\tau(\\widetilde{\\mathbb{G}}(x))$ for *x* ∈ A.\nIn order to set our notation, we now recall some relevant facts about the AF algebra A defined in and. We will use the same notation as in, in particular *p*(*n*, *k*), *q*(*n*, *k*) ∈ Z+ and $r(n,k)=\\frac{p(n,k)}{q(n,k)}$ for *n* ≥ 0 and 0 ≤ *k* ≤ 2*n* all have the same meaning and we will frequently refer to the relationships between them as defined on. Recall that A is the inductive limit of the finite dimensional *C*\\*-algebras,\n \nA*n* = ⨁0 ≤ *k* ≤ 2*n**M**q*(*n*, *k*). \nFor the convenience of the reader, and with thanks to F. Boca for supplying us with the code, we reproduce the Bratelli diagram of A from.\n0.6mm\n(0,120)(0,0) (-120,100)(-120,120)(120,100) (-120,80)(-120,100)(0,80)(120,100)(120,80) (-120,80)(-120,60) (-120,60)(-120,80)(-60,60)(0,80)(60,60)(120,80)(120,60) (0,80)(0,60)\n(-120,40)(-120,60)(-90,40)(-60,60)(-30,40)(0,60)(30,40)(60,60)(90,40)(120,60)(120,40) (-60,60)(-60,40) (0,60)(0,40) (60,60)(60,40) (-120,20)(-120,40)(-105,20)(-90,40)(-75,20)(-60,40)(-45,20)(-30,40)(-15,20)(0,40)(15,20) (30,40)(45,20)(60,40)(75,20)(90,40)(105,20)(120,40)(120,20) (-90,40)(-90,20) (-60,40)(-60,20) (-30,40)(-30,20) (0,40)(0,20) (30,40)(30,20) (60,40)(60,20) (90,40)(90,20)\n(-120,5)(-120,20)(-112.5,5)(-105,20)(-97.5,5)(-90,20)(-82.5,5)(-75,20)(-67.5,5)(-60,20) (-52.5,5)(-45,20)(-37.5,5)(-30,20)(-22.5,5)(-15,20)(-7.5,5)(0,20) (120,5)(120,20)(112.5,5)(105,20)(97.5,5)(90,20)(82.5,5)(75,20)(67.5,5)(60,20) (52.5,5)(45,20)(37.5,5)(30,20)(22.5,5)(15,20)(7.5,5)(0,20)\n(-105,20)(-105,5) (-90,20)(-90,5) (-75,20)(-75,5) (-60,20)(-60,5) (-45,20)(-45,5) (-30,20)(-30,5) (-15,20)(-15,5) (0,20)(0,5) (105,20)(105,5) (90,20)(90,5) (75,20)(75,5) (60,20)(60,5) (45,20)(45,5) (30,20)(30,5) (15,20)(15,5)\n(-125,100)(0,0)$\\frac{0}{1}$ (0,87)(0,0)$\\frac{1}{2}$ (125,100)(0,0)$\\frac{1}{1}$ (-125,80)(0,0)$\\frac{0}{1}$ (125,80)(0,0)$\\frac{1}{1}$ (-125,60)(0,0)$\\frac{0}{1}$ (-60,66)(0,0)$\\frac{1}{3}$ (-4,63)(0,0)$\\frac{1}{2}$ (60,66)(0,0)$\\frac{2}{3}$ (125,60)(0,0)$\\frac{1}{1}$ (-125,60)(0,0)$\\frac{0}{1}$\n(-125,40)(0,0)$\\frac{0}{1}$ (-90,46)(0,0)$\\frac{1}{4}$ (-65,40)(0,0)$\\frac{1}{3}$ (-30,46)(0,0)$\\frac{2}{5}$ (-5,40)(0,0)$\\frac{1}{2}$ (30,46)(0,0)$\\frac{3}{5}$ (65,40)(0,0)$\\frac{2}{3}$ (90,46)(0,0)$\\frac{3}{4}$ (125,40)(0,0)$\\frac{1}{1}$\n(-120,100)(0,0) • (120,100)(0,0) • (-120,80)(0,0) • (0,80)(0,0) • (120,80)(0,0) • (-120,60)(0,0) • (-60,60)(0,0) • (0,60)(0,0) • (60,60)(0,0) • (120,60)(0,0) • \n(-120,40)(0,0) • (-90,40)(0,0) • (-60,40)(0,0) • (-30,40)(0,0) • (0,40)(0,0) • (30,40)(0,0) • (60,40)(0,0) • (90,40)(0,0) • (120,40)(0,0) • \n(-125,20)(0,0)$\\frac{0}{1}$ (-105,27)(0,0)$\\frac{1}{5}$ (-95,20)(0,0)$\\frac{1}{4}$ (-75,27)(0,0)$\\frac{2}{7}$ (-65,20)(0,0)$\\frac{1}{3}$ (-45,27)(0,0)$\\frac{3}{8}$ (-35,20)(0,0)$\\frac{2}{5}$ (-15,27)(0,0)$\\frac{3}{7}$ (-5,20)(0,0)$\\frac{1}{2}$ (125,20)(0,0)$\\frac{1}{1}$ (105,27)(0,0)$\\frac{4}{5}$ (95,20)(0,0)$\\frac{3}{4}$ (75,27)(0,0)$\\frac{5}{7}$ (65,20)(0,0)$\\frac{2}{3}$ (45,27)(0,0)$\\frac{5}{8}$ (35,20)(0,0)$\\frac{3}{5}$ (15,27)(0,0)$\\frac{4}{7}$ (-120,120)(0,0) ⋆ \n(-120,20)(0,0) • (-105,20)(0,0) • (-90,20)(0,0) • (-75,20)(0,0) • (-60,20)(0,0) • (-45,20)(0,0) • (-30,20)(0,0) • (-15,20)(0,0) • (0,20)(0,0) • (120,20)(0,0) • (105,20)(0,0) • (90,20)(0,0) • (75,20)(0,0) • (60,20)(0,0) • (45,20)(0,0) • (30,20)(0,0) • (15,20)(0,0) • \n(-125,0)(0,0)$\\frac{0}{1}$ (-112.5,-2)(0,0)$\\frac{1}{6}$ (-105,-2)(0,0)$\\frac{1}{5}$ (-97.5,-2)(0,0)$\\frac{2}{9}$ (-90,-2)(0,0)$\\frac{1}{4}$ (-82.5,-2)(0,0)$\\frac{3}{11}$ (-75,-2)(0,0)$\\frac{2}{7}$ (-67.5,-2)(0,0)$\\frac{3}{10}$ (-60,-2)(0,0)$\\frac{1}{3}$ (-52.5,-2)(0,0)$\\frac{4}{11}$ (-45,-2)(0,0)$\\frac{3}{8}$ (-37.5,-2)(0,0)$\\frac{5}{13}$ (-30,-2)(0,0)$\\frac{2}{5}$ (-22.5,-2)(0,0)$\\frac{5}{12}$ (-15,-2)(0,0)$\\frac{3}{7}$ (-7.5,-2)(0,0)$\\frac{4}{9}$ (0,-2)(0,0)$\\frac{1}{2}$ (125,0)(0,0)$\\frac{1}{1}$ (112.5,-2)(0,0)$\\frac{5}{6}$ (105,-2)(0,0)$\\frac{4}{5}$ (97.5,-2)(0,0)$\\frac{7}{9}$ (90,-2)(0,0)$\\frac{3}{4}$ (82.5,-2)(0,0)$\\frac{8}{11}$ (75,-2)(0,0)$\\frac{5}{7}$ (67.5,-2)(0,0)$\\frac{7}{10}$ (60,-2)(0,0)$\\frac{2}{3}$ (52.5,-2)(0,0)$\\frac{7}{11}$ (45,-2)(0,0)$\\frac{5}{8}$ (37.5,-2)(0,0)$\\frac{8}{13}$ (30,-2)(0,0)$\\frac{3}{5}$ (22.5,-2)(0,0)$\\frac{7}{12}$ (15,-2)(0,0)$\\frac{4}{7}$ (7.5,-2)(0,0)$\\frac{5}{9}$\n(-120,5)(0,0) • (-112.5,5)(0,0) • (-105,5)(0,0) • (-97.5,5)(0,0) • (-90,5)(0,0) • (-82.5,5)(0,0) • (-75,5)(0,0) • (-67.5,5)(0,0) • (-60,5)(0,0) • (-52.5,5)(0,0) • (-45,5)(0,0) • (-37.5,5)(0,0) • (-30,5)(0,0) • (-22.5,5)(0,0) • (-15,5)(0,0) • (-7.5,5)(0,0) • (0,5)(0,0) • (120,5)(0,0) • (112.5,5)(0,0) • (105,5)(0,0) • (97.5,5)(0,0) • (90,5)(0,0) • (82.5,5)(0,0) • (75,5)(0,0) • (67.5,5)(0,0) • (60,5)(0,0) • (52.5,5)(0,0) • (45,5)(0,0) • (37.5,5)(0,0) • (30,5)(0,0) • (22.5,5)(0,0) • (15,5)(0,0) • (7.5,5)(0,0) • \n[Figure1]\nFor each *n* ≥ 0 let E*n* : A → A*n* be conditional expectations such that\n \nE*n*E*m* = E*m*E*n* for all *n*, *m* ≥ 0. \nThe existence of such conditional expectations is guaranteed by Arveson’s extension theorem, or since A is AF, one can construct such maps explicitly. Furthermore for 0 ≤ *k* ≤ 2*n* let E(*n*, *k*) : A → *M**q*(*n*, *k*) ⊆ A*n* be conditional expectations, such that\n \nE(*n*, *k*)E*n* = E*n*E(*n*, *k*) for *n* ≥ 0 and 0 ≤ *k* ≤ 2*n*. \nNote that for each *x* ∈ A we have\n \nlim*n* → ∞E*n*(*x*) = *x*. \nWe will use the following notation throughout: For a unital *C*\\*-algebra *A*, we let\n1. *Z*(*A*) denote the center of *A*,\n2. S(*A*) denote the state space of *A*,\n3. T(*A*) denote the set of all unital traces of *A*.\n4. *M**n* denote *n* × *n* matrices over C,\n5. *τ**n* the unital trace on *M**n* and 1*n* ∈ *M**n* the identity.\n6. For $\\frac{p}{q}\\in\\mathbb{Q}\\cap[0,1]$ in reduced form we define\n \n$$M\\_{\\frac{p}{q}}:=M\\_q\\quad \\tau\\_{\\frac{p}{q}}:=\\tau\\_q \\in \\mathcal{T}(M\\_q).$$\nState Extensions and Conditional Expectation onto *Z*(A)\n========================================================\nIn this section we will construct a conditional expectation from A onto *Z*(A) that preserves every *τ* ∈ T(A). This will provide the key step in the proof of Theorem [thm:statetrace].\n[defn:rationaltraces] Let *n* ≥ 0 and 0 ≤ *k* ≤ 2*n*. Define *τ*(*n*, *k*) ∈ T(A) as\n \n*τ*(*n*, *k*)(*x*) = *τ**q*(*n*, *k*) ∘ E(*n*, *k*)(*x*). \nThe following lemma is immediate from ([eq:commutingexpectations]) and ([eq:commutingfdexpectations]):\n[lem:welldefinedfx] For *n* ≥ 0, 0 ≤ *k* ≤ 2*n* and ℓ ≥ 0, we have\n \n*τ*(*n*, *k*) = *τ*(*n* + ℓ, 2ℓ*k*). \nLet *x* ∈ A. Define the function *f**x* : Q ∩ [0, 1] → C as\n \n*f**x*(*r*(*n*, *k*)) = *τ*(*n*, *k*)(*x*) for *n* ≥ 0and 0 ≤ *k* ≤ 2*n*. \nThen *f**x* is well-defined and extends to a continuous function on [0, 1].\nIf *r*(*n*, *k*) = *r*(*n*ʹ, *k*ʹ) with *n*ʹ ≥ *n* then there is an ℓ ≥ 0 such that *n*ʹ = *n* + ℓ and *k*ʹ = 2ℓ*k*. Hence *f**x* is well-defined by Lemma [lem:welldefinedfx].\nLet *n* ≥ 0 and 0 < *k* < 2*n*. By the relationships defined in, the following function is continuous and piecewise affine on [0, 1]:\n \n$$B\\_{(n,k)}(\\theta)=\\left\\{ \\begin{array}{ll}0 & \\textrm{ if }0\\leq \\theta\\leq r(n,k-1)\\\\\nq(n,k)\\Big(q(n,k-1)\\theta-p(n,k-1)\\Big) & \\textrm{ if }r(n,k-1)\\leq \\theta\\leq r(n,k)\\\\\nq(n,k)\\Big(p(n,k+1)-q(n,k+1)\\theta\\Big) &\\textrm{ if }r(n,k)\\leq \\theta\\leq r(n,k+1)\\\\\n0& \\textrm{ if }r(n,k+1)\\leq \\theta\\leq 1\n\\end{array} \\right.$$ \nWe first let *x* ∈ A*n* ⊂ A and prove that *f**x* extends to a continuous function on [0, 1]. Suppose first that 0 < 2*k* + 1 < 2*n* and E(*n*, 2*k* + 1)(*x*) = *x*. Without loss of generality suppose that *τ*(*n*, 2*k* + 1)(*x*) = 1. We show that *f**x* = *B*(*n*, 2*k* + 1).\nIt is clear that *f**x*∣[0, *r*(*n*, 2*k*)] ∪ [*r*(*n*, 2*k* + 2), 1] ≡ 0. We now show by induction on ℓ ≥ 0 that\n \n(∀ℓ ≥ 0)(∀2ℓ2*k* < *j* < 2ℓ(2*k* + 2))(*f**x*(*r*(*n* + ℓ, *j*)) = *B*(*n*, 2*k* + 1)(*r*(*n* + ℓ, *j*))). \nFor ℓ = 0, we have *f**x*(*r*(*n*, 2*k* + 1)) = *τ*(*n*, 2*k* + 1)(*x*) = 1 = *B*(*n*, 2*k* + 1)(*r*(*n*, 2*k* + 1)). Suppose now that ([eq:inductionhypo]) holds for ℓ ≥ 0 and prove ([eq:inductionhypo]) for ℓ + 1.\nIf *j* = 2*i* is even, then\n \n$$\\begin{aligned}\n &f\\_x(r(n+\\ell+1,2i))=\\tau\\_{(n+\\ell+1,2i)}(x)=\\tau\\_{(n+\\ell,i)}(x)=f\\_x(r(n+\\ell,i))\\\\\n=&B\\_{(n,2k+1)}(r(n+\\ell,i))=B\\_{(n,2k+1)}(r(n+\\ell+1,2i)).\\end{aligned}$$ \nIf *j* = 2*i* + 1 is odd, then\n \n$$\\begin{aligned}\n &f\\_x(r(n+\\ell+1,2i+1))=\\tau\\_{(n+\\ell+1,2i+1)}(x)\\\\\n&=\\frac{q(n+\\ell,i)}{q(n+\\ell+1,2i+1)}\\tau\\_{(n+\\ell,i)}(x)+\\frac{q(n+\\ell,i+1)}{q(n+\\ell+1,2i+1)}\\tau\\_{(n+\\ell,i+1)}(x)\\\\\n&=\\frac{q(n+\\ell,i)}{q(n+\\ell+1,2i+1)}B\\_{(n,2k+1)}(r(n+\\ell,i))+\\frac{q(n+\\ell,i+1)}{q(n+\\ell+1,2i+1)}B\\_{(n,2k+1)}(r(n+\\ell,i+1))\\\\\n&=B\\_{(n,2k+1)}(r(n+\\ell+1,2i+1)).\\end{aligned}$$ \nHere the last line follows by the relationships in and because *B*(*n*, 2*k* + 1) is piecewise affine. This shows that ([eq:inductionhypo]) holds, hence *f**x* extends to a continuous function on [0, 1].\nNow suppose that 0 < 2*m**k* < 2*n* with *k* odd and E(*n*, 2*m**k*)(*x*) = *x*. Then,\n \n*x* = E(*n* − *m*, *k*)(*x*) − E(*n* − *m* + 1, 2*k* − 1)(*x*) − E(*n* − *m* + 1, 2*k* + 1)(*x*). \nSo, by the first part of the proof it follows that *f**x* is continuous.\nFor *x* = 1 ⊕ 0 ⊕ ⋯ ⊕ 0, 0 ⊕ ⋯ ⊕ 0 ⊕ 1 ∈ A*n*, the proof that *f**x* is continuous is exactly the same as above, so we omit the proof. This shows that for every *n* ≥ 0 and each *x* ∈ A*n* that *f**x* is continuous. Moreover note that the linear map *x* ↦ *f**x* defined on ⋃*n* = 1∞A*n* is contractive, hence *f**x* is continuous for every *x* ∈ A.\nAs observed in, *Z*(A) ≅ *C*[0, 1]. We now construct an explicit isomorphism. For each *n* ≥ 0, define Z*n* : *C*[0, 1] → *Z*(A*n*) ⊂ A by\n \nZ*n*(*f*) = ⨁0 ≤ *k* ≤ 2*n**f*(*r*(*n*, *k*))1*q*(*n*, *k*) \nBy, for each *n* ≥ 0 we have max{∣*r*(*n*, *k*) − *r*(*n*, *k* + 1)∣ : 0 ≤ *k* < 2*n*} = 1/(*n* + 1). Hence for *m* ≥ *n* we have\n \n‖Z*n*(*f*) − Z*m*(*f*)‖ ≤ sup{∣*f*(*θ*) − *f*(*θ*ʹ)∣ : ∣*θ* − *θ*ʹ∣ ≤ 1/(*n* + 1)}. \nTherefore Z*n*(*f*) is a Cauchy sequence in A because *f* is uniformly continuous on [0, 1].\nDefine Z : *C*[0, 1] → A by\n \nZ(*f*) = lim*n* → ∞Z*n*(*f*). \n[thm:conditionalexpectationdefn] The map Z : *C*[0, 1] → *Z*(A) is a \\*-isomorphism. Moreover the map E*Z* : A → *Z*(A) defined by\n \nE*Z*(*x*) = Z(*f**x*) \nis a conditional expectation such that\n \n*τ*(E*Z*(*x*)) = *τ*(*x*) for every *τ* ∈ T(A). \nBy ([eq:Tdefn]) it is clear that Z is a \\*-monomorphism, and since Z*n*(*f*) ∈ *Z*(A*n*) for each *n* ≥ 0, it follows that Z(*f*) ∈ *Z*(A). We now show that Z is surjective. Let *n* ≥ 0 and *y* ∈ A*n*. Then\n \n*y* ∈ *Z*(A*n*) if and only if *y* = ⨁0 ≤ *k* ≤ 2*n**τ*(*n*, *k*)(*y*)1*q*(*n*, *k*). \nLet *x* ∈ *Z*(A). By ([eq:defnoffx]) and ([eq:defofZn]) it follows that\n \nZ*n*(*f**x*) = ⨁0 ≤ *k* ≤ 2*n**τ*(*n*, *k*)(*x*)1*q*(*n*, *k*) ∈ *Z*(A*n*). \nSince *x* ∈ *Z*(A), it follows from ([eq:convergencethroughexpectations]) that\n \nlim*n* → ∞*d**i**s**t*(E*n*(*x*), *Z*(A*n*)) = 0, \nfrom which we deduce by ([eq:charofcenterinfdalgebras]) that Z*n*(*f**x*) → *x*. Therefore\n \nE*Z*(*x*) = Z(*f**x*) = lim*n* → ∞Z*n*(*f**x*) = *x*. \nThis shows that Z is surjective and also that E*Z* is a conditional expectation. We now show that E*Z* preserves every trace of A. Let *τ* ∈ T(A). By ([eq:convergencethroughexpectations]) it follows that *τ* is the weak\\*-limit of *τ* ∘ E*n*. Since *τ* ∘ E*n*∣A*n* ∈ T(A*n*), there is a convex combination of scalars (*λ*(*n*, *k*))0 ≤ *k* ≤ 2*n* such that\n \n*τ* ∘ E*n* = ∑0 ≤ *k* ≤ 2*n**λ*(*n*, *k*)*τ*(*n*, *k*). \nIt follows that T(A) equals the weak\\* closure of the convex hull of the set {*τ*(*n*, *k*) : *n* ≥ 0, 0 ≤ *k* ≤ 2*n*}. Therefore, we only need to check ([eq:tracepreserving]) for the traces *τ*(*n*, *k*). To this end, let *x* ∈ A then\n \n*τ*(*n*, *k*)(*x*) = *f**x*(*r*(*n*, *k*)) = *τ*(*n*, *k*)(Z*n*(*f**x*)) = *τ*(*n*, *k*)(Z(*f**x*)) = *τ*(*n*, *k*)(E*Z*(*x*)). \n[thm:statetrace] The restriction map *τ* ↦ *τ*∣*Z*(A) defines a weak\\* homeomorphism from T(A) onto S(*C*[0, 1]). In particular, every state on *Z*(A) has a unique tracial extension to A.\nInjectivity and weak\\*-continuity of the inverse both follow from ([eq:tracepreserving]). By ([eq:diracpointmassesabound]) it follows that the restriction of *τ*(*n*, *k*) to *Z*(A) ≅ *C*[0, 1] is the Dirac measure *δ*{*r*(*n*, *k*)}, which shows surjectivity.\nIdeals of A and traces of A\n===========================\nFix *θ* ∈ [0, 1]. Define *τ**θ*A ∈ T(A) as the unique tracial extension of the Dirac measure *δ*{*θ*} ∈ S(*C*[0, 1]) given by Theorem [thm:statetrace]. Note that for each *n* ≥ 0 and 0 ≤ *k* ≤ 2*n*, we have *τ**r*(*n*, *k*)A = *τ*(*n*, *k*) from Definition [defn:rationaltraces].\nFor each *θ* ∈ [0, 1], we recall the maximal ideals *I**θ* ⊂ A defined in. The following is a consequence of the proof of and the correspondence made in Theorem [thm:statetrace].\n[cor:traceideal] Fix *θ* ∈ [0, 1]. Then\n \n*I**θ* = {*x* ∈ A : *τ**θ*A(*x*\\**x*) = 0}. \nFix $\\frac{p(n,k)}{q(n,k)}=\\frac{p}{q}\\in\\mathbb{Q}\\cap (0,1)$ in reduced form. We define the \\*-homomorphism\n \n$$\\pi\\_{\\frac{p}{q}}:{\\mathfrak{A}}\\rightarrow M\\_{\\frac{p}{q}} \\label{eq:rationalhomo}$$ \nas “evaluation along the path *r*(*n*, *k*), *r*(*n* + 1, 2*k*), ..., *r*(*n* + ℓ, 2ℓ*k*), ... in the Bratteli diagram.” In particular, $\\textrm{ker}(\\pi\\_{\\frac{p}{q}})=I\\_{\\frac{p}{q}}$ (see for details). We note that\n \n$$\\tau^{\\mathfrak{A}}\\_{\\frac{p}{q}}(x)=\\tau\\_{\\frac{p}{q}}(\\pi\\_{\\frac{p}{q}}(x))\\quad \\textrm{ for every }x\\in{\\mathfrak{A}}. \\label{eq:tracesthroughrepresentations}$$ \nConstruction of $\\widetilde{\\mathbb{G}}$\n========================================\nIn this section we construct our noncommutative Gauss map $\\widetilde{\\mathbb{G}}:{\\mathfrak{A}}\\rightarrow {\\mathfrak{A}}.$ Let *s* ≥ 1. As in we define\n \n$$\\mathcal{J}\\_s:=\\mathcal{J}([\\frac{1}{s+1},\\frac{1}{s}])=\\bigcap\\_{\\theta\\in [\\frac{1}{s+1},\\frac{1}{s}]}I\\_\\theta.$$ \nBy Theorem [thm:statetrace] and Section [sec:ideals] we have\n \n$$\\mathcal{J}\\_s=\\textrm{ker}\\Big(\\bigoplus\\_{\\frac{1}{s+1}<\\frac{p}{q}\\in\\mathbb{Q}<\\frac{1}{s}}\\pi\\_{\\frac{p}{q}}\\Big). \\label{eq:anotherdescriptionJ\\_s}$$ \nFor each *s* ≥ 1 the Bratelli diagram of A/J*s* is the subdiagram of the Bratelli diagram of A obtained by deleting all of the nodes\n \n{*r*(*n*, *k*) : *r*(*n*, *k*) ∉ [1/(*s* + 1), 1/*s*]} ∪ {*r*(*n*, *k*) : *n* < *s*}, \nand deleting all edges connected to any of these nodes. See Figure [Figure2] for the Bratelli diagram of A/J2.\n0.6mm\n(-30,60)(0,0)\n(-60,60)(-30,40)(0,60) (-60,60)(-60,40) (0,60)(0,40) (-60,40)(-45,20)(-30,40)(-15,20)(0,40) (-60,40)(-60,20) (-30,40)(-30,20) (0,40)(0,20)\n(-60,20) (-52.5,5)(-45,20)(-37.5,5)(-30,20)(-22.5,5)(-15,20)(-7.5,5)(0,20)\n(-60,20)(-60,5) (-45,20)(-45,5) (-30,20)(-30,5) (-15,20)(-15,5) (0,20)(0,5)\n(-65,60)(0,0)$\\frac{1}{3}$ (5,60)(0,0)$\\frac{1}{2}$\n(-65,40)(0,0)$\\frac{1}{3}$ (-30,46)(0,0)$\\frac{2}{5}$ (5,40)(0,0)$\\frac{1}{2}$\n(-60,60)(0,0) • (0,60)(0,0) • \n(-60,40)(0,0) • (-30,40)(0,0) • (0,40)(0,0) • \n(-65,20)(0,0)$\\frac{1}{3}$ (-45,27)(0,0)$\\frac{3}{8}$ (-35,20)(0,0)$\\frac{2}{5}$ (-15,27)(0,0)$\\frac{3}{7}$ (5,20)(0,0)$\\frac{1}{2}$\n(-60,20)(0,0) • (-45,20)(0,0) • (-30,20)(0,0) • (-15,20)(0,0) • (0,20)(0,0) • \n(-60,-2)(0,0)$\\frac{1}{3}$ (-52.5,-2)(0,0)$\\frac{4}{11}$ (-45,-2)(0,0)$\\frac{3}{8}$ (-37.5,-2)(0,0)$\\frac{5}{13}$ (-30,-2)(0,0)$\\frac{2}{5}$ (-22.5,-2)(0,0)$\\frac{5}{12}$ (-15,-2)(0,0)$\\frac{3}{7}$ (-7.5,-2)(0,0)$\\frac{4}{9}$ (0,-2)(0,0)$\\frac{1}{2}$\n(-60,5)(0,0) • (-52.5,5)(0,0) • (-45,5)(0,0) • (-37.5,5)(0,0) • (-30,5)(0,0) • (-22.5,5)(0,0) • (-15,5)(0,0) • (-7.5,5)(0,0) • (0,5)(0,0) • \n[Figure2]\nFor each *s* ≥ 1, define the homeomorphism *g**s* : [0, 1] → [1/(*s* + 1), 1/*s*] as\n \n$$g\\_s(\\theta)=\\frac{1}{\\theta+s}, \\label{eq:defng\\_s}$$ \nand recall that these maps are the building blocks for the commutative Gauss map G : *C*[0, 1] → *C*[0, 1] defined in ([eq:VG\\*definition]). Then consider the induced isomorphism\n \n(*g**s*)\\* : *C*[1/(*s* + 1), 1/*s*] → *C*[0, 1] defined by (*g**s*)\\*(*f*) = *f* ∘ *g**s*. \nSince our goal is to extend G to a map on A, we first consider extensions of the maps (*g**s*)\\* as maps from A/J*s* into A. Unfortunately, there is no hope for these extensions to also be isomorphisms. Indeed, by considering the Bratelli diagrams of A and A/J*s* it is clear that *K*0(A) ≅ *K*0(A/J*s*), but there is no unit-preserving, positive homomorphism that implements this isomorphism. Hence ${\\mathfrak{A}}\\not\\cong {\\mathfrak{A}}/\\mathcal{J}\\_s.$ We do the next best thing by defining a (non-unital) \\*-monomorphism *H**s* : A → A/J*s* and a unital completely positive (UCP for short) map *G**s* : A/J*s* → A such that *G**s**H**s* = *i**d*A, and such that *G**s* is an extension of (*g**s*)\\*. More importantly, the maps *G**s* and *H**s* will provide a nice relationship (see ([eq:maintracerelationshipall])) between T(A) and T(A/J*s*).\nFor *n* ≥ 0, let *A**n* ∈ *M*2*n* + 1 + 1, 2*n* + 1(Z+) be the connecting homomorphisms from A*n* into A*n* + 1 such that\n \n$${\\mathfrak{A}}=\\varinjlim({\\mathfrak{A}}\\_n, A\\_n) \\label{eq:Fareyasinductivelimit}$$ \nFor example we have,\n \n$$A\\_0=\\left[ \\begin{array}{cc} 1&0\\\\ 1&1\\\\ 0&1\\\\ \\end{array}\\right]\\in M\\_{3,2}\\textrm{ },\\quad A\\_1=\\left[ \\begin{array}{ccc} 1&0&0\\\\ 1&1&0 \\\\ 0&1&0\\\\ 0&1&1\\\\ 0&0&1\\\\ \\end{array}\\right]\\in M\\_{5,3}\\textrm{ }, \\cdots$$ \nFor *n* ≥ 0, we define\n \n$$({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n:= \\bigoplus\\_{0\\leq k\\leq 2^n}M\\_{\\frac{q(n,k)}{p(n,k)+sq(n,k)}}=\\bigoplus\\_{0\\leq k\\leq 2^n}M\\_{g\\_s(r(n,k))}.$$ \nBy the description of the Bratteli diagram of A/J*s* (see also Figure [Figure2]) given above it follows that\n \n$${\\mathfrak{A}}/\\mathcal{J}\\_s=\\varinjlim(({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n, A\\_n) \\label{eq:quotientasinductivelimit}$$ \nLet ℓ∞(*s*) denote the *s* dimensional, commutative *C*\\*-algebra. Consider the *C*\\*-algebra,\n \n$$\\ell^\\infty(s)\\otimes {\\mathfrak{A}}=\\varinjlim (\\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_n, id\\_{\\ell^\\infty(s)}\\otimes A\\_n)$$ \nDefine $S=\\left[ \\begin{array}{llll} 1&1&\\cdot&1 \\end{array}\\right]\\in M\\_{1,s}.$ It is easy to see (using only the fact that *A**n* ∈ *M*2*n* + 1 + 1, 2*n* + 1(Z+)) that\n \n*A**n*(*S* ⊗ 12*n* + 1) = (*S* ⊗ 12*n* + 1 + 1)1*s* ⊗ *A**n* for every *n* ≥ 0. \nHence, for each *n* ≥ 0 we are able to define a \\*-homomorphism\n \n$$\\sigma\\_n: \\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_n=\\bigoplus\\_{0\\leq k\\leq 2^n}\\ell^\\infty(s)\\otimes M\\_{\\frac{p(n,k)}{q(n,k)}}\\rightarrow \\bigoplus\\_{0\\leq k\\leq 2^n}M\\_{\\frac{q(n,k)}{p(n,k)+sq(n,k)}}=({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n$$ \ngiven by the matrix *S* ⊗ 12*n* + 1 ∈ *M*2*n* + 1, *s*(2*n* + 1) such that the following diagram commutes for every *n* ≥ 0\n \n$$\\xymatrix{ ({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n \\ar[r]^{A\\_n} & ({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_{n+1} \\\\\n\\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_n \\ar[u]^{\\sigma\\_n} \\ar[r]^{1\\_s\\otimes A\\_n} & \\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_{n+1} \\ar[u]^{\\sigma\\_{n+1}} } \\label{eq:commutingsquare1}$$ \nLet *e*1, ..., *e**s* ∈ ℓ∞(*s*) denote the standard basis. For each *n* ≥ 0 define the UCP map *V**n* : (A/J)*n* → ℓ∞(*s*) ⊗ A*n* by\n \n$$V\\_n(x)=\\sigma\\_n^{-1}\\Big( \\sum\\_{i=1}^s \\sigma\\_n(e\\_i\\otimes 1\\_{{\\mathfrak{A}}\\_n})x\\sigma\\_n(e\\_i\\otimes 1\\_{{\\mathfrak{A}}\\_n})\\Big)$$ \nDefine *ψ**s* ∈ S(ℓ∞(*s*)) by\n \n$$\\psi\\_s\\Big(\\sum\\_{i=1}^s \\alpha\\_ie\\_i\\Big)=\\frac{1}{s}\\sum\\_{i=1}^s \\alpha\\_i.$$ \nIt now follows from ([eq:commutingsquare1]) that the following diagram commutes for all *n* ≥ 0 : \n \n$$\\xymatrix @R=.7in @C=2in { ({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n \\ar[r]^{A\\_n} \\ar@<1ex>[d]^{V\\_n} & ({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_{n+1} \\ar@<1ex>[d]^{V\\_{n+1}} \\\\\n\\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_n \\ar@<1ex>[u]^{\\sigma\\_n} \\ar[r]^{1\\_s\\otimes A\\_n} \\ar@<1ex>[d]^{\\psi\\_s\\otimes id\\_{{\\mathfrak{A}}\\_n}} & \\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_{n+1} \\ar@<1ex>[u]^{\\sigma\\_{n+1}} \\ar@<1ex>[d]^{\\psi\\_s\\otimes id\\_{{\\mathfrak{A}}\\_{n+1}}}\\\\\n{\\mathfrak{A}}\\_n \\ar@<1ex>[u]^{1\\_s\\otimes id\\_{{\\mathfrak{A}}\\_n}} \\ar[r]^{A\\_n} & {\\mathfrak{A}}\\_{n+1} \\ar@<1ex>[u]^{1\\_s\\otimes id\\_{{\\mathfrak{A}}\\_{n+1}}} } \\label{eq:commutingsquare2}$$ \nFurthermore,\n \n(*ψ**s* ⊗ *i**d*A*n*) ∘ *V**n* ∘ *σ**n* ∘ (1*s* ⊗ *i**d*A*n*) = *i**d*A*n* for all *n* ≥ 0. \nNow, let $x\\in M\\_{\\frac{p(n,k)}{q(n,k)}}\\subset {\\mathfrak{A}}\\_n$ and $y\\in M\\_{\\frac{q(n,k)}{p(n,k)+sq(n,k)}}\\subset ({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n.$ Set *p* = *p*(*n*, *k*) and *q* = *q*(*n*, *k*). Then, by basic properties of the trace it follows that\n \n$$\\begin{aligned}\n&\\tau\\_{\\frac{q}{p+sq}}\\Big( \\sigma\\_n(1\\_s\\otimes x)y\\Big) \\notag \\\\ \n&= \\tau\\_{\\frac{q}{p+sq}}\\Big( \\sigma\\_n(1\\_s\\otimes x)\\sum\\_{i=1}^s \\sigma\\_n(e\\_i\\otimes 1\\_{{\\mathfrak{A}}\\_n})y\\sigma\\_n(e\\_i\\otimes 1\\_{{\\mathfrak{A}}\\_n})\\Big) \\notag \\\\\n&= \\frac{sq}{p+sq}\\psi\\_s\\otimes \\tau\\_{\\frac{p}{q}}\\Big( \\sigma\\_n^{-1}\\Big( \\sigma\\_n(1\\_s\\otimes x)\\sum\\_{i=1}^s \\sigma\\_n(e\\_i\\otimes 1\\_{{\\mathfrak{A}}\\_n})y\\sigma\\_n(e\\_i\\otimes 1\\_{{\\mathfrak{A}}\\_n}) \\Big)\\Big) \\notag \\\\\n&= \\frac{sq}{p+sq}\\psi\\_s\\otimes \\tau\\_{\\frac{p}{q}}\\Big( (1\\_s\\otimes x)V\\_n(y)\\Big) \\notag \\\\\n&= \\frac{sq}{p+sq}\\tau\\_{\\frac{p}{q}}\\Big( x(\\psi\\_s\\otimes id\\_{{\\mathfrak{A}}\\_n}(V\\_n(y)))\\Big) \\label{align:tracialrelate}\\end{aligned}$$ \nWe now let $\\widetilde{V}\\_s:{\\mathfrak{A}}/\\mathcal{J}\\_s\\rightarrow {\\mathfrak{A}}$ be the inductive limit of the maps (*ψ**s* ⊗ *i**d*A*n*) ∘ *V**n*, which is well-defined by ([eq:commutingsquare2]). We also let $\\widetilde{\\sigma}\\_s:{\\mathfrak{A}}\\rightarrow {\\mathfrak{A}}/\\mathcal{J}\\_s$ be the inductive limit of the maps *σ**n* ∘ (1*s* ⊗ *i**d*A*n*), which again are well-defined by ([eq:commutingsquare2]).\nFigure [Figure3] (graciously provided by F. Boca) displays the mapping $\\widetilde{\\sigma}\\_1$ in terms of the Bratelli diagrams of A and A/J1.\n0.35mm\n(400,120)(0,0) (-20,0)(-20,90)(60,60)(140,90)(140,0)(-20,0) (340,0)(420,0)(420,60)(380,30)(340,60)(340,0)\n(230,47)(235,45)(230,43) (-22,48)(-20,43)(-18,48) (12.5,75)(20,75)(14,80) (-22,78)(-20,73)(-18,78) (408,47)(400,45)(405,52) (410.5,20.5)(408,12)(415,18) (422,17)(420,12)(418,17) (422,47)(420,42)(418,47) (-4,51)(2,43.5)(-6,47)\n(40,47)(35,41)(42,44) (46,52)(41,46)(48,49) (80,47)(85,41)(78,44) (74,52)(79,46)(72,49) (58,45)(60,40)(62,45) (58,50)(60,45)(62,50)\n(378,12)(380,7)(382,12) (378,17)(380,12)(382,17) (367,15.5)(365,7.2)(371.5,13) (371,21.4)(369,13.1)(375.5,18.9) (389,21.4)(391,13.1)(384.5,18.9) (393,15.5)(395,7.2)(388.5,13)\n(97,76.2)(90,71.2)(98,72.2) (103,78.5)(96,73.5)(104,74.5) (109,80.8)(102,75.8)(110,76.8)\n(119,47)(113,40)(121,43) (124,50.7)(118,43.7)(126,46.7) (129,54.4)(123,47.4)(131,50.4)\n(138,72)(140,67)(142,72) (138,77)(140,72)(142,77) (138,82)(140,77)(142,82)\n(138,42)(140,37)(142,42) (138,47)(140,42)(142,47) (138,52)(140,47)(142,52)\n(338,12)(340,7)(342,12) (338,17)(340,12)(342,17) (338,22)(340,17)(342,22)\n(338,42)(340,37)(342,42) (338,47)(340,42)(342,47) (338,52)(340,47)(342,52)\n(349.5,20.5)(352,12)(345,18) (352.5,16)(355,7.5)(348,13.5) (355.5,11.5)(358,3)(351,9)\n(352,47.2)(360,45.2)(355,52.2)(357,43.3)(365,41.3)(360,48.3) (362,39.3)(370,37.3)(365,44.3)\n(-20,0)(-20,120)(140,90)(140,0) (-20,90)(60,60)(140,90) (60,0)(60,60) (20,0)(20,30) (100,0)(100,30) (-20,60)(20,30)(60,60)(100,30)(140,60)\n(260,0)(260,90)(420,120)(420,0) (260,90)(340,60)(420,90) (340,0)(340,60) (300,0)(300,30) (380,0)(380,30) (260,60)(300,30)(340,60)(380,30)(420,60) (-20,30)(0,0)(20,30)(40,0)(60,30)(80,0)(100,30)(120,0)(140,30) (260,30)(280,0)(300,30)(320,0)(340,30)(360,0)(380,30)(400,0)(420,30)\n(-20,0)(0,0) • (0,0)(0,0) • (20,0)(0,0) • (40,0)(0,0) • (60,0)(0,0) • (80,0)(0,0) • (100,0)(0,0) • (120,0)(0,0) • (140,0)(0,0) • \n(-20,30)(0,0) • (20,30)(0,0) • (60,30)(0,0) • (100,30)(0,0) • (140,30)(0,0) • (260,30)(0,0) • (300,30)(0,0) • (340,30)(0,0) • (380,30)(0,0) ⊙ (420,30)(0,0) • \n(-20,60)(0,0) • (60,60)(0,0) ⊙ (140,60)(0,0) • (260,60)(0,0) • (340,60)(0,0)$\\boxdot$ (420,60)(0,0) ⋆ (-20,90)(0,0) ⋆ (140,90)(0,0)$\\boxdot$ (260,90)(0,0) • (420,90)(0,0) • (-20,120)(0,0) • (420,120)(0,0) • \n(260,0)(0,0) • (280,0)(0,0) • (300,0)(0,0) • (320,0)(0,0) • (340,0)(0,0) • (360,0)(0,0) • (380,0)(0,0) • (400,0)(0,0) • (420,0)(0,0) • \n(-28,90)(0,0)$\\frac{0}{1}$ (-28,60)(0,0)$\\frac{0}{1}$\n(148,90)(0,0)$\\frac{1}{1}$ (148,60)(0,0)$\\frac{1}{1}$\n(60,72)(0,0)$\\frac{1}{2}$ (20,40)(0,0)$\\frac{1}{3}$ (100,40)(0,0)$\\frac{2}{3}$\n(252,90)(0,0)$\\frac{0}{1}$ (252,60)(0,0)$\\frac{0}{1}$\n(428,90)(0,0)$\\frac{1}{1}$ (428,60)(0,0)$\\frac{1}{1}$\n(340,72)(0,0)$\\frac{1}{2}$ (300,40)(0,0)$\\frac{1}{3}$ (380,40)(0,0)$\\frac{2}{3}$\n(200,57)(0,0)$\\widetilde{\\sigma}\\_1:{\\mathfrak{A}}\\rightarrow\n{\\mathfrak{A}}/\\mathcal{J}\\_1$\n(165,45)(235,45)\n[Figure3]\nSet $\\pi=\\oplus\\_{\\frac{1}{s+1}<\\frac{p}{q}<\\frac{1}{s}}\\pi\\_{\\frac{p}{q}}$ and identify A/J*s* with *π*(A) by ([eq:anotherdescriptionJs]). By the Choi-Effros lifting theorem in, there is a UCP lifting *ϕ* : A/J*s* → A of *π*. Then let\n \n$$G\\_s:=\\widetilde{V}\\_s\\circ \\pi :{\\mathfrak{A}}\\rightarrow{\\mathfrak{A}}\\quad \\textrm{ and }H\\_s:=\\phi\\circ \\widetilde{\\sigma}\\_s:{\\mathfrak{A}}\\rightarrow {\\mathfrak{A}}.$$ \nBy ([eq:paritalinversesonfd]), it follows that\n \n*G**s**H**s* = *i**d*A. \nIt is also routine to verify, using the definitions of *σ**n* and *V**n*, that\n \n*G**s*(*x**H**s*(*y*)) = *G**s*(*x*)*y* for every *x*, *y* ∈ A. \nBy ([align:tracialrelate]), we have the following relationship for every *x*, *y* ∈ A, and $\\frac{p}{q}\\in\\mathbb{Q}\\cap[0,1]:$\n \n$$\\begin{aligned}\n&=\\tau\\_{g\\_s(p/q)}^{\\mathfrak{A}}(H\\_s(x)y)=\\tau\\_{\\frac{q}{p+sq}}^{\\mathfrak{A}}(H\\_s(x)y)=\\tau\\_{\\frac{q}{p+sq}}\\Big( \\pi\\_{\\frac{q}{p+sq}}(H\\_s(x)y)\\Big)\\\\\n&=\\frac{sq}{p+sq}\\tau\\_{\\frac{p}{q}}\\Big( \\pi\\_{\\frac{p}{q}}(xG\\_s(y))\\Big)=\\frac{sq}{p+sq}\\tau\\_{\\frac{p}{q}}^{\\mathfrak{A}}(xG\\_s(y))=sg\\_s(p/q)\\tau^{\\mathfrak{A}}\\_{\\frac{p}{q}}(xG\\_s(y))\\end{aligned}$$ \nTherefore, by Theorem [thm:statetrace], for any *θ* ∈ [0, 1] we have\n \n*τ**g**s*(*θ*)A(*H**s*(*x*)*y*) = *s**g**s*(*θ*)*τ**θ*A(*x**G**s*(*y*)) \nTherefore, by Corollary [cor:traceideal], it follows that for any *θ* ∈ [0, 1], we have\n \n*G**s*(*I**g**s*(*θ*)) = *I**θ*. \nMoreover, by the description of *Z*(A) given in Theorem [thm:conditionalexpectationdefn] and ([eq:maintracerelationshipall]) it is clear that\n \n*G**s*(*f*) = *f* ∘ *g**s* for every *f* ∈ *C*[0, 1]. \nFor each *s* ≥ 1, define *f**s* ∈ *Z*(A) ≅ *C*[0, 1] as\n \n$$f\\_s(\\theta)=\\frac{\\theta+1}{(\\theta+s)(\\theta+s+1)} \\label{eq:defnofthef\\_s}$$ \nLet us now define $\\widetilde{\\mathbb{G}}:{\\mathfrak{A}}\\rightarrow {\\mathfrak{A}}$ as\n \n$$\\widetilde{\\mathbb{G}}(x)=\\sum\\_{s=1}^\\infty G\\_s(x)f\\_s.$$ \nProof of Theorem [thm:maintheorem]\n==================================\nIn this section we will prove the 5 assertions from Theorem [thm:maintheorem]. First note that Theorem [thm:maintheorem](1) follows from ([eq:VG\\*definition]) and ([eq:Gsandcomposition]), and (2) follows from ([eq:idealcorrespondence]).\nFirst define *ϕ*0 : = *τ*0A. Then, let *θ* ∈ [0, 1] with $\\frac{1}{s+1}<\\theta\\leq \\frac{1}{s}$ for some *s* ≥ 1. Then define\n \n$$\\phi\\_\\theta(x)=\\tau\\_\\theta^{\\mathfrak{A}}(H\\_s(1))^{-1}\\tau\\_\\theta^{\\mathfrak{A}}(H\\_s(1)x)=\\frac{1}{s\\theta}\\tau\\_\\theta^{\\mathfrak{A}}(H\\_s(1)x) \\quad \\textrm{ for every }\\quad x\\in{\\mathfrak{A}}.$$ \nRecall that Gauss measure *μ* on [0, 1] is defined as the probability measure $d\\mu=\\frac{d\\theta}{\\ln 2(\\theta+1)}$, where *d**θ* is Lebesgue measure. Let *ϕ* ∈ S(A) be the direct integral of the states *ϕ**θ* over *μ*, i.e.\n \n*ϕ*(*x*) = ∫01*ϕ**θ*(*x*)*d**μ*(*θ*). \nLet *τ* ∈ T(A) be the unique tracial extension of *μ* provided by Theorem [thm:statetrace]. By uniqueness we have\n \n*τ* = ∫01*τ**θ*A*d**μ*(*θ*). \nNotice that for every *f* ∈ *C*[0, 1] and *x* ∈ A, we have\n \n*ϕ**θ*(*f**x*) = *f*(*θ*)*ϕ**θ*(*x*) and *τ**θ*A(*f**x*) = *f*(*θ*)*ϕ**θ*(*x*) \nIt also follows from ([eq:Gaussmeasurerestriction]) that *ϕ* restricted to *C*[0, 1] is Gauss measure *μ*.\nFor any state *ψ* ∈ S(A), let (*L*2(A, *ψ*), *π**ψ*) denote the GNS representation of *ψ* and ⟨ ⋅ , ⋅ ⟩*ψ* the inner product on *L*2(A, *ψ*). For *x* ∈ A, we will denote by *x**ψ* the image of *x* in *L*2(A, *ψ*) and denote by A*ψ* the dense subspace of *L*2(A, *ψ*) consisting of the *x**ψ*.\nBy the definitions of *ϕ* and *τ*, we can decompose\n \n*L*2(A, *ϕ*) = ∫01*L*2(A, *ϕ**θ*)*d**μ*(*θ*) and *L*2(A, *τ*) = ∫01*L*2(A, *τ**θ*A)*d**μ*(*θ*). \nFurthermore, by ([eq:Gaussmeasurerestriction]) we have\n \n*L*2(*μ*) ⊂ *L*2(A, *ϕ*) and *L*2(*μ*) ⊂ *L*2(A, *τ*) \nas\n \n*f**ϕ* = ∫01*f*(*θ*)1*ϕ**θ**d**μ*(*θ*) and *f**τ* = ∫01*f*(*θ*)1*τ**θ*A*d**μ*(*θ*). \nWe now define an isometry $\\widetilde{V}\\_G:L^2({\\mathfrak{A}},\\tau)\\rightarrow L^2({\\mathfrak{A}},\\phi)$ that satisfies (3)-(5) in Theorem [thm:maintheorem]. As short hand notation, for each vector *η* ∈ *L*2(A, *ϕ*) and Borel set *E* ⊂ [0, 1] we will write\n \n*η*1*E* : = ∫*E**η*(*θ*)*d**μ*(*θ*) ∈ ∫01*L*2(A, *ϕ**θ*)*d**μ*(*θ*) \nFor each *s* ≥ 1, define operators on A*ϕ* and A*τ* respectively as\n \n$$\\widetilde{H}\\_s(x\\_\\tau)=H\\_s(x)\\_\\phi 1\\_{[\\frac{1}{s},\\frac{1}{s+1}]}\\quad \\textrm{and }\\widetilde{G}\\_s(x\\_\\phi)=(G\\_s(x)f\\_s)\\_\\tau.$$ \nClearly these maps are contractive, so they extend to operators on *L*2(A, *ϕ*) and *L*2(A, *τ*) respectively. Now define\n \n$$\\widetilde{V}\\_G=WOT-\\sum\\_{s=1}^\\infty \\widetilde{H}\\_s.$$ \nWe now show that $\\widetilde{V}\\_G$ is an isometry. Let us first recall *f**s* from ([eq:defnofthefs]) and note that\n \n∑*s* = 1∞*f**s*(*θ*) = 1 for every *θ* ∈ [0, 1]. \nWe will implicitly use this fact throughout the rest of the proof of Theorem [thm:maintheorem]. We have,\n \n$$\\begin{aligned}\n {\\langle}\\widetilde{V}\\_G(x\\_\\tau),\\widetilde{V}\\_G(x\\_\\tau){\\rangle}\\_{\\phi}&= \\sum\\_{s=1}^\\infty\\int\\_{\\frac{1}{s+1}}^{\\frac{1}{s}}\\phi\\_\\theta(H\\_s(x)^\\*H\\_s(x))d\\mu(\\theta) \\notag \\\\\n&=\\sum\\_{s=1}^\\infty\\int\\_{\\frac{1}{s+1}}^{\\frac{1}{s}} \\frac{1}{s\\theta}\\tau^{\\mathfrak{A}}\\_\\theta(H\\_s(x^\\*x))d\\mu(\\theta)\\notag \\\\\n&=\\sum\\_{s=1}^\\infty\\int\\_{\\frac{1}{s+1}}^{\\frac{1}{s}} \\tau^{\\mathfrak{A}}\\_{\\frac{1}{\\theta}-s}(x^\\*x)d\\mu(\\theta)\\quad (\\textup{by (\\ref{eq:maintracerelationshipall})}) \\notag \\\\\n&=\\frac{1}{\\ln 2}\\sum\\_{s=1}^\\infty\\int\\_0^1 \\tau\\_u^{\\mathfrak{A}}(x^\\*x)\\frac{u+1}{(u+s)(u+s+1)}\\frac{du}{u+1}\\quad (\\textup{with }u=\\frac{1}{\\theta}-s) \\label{align:V\\_Gisometry} \\\\\n&=\\int\\_0^1\\tau\\_u^{\\mathfrak{A}}(x^\\*x)\\Big(\\sum\\_{s=1}^\\infty f\\_s(u) \\Big)d\\mu(u) \\notag \\\\\n&={\\langle}x\\_\\tau,x\\_\\tau{\\rangle}\\_{\\tau}. \\notag\\end{aligned}$$ \nWe now calculate $\\widetilde{V}\\_G^\\*.$ Let *x*, *y* ∈ A, then\n \n$$\\begin{aligned}\n{\\langle}\\widetilde{V}\\_G(x\\_\\tau),y\\_\\phi{\\rangle}\\_\\phi&=\\sum\\_{s=1}^\\infty \\int\\_{\\frac{1}{s+1}}^{\\frac{1}{s}}\\phi\\_\\theta(y^\\*H\\_s(x))d\\mu(\\theta)\\\\\n&=\\sum\\_{s=1}^\\infty \\int\\_{\\frac{1}{s+1}}^{\\frac{1}{s}}\\frac{1}{s\\theta}\\tau\\_\\theta^{\\mathfrak{A}}(y^\\*H\\_s(x))d\\mu(\\theta)\\\\\n&=\\sum\\_{s=1}^\\infty \\int\\_{\\frac{1}{s+1}}^{\\frac{1}{s}}\\tau\\_{\\frac{1}{\\theta}-s}^{\\mathfrak{A}}(G\\_s(y)^\\*x)d\\mu(\\theta)\\quad (\\textup{by (\\ref{eq:maintracerelationshipall})})\\\\\n&=\\sum\\_{s=1}^\\infty \\int\\_0^1\\tau\\_\\theta^{\\mathfrak{A}}(G\\_s(y)^\\*x)f\\_s(\\theta)d\\mu(\\theta)\\quad (\\textup{Reasoning as in (\\ref{align:V\\_Gisometry})})\\\\\n&=\\sum\\_{s=1}^\\infty \\int\\_0^1\\tau\\_\\theta^{\\mathfrak{A}}(G\\_s(y)^\\*f\\_s x)d\\mu(\\theta)\\quad (\\textup{By (\\ref{eq:Gaussmeasurerestriction})})\\\\\n&={\\langle}x\\_\\tau, \\sum \\widetilde{G}\\_s(y\\_\\phi){\\rangle}\\_\\tau.\\\\\n&={\\langle}x\\_\\tau,\\widetilde{\\mathbb{G}}(y)\\_\\tau{\\rangle}\\_\\tau.\\end{aligned}$$ \nWe now show (3). Let *f* = ∫01*f*(*θ*)1*τ**θ**d**μ*(*θ*) ∈ *L*2(*μ*) ⊂ *L*2(A, *τ*). Then\n \n$$\\begin{aligned}\n\\widetilde{V}\\_G(f)&=\\sum\\_{s=1}^\\infty\\int\\_{\\frac{1}{s+1}}^{\\frac{1}{s}}f(\\frac{1}{\\theta}-s)(H\\_s(1))\\_{\\phi\\_\\theta}d\\mu(\\theta) \\notag \\\\\n&=\\sum\\_{s=1}^\\infty\\int\\_{\\frac{1}{s+1}}^{\\frac{1}{s}}f(\\frac{1}{\\theta}-s)1\\_{\\phi\\_\\theta}d\\mu(\\theta)\\notag \\\\\n&=f\\circ G \\in L^2({\\mathfrak{A}},\\phi).\\label{align:V\\_goncommutative}\\end{aligned}$$ \nSimilarly, one shows that $\\widetilde{V}\\_G^\\*|\\_{L\\_2(\\mu)}=V\\_G^\\*$. This proves (3).\nWe now show (4). It follows from the definition of *G**s* that for every *s* ≥ 1 we have\n \n$$\\int\\_{[\\frac{1}{s+1},\\frac{1}{s}]^c}L^2({\\mathfrak{A}},\\phi\\_\\theta)d\\mu(\\theta)\\subset \\textrm{ker}(\\widetilde{G}\\_s)$$ \nFrom this and ([eq:shiftingGandH]) it follows that for every *x*, *y* ∈ A we have\n \n$$\\begin{aligned}\n \\widetilde{V}\\_G^\\*\\pi\\_\\phi(x)\\widetilde{V}\\_G(y\\_\\tau)&=\\widetilde{V}\\_G^\\*\\Big(\\sum\\_{s=1}^\\infty (xH\\_s(y))\\_\\phi 1\\_{[\\frac{1}{s+1},\\frac{1}{s}]}\\Big)\\\\\n&=\\sum\\_{s=1}^\\infty(G\\_s(xH\\_s(y))f\\_s)\\_\\tau\\\\\n&=\\sum\\_{s=1}^\\infty(G\\_s(x)yf\\_s)\\_\\tau\\quad (\\textup{By (\\ref{eq:shiftingGandH})})\\\\\n&=\\pi\\_\\tau(\\widetilde{\\mathbb{G}}(x))y\\_\\tau.\\end{aligned}$$ \nBy ([align:Vgoncommutative]) we have $\\widetilde{V}\\_G(1\\_\\tau)=1\\_\\phi$, from which it follows that\n \n$$\\phi(x)={\\langle}x\\_\\phi, 1\\_\\phi{\\rangle}={\\langle}x\\_\\phi,\\widetilde{V}\\_G(1\\_\\tau){\\rangle}={\\langle}\\widetilde{\\mathbb{G}}(x)\\_\\tau,1\\_\\tau{\\rangle}=\\tau(\\widetilde{\\mathbb{G}}(x)).$$ \nThis proves (5) and finishes the proof of Theorem [thm:maintheorem].\nAcknowledgment\n==============\nMany thanks are due to Florin Boca for suggesting this problem to me, several helpful conversations, and for providing code for the figures in this article.\n\nA Noncommutative Gauss Map\n==========================\nThe aim of this paper is to transfer the Gauss map, which is a Bernoulli shift for continued fractions, to the noncommutative setting. We feel that a natural place for such a map to act is on the AF algebra A considered separately by F. Boca and D. Mundici. The center of A is isomorphic to *C*[0, 1], so we first consider the action of the Gauss map on *C*[0, 1] and then extend the map to A and show that the extension inherits many desirable properties.\nIntroduction and Notation\n=========================\nFlorin Boca in and Daniele Mundici in separately considered an AF algebra A that is associated with the Farey tessellation. The algebra A exhibits many interesting properties, not the least of which is the connection between A and the unit interval [0, 1]. This connection is not merely topological, but also number theoretic. We briefly explain this connection\nLet *Z*(A) denote the center of A. As noted in, we have *C*[0, 1] ≅ *Z*(A). Moreover, the maximal ideal space of A is homeomorphic (when equipped with the topology induced by Prim(A)) to [0, 1] in a natural way. For each irrational 0 < *θ* < 1, let J*θ* denote the maximal ideal of A associated to *θ*. It was shown in that A/J*θ* ≅ F*θ*, the Effros-Shen algebra, defined in, associated with the continued fraction expansion of *θ*.\nIn other words, if we employ the topological decomposition theory of *C*\\*-algebras and visualize A as continuous, operator-valued functions on its maximal ideal space (a visualization which is usually “incorrect, but fruitful” ), then each function evaluated at *θ* takes values in the Effros-Shen algebra F*θ*. So it is not simply the topology around *θ* that determines this visualization, but also the continued fraction expansion of *θ*.\nGiven the close connection between A and the continued fraction expansions of numbers in [0, 1], it is natural to try and extend important functions from number theory (especially those related to continued fractions) to the *C*\\*-algebra A. The Gauss map might be the most fundamental such function, hence we take it as our starting point\nRecall the Gauss map *G* : [0, 1] → [0, 1] defined by *G*(0) = 0 and *G*(*x*) = 1/*x* − ⌊1/*x*⌋ if *x* ≠ 0, where ⌊ ⋅ ⌋ denotes the greatest integer function. One can think of *G* as the Bernoulli shift for continued fractions. Indeed, given *θ* ∈ [0, 1] with continued fraction expansion *θ* = [*a*1, *a*2, ...] then *G*([*a*1, *a*2, ...]) = [*a*2, *a*3, ...]. One can also recover the continued fraction expansion of *θ* by implementations of *G* and ⌊ ⋅ ⌋.\nWe first consider the induced action of *G* on *Z*(A) = *C*[0, 1]. First note that *f* ∘ *G* ∈ *C*[0, 1] if and only if *f* is a constant function. Therefore we will consider the “adjoint” action of the Gauss map on *C*[0, 1]. Let *μ* denote Gauss measure on [0, 1] defined by $d\\mu=\\frac{d\\theta}{\\ln 2 (\\theta+1)}$, where *d**θ* denotes Lebesgue measure. Then *G* is *μ*-invariant, i.e. *μ*(*G*− 1(*E*)) = *μ*(*E*) for every Borel set *E* ⊆ [0, 1] (see for details). From this it follows that the map\n \n*V**G*(*f*)(*θ*) = *f*(*G*(*θ*)) for *f* ∈ *L*2(*μ*), *θ* ∈ [0, 1] \nis an isometry. A standard calculation reveals that\n \n$$V\\_G^\\*(f)(\\theta)=\\sum\\_{s=1}^\\infty f\\Big( \\frac{1}{\\theta+s}\\Big) \\frac{1+\\theta}{(\\theta+s)(\\theta+s+1)} \\label{eq:V\\_G\\*definition}$$ \nand it is routine to verify that *V**G*\\*(*f*) ∈ *C*[0, 1] when *f* ∈ *C*[0, 1]. We mention that, symbolically, *V**G*\\* is the Perron-Frobenius operator of *G* under *μ* and refer the reader to for details about Perron-Frobenius operators and their connections to continued fractions.\nFurthermore, if we embed *C*[0, 1] into *B*(*L*2(*μ*)) as *f* ↦ M*f* where M*f*(*g*) = *f**g* for all *g* ∈ *L*2(*μ*), then\n \n*V**G*\\*M*f**V**G* = M*V**G*\\*(*f*). \nThis defines a unital completely positive map, which we will henceforth denote by G, on *C*[0, 1] that not only respects the action of *G* on its maximal ideal space, i.e. for each *E* ⊆ [0, 1] let *J**E* denote the ideal of *C*[0, 1] consisting of those functions that vanish on *E*, then\n \nG(*J**G*− 1(*E*)) ⊆ *J**E*, \nbut is also *μ* − invariant, i.e.\n \n∫*f**d**μ* = ∫G(*f*)*d**μ* for all *f* ∈ *C*[0, 1]. \nHence we are looking for an extension of G to A that satisfies the natural analogs of ([eq:conjugationproperty])-([eq:muinvariant]). In order to do this we must first consider what Gauss measure should mean on A. Our first step is proving that every state on *C*[0, 1] has a unique extension to a trace on A (Theorem [thm:statetrace]). D. Mundici showed that the state space of *C*[0, 1] and the space of tracial states on A are affinely, weak\\* homeomorphic. But for our purposes, we will need the extension property from Theorem [thm:statetrace].\nFor reasons that will become clear, we have to slightly modify the natural analogs of ([eq:conjugationproperty]) and ([eq:muinvariant]). In particular, we will use Theorem [thm:statetrace] to define two separate state extensions, *ϕ* and *τ*, of *μ* and intertwine between these two GNS representations to obtain analogs of ([eq:conjugationproperty]) and ([eq:muinvariant]). Let (*π**ϕ*, *L*2(A, *ϕ*)) and (*π**τ*, *L*2(A, *τ*)) be the GNS representations of A associated with *ϕ* and *τ*. Since *ϕ* and *τ* are extensions of *μ*, it follows that *L*2(*μ*) ⊆ *L*2(A, *ϕ*), *L*2(A, *τ*) and\n \n*π**ϕ*(*f*)∣*L*2(*μ*) = M*f* for every *f* ∈ *Z*(A) ≅ *C*[0, 1]. \nThis allows us to prove the main theorem:\n[thm:maintheorem] There is a unital completely positive map $\\widetilde{\\mathbb{G}}:{\\mathfrak{A}}\\rightarrow{\\mathfrak{A}}$ and an isometry $\\widetilde{V}\\_G: L^2({\\mathfrak{A}},\\tau)\\rightarrow L^2({\\mathfrak{A}},\\phi)$ such that\n1. $\\widetilde{\\mathbb{G}}|\\_{C[0,1]}=\\mathbb{G}.$\n2. $\\widetilde{\\mathbb{G}}(\\mathcal{J}(G^{-1}(E)))\\subseteq \\mathcal{J}(E),$ for each *E* ⊂ [0, 1] J*E* are the ideals of A defined in.\n3. $\\widetilde{V}\\_G|\\_{L^2(\\mu)}=V\\_G$ and $\\widetilde{V}\\_G^\\*|\\_{L^2(\\mu)}=V\\_G^\\*.$\n4. $\\widetilde{V}\\_G^\\*\\pi\\_\\phi(x)\\widetilde{V}\\_G=\\pi\\_\\tau(\\widetilde{\\mathbb{G}}(x))$ for *x* ∈ A. Hence $\\widetilde{V}\\_G^\\*\\pi\\_\\phi(f)\\widetilde{V}\\_G|\\_{L^2(\\mu)}=\\mathcal{M}\\_{\\mathbb{G}(f)}$ for *f* ∈ *C*[0, 1].\n5. $\\phi(x)=\\tau(\\widetilde{\\mathbb{G}}(x))$ for *x* ∈ A.\nIn order to set our notation, we now recall some relevant facts about the AF algebra A defined in and. We will use the same notation as in, in particular *p*(*n*, *k*), *q*(*n*, *k*) ∈ Z+ and $r(n,k)=\\frac{p(n,k)}{q(n,k)}$ for *n* ≥ 0 and 0 ≤ *k* ≤ 2*n* all have the same meaning and we will frequently refer to the relationships between them as defined on. Recall that A is the inductive limit of the finite dimensional *C*\\*-algebras,\n \nA*n* = ⨁0 ≤ *k* ≤ 2*n**M**q*(*n*, *k*). \nFor the convenience of the reader, and with thanks to F. Boca for supplying us with the code, we reproduce the Bratelli diagram of A from.\n0.6mm\n(0,120)(0,0) (-120,100)(-120,120)(120,100) (-120,80)(-120,100)(0,80)(120,100)(120,80) (-120,80)(-120,60) (-120,60)(-120,80)(-60,60)(0,80)(60,60)(120,80)(120,60) (0,80)(0,60)\n(-120,40)(-120,60)(-90,40)(-60,60)(-30,40)(0,60)(30,40)(60,60)(90,40)(120,60)(120,40) (-60,60)(-60,40) (0,60)(0,40) (60,60)(60,40) (-120,20)(-120,40)(-105,20)(-90,40)(-75,20)(-60,40)(-45,20)(-30,40)(-15,20)(0,40)(15,20) (30,40)(45,20)(60,40)(75,20)(90,40)(105,20)(120,40)(120,20) (-90,40)(-90,20) (-60,40)(-60,20) (-30,40)(-30,20) (0,40)(0,20) (30,40)(30,20) (60,40)(60,20) (90,40)(90,20)\n(-120,5)(-120,20)(-112.5,5)(-105,20)(-97.5,5)(-90,20)(-82.5,5)(-75,20)(-67.5,5)(-60,20) (-52.5,5)(-45,20)(-37.5,5)(-30,20)(-22.5,5)(-15,20)(-7.5,5)(0,20) (120,5)(120,20)(112.5,5)(105,20)(97.5,5)(90,20)(82.5,5)(75,20)(67.5,5)(60,20) (52.5,5)(45,20)(37.5,5)(30,20)(22.5,5)(15,20)(7.5,5)(0,20)\n(-105,20)(-105,5) (-90,20)(-90,5) (-75,20)(-75,5) (-60,20)(-60,5) (-45,20)(-45,5) (-30,20)(-30,5) (-15,20)(-15,5) (0,20)(0,5) (105,20)(105,5) (90,20)(90,5) (75,20)(75,5) (60,20)(60,5) (45,20)(45,5) (30,20)(30,5) (15,20)(15,5)\n(-125,100)(0,0)$\\frac{0}{1}$ (0,87)(0,0)$\\frac{1}{2}$ (125,100)(0,0)$\\frac{1}{1}$ (-125,80)(0,0)$\\frac{0}{1}$ (125,80)(0,0)$\\frac{1}{1}$ (-125,60)(0,0)$\\frac{0}{1}$ (-60,66)(0,0)$\\frac{1}{3}$ (-4,63)(0,0)$\\frac{1}{2}$ (60,66)(0,0)$\\frac{2}{3}$ (125,60)(0,0)$\\frac{1}{1}$ (-125,60)(0,0)$\\frac{0}{1}$\n(-125,40)(0,0)$\\frac{0}{1}$ (-90,46)(0,0)$\\frac{1}{4}$ (-65,40)(0,0)$\\frac{1}{3}$ (-30,46)(0,0)$\\frac{2}{5}$ (-5,40)(0,0)$\\frac{1}{2}$ (30,46)(0,0)$\\frac{3}{5}$ (65,40)(0,0)$\\frac{2}{3}$ (90,46)(0,0)$\\frac{3}{4}$ (125,40)(0,0)$\\frac{1}{1}$\n(-120,100)(0,0) • (120,100)(0,0) • (-120,80)(0,0) • (0,80)(0,0) • (120,80)(0,0) • (-120,60)(0,0) • (-60,60)(0,0) • (0,60)(0,0) • (60,60)(0,0) • (120,60)(0,0) • \n(-120,40)(0,0) • (-90,40)(0,0) • (-60,40)(0,0) • (-30,40)(0,0) • (0,40)(0,0) • (30,40)(0,0) • (60,40)(0,0) • (90,40)(0,0) • (120,40)(0,0) • \n(-125,20)(0,0)$\\frac{0}{1}$ (-105,27)(0,0)$\\frac{1}{5}$ (-95,20)(0,0)$\\frac{1}{4}$ (-75,27)(0,0)$\\frac{2}{7}$ (-65,20)(0,0)$\\frac{1}{3}$ (-45,27)(0,0)$\\frac{3}{8}$ (-35,20)(0,0)$\\frac{2}{5}$ (-15,27)(0,0)$\\frac{3}{7}$ (-5,20)(0,0)$\\frac{1}{2}$ (125,20)(0,0)$\\frac{1}{1}$ (105,27)(0,0)$\\frac{4}{5}$ (95,20)(0,0)$\\frac{3}{4}$ (75,27)(0,0)$\\frac{5}{7}$ (65,20)(0,0)$\\frac{2}{3}$ (45,27)(0,0)$\\frac{5}{8}$ (35,20)(0,0)$\\frac{3}{5}$ (15,27)(0,0)$\\frac{4}{7}$ (-120,120)(0,0) ⋆ \n(-120,20)(0,0) • (-105,20)(0,0) • (-90,20)(0,0) • (-75,20)(0,0) • (-60,20)(0,0) • (-45,20)(0,0) • (-30,20)(0,0) • (-15,20)(0,0) • (0,20)(0,0) • (120,20)(0,0) • (105,20)(0,0) • (90,20)(0,0) • (75,20)(0,0) • (60,20)(0,0) • (45,20)(0,0) • (30,20)(0,0) • (15,20)(0,0) • \n(-125,0)(0,0)$\\frac{0}{1}$ (-112.5,-2)(0,0)$\\frac{1}{6}$ (-105,-2)(0,0)$\\frac{1}{5}$ (-97.5,-2)(0,0)$\\frac{2}{9}$ (-90,-2)(0,0)$\\frac{1}{4}$ (-82.5,-2)(0,0)$\\frac{3}{11}$ (-75,-2)(0,0)$\\frac{2}{7}$ (-67.5,-2)(0,0)$\\frac{3}{10}$ (-60,-2)(0,0)$\\frac{1}{3}$ (-52.5,-2)(0,0)$\\frac{4}{11}$ (-45,-2)(0,0)$\\frac{3}{8}$ (-37.5,-2)(0,0)$\\frac{5}{13}$ (-30,-2)(0,0)$\\frac{2}{5}$ (-22.5,-2)(0,0)$\\frac{5}{12}$ (-15,-2)(0,0)$\\frac{3}{7}$ (-7.5,-2)(0,0)$\\frac{4}{9}$ (0,-2)(0,0)$\\frac{1}{2}$ (125,0)(0,0)$\\frac{1}{1}$ (112.5,-2)(0,0)$\\frac{5}{6}$ (105,-2)(0,0)$\\frac{4}{5}$ (97.5,-2)(0,0)$\\frac{7}{9}$ (90,-2)(0,0)$\\frac{3}{4}$ (82.5,-2)(0,0)$\\frac{8}{11}$ (75,-2)(0,0)$\\frac{5}{7}$ (67.5,-2)(0,0)$\\frac{7}{10}$ (60,-2)(0,0)$\\frac{2}{3}$ (52.5,-2)(0,0)$\\frac{7}{11}$ (45,-2)(0,0)$\\frac{5}{8}$ (37.5,-2)(0,0)$\\frac{8}{13}$ (30,-2)(0,0)$\\frac{3}{5}$ (22.5,-2)(0,0)$\\frac{7}{12}$ (15,-2)(0,0)$\\frac{4}{7}$ (7.5,-2)(0,0)$\\frac{5}{9}$\n(-120,5)(0,0) • (-112.5,5)(0,0) • (-105,5)(0,0) • (-97.5,5)(0,0) • (-90,5)(0,0) • (-82.5,5)(0,0) • (-75,5)(0,0) • (-67.5,5)(0,0) • (-60,5)(0,0) • (-52.5,5)(0,0) • (-45,5)(0,0) • (-37.5,5)(0,0) • (-30,5)(0,0) • (-22.5,5)(0,0) • (-15,5)(0,0) • (-7.5,5)(0,0) • (0,5)(0,0) • (120,5)(0,0) • (112.5,5)(0,0) • (105,5)(0,0) • (97.5,5)(0,0) • (90,5)(0,0) • (82.5,5)(0,0) • (75,5)(0,0) • (67.5,5)(0,0) • (60,5)(0,0) • (52.5,5)(0,0) • (45,5)(0,0) • (37.5,5)(0,0) • (30,5)(0,0) • (22.5,5)(0,0) • (15,5)(0,0) • (7.5,5)(0,0) • \n[Figure1]\nFor each *n* ≥ 0 let E*n* : A → A*n* be conditional expectations such that\n \nE*n*E*m* = E*m*E*n* for all *n*, *m* ≥ 0. \nThe existence of such conditional expectations is guaranteed by Arveson’s extension theorem, or since A is AF, one can construct such maps explicitly. Furthermore for 0 ≤ *k* ≤ 2*n* let E(*n*, *k*) : A → *M**q*(*n*, *k*) ⊆ A*n* be conditional expectations, such that\n \nE(*n*, *k*)E*n* = E*n*E(*n*, *k*) for *n* ≥ 0 and 0 ≤ *k* ≤ 2*n*. \nNote that for each *x* ∈ A we have\n \nlim*n* → ∞E*n*(*x*) = *x*. \nWe will use the following notation throughout: For a unital *C*\\*-algebra *A*, we let\n1. *Z*(*A*) denote the center of *A*,\n2. S(*A*) denote the state space of *A*,\n3. T(*A*) denote the set of all unital traces of *A*.\n4. *M**n* denote *n* × *n* matrices over C,\n5. *τ**n* the unital trace on *M**n* and 1*n* ∈ *M**n* the identity.\n6. For $\\frac{p}{q}\\in\\mathbb{Q}\\cap[0,1]$ in reduced form we define\n \n$$M\\_{\\frac{p}{q}}:=M\\_q\\quad \\tau\\_{\\frac{p}{q}}:=\\tau\\_q \\in \\mathcal{T}(M\\_q).$$\nState Extensions and Conditional Expectation onto *Z*(A)\n========================================================\nIn this section we will construct a conditional expectation from A onto *Z*(A) that preserves every *τ* ∈ T(A). This will provide the key step in the proof of Theorem [thm:statetrace].\n[defn:rationaltraces] Let *n* ≥ 0 and 0 ≤ *k* ≤ 2*n*. Define *τ*(*n*, *k*) ∈ T(A) as\n \n*τ*(*n*, *k*)(*x*) = *τ**q*(*n*, *k*) ∘ E(*n*, *k*)(*x*). \nThe following lemma is immediate from ([eq:commutingexpectations]) and ([eq:commutingfdexpectations]):\n[lem:welldefinedfx] For *n* ≥ 0, 0 ≤ *k* ≤ 2*n* and ℓ ≥ 0, we have\n \n*τ*(*n*, *k*) = *τ*(*n* + ℓ, 2ℓ*k*). \nLet *x* ∈ A. Define the function *f**x* : Q ∩ [0, 1] → C as\n \n*f**x*(*r*(*n*, *k*)) = *τ*(*n*, *k*)(*x*) for *n* ≥ 0and 0 ≤ *k* ≤ 2*n*. \nThen *f**x* is well-defined and extends to a continuous function on [0, 1].\nIf *r*(*n*, *k*) = *r*(*n*ʹ, *k*ʹ) with *n*ʹ ≥ *n* then there is an ℓ ≥ 0 such that *n*ʹ = *n* + ℓ and *k*ʹ = 2ℓ*k*. Hence *f**x* is well-defined by Lemma [lem:welldefinedfx].\nLet *n* ≥ 0 and 0 < *k* < 2*n*. By the relationships defined in, the following function is continuous and piecewise affine on [0, 1]:\n \n$$B\\_{(n,k)}(\\theta)=\\left\\{ \\begin{array}{ll}0 & \\textrm{ if }0\\leq \\theta\\leq r(n,k-1)\\\\\nq(n,k)\\Big(q(n,k-1)\\theta-p(n,k-1)\\Big) & \\textrm{ if }r(n,k-1)\\leq \\theta\\leq r(n,k)\\\\\nq(n,k)\\Big(p(n,k+1)-q(n,k+1)\\theta\\Big) &\\textrm{ if }r(n,k)\\leq \\theta\\leq r(n,k+1)\\\\\n0& \\textrm{ if }r(n,k+1)\\leq \\theta\\leq 1\n\\end{array} \\right.$$ \nWe first let *x* ∈ A*n* ⊂ A and prove that *f**x* extends to a continuous function on [0, 1]. Suppose first that 0 < 2*k* + 1 < 2*n* and E(*n*, 2*k* + 1)(*x*) = *x*. Without loss of generality suppose that *τ*(*n*, 2*k* + 1)(*x*) = 1. We show that *f**x* = *B*(*n*, 2*k* + 1).\nIt is clear that *f**x*∣[0, *r*(*n*, 2*k*)] ∪ [*r*(*n*, 2*k* + 2), 1] ≡ 0. We now show by induction on ℓ ≥ 0 that\n \n(∀ℓ ≥ 0)(∀2ℓ2*k* < *j* < 2ℓ(2*k* + 2))(*f**x*(*r*(*n* + ℓ, *j*)) = *B*(*n*, 2*k* + 1)(*r*(*n* + ℓ, *j*))). \nFor ℓ = 0, we have *f**x*(*r*(*n*, 2*k* + 1)) = *τ*(*n*, 2*k* + 1)(*x*) = 1 = *B*(*n*, 2*k* + 1)(*r*(*n*, 2*k* + 1)). Suppose now that ([eq:inductionhypo]) holds for ℓ ≥ 0 and prove ([eq:inductionhypo]) for ℓ + 1.\nIf *j* = 2*i* is even, then\n \n$$\\begin{aligned}\n &f\\_x(r(n+\\ell+1,2i))=\\tau\\_{(n+\\ell+1,2i)}(x)=\\tau\\_{(n+\\ell,i)}(x)=f\\_x(r(n+\\ell,i))\\\\\n=&B\\_{(n,2k+1)}(r(n+\\ell,i))=B\\_{(n,2k+1)}(r(n+\\ell+1,2i)).\\end{aligned}$$ \nIf *j* = 2*i* + 1 is odd, then\n \n$$\\begin{aligned}\n &f\\_x(r(n+\\ell+1,2i+1))=\\tau\\_{(n+\\ell+1,2i+1)}(x)\\\\\n&=\\frac{q(n+\\ell,i)}{q(n+\\ell+1,2i+1)}\\tau\\_{(n+\\ell,i)}(x)+\\frac{q(n+\\ell,i+1)}{q(n+\\ell+1,2i+1)}\\tau\\_{(n+\\ell,i+1)}(x)\\\\\n&=\\frac{q(n+\\ell,i)}{q(n+\\ell+1,2i+1)}B\\_{(n,2k+1)}(r(n+\\ell,i))+\\frac{q(n+\\ell,i+1)}{q(n+\\ell+1,2i+1)}B\\_{(n,2k+1)}(r(n+\\ell,i+1))\\\\\n&=B\\_{(n,2k+1)}(r(n+\\ell+1,2i+1)).\\end{aligned}$$ \nHere the last line follows by the relationships in and because *B*(*n*, 2*k* + 1) is piecewise affine. This shows that ([eq:inductionhypo]) holds, hence *f**x* extends to a continuous function on [0, 1].\nNow suppose that 0 < 2*m**k* < 2*n* with *k* odd and E(*n*, 2*m**k*)(*x*) = *x*. Then,\n \n*x* = E(*n* − *m*, *k*)(*x*) − E(*n* − *m* + 1, 2*k* − 1)(*x*) − E(*n* − *m* + 1, 2*k* + 1)(*x*). \nSo, by the first part of the proof it follows that *f**x* is continuous.\nFor *x* = 1 ⊕ 0 ⊕ ⋯ ⊕ 0, 0 ⊕ ⋯ ⊕ 0 ⊕ 1 ∈ A*n*, the proof that *f**x* is continuous is exactly the same as above, so we omit the proof. This shows that for every *n* ≥ 0 and each *x* ∈ A*n* that *f**x* is continuous. Moreover note that the linear map *x* ↦ *f**x* defined on ⋃*n* = 1∞A*n* is contractive, hence *f**x* is continuous for every *x* ∈ A.\nAs observed in, *Z*(A) ≅ *C*[0, 1]. We now construct an explicit isomorphism. For each *n* ≥ 0, define Z*n* : *C*[0, 1] → *Z*(A*n*) ⊂ A by\n \nZ*n*(*f*) = ⨁0 ≤ *k* ≤ 2*n**f*(*r*(*n*, *k*))1*q*(*n*, *k*) \nBy, for each *n* ≥ 0 we have max{∣*r*(*n*, *k*) − *r*(*n*, *k* + 1)∣ : 0 ≤ *k* < 2*n*} = 1/(*n* + 1). Hence for *m* ≥ *n* we have\n \n‖Z*n*(*f*) − Z*m*(*f*)‖ ≤ sup{∣*f*(*θ*) − *f*(*θ*ʹ)∣ : ∣*θ* − *θ*ʹ∣ ≤ 1/(*n* + 1)}. \nTherefore Z*n*(*f*) is a Cauchy sequence in A because *f* is uniformly continuous on [0, 1].\nDefine Z : *C*[0, 1] → A by\n \nZ(*f*) = lim*n* → ∞Z*n*(*f*). \n[thm:conditionalexpectationdefn] The map Z : *C*[0, 1] → *Z*(A) is a \\*-isomorphism. Moreover the map E*Z* : A → *Z*(A) defined by\n \nE*Z*(*x*) = Z(*f**x*) \nis a conditional expectation such that\n \n*τ*(E*Z*(*x*)) = *τ*(*x*) for every *τ* ∈ T(A). \nBy ([eq:Tdefn]) it is clear that Z is a \\*-monomorphism, and since Z*n*(*f*) ∈ *Z*(A*n*) for each *n* ≥ 0, it follows that Z(*f*) ∈ *Z*(A). We now show that Z is surjective. Let *n* ≥ 0 and *y* ∈ A*n*. Then\n \n*y* ∈ *Z*(A*n*) if and only if *y* = ⨁0 ≤ *k* ≤ 2*n**τ*(*n*, *k*)(*y*)1*q*(*n*, *k*). \nLet *x* ∈ *Z*(A). By ([eq:defnoffx]) and ([eq:defofZn]) it follows that\n \nZ*n*(*f**x*) = ⨁0 ≤ *k* ≤ 2*n**τ*(*n*, *k*)(*x*)1*q*(*n*, *k*) ∈ *Z*(A*n*). \nSince *x* ∈ *Z*(A), it follows from ([eq:convergencethroughexpectations]) that\n \nlim*n* → ∞*d**i**s**t*(E*n*(*x*), *Z*(A*n*)) = 0, \nfrom which we deduce by ([eq:charofcenterinfdalgebras]) that Z*n*(*f**x*) → *x*. Therefore\n \nE*Z*(*x*) = Z(*f**x*) = lim*n* → ∞Z*n*(*f**x*) = *x*. \nThis shows that Z is surjective and also that E*Z* is a conditional expectation. We now show that E*Z* preserves every trace of A. Let *τ* ∈ T(A). By ([eq:convergencethroughexpectations]) it follows that *τ* is the weak\\*-limit of *τ* ∘ E*n*. Since *τ* ∘ E*n*∣A*n* ∈ T(A*n*), there is a convex combination of scalars (*λ*(*n*, *k*))0 ≤ *k* ≤ 2*n* such that\n \n*τ* ∘ E*n* = ∑0 ≤ *k* ≤ 2*n**λ*(*n*, *k*)*τ*(*n*, *k*). \nIt follows that T(A) equals the weak\\* closure of the convex hull of the set {*τ*(*n*, *k*) : *n* ≥ 0, 0 ≤ *k* ≤ 2*n*}. Therefore, we only need to check ([eq:tracepreserving]) for the traces *τ*(*n*, *k*). To this end, let *x* ∈ A then\n \n*τ*(*n*, *k*)(*x*) = *f**x*(*r*(*n*, *k*)) = *τ*(*n*, *k*)(Z*n*(*f**x*)) = *τ*(*n*, *k*)(Z(*f**x*)) = *τ*(*n*, *k*)(E*Z*(*x*)). \n[thm:statetrace] The restriction map *τ* ↦ *τ*∣*Z*(A) defines a weak\\* homeomorphism from T(A) onto S(*C*[0, 1]). In particular, every state on *Z*(A) has a unique tracial extension to A.\nInjectivity and weak\\*-continuity of the inverse both follow from ([eq:tracepreserving]). By ([eq:diracpointmassesabound]) it follows that the restriction of *τ*(*n*, *k*) to *Z*(A) ≅ *C*[0, 1] is the Dirac measure *δ*{*r*(*n*, *k*)}, which shows surjectivity.\nIdeals of A and traces of A\n===========================\nFix *θ* ∈ [0, 1]. Define *τ**θ*A ∈ T(A) as the unique tracial extension of the Dirac measure *δ*{*θ*} ∈ S(*C*[0, 1]) given by Theorem [thm:statetrace]. Note that for each *n* ≥ 0 and 0 ≤ *k* ≤ 2*n*, we have *τ**r*(*n*, *k*)A = *τ*(*n*, *k*) from Definition [defn:rationaltraces].\nFor each *θ* ∈ [0, 1], we recall the maximal ideals *I**θ* ⊂ A defined in. The following is a consequence of the proof of and the correspondence made in Theorem [thm:statetrace].\n[cor:traceideal] Fix *θ* ∈ [0, 1]. Then\n \n*I**θ* = {*x* ∈ A : *τ**θ*A(*x*\\**x*) = 0}. \nFix $\\frac{p(n,k)}{q(n,k)}=\\frac{p}{q}\\in\\mathbb{Q}\\cap (0,1)$ in reduced form. We define the \\*-homomorphism\n \n$$\\pi\\_{\\frac{p}{q}}:{\\mathfrak{A}}\\rightarrow M\\_{\\frac{p}{q}} \\label{eq:rationalhomo}$$ \nas “evaluation along the path *r*(*n*, *k*), *r*(*n* + 1, 2*k*), ..., *r*(*n* + ℓ, 2ℓ*k*), ... in the Bratteli diagram.” In particular, $\\textrm{ker}(\\pi\\_{\\frac{p}{q}})=I\\_{\\frac{p}{q}}$ (see for details). We note that\n \n$$\\tau^{\\mathfrak{A}}\\_{\\frac{p}{q}}(x)=\\tau\\_{\\frac{p}{q}}(\\pi\\_{\\frac{p}{q}}(x))\\quad \\textrm{ for every }x\\in{\\mathfrak{A}}. \\label{eq:tracesthroughrepresentations}$$ \nConstruction of $\\widetilde{\\mathbb{G}}$\n========================================\nIn this section we construct our noncommutative Gauss map $\\widetilde{\\mathbb{G}}:{\\mathfrak{A}}\\rightarrow {\\mathfrak{A}}.$ Let *s* ≥ 1. As in we define\n \n$$\\mathcal{J}\\_s:=\\mathcal{J}([\\frac{1}{s+1},\\frac{1}{s}])=\\bigcap\\_{\\theta\\in [\\frac{1}{s+1},\\frac{1}{s}]}I\\_\\theta.$$ \nBy Theorem [thm:statetrace] and Section [sec:ideals] we have\n \n$$\\mathcal{J}\\_s=\\textrm{ker}\\Big(\\bigoplus\\_{\\frac{1}{s+1}<\\frac{p}{q}\\in\\mathbb{Q}<\\frac{1}{s}}\\pi\\_{\\frac{p}{q}}\\Big). \\label{eq:anotherdescriptionJ\\_s}$$ \nFor each *s* ≥ 1 the Bratelli diagram of A/J*s* is the subdiagram of the Bratelli diagram of A obtained by deleting all of the nodes\n \n{*r*(*n*, *k*) : *r*(*n*, *k*) ∉ [1/(*s* + 1), 1/*s*]} ∪ {*r*(*n*, *k*) : *n* < *s*}, \nand deleting all edges connected to any of these nodes. See Figure [Figure2] for the Bratelli diagram of A/J2.\n0.6mm\n(-30,60)(0,0)\n(-60,60)(-30,40)(0,60) (-60,60)(-60,40) (0,60)(0,40) (-60,40)(-45,20)(-30,40)(-15,20)(0,40) (-60,40)(-60,20) (-30,40)(-30,20) (0,40)(0,20)\n(-60,20) (-52.5,5)(-45,20)(-37.5,5)(-30,20)(-22.5,5)(-15,20)(-7.5,5)(0,20)\n(-60,20)(-60,5) (-45,20)(-45,5) (-30,20)(-30,5) (-15,20)(-15,5) (0,20)(0,5)\n(-65,60)(0,0)$\\frac{1}{3}$ (5,60)(0,0)$\\frac{1}{2}$\n(-65,40)(0,0)$\\frac{1}{3}$ (-30,46)(0,0)$\\frac{2}{5}$ (5,40)(0,0)$\\frac{1}{2}$\n(-60,60)(0,0) • (0,60)(0,0) • \n(-60,40)(0,0) • (-30,40)(0,0) • (0,40)(0,0) • \n(-65,20)(0,0)$\\frac{1}{3}$ (-45,27)(0,0)$\\frac{3}{8}$ (-35,20)(0,0)$\\frac{2}{5}$ (-15,27)(0,0)$\\frac{3}{7}$ (5,20)(0,0)$\\frac{1}{2}$\n(-60,20)(0,0) • (-45,20)(0,0) • (-30,20)(0,0) • (-15,20)(0,0) • (0,20)(0,0) • \n(-60,-2)(0,0)$\\frac{1}{3}$ (-52.5,-2)(0,0)$\\frac{4}{11}$ (-45,-2)(0,0)$\\frac{3}{8}$ (-37.5,-2)(0,0)$\\frac{5}{13}$ (-30,-2)(0,0)$\\frac{2}{5}$ (-22.5,-2)(0,0)$\\frac{5}{12}$ (-15,-2)(0,0)$\\frac{3}{7}$ (-7.5,-2)(0,0)$\\frac{4}{9}$ (0,-2)(0,0)$\\frac{1}{2}$\n(-60,5)(0,0) • (-52.5,5)(0,0) • (-45,5)(0,0) • (-37.5,5)(0,0) • (-30,5)(0,0) • (-22.5,5)(0,0) • (-15,5)(0,0) • (-7.5,5)(0,0) • (0,5)(0,0) • \n[Figure2]\nFor each *s* ≥ 1, define the homeomorphism *g**s* : [0, 1] → [1/(*s* + 1), 1/*s*] as\n \n$$g\\_s(\\theta)=\\frac{1}{\\theta+s}, \\label{eq:defng\\_s}$$ \nand recall that these maps are the building blocks for the commutative Gauss map G : *C*[0, 1] → *C*[0, 1] defined in ([eq:VG\\*definition]). Then consider the induced isomorphism\n \n(*g**s*)\\* : *C*[1/(*s* + 1), 1/*s*] → *C*[0, 1] defined by (*g**s*)\\*(*f*) = *f* ∘ *g**s*. \nSince our goal is to extend G to a map on A, we first consider extensions of the maps (*g**s*)\\* as maps from A/J*s* into A. Unfortunately, there is no hope for these extensions to also be isomorphisms. Indeed, by considering the Bratelli diagrams of A and A/J*s* it is clear that *K*0(A) ≅ *K*0(A/J*s*), but there is no unit-preserving, positive homomorphism that implements this isomorphism. Hence ${\\mathfrak{A}}\\not\\cong {\\mathfrak{A}}/\\mathcal{J}\\_s.$ We do the next best thing by defining a (non-unital) \\*-monomorphism *H**s* : A → A/J*s* and a unital completely positive (UCP for short) map *G**s* : A/J*s* → A such that *G**s**H**s* = *i**d*A, and such that *G**s* is an extension of (*g**s*)\\*. More importantly, the maps *G**s* and *H**s* will provide a nice relationship (see ([eq:maintracerelationshipall])) between T(A) and T(A/J*s*).\nFor *n* ≥ 0, let *A**n* ∈ *M*2*n* + 1 + 1, 2*n* + 1(Z+) be the connecting homomorphisms from A*n* into A*n* + 1 such that\n \n$${\\mathfrak{A}}=\\varinjlim({\\mathfrak{A}}\\_n, A\\_n) \\label{eq:Fareyasinductivelimit}$$ \nFor example we have,\n \n$$A\\_0=\\left[ \\begin{array}{cc} 1&0\\\\ 1&1\\\\ 0&1\\\\ \\end{array}\\right]\\in M\\_{3,2}\\textrm{ },\\quad A\\_1=\\left[ \\begin{array}{ccc} 1&0&0\\\\ 1&1&0 \\\\ 0&1&0\\\\ 0&1&1\\\\ 0&0&1\\\\ \\end{array}\\right]\\in M\\_{5,3}\\textrm{ }, \\cdots$$ \nFor *n* ≥ 0, we define\n \n$$({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n:= \\bigoplus\\_{0\\leq k\\leq 2^n}M\\_{\\frac{q(n,k)}{p(n,k)+sq(n,k)}}=\\bigoplus\\_{0\\leq k\\leq 2^n}M\\_{g\\_s(r(n,k))}.$$ \nBy the description of the Bratteli diagram of A/J*s* (see also Figure [Figure2]) given above it follows that\n \n$${\\mathfrak{A}}/\\mathcal{J}\\_s=\\varinjlim(({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n, A\\_n) \\label{eq:quotientasinductivelimit}$$ \nLet ℓ∞(*s*) denote the *s* dimensional, commutative *C*\\*-algebra. Consider the *C*\\*-algebra,\n \n$$\\ell^\\infty(s)\\otimes {\\mathfrak{A}}=\\varinjlim (\\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_n, id\\_{\\ell^\\infty(s)}\\otimes A\\_n)$$ \nDefine $S=\\left[ \\begin{array}{llll} 1&1&\\cdot&1 \\end{array}\\right]\\in M\\_{1,s}.$ It is easy to see (using only the fact that *A**n* ∈ *M*2*n* + 1 + 1, 2*n* + 1(Z+)) that\n \n*A**n*(*S* ⊗ 12*n* + 1) = (*S* ⊗ 12*n* + 1 + 1)1*s* ⊗ *A**n* for every *n* ≥ 0. \nHence, for each *n* ≥ 0 we are able to define a \\*-homomorphism\n \n$$\\sigma\\_n: \\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_n=\\bigoplus\\_{0\\leq k\\leq 2^n}\\ell^\\infty(s)\\otimes M\\_{\\frac{p(n,k)}{q(n,k)}}\\rightarrow \\bigoplus\\_{0\\leq k\\leq 2^n}M\\_{\\frac{q(n,k)}{p(n,k)+sq(n,k)}}=({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n$$ \ngiven by the matrix *S* ⊗ 12*n* + 1 ∈ *M*2*n* + 1, *s*(2*n* + 1) such that the following diagram commutes for every *n* ≥ 0\n \n$$\\xymatrix{ ({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_n \\ar[r]^{A\\_n} & ({\\mathfrak{A}}/\\mathcal{J}\\_s)\\_{n+1} \\\\\n\\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_n \\ar[u]^{\\sigma\\_n} \\ar[r]^{1\\_s\\otimes A\\_n} & \\ell^\\infty(s)\\otimes {\\mathfrak{A}}\\_{n+1} \\ar[u]^{\\sigma\\_{n+1}} } \\label{eq:commutingsquare1}$$ \nLet *e*1, ..., *e**s* ∈ ℓ∞(*s*) denote the standard basis. For each *n* ≥ 0 define the UCP map *V**n* : (A/J)*n* → ℓ∞(*s*) ⊗ A*n* by\n \n$$V\\_n(x)=\\sigma\\_n^{-1}\\Big( \\sum\\_{i=1}^s \\sigma\\_n(e\\_i\\otimes 1\\_{{\\mathfrak{A}}\\_n})x\\sigma\\_n(e\\_i\\otimes 1\\_{{\\mathfrak{A}}\\_n})\\Big)$$ \nDefine *ψ**s* ∈ S(ℓ∞(*s*)) by\n \n$$\\psi\\_s\\Big(\\sum\\_{i=1}^s \\alpha\\_ie\\_i\\Big)=\\frac{1}{s}\\sum\\_{i=1}^s \\alpha\\_i.$$ \nIt now follows from ([eq:commutingsquare1]) that the following diagram commutes for all *n*�"},"data_id":{"kind":"string","value":"arxiv_0002436"}}},{"rowIdx":590,"cells":{"content":{"kind":"string","value":"Two-way sparsity for time-varying networks, with applications in genomics\n=========================================================================\n,\nWe propose a novel way of modelling time-varying networks, by inducing two-way sparsity on local models of node connectivity. This two-way sparsity separately promotes sparsity across time and sparsity across variables (within time). Separation of these two types of sparsity is achieved through a novel prior structure, which draws on ideas from the Bayesian lasso and from copula modelling. We provide an efficient implementation of the proposed model via a Gibbs sampler, and we apply the model to data from neural development. In doing so, we demonstrate that the proposed model is able to identify changes in genomic network structure that match current biological knowledge. Such changes in genomic network structure can then be used by neuro-biologists to identify potential targets for further experimental investigation.\nIntroduction\n============\nNetwork models have become an important topic in modern statistics, and the evolution of network structure over time (illustrated in Figure [dynNetFig]) is an important area of study. Network structures that evolve over time naturally occur in a range of applications. Examples of recent applications include evolving patterns of human interaction such as in social networks, time-varying patterns of interaction between genes and their protein-products in biological networks, and time-varying patterns of connectivity in the brain. However, network models with temporal structure have only recently begun to be studied in detail in statistical research.\n\n[dynNetFig]\nAn important application area of statistical network models is genomics. Network models are a natural way to describe and analyse patterns of interactions (represented by network edges) between genes and their protein-products (represented by network nodes). An important interaction of this type is gene regulation, in which the protein-product of one gene influences the output level of the protein-product of another gene. Much gene regulation is characteristic of a particular cell type, so that a cell knows its role within the organism. These specific regulatory network structures that are characteristic of particular cell-types are established during embryonic development. Changes in normal gene regulation are also inherent to cancer progression, so that cells forget how they should act, taking on pathological roles (regulatory network re-wiring). However, whilst network models are well established in genomics, historically these models have typically been static, ignoring the fact that genomic processes are inherently time-varying.\nThere are many examples of recent work on models of time-varying networks. In statistics, this work covers methods based on Markov processes, on dynamic Erds-Rényi graphs, and on sparse regression methods. It also includes work on time-varying community structure, on methods extending the stochastic block model, and related non-parametric graphon-based methods, as well as non-parametric methods for dynamic link prediction and methods from Bayesian nonparametrics. Other related work includes sparse graphical models that can take account of different time-points.\nMotivated by genomics applications, we propose a novel framework for modelling time-varying networks, by inducing *two-way sparsity* on local models of the connectivity of each node to all the others. This is achieved as follows. We start with a regression likelihood function that assumes that observations are mutually independent over time. Dependence is then induced through a novel prior structure that promotes sparsity in a two ways: across time, and within time. This decoupling of the induced sparsity is achieved through a copula specification for the parameters in the likelihood function. Specifically, the regression coefficients for one node across different time-points are jointly distributed according to a Gaussian copula with Laplace marginal distributions. The correlation matrix of the Gaussian copula is formed by assuming that the correlation between time-points decays with time in a structured, parsimonious way that also ensures its positive definiteness. In this correlation matrix, the only free parameter is the correlation between consecutive time-points, which is given a reverse-exponential prior distribution with support in [0, 1). This prior on the correlation across time discourages large differences in the regression coefficients between consecutive time-points and, as a consequence, also discourages large changes in the inferred structure of the network.\nThe decoupling of the marginal and dependence structure that is facilitated by the copula specification, and the particular form of the correlation matrix, allow for precise control of marginal priors. This decoupling also makes the adoption of generalisations of the Bayesian lasso, such as the horseshoe, easy to implement in place of the marginal Bayesian lasso prior that we use. The prior dependence among parameters across time can also be viewed as a Bayesian version of the fused lasso, while within each time-slice we directly utilise existing work on the Bayesian lasso. In fact, the proposed modelling framework has the Bayesian lasso as a special case, when the correlation between time-points is set to zero. From a frequentist point of view, the sparsity structure we propose would fall within the remit of the generalised lasso, which has the fused lasso as a special case. Bayesian versions of the fused lasso have also been proposed. However, a key difference between those methods and the modelling framework we propose, is the formal decoupling of sparsity across time (which the fused lasso induces), from sparsity within time. Importantly, we are able to apply this proposed modelling framework locally to each network node, as previous authors have done. Because these local model fits are mutually independent they can easily be carried out sequentially or in parallel, meaning that in practice, we are able to work with large networks of tens of thousands of nodes. The novel prior structure proposed, which enables the time-varying network inference, is also of interest more generally beyond the context of network science. This novel prior structure is relevant in any context where sparse regression with time-varying regression parameters is desirable.\nThe rest of the paper is structured as follows. In Section [methodsSec], we set up notation, and specify the model. Then, in Section [simSec] we present the results of fitting the model to simulated data, and in Section [resSec] we present the results of fitting the model to single-cell transcriptome data. Finally, in Section [discussSect], we summarise our findings and discuss their broader context. The Supplementary Material we provide all proofs and derivations, data pre-processing details, and Supplementary Figures, as well as a freely available software implementation of our proposed model and algorithm.\nProposed methodology\n====================\nData description\n----------------\nThe two-way sparsity that is induced by the proposed modelling framework is motivated by the problem of inferring time-varying structure in genomic networks. In these networks nodes represent genes: for each node there are observations or measurements of the activity level of the corresponding gene (the ‘gene-expression level’). These node-specific observations make up the data-set. The expression-level of a gene is generally influenced by the expression-level of several other genes (in a process called ‘gene regulation’). Hence, a natural application for models of time-varying networks is understanding dynamic patterns of gene-regulation in biological processes, such as neural development. Measurements of gene transcript counts are often used as a surrogate for gene expression level in RNA sequencing data, and hence we base our real-data example on single-cell transcriptomic data. We note that single-cell transcriptomic data is a type of single-cell gene-expression data.\nSingle-cell gene-expression data are ideal for this application, because data from a study of this type will typically be obtained from a heterogeneous mixture of cells, each of which may be at a different point on a trajectory through the biological process under investigation. For example, in the context of neural development, some of these cells may be stem-cells, whereas some may be fully differentiated cells (e.g., neurons), with a whole spectrum of cells in between. Each cell can be thought of as an independent sample from the underlying latent biological process; in this example, that process is neural development. Thus, we can think of the progression of a cell through this process of neural development in terms of a ‘developmental trajectory’. The progression along such a developmental trajectory can be quantified in terms of ‘developmental time’, which is simply a measure of a temporally-ordered progression through the process of cellular development. For each of the cell-samples in the data, no information is available other than its high-dimensional gene-expression measurements. Hence, it is necessary to first infer the ‘developmental time’ of the cell-samples before fitting any time-varying network model. This leads to an ordered sequence of pseudo-temporal measurements *x*1, *i*, *x*2, *i*, ...*x**t*, *i*, ..., *x**T*, *i* of the log-expression level for gene *i*. Importantly, the *x*1, *i*, *x*2, *i*, ...etc are taken from different cell samples for each pseudo-time point, and are hence independent. Inference like this is more generally referred to as ‘pseudo-time’ inference, and several methods exist to carry it out: see for example work by and.\nModel overview\n--------------\nWe develop a model for each target-node conditional on all the other nodes, and then we apply this model to several target-nodes of interest. This is different from, for example, the work of and who consider the modelling of all nodes jointly. Such a target-node approach has been used previously by, and it allows the network structure to be inferred independently around each target-node *i* ∈ {1, ..., *p*}. This strategy has several advantages. Firstly, variable screening can be applied before model fitting. This allows the dimensionality of the problem to be reduced from *p* of the order of tens of thousands down to *p*ʹ of the order of a few hundred for each parallel model fit around a target-node, whilst still allowing the global network structure to be estimated over tens of thousands of target-nodes, if required. Our modelling strategy also allows the local network structure to be estimated around only a small number of target-nodes if required, controlling computational expense, whilst still inferring the connected node-sets from tens of thousands of nodes. Inference is carried out with a sparse linear model, taking the observations for node *i* at time *t* as the response, and the observations for all nodes *j* ≠ *i* at time *t* as potential predictors. From these potential predictors, the set of predictors ‘chosen’ by the sparse model fit are then used to infer the network structure. Specifically, we want to infer the network structure around a fixed set of nodes with a set of edges that varies with time. In this scenario, only the patterns of interconnectivity change as the network evolves (Figure [dynNetFig]), which is the scenario most relevant to genomics applications. Such a network can be represented with a time-varying adjacency matrix **A**, where *A**i*, *j*, *t* denotes the absence (*A**i*, *j*, *t* = 0) or presence (*A**i*, *j*, *t* > 0) of an edge between nodes *i* and *j* at time *t*. We note that under this scheme, the local model fit (which is responsible for the computational load) does not depend on the network estimation (which takes place subsequently). The inferred network is a particular summary of the posteriors that are obtained from several of our model fits. We propose a model for node-wise regression, and we suggest how to summarise these models over several nodes of a network.\nModel likelihood\n----------------\nWe assume a likelihood function where observations are mutually independent over time. This is an assumption that is compatible with high-dimensional gene-expression data, where no single cell can be measured at more than one time-point. We note that this implies that observations are independent at different time-points. Let **X** represent the full data-set for the nodes shown in Figure [dynNetFig], with time varying down the rows, and with each node corresponding to a different column. Then, *x**t*, *i* denotes the value for some node in the system at time *t* ∈ {1, ..., *T*}, for *i* ∈ {1, ..., *p*}, and the row-vector $\\mathbf{x}\\_{t,\\fgebackslash i}$ denotes the values for the other *p* − 1 nodes at time *t*. We model the dependence of *x**t*, *i* on $\\mathbf{x}\\_{t,\\fgebackslash i}$ as:\n \n$$x\\_{t,i}=a\\_i+\\mathbf{b}^{(i)}\\_{t,:}\\ \\mathbf{x}\\_{t,\\fgebackslash i}^\\top+\\epsilon\\_{t,i},\\label{mainLinModelEq}$$ \nwhere **b***t*, :(*i*) is a vector of linear model parameters, and *ε**t*, *i* ∼ N(0, *τ**i*− 1).\nThe response variable *x**t*, *i* corresponds to the observations for a ‘target’ node around which we are modelling the local network structure, whereas the variables represented by $\\mathbf{x}\\_{t,\\fgebackslash i}$ correspond to the observations for all the other nodes of the network. To model the whole network, we must fit model ([mainLinModelEq]) around each target-node in turn. We note that here we make an assumption about the existence of a global undirected Markov network that explains the independence constraints in the model. This assumption has also been used previously by in an equivalent context. We note that our approach does not enforce hard constraints, such as **b***t*, *j*(*i*) = **b***t*, *i*(*j*). However, it is computationally very expensive to work with a global, coherent model, where such constraints can be enforced. In this work, we have opted to sacrifice some coherence for the sake of computational efficiency. This enables us to estimate quantities of interest in a computationally-efficient manner through an overparameterized representation of a joint model. It also enables us to focus on a particular subset of nodes of interest without having to go through an overly-expensive computation for the estimation of a global, coherent model.\nUsing **b**: , *j*(*i*) to denote the column-vector of model parameters for covariate *j* for *t* ∈ {1, ..., *T*}, we collect parameters in matrix **B**(*i*) = [**b**: , 1(*i*), **b**: , 2(*i*), ..., **b**: , *p* − 1(*i*)]. In the next section, we postulate a prior for dependencies within each column *j* of **B**(*i*), whilst noting that the columns of **B**(*i*) (each corresponding to a different node as covariate) are independent of each other. We also introduce the notation *x**t*, *i*, *k* and $\\mathbf{x}\\_{t,\\fgebackslash i,k}$ to represent observations of *x**t*, *i* and $\\mathbf{x}\\_{t,\\fgebackslash i}$ for sample *k* ∈ {1, ..., *n**t*} at time *t*.\nWe denote **x**: , *i* = [*x*1, *i*, 1, ..., *x*1, *i*, *n*1, ..., *x**t*, *i*, 1, ..., *x**t*, *i*, *n**t*, ..., *x**T*, *i*, 1, ..., *x**T*, *i*, *n**T*]⊤ and \n$\\mathbf{X}\\_{:,\\fgebackslash i}=[\\mathbf{x}\\_{1,\\fgebackslash i,1}^\\top,\\mathbf{x}\\_{1,\\fgebackslash i,n\\_1}^\\top,...,\\mathbf{x}\\_{t,\\fgebackslash i,1}^\\top,...,\\mathbf{x}\\_{t,\\fgebackslash i,n\\_t}^\\top,...,\\mathbf{x}\\_{T,\\fgebackslash i,1}^\\top,...,\\mathbf{x}\\_{T,\\fgebackslash i,n\\_T}^\\top]^\\top$, where **x**: , *i* is column *i* of data-matrix **X**, and $\\mathbf{X}\\_{:,\\fgebackslash i}$ is data-matrix **X** without column *i*. Hence, we can write the model likelihood for the target-node *i* as:\n \n$$P(\\mathbf{x}\\_{:,i}|\\mathbf{X}\\_{:,\\fgebackslash i},\\mathbf{B}^{(i)},a\\_i,\\tau\\_i)=\\prod\\_{t=1}^T\\prod\\_{k=1}^{n\\_t}\\sqrt{\\frac{\\tau\\_i}{2\\pi}}e^{-\\tau\\_i(x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,:}\\cdot\\mathbf{x}\\_{t,\\fgebackslash i,k}^\\top-a\\_i)^2/2}.{\\addtocounter{equation}{1}\\tag{\\theequation}}\\label{dataCondParams}$$ \nWe note that we consider likelihoods in the form of equation ([dataCondParams]) for each target-node *i*.\nPriors with decoupled two-way sparsity\n--------------------------------------\nWe model the regression coefficients **b**: , *j*(*i*) across time-points *t* = 1, ..., *T* with a Gaussian copula with Laplace marginal distributions, as follows. The elements of **b***t*, :(*i*) (*t* = 1, ..., *T*) are marginally distributed as *b**t*, *j*(*i*) ∼ Laplace (1/*λ*), with probability density function $\\frac{\\lambda}{2}e^{-\\lambda|\\cdot|}$ and cumulative distribution function *F*L[*b**t*, *j*(*i*)], for *t* ∈ {1, ..., *T*} and *j* ∈ {1, ..., *p* − 1}. Hence, Φ− 1{*F*L[*b**t*, *j*(*i*)]} follows a Gaussian distribution for *t* ∈ {1, ..., *T*} and *j* ∈ {1, ..., *p* − 1}, where Φ is the standard-normal cumulative distribution function. The dependencies between Φ− 1{*F*L[*b**t*, *j*(*i*)]} and Φ− 1{*F*L[*b**t* + 1, *j*(*i*)]} are then modelled through their joint distribution as:\n \n$${\\footnotesize\n\\left[\\begin{array}{c}\\vspace{1ex} \\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{1,j}]\\right\\} \\\\\\vspace{1ex}\\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{2,j}]\\right\\} \\\\\\vspace{1ex} \\vdots \\\\\\vspace{1ex} \\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{T,j}]\\right\\} \\end{array} \\right]\\sim\\mathcal{N}\\left(\\mathbf{0},\\boldsymbol{\\Sigma}^{(i)}\\_j\\right),\\enspace\\text{with}\\enspace\\boldsymbol{\\Sigma}^{(i)}\\_j=\\left[\\begin{array}{ccccc}\\vspace{1.3ex}1&\\rho^{(i)}\\_j&\\left(\\rho^{(i)}\\_j\\right)^2&\\cdots&\\left(\\rho^{(i)}\\_j\\right)^T\\\\\\vspace{1.3ex}\\rho^{(i)}\\_j&1&\\rho^{(i)}\\_j&\\cdots&\\left(\\rho^{(i)}\\_j\\right)^{T-1}\\\\\\vspace{1.3ex}\\left(\\rho^{(i)}\\_j\\right)^2&\\rho^{(i)}\\_j&1&\\cdots&\\left(\\rho^{(i)}\\_j\\right)^{T-2}\\\\\\vspace{1.3ex}\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\\\vspace{1.3ex}\\left(\\rho^{(i)}\\_j\\right)^T&\\left(\\rho^{(i)}\\_j\\right)^{T-1}&\\left(\\rho^{(i)}\\_j\\right)^{T-2}&\\cdots&1\\end{array}\\right],{\\addtocounter{equation}{1}\\tag{\\theequation}}\\label{fullCovSpec}\n}$$ \nand hence the regression coefficients are modelled as a Gaussian copula:\n \n$$F\\left[\\mathbf{b}^{(i)}\\_{:,j}\\right]=\\Phi\\_p\\left[\\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{1,j}]\\right\\},\\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{2,j}]\\right\\},...,\\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{T,j}]\\right\\};\\boldsymbol{\\Sigma}^{(i)}\\_j\\right].$$ \nThe correlation parameter *ρ**j*(*i*) is assumed to have a reverse-exponential distribution with support [0, 1) and density\n \n*f*rexp[*ρ**j*(*i*)] ∼ *k**e**k**ρ**j*(*i*)/(*e**k* − 1). \n[graphicalModel]\nThe structure of $\\boldsymbol{\\Sigma}^{(i)}\\_j$ is such that transformed model parameters at adjacent points in time, such as Φ− 1{*F*L[*b**t*, *j*(*i*)]} and Φ− 1{*F*L[*b**t* + 1, *j*(*i*)]}, have correlation *ρ**j*(*i*) (Figure [graphicalModel]). Then, the transformed parameters separated by two time-points have correlation (*ρ**j*(*i*))2, etc. Thus, also denoting the sequence of transformed model parameters Φ− 1{*F*L[*b*1, *j*(*i*)]}, Φ− 1{*F*L[*b*2, *j*(*i*)]},...,Φ− 1{*F*L[*b**t*, *j*(*i*)]} forms a Markov chain, meaning that $\\boldsymbol{\\Sigma}^{(i)}\\_j$ is guaranteed to be positive-definite for *ρ**j*(*i*) ∈ [0, 1), and by construction\n \n*b**t* + 1, *j*(*i*) ⊥ *b**t* − 1, *j*(*i*), *b**t* − 2, *j*(*i*), ...∣*b**t*, *j*(*i*). \n[priorTrajSamp]\nSuch a construction for $\\boldsymbol{\\Sigma}^{(i)}\\_j$ discourages differences in the regression coefficients for the same covariate between adjacent time-points, and hence also discourages changes in the network structure over time, resulting in sparsity across time. Then, transforming the Φ− 1{*F*L[*b**t*, *j*(*i*)]} back to *b**t*, *j*(*i*), where the *b**t*, *j*(*i*) are marginally Laplace distributed, achieves sparsity within time by discouraging regression coefficients from taking non-zero values, hence also encouraging discovery of sparse network structures. Figure [priorTrajSamp] shows ten samples from our proposed prior on *b**t*, *j*(*i*) plotted against *t*, and demonstrates the correlation structure enforced by the prior over time. We again note that while *b**t*, *j*(*i*) and *b**t* + 1, *j*(*i*) are correlated, *b**t*, *j*(*i*) and *b**t*, *j*ʹ(*i*) are independent.\nRecent work by that takes a Bayesian approach to generalising the fused lasso could be used similarly to the approach we propose, by modelling the same set of covariates at multiple time-points whilst enforcing smooth changes across time as well as sparsity overall. However, achieve their result by simply multiplying together separate frequentist-inspired priors for smoothness across time and for sparsity. Specifically, they multiply together a Laplace prior to penalise individual non-zero model parameters, with the ultra-sparse negative-exponential-gamma (NEG) prior to penalise non-zero differences in parameters. The Laplace-NEG prior is defined (choosing notation to be consistent with that of our model) as:\n \n*P*(**b**: , *j*(*i*)) ∝ ∏*t* = 1*T*Laplace(*b**t*, *j*(*i*)∣*λ*)∏*t* = 2*T*NEG(*b**t*, *j*(*i*) − *b**t* − 1, *j*(*i*)∣*λ*†, *γ*), \nwhere the Laplace density is defined as $\\frac{\\lambda}{2}e^{-\\lambda|\\cdot|}$, and\n \nNEG( ⋅ ∣*λ*†, *γ*) = ∫0∞∫0∞*f*N( ⋅ ∣0, *τ*2)*f*\\gamma(*τ*2∣1, 1/*ψ*)*f*\\gamma(*ψ*∣*λ*†, 1/*γ*2)*d**τ*2*d**ψ*, \nwhere *f*N and *f*\\gamma are the Normal and Gamma densities, respectively. Sampling from the distribution of equation ([laplNegDist]) is done by simulating exponential and gamma random variables, which are then used to form the precision matrix of a multivariate normal distribution, as specified by. In contrast to the Laplace-NEG prior, the model we propose retains the property that, marginally, each parameter still follows the Bayesian lasso prior. In particular, if we set *ρ**j*(*i*) = 0 (for *j* = 1, 2, ..., *p* − 1), then the model we propose is exactly the same as the Bayesian lasso. This is important because it makes it easier to set priors, including variants of the Bayesian lasso that avoid its well-known shortcomings (see for example the work by and ). Although we will not consider such variants here, they follow directly by mimicking the construction using the Bayesian lasso.\n[priorFig]\nThe novel prior we use on *ρ**j*(*i*) is a ‘reverse exponential prior’ (equation ([revExpPriorDef])). Figure [priorFig]a shows the probability density function of the reverse-exponential prior for different values of hyper-parameter *k*. Figure [priorFig]b then shows heatmaps of the bivariate density distributions of samples from the decoupled-sparsity prior for a parameter *j* over two time-points, i.e., **b**: , *j*(*i*) = [*b*1, *j*(*i*), *b*2, *j*(*i*)]⊤, for a range of values of *λ* and *k* (the corresponding marginal densities are shown in Figures S6 and S7 in the Supplementary Information). For comparison, Figures S3 - S5 in Supplement D show samples from the Laplace-NEG prior as defined in equation ([laplNegDist]), for various values of *λ* (which acts equivalently to *λ* in our model, controlling sparsity of individual model parameters), and various values of *λ*† and *γ* (which both act equivalently to *k* in our model, controlling sparsity of differences between model parameters). The main difference between these priors is that our decoupled-sparsity prior still marginally follows the Bayesian lasso prior, and is hence a direct generalisation of the Bayesian lasso to this setting with time-varying model parameters. In other words, our prior does what the Laplace-NEG does, but with the added benefit that we generalise the Bayesian lasso.\nPosterior inference\n-------------------\nThe order-1 Markovian relations specified by equation ([thetaCondIndepState]) are also computationally attractive, because they result in models with banded precision matrices. From equation ([thetaCondIndepState]), and denoting *θ**t*, *j*(*i*) = Φ− 1{*F*L[*b**t*, *j*(*i*)]} it follows that the partial correlation of *θ**t* + *m*, *j*(*i*) with *θ**t* + *l*, *j*(*i*) will be zero for all ∣*m* − *l*∣ > 1. Hence, all entries of the precision matrix $[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}$ will be zero except the diagonal and the elements immediately adjacent to it (i.e., the sub- and super-diagonals). These relationships allow all the entries of this precision matrix to be found easily in terms of *ρ**j*(*i*) by solving $[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\boldsymbol{\\Sigma}^{(i)}\\_j=\\mathbb{I}$, which gives:\n \n$$\\left(\\left[\\boldsymbol{\\Sigma}^{(i)}\\_j\\right]^{-1}\\right)\\_{t,t^\\prime}=\n\\begin{cases}\n\\vspace\\*{1ex}\n 1/(1-[\\rho^{(i)}\\_j]^2), & \\text{if}\\ t^\\prime=t=1\\ \\text{or}\\ t^\\prime=t=T, \\\\\n\\vspace\\*{1ex}\n (1+[\\rho^{(i)}\\_j]^2)/(1-[\\rho^{(i)}\\_j]^2), & \\text{if}\\ t^\\prime=t>1\\ \\text{and}\\ t^\\prime=t<T, \\\\\n\\vspace{1ex}\n -\\rho^{(i)}\\_j/(1-[\\rho^{(i)}\\_j]^2), & \\text{if}\\ t^\\prime=t+1\\ \\text{or}\\ t^\\prime=t-1, \\\\\n 0, & \\text{otherwise}\\label{precMatSpec}\n\\end{cases}$$ \nwhere $\\left(\\left[\\boldsymbol{\\Sigma}^{(i)}\\_j\\right]^{-1}\\right)\\_{t,t^\\prime}$ represents the (*t*, *t*′) element of the precision matrix $[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}$. A full derivation of equation ([precMatSpec]) is given in Supplement A.\nThe model parameters **B**(*i*) can be sampled directly from multivariate Normal distributions, without needing the intermediate transformation to the marginally Laplace-distributed variables described in Section [priorDef]. This can be achieved with an algebraic manipulation which is an extension from the Bayesian lasso, as follows. The Laplace distribution can be written as an uncountable mixture of zero-mean Normal distributions, with the variances of the mixture components distributed as $\\text{Exp}(\\frac{\\lambda^2}{2})$. Specifically,\n \n$$P(b^{(i)}\\_{t,j}|\\lambda)=\\frac{\\lambda}{2}e^{-\\lambda|b^{(i)}\\_{t,j}|}=\\int\\_0^\\infty P(b^{(i)}\\_{t,j},s^{(i)}\\_j|\\lambda)ds^{(i)}\\_j,$$ \nwhere\n \n$$P(b^{(i)}\\_{t,j},s^{(i)}\\_j|\\lambda)=\\frac{1}{\\sqrt{2\\pi s^{(i)}\\_j}}e^{-[b^{(i)}\\_{t,j}]^2/[2s^{(i)}\\_j]}\\frac{\\lambda^2}{2}e^{-\\lambda^2s^{(i)}\\_j/2},$$ \nfor $s^{(i)}\\_j\\sim\\text{Exp}(\\frac{2}{\\lambda^2})$. This says that we will achieve *b**t*, *j*(*i*) being marginally Laplace distributed by sampling these *s**j*(*i*) from the $\\text{Exp}(\\frac{2}{\\lambda^2})$ prior, and then sampling the *b**t*, *j*(*i*) from zero-mean Normal distributions with variances *s**j*(*i*). Hence\n \n$$P(b^{(i)}\\_{t,j}|s^{(i)}\\_j)=\\frac{1}{\\sqrt{2\\pi s^{(i)}\\_j}}e^{-[b^{(i)}\\_{t,j}]^2/[2s^{(i)}\\_j]},$$ \nand so **b**: , *j*(*i*) has the same Normal distribution as Φ− 1{*F*L[*b**t*, *j*(*i*)]} but with the variances and covariances scaled up by *s**j*(*i*), with $s^{(i)}\\_j\\sim\\text{Exp}(\\frac{2}{\\lambda^2})$. Therefore, also referring back to equation ([fullCovSpec]), it follows that\n \n$$\\begin{aligned}\nP(\\mathbf{b}^{(i)}\\_{:,j},s^{(i)}\\_j|\\rho^{(i)}\\_j,\\lambda)=&\\frac{\\lambda^2}{2}e^{-\\lambda^2s^{(i)}\\_j/2}\\frac{1}{(2\\pi)^{T/2}[s^{(i)}\\_j]^{1/2}|\\boldsymbol{\\Sigma}^{(i)}\\_j|^{1/2}}e^{-{\\mathbf{b}^{(i)}\\_{:,j}}^\\top [s^{(i)}\\_j]^{-1}[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}/2}.\\end{aligned}$$ \nTo make sampling easier, at this stage we let *s**j*(*i*) = [*ν**j*(*i*)]− 1, leading to the density\n \n$$\\begin{aligned}\nP(\\mathbf{b}^{(i)}\\_{:,j},\\nu^{(i)}\\_j|\\rho^{(i)}\\_j,\\lambda)=&\\frac{1}{[\\nu^{(i)}\\_j]^2}\\frac{\\lambda^2}{2}e^{-\\lambda^2/(2\\nu^{(i)}\\_j)}\\frac{[\\nu^{(i)}\\_j]^{1/2}}{(2\\pi)^{T/2}|\\boldsymbol{\\Sigma}^{(i)}\\_j|^{1/2}}e^{-{\\mathbf{b}^{(i)}\\_{:,j}}^\\top \\nu^{(i)}\\_j[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}/2}\\\\\n=&\\frac{\\lambda^2}{2}e^{-\\lambda^2/(2\\nu^{(i)}\\_j)}\\frac{[\\nu^{(i)}\\_j]^{-3/2}}{(2\\pi)^{T/2}|\\boldsymbol{\\Sigma}^{(i)}\\_j|^{1/2}}e^{-{\\mathbf{b}^{(i)}\\_{:,j}}^\\top[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}\\nu^{(i)}\\_j/2},{\\addtocounter{equation}{1}\\tag{\\theequation}}\\label{bNuPriorFinal}\\end{aligned}$$ \nwhere the extra factor of 1/[*ν**j*(*i*)]2 is the factor ∣*d*{[*ν**j*(*i*)]− 1}/*d**ν**j*(*i*)∣ due to the change of variable. Assuming that the model will be fit to data standardised to have unit variance, we set the prior on the intercept as *a* ∼ N(0, 1), and we set the prior on the model precision as *τ**i* ∼ Gamma(1, 1) (which has prior mean 1, with 95% of the prior mass between 0.025 and 3.7, which we believe is reasonable for these data). Now combining equation ([bNuPriorFinal]) with these prior specifications, and $P(\\rho^{(i)}\\_j|k)=\\frac{k}{e^k-1}e^{k\\rho^{(i)}\\_j}$ (for 0 ≤ *ρ**j*(*i*) ≤ 1), as well as with the model likelihood (equation ([dataCondParams])), we get:\n \n$$\\begin{aligned}\nP&(\\mathbf{x}\\_{:,i},\\mathbf{B}^{(i)},\\boldsymbol{\\rho}^{(i)},\\boldsymbol{\\nu}^{(i)},a\\_i,\\tau\\_i|\\mathbf{X}\\_{:,\\fgebackslash i},\\lambda,k)=\\left\\{\\prod\\_{t=1}^T\\prod\\_{k=1}^{n\\_t}\\sqrt{\\frac{\\tau\\_i}{2\\pi}}e^{-\\tau\\_i\\left(x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,:}\\cdot\\mathbf{x}\\_{t,\\fgebackslash i,k}^\\top-a\\_i\\right)^2/2}\\right\\}\\\\\n&\\frac{1}{\\sqrt{2\\pi}}e^{-\\left\\{\\tau\\_i+a\\_i^2/2\\right\\}}\\prod\\_{j=1}^{p-1}\\left\\{\\frac{k}{e^k-1}e^{k\\rho^{(i)}\\_j}\\frac{\\lambda^2}{2}e^{-\\lambda^2/(2\\nu^{(i)}\\_j)}\\frac{[\\nu^{(i)}\\_j]^{-3/2}}{(2\\pi)^{T/2}|\\boldsymbol{\\Sigma}^{(i)}\\_j|^{1/2}}e^{-{\\mathbf{b}^{(i)}\\_{:,j}}^\\top[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}\\nu^{(i)}\\_j/2}\\right\\}.{\\addtocounter{equation}{1}\\tag{\\theequation}}\\label{fullJdist}\\end{aligned}$$ \nFollowing equation ([fullJdist]), posterior sampling for the model described in Sections [mainModel] and [priorDef] can be implemented through a Gibbs sampler with the steps given in Algorithm [gibbsSampEqs]. We note that Algorithm [gibbsSampEqs] has a relatively low computational cost, because each of the steps (with the exception of step 4) involves sampling from a known distribution for which the parameters can be easily calculated. Then for step 4, we can simply use a slice-sampler to sample *ρ**j*(*i*), which has finite support *ρ**j*(*i*) ∈ [0, 1). The full derivations of each step of Algorithm [gibbsSampEqs] appear in Supplement B.\n[gibbsSampEqs] A Gibbs sampler with the following steps:\n1. **Sample:** *a**i* from:\n \nwhere *f*N is the Normal density, *σ**a*− 2 = 1 + *n**τ**i* and $\\mu\\_{a}=\\sigma\\_{a}^2\\tau\\_i\\sum\\_{t=1}^T\\sum\\_{k=1}^{n\\_t}\\{x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,:}\\cdot\\mathbf{x}\\_{t,\\fgebackslash i,k}^\\top\\}$.\n2. **Sample:** *τ**i* from:\n \nwhere *f**γ* is the density of the gamma distribution with $k\\_\\tau=1+\\frac{\\sum\\limits\\_{t=1}^Tn\\_t}{2}$ and\n \n$$\\theta\\_\\tau=1/\\{1+\\sum\\limits\\_{t=1}^T\\sum\\limits\\_{k=1}^{n\\_t}(x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,:}\\cdot\\mathbf{x}\\_{t,\\fgebackslash i,k}^\\top-a\\_i)^2/2\\}.$$\n3. **Sample:** *ν**j*(*i*) from: \nwhere *f**I**G* is the density of the inverse Gaussian distribution with parameters \n*λ**ν* = *λ*2 and $\\mu\\_\\nu=\\lambda\\bigg/\\sqrt{{\\mathbf{b}^{(i)}\\_{:,j}}^\\top[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}}$.\n4. **Sample:** *ρ**j*(*i*) from:\n5. **Sample:** **b**: , *j*(*i*) from:\n \nwhere *f*N is the multivariate Normal density, $[\\widetilde{\\boldsymbol{\\Sigma}}^{(i)}\\_j]^{-1}=\\nu^{(i)}\\_j[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}+[\\mathbf{V}^{(i)}\\_j]^{-1}$, \nand $\\widetilde{\\mathbf{m}}^{(i)}\\_j=\\widetilde{\\boldsymbol{\\Sigma}}^{(i)}\\_j[\\mathbf{V}^{(i)}\\_j]^{-1}\\mathbf{m}^{(i)}\\_j$, where the *t*th element of the vector **m***j*(*i*) is\n \n$$m^{(i)}\\_{t,j}=\\sum\\_{t=1}^T\\sum\\_{k=1}^{n\\_t}x\\_{t,j,k}\\left\\{x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,\\fgebackslash j}\\left(\\mathbf{x}\\_{t,\\fgebackslash i,k}\\right)\\_{\\fgebackslash j}^\\top-a\\_i\\right\\}/\\sum\\_{t=1}^T\\sum\\_{k=1}^{n\\_t}x\\_{t,j,k}^2,$$ \nwhere $\\mathbf{b}^{(i)}\\_{t,\\fgebackslash j}$ and $\\left(\\mathbf{x}\\_{t,\\fgebackslash i,k}\\right)\\_{\\fgebackslash j}$ represent **b***t*, :(*i*) and $\\mathbf{x}\\_{t,\\fgebackslash i,k}$ without the $j\\textsuperscript{th}$ elements, respectively, and **V***j*(*i*) is a diagonal matrix, with the *t*th diagonal element equal to 1/{*τ**i*∑*t* = 1*T*∑*k* = 1*n**t**x**t*, *j*, *k*2}.\nSimulation study\n================\nIn this section, we present the results from a simulation study, to test how accurately our model can recover network structure which we know in advance. We generate simulated data with structure that we expect to be typical of real data, and then fit the proposed model to the simulated data. To generate the data, the observations *x**t*, *i* for each node *i* are generated such that they follow a mean time-series of one of four types (illustrated in Figure [simDataEG]), as follows:\n1. Monotonic; decreasing to no signal.\n2. Monotonic; increasing from no signal.\n3. Maximum: increasing from and decreasing to no signal.\n4. Null: random noise.\n[simDataEG]\nTypes (a) and (b) represent node-types of interest to the biological setting, as follows. Type (a) corresponds to genes that are activated (i.e., *x**t*, *i* > 0) early in the time-series before becoming de-activated (as we would expect of genes which are important for stem-like cell identity). Type (b) corresponds to genes which only become activated later in the time-series (as we would expect of genes which are important for the identity of mature cells, such as neurons). Types (c) and (d) make the simulated data closer to what we would expect of the real data, by mixing in nodes with other sorts of signals: type (c) corresponds to genes which are active in the middle of the time-series only, and type (d) are null nodes (with random activation).\nAfter generating each characteristic mean time-series according to fixed types (a)-(d), we then time-stretch the particular characteristic mean time-series chosen for each node *i* by a random amount. We do this to reflect the fact that developmental events (as represented by gene-expression measurements) occur at different times in different cells. We achieve this effect by changing the length of the time-series to have a random but uniformly-distributed period *T*ʹ ∼ U[*T* − 3, *T*], before zero-padding to return the time-series to its original period *T*. This gives a mean profile *x**t*, *i*, *t* = 1, ..., *T* for each node that is distinct from all other nodes of the same type. We then generate observations *x**t*, *i*, *k* for each node according to equation ([mainLinModelEq]) based on these mean profiles, also setting the intercept parameter *a**i* to 0, and sampling via the Markov chain described in Algorithm [simDataGen]:\n[simDataGen] A Markov chain: \n**Loop**: *t*in 1 : *T* \n*X**t*, : , 1 ← 0\n// Initialize Markov chain at 0 \n: *r*in 2 : *R* \n**S** ← **X***t*, : , *r*–1 \n: *i*in 1 : *p* \n: *S**i* ∼ *N*(**S**∖ *i*.**b***t*, :(*i*), *σ*2) \n \n**X***t*, : , *r* ← **s** \n \n**end loop**\nwhere **X** is a *T* × *p* × *R* array containing the sampled data, **S** is a vector of length *p* which temporarily stores intermediate results, and the elements of **b**(*i*) are specified as:\n \n$$b^{(i)}\\_{t,j}=\n\\begin{cases}\n\\vspace\\*{1ex}\n1/{p'}, & \\text{if nodes $i$ and $j$ are of the same type}\\ \\\\\n\\vspace\\*{1ex}\n0, & \\text{otherwise}\\\\\n\\end{cases}$$ \nwhere *p*ʹ is the number of nodes *j* of the same type as *i*. The number of MCMC samples in the Markov chain specified in Algorithm [simDataGen] is given by the variable *R*: we use *R* = 104, and after thinning to take one sample in every 100, we choose the final 25 (thinned) samples to pass forward to the model fitting after adding the mean characteristic profiles. That is, we have 25 samples per time-point, i.e., *n**t* = 25, where autocorrelation analysis and an experimentation with burn-in times show no evidence against them being independently and identically distributed at each time-point group. We note that in the simulation we specify means that vary with time but our model has constant mean. In practice this does not make any difference as the data are always standardised before model fitting, but the user can easily make the intercept time-varying if this is a concern. The procedure for generating the simulated data is also illustrated in Figure [simDataExplain].\n[simDataExplain]\nWe generate each time-series with *T* = 8, *n**t* = 25 (constant for all values of *t*), and *p*ʹ = 10, adding noise with standard deviations *τ**i*− 1/2 = *τ*− 1/2 ∈ {0.1, 0.2, 0.3}. Then, we apply Algorithm [gibbsSampEqs], and calculate each *b̂**t*, *j*(*i*) from the median of the corresponding posterior. We infer an edge between nodes *i* and *j* if *b̂**t*, *j*(*i*) ≠ 0, after thresholding the *b̂**t*, *j*(*i*) to remove trivially small values, i.e., if ∣*b̂**t*, *j*(*i*)∣ ≥ *ϕ*. We generate ROC (receiver-operator characteristic) curves as this threshold *ϕ* is decreased to 0 from max∣*b̂**t*, *j*(*i*)∣ (for *t* ∈ {1, ..., *T*} and all *j*). We generate these curves from the true-positives (TP) and false-positives (FP) which we calculate from the ground-truth network edges *b**t*, *j*(*i*) and estimated network edges *b̂**t*, *j*(*i*) as follows:\n \n$$\\begin{aligned}\n|\\hat{b}^{(i)}\\_{t,j}|>0\\enspace\\text{for}\\enspace|b^{(i)}\\_{t,j}|>0\\implies\\enspace&\\text{TP}\\\\\n\\text{and}\\enspace\\enspace|\\hat{b}^{(i)}\\_{t,j}|>0\\enspace\\text{for}\\enspace|b^{(i)}\\_{t,j}|=0\\implies\\enspace&\\text{FP}\\end{aligned}$$ \nWe generate an average ROC curve over 1000 repetitions of this procedure, and then calculate an AUC (area under curve) statistic for this average ROC curve.\n[simDataNullParam]\n[simDataRes]\nWe assess the performance of our full model using both sparsity within and sparsity across time, compared with the scenarios when one of these priors is excluded from the model. To exclude sparsity across time, we enforce *ρ* = 0, and to exclude sparsity within time, we use *λ* → 0: these results are shown in Figure [simDataNullParam]. With *ρ* = 0, there is no correlation of the model parameters across time, and so we see the effect of inferring the networks separately for each time-point; i.e., sparsity across time is removed: in this case, AUC = 0.5 indicates that none of the intended structure in the data is being detected. Alternatively, as *λ* → 0, the prior becomes flat or uninformative, and so in this case we see the effect of fitting the model without the sparsity within time. We again note that decoupling these types of sparsity is made possible by design with the model structure we propose, unlike alternatives such as Laplace-NEG. Then for the full model (which includes the priors to enforce both the sparsity within time and across time), we repeated the simulation for various values of sparsity parameter *λ*: Figure [simDataRes] shows the results (with hyperparameter *k* = 20, equivalent results with *k* = 10 and *k* = 50 are shown in Figures S10 and S11 in Supplement D). When we include the priors for both sparsity within and across time, we can achieve AUC of 0.9 or more, as long as the sparsity parameter *λ* is large enough. This result demonstrates that our priors are responsible for good detection of network edges with respect to the ground-truth in these simulated data. We also found that these results were not very sensitive to *p*ʹ, the number of covariates included in the simulated data. Figure S12 shows equivalent results to Figure [simDataRes], except with the number of covariates halved to *p*ʹ = 5. In this case we found that the network inference is a bit more accurate, as would be expected with a smaller number of variables to predict; although the difference is minimal as long as the sparsity is great enough.\n[simDataResDrop]\nDropouts, or missing values, are a well known source of technical noise in single-cell transcriptome data. These missing values are replaced by zeros, leading to ‘zero inflation’. A characteristic of this dropout effect is that data-values which are already small are more likely to drop out (i.e., get missed out), than values which are larger in magnitude. This is data missing not-at-random, an effect that can be challenging to model. Dropout rates (i.e., the proportion of data-values missing from the data-set) are often over 60% in typical single-cell transcriptome data-sets that we have seen, such as the one analysed in Section [resSec]. To test the robustness of our method to dropouts, we used a well known and effective model of the dropout effect, published previously by. This model of dropouts specifies the probability of an observed data-value dropping out as\n \n*p**t*, *i*, *k* = exp( − *ω**x̃**t*, *i*, *k*2), \nwhere the parameter *ω* controls the dropout rate (decreasing *ω* increases the number of dropouts), and *x̃**t*, *i*, *k* is the data-value that would have been present without the dropout effect. This is essentially a hurdle model, with *h**t*, *i*, *k* ∼ Bernouilli(*p**t*, *i*, *k*), so that the observed data *x**t*, *i*, *k* (i.e., with dropouts included) is modelled as *x**t*, *i*, *k* = *h**t*, *i*, *k* ⋅ *x̃**t*, *i*, *k*. We found that under this model, *ω* = 2 leads to a dropout rate of around 66% in data-sets generated according to the data-simulation procedure presented earlier in this section. We used this value of *ω* = 2, and repeated our simulation study now with the addition of this dropout effect, carrying out out the same ROC-curve analysis as before. The results of this analysis (again with *λ* = 50 and *k* = 20) are shown in Figure [simDataResDrop]: we found that our method is quite resilient to dropouts, with only a moderate decrease in performance compared to the results shown in Figure [simDataRes]. The time-varying aspect of the model apparently helps to maintain performance when many dropouts are present, because when some values in the time-series are missing, sparsity across time encourages interpolation over the missing values. Interestingly, in very sparse cases, the dropout effect may even be helpful, possibly via a de-noising mechanism, as follows. Referring to the generative model of equation ([dropEq]), it’s clear that the dropouts mostly take place for small values of *x**t*, *i*, *k*. As these are much more likely to correspond to noise than larger values do, this leads to a strong de-noising effect. Finally, we note that to include a hurdle model or dropout effect in a model likelihood such as the one proposed in Section [mainModel] would result in a much more computationally intensive model fitting procedure than the one we propose in our Algorithm [gibbsSampEqs] of Section [postInf].\nSingle-cell gene-expression data\n================================\nIn this section we present an example application of our proposed methodology, to single-cell gene-expression data. These data have been published previously by, and are publicly available from the NCBI database of genotypes and phenotypes (dbGaP), under accession number phs000989.v3 \nIn this context, *x**t*, *i*, *k* represents the log-expression of gene *i*, defined as log(transcript counts + 1), in sample *k* from time *t*. For the pseudo-time assignments for each cell, we use cell-type classifications provided with the data, together with an ordering for these cell-types according to the developmental lineage (for full details see Supplement C). We fitted the model to *n* = 1557 cell samples, and *p* = 22988 genes/nodes, reduced to *p* = 212 for each individual model fit by variable screening. For the fitting we used values of *λ* = 20 and *k* = 1: these values were chosen by grid-search stochastic EM (Figure S13 in Supplement D). Fitting the model as described, we obtained posterior distributions for each model parameter *b**t*, *j*(*i*), and we used the posterior medians as posterior summaries, *b̂**t*, *j*(*i*). To fit each model, we ran the Gibbs’ sampler proposed in Algorithm [gibbsSampEqs] for 1 × 104 samples (after 1 × 103 samples burn-in), which took 1.4 hours for each target-node on one core of a Macbook Pro laptop (mid 2015, 2.8 GHz, 16GB RAM).\n[bestRegs]\nThe model was fitted initially to a panel of 25 genes, as target-nodes: these genes were chosen in an unbiased way by searching the biological sciences literature for genes that are important in this biological setting, and then analysing those that were present in this data-set after quality control. Estimated model parameters *b̂**t*, *j*(*i*) for a selection of these genes are shown in Figure [bestRegs], and the full panel is shown in Figures S14 and S15.\n[convPlotPs]\nWe carried out Geweke and Heidel convergence tests, using the R package CODA, for the sampler outputs for all the parameters *b**t*, *j*(*i*) shown in Figures [bestRegs], S14 and S15. These convergence test results appear in Figure [convPlotPs]. In convergence tests such as these, if an individual *p*-value is significantly small, it can be taken as evidence that the chain has not yet converged. Hence, the uniform distributions of *p*-values shown in Figure [convPlotPs], in which these *p*-values are aggregated over all the test results, indicate that the MCMC sampler has converged for these target-nodes. Then, to give an indication of how ‘stiff’ or ‘sloppy’ these parameters are, we estimated the standard-deviations of these posterior distributions for this panel of genes: these are plotted against the corresponding posterior averages in Figure [postSDs]. These posterior standard deviations are typically much smaller in magnitude than the posterior averages, demonstrating that the posteriors are not ‘sloppy’, and indicating that the estimates from our model are reliable. We also wanted to make sure that our results are not driven by a few outlier cells. So we repeated the inference for this same panel of genes, but now using only a random sample of 50% of the cells originally used, i.e., *n* = 779. The results of this analysis are plotted in Figure S16, for the same genes as are shown in Figure [bestRegs]. The results shown in these figures are clearly very similar, and therefore we conclude that our results here are not driven by outliers.\nFigure [bestRegs]a shows inferred model parameters *b̂**t*, *j*(*i*), for a selection of nodes/genes which are characteristic of stem cells, and of neurons (i.e., mature cells), selected from the full panel of 25 genes. We expect stem cells to predominate at earlier times, and hence we expect to see decreasing time-series for genes which are characteristic of this type of cell. On the other hand, we expect mature cells such as neurons to predominate at later times, and so we expect to see increasing time-series for genes characteristic of this type of cell. As would be expected for stem-cell genes, important model parameters *b**t*, *j*(*i*) tend to decrease in magnitude during the developmental trajectory as cells go from stem-cell to mature cell types (e.g., gene transcript MOXD1). Figure [bestRegs]b then shows, as would be expected, that important model parameters *b̂**t*, *j*(*i*) become non-zero (corresponding to network edges appearing) late in the developmental trajectory, when the cells become neurons and hence their characteristic gene regulatory program is activated (e.g., for SATB2). Equivalent results to Figure [bestRegs]a-b for the full panel of 25 genes analysed then appear in Figures S14 and S15 respectively in Supplement D. In these figures, we also see similar results: for genes that tend to be active in stem-cells, model parameters *b**t*, *j*(*i*) tend to decrease in magnitude during the developmental trajectory as cells go from stem-cell to mature cell types (Figure S14), and *vice-versa* for genes which are important to mature cells such as neurons (Figure S15).\nWe wish to infer a network edge between nodes *i* and *j* if ∣*b̂**t*, *j*(*i*)∣ > 0. We estimate these *b̂**t*, *j*(*i*) from the posterior medians, but because we find that many of these medians are close to, but not exactly zero, we set *b̂**t*, *j*(*i*) to zero in such cases by thresholding. Therefore, we infer ‘no edge’ between nodes *i* and *j* when the posterior median is close to zero. Hence, if (and only if) ∣*b̂**t*, *j*(*i*)∣ > *ϕ*, where *ϕ* is the threshold parameter, we would infer a network edge between nodes *i* and *j* at time *t* (for the model fit around node *i*). We note that the local model fitting (equation ([mainLinModelEq])) does not depend on this network estimation. Hence, this thresholding can take place independently of the computationally-intensive MCMC sampling. Thus, we leave *ϕ* as a tuning parameter, which can be varied by the user in real time to interpret results, equivalently to changing the resolution or granularity in a visualisation. We recommend the user does a full sweep through *ϕ* ∈ [0, ∞] to interpret the results. We also note that if ∣*b̂**t*, *i*(*j*)∣ > *ϕ* (for the independent model fit around node *j* rather than node *i*), we would independently infer an edge between nodes *i* and *j* at time *t*. Thus, some inconsistency may arise, due to these independent model fits around nodes *i* and *j*. To deal with this, we use the ‘min\\_symmetrisation’ scheme of, inferring an edge between nodes *i* and *j* at time *t*, i.e., *Â**i*, *j*, *t* ≠ 0, if and only if ∣*b̂**t*, *j*(*i*)∣ > *ϕ* and ∣*b̂**t*, *i*(*j*)∣ > *ϕ*.\n[postSDs]\nPlots of the inferred network structure around an example of a gene shown in Figure [bestRegs]b, namely SATB2, are shown in Figure [SATB2netPlot], after ‘min\\_symmetrisation’ (Section [modelOverviewSec]) with *ϕ* = 0.05. In addition to the neuronal identity gene SATB2, several of the genes shown in Figure [SATB2netPlot] are already known to be important in neuronal development, including NEUROD1 which initiates the programme of neuronal development and RUNX1T1 which regulates the differentiation of neurons from neural stem cells, as well as the neuronal circuit-formation gene BCL11A. Intriguingly, this network structure also includes MIR133A1HG and LINC00478, which are (respectively) examples of micro-RNA (miRNA) and long non-coding RNA (lncRNA). Non-coding RNA transcripts such as these do not get translated into proteins, as would usually be the case for a transcript from a region of DNA which codes for a gene. Instead, non-coding RNA transcripts are known to play an important role in gene regulation. However, we still only understand a small amount about their function, and gene regulation involving these sorts of non-coding RNA is an important research topic. We note that MIR133A1HG and LINC00478 are promising candidates for for further experimental investigation which have been identified using our proposed methodology.\n[SATB2netPlot]\nNext, we compared our method with alternative network inference method for single-cell transcriptome data. Methods for inferring time-varying network structure in data of this type include alternatives designed for many fewer nodes than our method can handle, such as the work of, which is designed to infer structure in networks with fewer than 100 nodes. However, there is also a static network inference method available for single-cell transcriptome data called ‘SCENIC’, that can be used to infer structure in large networks of 20000 or more nodes, and that is therefore also appropriate for the data-set analysed here. For comparison with our proposed methodology, we ran the SCENIC method on the same single-cell gene-expression data-set already analysed. Equivalently to our proposed method, SCENIC returns fitted model parameters that indicate the strength of the network connection between a pair of nodes, or genes: maintaining equivalent notation, we label these SCENIC model parameters *b**j*′(*i*). Thus, by again choosing a threshold *ϕ*′, it is possible to infer network structure by inferring edges between the pair of nodes *i* and *j* if the corresponding fitted model parameter ∣*b̂**j*′(*i*)∣ > *ϕ*′. We choose *ϕ*′ so as to maintain the same number of connections to each target in the network inferred by the SCENIC method, as compared with our method. Table [SCENICtab] shows the genes inferred in the network structure around the neuronal identity gene SATB2 (summarised from Figure [SATB2netPlot]), together with the genes equivalently found from the SCENIC method (setting *ϕ*′ to maintain the same number of connections).\n| Our method | SCENIC |\n| --- | --- |\n| BCL11A | ARPP21 |\n| DDX1 | CHL1 |\n| LINC00478 | KIAA1598 |\n| MIR133A1HG | MEF2C |\n| NEUROD1 | NFIA |\n| RUNX1T1 | RUNX1T1 |\n[SCENICtab]\nOf the genes shown in Table [SCENICtab], just as with those found by our method, those found by the SCENIC method are mostly already known to be involved in neural development, as follows. ARPP21 is involved with branching of dendrites, CHL1 and KIAA1598 are thought to be involved in neuronal migration and axon formation, and MEF2C and NFIA are known to be important for neural stem and progenitor cell differentiation. However, we note that the SCENIC method is not able to infer time-varying network structure, as our method can: to make the comparison shown Table [SCENICtab], the time-varying aspect of the network structure inferred by our method had to be ‘flattened out’.\nIt is challenging to visualise in a meaningful way the entire structure of a large network, such as the full genome-wide network inferred here, if it is inferred for all 22989 nodes. This challenge becomes even larger when the dimension of time is added. After fitting the\n\n[fullNetPlot]\nmodel to all 22989 target-nodes on a high-performance computing cluster, we inferred the structure of this network, and found the fully connected component (11133 nodes), which is shown in Figure [fullNetPlot]. As a minimum, it can be seen from this figure that the network structure changes gradually rather than suddenly with time, as we would expect from our proposed methodology.\nDiscussion\n==========\nIn this paper, we have proposed a new model to infer time-varying network structure. This model makes use of a novel prior structure we introduce here, which extends the Bayesian lasso to the time-varying case. The novel structure of this prior allows for effective modelling of time-varying network structure even in situations where there are very few time-points, as is typical in cell-biological (i.e., `omics) data. We also found that the model fitting and inference procedure we have proposed works well even in with large networks of over 20000 nodes, which compares very well with alternatives (see for example the work by ).\nWe used simulated data to assess the ability of the proposed model to accurately infer time-varying network structure, and we showed that the model is effective in inferring time-varying genomic network structure from single-cell gene-expression data. However, we note that genomic network structure which is inferred from only gene-expression data (as we do here) is not guaranteed to correspond to true gene regulatory patterns. To strengthen any belief that the inferred genomic network structure corresponds to true gene regulatory patterns rather than simply gene co-expression patterns, evidence from, for example, chromatin binding and epigenomic data could also be incorporated into the model. We intend to incorporate such data as the next stage of the development of this model. Specifically, we will do this by allowing the sparsity parameter *λ* to vary for each pair of nodes *i* and *j*, depending on any prior evidence of a physical interaction between the protein-product of gene *j* with the DNA or surrounding chromatin of gene *i*.\nAnother characteristic of the single-cell transcriptome data analysed here is that the data are zero-inflated. This is a case of data missing-not-at-random, because the dropout events which lead to the extra zeros in the data are more likely to occur when the true transcriptome level is low. As part of the next stage of the development of this model, we intend to account for dropouts as other authors have done, for example by explicitly including the dropout events in the model likelihood. We also note that existing time-inference methods for data such as those presented here are algorithmic, rather than model-based. Hence it is not easy to obtain uncertainties on the inferred times when using these methods. Thus, we would like to develop a model-based time-inference method that will provide such uncertainties, and then feed these uncertainties directly into the time-varying network model we have proposed. We also note that in other contexts, it could complicate the analysis if there is uneven time-sampling. For example, if we expect highly deterministic behaviour with little noise, but have data with time-sampling at known but uneven time-points, the method might need to be adapted. Specifically, in that context we would expect to see larger changes in parameters over larger time-intervals: this structure is not explicitly captured by our model, in its current form.\nUnderstanding interactions between genes and their transcriptional regulators is a fundamental question in genomics, and network models are a natural way to represent and analyse groups of interactions between genes and their regulators. Biomedical science in the high-throughput genomic age has been developing ever more innovative ways to collect increasingly vast quantities of data. However, the statistical techniques to represent, analyse and interpret such data still lag behind the means to generate them. In particular, there is currently a lack of good computational statistical methodology to represent and analyse changes in gene-regulatory interactions as cells are specified and change state - an issue we address with the time-varying network model that we propose here. The computational-statistical tools that we are developing allow novel characterisation of genomic interactions in important settings, adding to knowledge of fundamental biological principles, and motivating further investigation by targeted experiments.\nAcknowledgements\n================\nWe are grateful to Aaron Diaz, Tom Nowakowski, Alex Pollen, and Aparna Bhaduri, for helpful discussions, insightful comments, and useful advice throughout this project, and for providing early access to the data. The work of the first author was supported by the MRC grant MR/P014070/1. The work of the second and third author was partially supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1.\n[id=supp]\n[id=soft]\n\nTwo-way sparsity for time-varying networks, with applications in genomics\n=========================================================================\n,\nWe propose a novel way of modelling time-varying networks, by inducing two-way sparsity on local models of node connectivity. This two-way sparsity separately promotes sparsity across time and sparsity across variables (within time). Separation of these two types of sparsity is achieved through a novel prior structure, which draws on ideas from the Bayesian lasso and from copula modelling. We provide an efficient implementation of the proposed model via a Gibbs sampler, and we apply the model to data from neural development. In doing so, we demonstrate that the proposed model is able to identify changes in genomic network structure that match current biological knowledge. Such changes in genomic network structure can then be used by neuro-biologists to identify potential targets for further experimental investigation.\nIntroduction\n============\nNetwork models have become an important topic in modern statistics, and the evolution of network structure over time (illustrated in Figure [dynNetFig]) is an important area of study. Network structures that evolve over time naturally occur in a range of applications. Examples of recent applications include evolving patterns of human interaction such as in social networks, time-varying patterns of interaction between genes and their protein-products in biological networks, and time-varying patterns of connectivity in the brain. However, network models with temporal structure have only recently begun to be studied in detail in statistical research.\n\n[dynNetFig]\nAn important application area of statistical network models is genomics. Network models are a natural way to describe and analyse patterns of interactions (represented by network edges) between genes and their protein-products (represented by network nodes). An important interaction of this type is gene regulation, in which the protein-product of one gene influences the output level of the protein-product of another gene. Much gene regulation is characteristic of a particular cell type, so that a cell knows its role within the organism. These specific regulatory network structures that are characteristic of particular cell-types are established during embryonic development. Changes in normal gene regulation are also inherent to cancer progression, so that cells forget how they should act, taking on pathological roles (regulatory network re-wiring). However, whilst network models are well established in genomics, historically these models have typically been static, ignoring the fact that genomic processes are inherently time-varying.\nThere are many examples of recent work on models of time-varying networks. In statistics, this work covers methods based on Markov processes, on dynamic Erds-Rényi graphs, and on sparse regression methods. It also includes work on time-varying community structure, on methods extending the stochastic block model, and related non-parametric graphon-based methods, as well as non-parametric methods for dynamic link prediction and methods from Bayesian nonparametrics. Other related work includes sparse graphical models that can take account of different time-points.\nMotivated by genomics applications, we propose a novel framework for modelling time-varying networks, by inducing *two-way sparsity* on local models of the connectivity of each node to all the others. This is achieved as follows. We start with a regression likelihood function that assumes that observations are mutually independent over time. Dependence is then induced through a novel prior structure that promotes sparsity in a two ways: across time, and within time. This decoupling of the induced sparsity is achieved through a copula specification for the parameters in the likelihood function. Specifically, the regression coefficients for one node across different time-points are jointly distributed according to a Gaussian copula with Laplace marginal distributions. The correlation matrix of the Gaussian copula is formed by assuming that the correlation between time-points decays with time in a structured, parsimonious way that also ensures its positive definiteness. In this correlation matrix, the only free parameter is the correlation between consecutive time-points, which is given a reverse-exponential prior distribution with support in [0, 1). This prior on the correlation across time discourages large differences in the regression coefficients between consecutive time-points and, as a consequence, also discourages large changes in the inferred structure of the network.\nThe decoupling of the marginal and dependence structure that is facilitated by the copula specification, and the particular form of the correlation matrix, allow for precise control of marginal priors. This decoupling also makes the adoption of generalisations of the Bayesian lasso, such as the horseshoe, easy to implement in place of the marginal Bayesian lasso prior that we use. The prior dependence among parameters across time can also be viewed as a Bayesian version of the fused lasso, while within each time-slice we directly utilise existing work on the Bayesian lasso. In fact, the proposed modelling framework has the Bayesian lasso as a special case, when the correlation between time-points is set to zero. From a frequentist point of view, the sparsity structure we propose would fall within the remit of the generalised lasso, which has the fused lasso as a special case. Bayesian versions of the fused lasso have also been proposed. However, a key difference between those methods and the modelling framework we propose, is the formal decoupling of sparsity across time (which the fused lasso induces), from sparsity within time. Importantly, we are able to apply this proposed modelling framework locally to each network node, as previous authors have done. Because these local model fits are mutually independent they can easily be carried out sequentially or in parallel, meaning that in practice, we are able to work with large networks of tens of thousands of nodes. The novel prior structure proposed, which enables the time-varying network inference, is also of interest more generally beyond the context of network science. This novel prior structure is relevant in any context where sparse regression with time-varying regression parameters is desirable.\nThe rest of the paper is structured as follows. In Section [methodsSec], we set up notation, and specify the model. Then, in Section [simSec] we present the results of fitting the model to simulated data, and in Section [resSec] we present the results of fitting the model to single-cell transcriptome data. Finally, in Section [discussSect], we summarise our findings and discuss their broader context. The Supplementary Material we provide all proofs and derivations, data pre-processing details, and Supplementary Figures, as well as a freely available software implementation of our proposed model and algorithm.\nProposed methodology\n====================\nData description\n----------------\nThe two-way sparsity that is induced by the proposed modelling framework is motivated by the problem of inferring time-varying structure in genomic networks. In these networks nodes represent genes: for each node there are observations or measurements of the activity level of the corresponding gene (the ‘gene-expression level’). These node-specific observations make up the data-set. The expression-level of a gene is generally influenced by the expression-level of several other genes (in a process called ‘gene regulation’). Hence, a natural application for models of time-varying networks is understanding dynamic patterns of gene-regulation in biological processes, such as neural development. Measurements of gene transcript counts are often used as a surrogate for gene expression level in RNA sequencing data, and hence we base our real-data example on single-cell transcriptomic data. We note that single-cell transcriptomic data is a type of single-cell gene-expression data.\nSingle-cell gene-expression data are ideal for this application, because data from a study of this type will typically be obtained from a heterogeneous mixture of cells, each of which may be at a different point on a trajectory through the biological process under investigation. For example, in the context of neural development, some of these cells may be stem-cells, whereas some may be fully differentiated cells (e.g., neurons), with a whole spectrum of cells in between. Each cell can be thought of as an independent sample from the underlying latent biological process; in this example, that process is neural development. Thus, we can think of the progression of a cell through this process of neural development in terms of a ‘developmental trajectory’. The progression along such a developmental trajectory can be quantified in terms of ‘developmental time’, which is simply a measure of a temporally-ordered progression through the process of cellular development. For each of the cell-samples in the data, no information is available other than its high-dimensional gene-expression measurements. Hence, it is necessary to first infer the ‘developmental time’ of the cell-samples before fitting any time-varying network model. This leads to an ordered sequence of pseudo-temporal measurements *x*1, *i*, *x*2, *i*, ...*x**t*, *i*, ..., *x**T*, *i* of the log-expression level for gene *i*. Importantly, the *x*1, *i*, *x*2, *i*, ...etc are taken from different cell samples for each pseudo-time point, and are hence independent. Inference like this is more generally referred to as ‘pseudo-time’ inference, and several methods exist to carry it out: see for example work by and.\nModel overview\n--------------\nWe develop a model for each target-node conditional on all the other nodes, and then we apply this model to several target-nodes of interest. This is different from, for example, the work of and who consider the modelling of all nodes jointly. Such a target-node approach has been used previously by, and it allows the network structure to be inferred independently around each target-node *i* ∈ {1, ..., *p*}. This strategy has several advantages. Firstly, variable screening can be applied before model fitting. This allows the dimensionality of the problem to be reduced from *p* of the order of tens of thousands down to *p*ʹ of the order of a few hundred for each parallel model fit around a target-node, whilst still allowing the global network structure to be estimated over tens of thousands of target-nodes, if required. Our modelling strategy also allows the local network structure to be estimated around only a small number of target-nodes if required, controlling computational expense, whilst still inferring the connected node-sets from tens of thousands of nodes. Inference is carried out with a sparse linear model, taking the observations for node *i* at time *t* as the response, and the observations for all nodes *j* ≠ *i* at time *t* as potential predictors. From these potential predictors, the set of predictors ‘chosen’ by the sparse model fit are then used to infer the network structure. Specifically, we want to infer the network structure around a fixed set of nodes with a set of edges that varies with time. In this scenario, only the patterns of interconnectivity change as the network evolves (Figure [dynNetFig]), which is the scenario most relevant to genomics applications. Such a network can be represented with a time-varying adjacency matrix **A**, where *A**i*, *j*, *t* denotes the absence (*A**i*, *j*, *t* = 0) or presence (*A**i*, *j*, *t* > 0) of an edge between nodes *i* and *j* at time *t*. We note that under this scheme, the local model fit (which is responsible for the computational load) does not depend on the network estimation (which takes place subsequently). The inferred network is a particular summary of the posteriors that are obtained from several of our model fits. We propose a model for node-wise regression, and we suggest how to summarise these models over several nodes of a network.\nModel likelihood\n----------------\nWe assume a likelihood function where observations are mutually independent over time. This is an assumption that is compatible with high-dimensional gene-expression data, where no single cell can be measured at more than one time-point. We note that this implies that observations are independent at different time-points. Let **X** represent the full data-set for the nodes shown in Figure [dynNetFig], with time varying down the rows, and with each node corresponding to a different column. Then, *x**t*, *i* denotes the value for some node in the system at time *t* ∈ {1, ..., *T*}, for *i* ∈ {1, ..., *p*}, and the row-vector $\\mathbf{x}\\_{t,\\fgebackslash i}$ denotes the values for the other *p* − 1 nodes at time *t*. We model the dependence of *x**t*, *i* on $\\mathbf{x}\\_{t,\\fgebackslash i}$ as:\n \n$$x\\_{t,i}=a\\_i+\\mathbf{b}^{(i)}\\_{t,:}\\ \\mathbf{x}\\_{t,\\fgebackslash i}^\\top+\\epsilon\\_{t,i},\\label{mainLinModelEq}$$ \nwhere **b***t*, :(*i*) is a vector of linear model parameters, and *ε**t*, *i* ∼ N(0, *τ**i*− 1).\nThe response variable *x**t*, *i* corresponds to the observations for a ‘target’ node around which we are modelling the local network structure, whereas the variables represented by $\\mathbf{x}\\_{t,\\fgebackslash i}$ correspond to the observations for all the other nodes of the network. To model the whole network, we must fit model ([mainLinModelEq]) around each target-node in turn. We note that here we make an assumption about the existence of a global undirected Markov network that explains the independence constraints in the model. This assumption has also been used previously by in an equivalent context. We note that our approach does not enforce hard constraints, such as **b***t*, *j*(*i*) = **b***t*, *i*(*j*). However, it is computationally very expensive to work with a global, coherent model, where such constraints can be enforced. In this work, we have opted to sacrifice some coherence for the sake of computational efficiency. This enables us to estimate quantities of interest in a computationally-efficient manner through an overparameterized representation of a joint model. It also enables us to focus on a particular subset of nodes of interest without having to go through an overly-expensive computation for the estimation of a global, coherent model.\nUsing **b**: , *j*(*i*) to denote the column-vector of model parameters for covariate *j* for *t* ∈ {1, ..., *T*}, we collect parameters in matrix **B**(*i*) = [**b**: , 1(*i*), **b**: , 2(*i*), ..., **b**: , *p* − 1(*i*)]. In the next section, we postulate a prior for dependencies within each column *j* of **B**(*i*), whilst noting that the columns of **B**(*i*) (each corresponding to a different node as covariate) are independent of each other. We also introduce the notation *x**t*, *i*, *k* and $\\mathbf{x}\\_{t,\\fgebackslash i,k}$ to represent observations of *x**t*, *i* and $\\mathbf{x}\\_{t,\\fgebackslash i}$ for sample *k* ∈ {1, ..., *n**t*} at time *t*.\nWe denote **x**: , *i* = [*x*1, *i*, 1, ..., *x*1, *i*, *n*1, ..., *x**t*, *i*, 1, ..., *x**t*, *i*, *n**t*, ..., *x**T*, *i*, 1, ..., *x**T*, *i*, *n**T*]⊤ and \n$\\mathbf{X}\\_{:,\\fgebackslash i}=[\\mathbf{x}\\_{1,\\fgebackslash i,1}^\\top,\\mathbf{x}\\_{1,\\fgebackslash i,n\\_1}^\\top,...,\\mathbf{x}\\_{t,\\fgebackslash i,1}^\\top,...,\\mathbf{x}\\_{t,\\fgebackslash i,n\\_t}^\\top,...,\\mathbf{x}\\_{T,\\fgebackslash i,1}^\\top,...,\\mathbf{x}\\_{T,\\fgebackslash i,n\\_T}^\\top]^\\top$, where **x**: , *i* is column *i* of data-matrix **X**, and $\\mathbf{X}\\_{:,\\fgebackslash i}$ is data-matrix **X** without column *i*. Hence, we can write the model likelihood for the target-node *i* as:\n \n$$P(\\mathbf{x}\\_{:,i}|\\mathbf{X}\\_{:,\\fgebackslash i},\\mathbf{B}^{(i)},a\\_i,\\tau\\_i)=\\prod\\_{t=1}^T\\prod\\_{k=1}^{n\\_t}\\sqrt{\\frac{\\tau\\_i}{2\\pi}}e^{-\\tau\\_i(x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,:}\\cdot\\mathbf{x}\\_{t,\\fgebackslash i,k}^\\top-a\\_i)^2/2}.{\\addtocounter{equation}{1}\\tag{\\theequation}}\\label{dataCondParams}$$ \nWe note that we consider likelihoods in the form of equation ([dataCondParams]) for each target-node *i*.\nPriors with decoupled two-way sparsity\n--------------------------------------\nWe model the regression coefficients **b**: , *j*(*i*) across time-points *t* = 1, ..., *T* with a Gaussian copula with Laplace marginal distributions, as follows. The elements of **b***t*, :(*i*) (*t* = 1, ..., *T*) are marginally distributed as *b**t*, *j*(*i*) ∼ Laplace (1/*λ*), with probability density function $\\frac{\\lambda}{2}e^{-\\lambda|\\cdot|}$ and cumulative distribution function *F*L[*b**t*, *j*(*i*)], for *t* ∈ {1, ..., *T*} and *j* ∈ {1, ..., *p* − 1}. Hence, Φ− 1{*F*L[*b**t*, *j*(*i*)]} follows a Gaussian distribution for *t* ∈ {1, ..., *T*} and *j* ∈ {1, ..., *p* − 1}, where Φ is the standard-normal cumulative distribution function. The dependencies between Φ− 1{*F*L[*b**t*, *j*(*i*)]} and Φ− 1{*F*L[*b**t* + 1, *j*(*i*)]} are then modelled through their joint distribution as:\n \n$${\\footnotesize\n\\left[\\begin{array}{c}\\vspace{1ex} \\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{1,j}]\\right\\} \\\\\\vspace{1ex}\\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{2,j}]\\right\\} \\\\\\vspace{1ex} \\vdots \\\\\\vspace{1ex} \\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{T,j}]\\right\\} \\end{array} \\right]\\sim\\mathcal{N}\\left(\\mathbf{0},\\boldsymbol{\\Sigma}^{(i)}\\_j\\right),\\enspace\\text{with}\\enspace\\boldsymbol{\\Sigma}^{(i)}\\_j=\\left[\\begin{array}{ccccc}\\vspace{1.3ex}1&\\rho^{(i)}\\_j&\\left(\\rho^{(i)}\\_j\\right)^2&\\cdots&\\left(\\rho^{(i)}\\_j\\right)^T\\\\\\vspace{1.3ex}\\rho^{(i)}\\_j&1&\\rho^{(i)}\\_j&\\cdots&\\left(\\rho^{(i)}\\_j\\right)^{T-1}\\\\\\vspace{1.3ex}\\left(\\rho^{(i)}\\_j\\right)^2&\\rho^{(i)}\\_j&1&\\cdots&\\left(\\rho^{(i)}\\_j\\right)^{T-2}\\\\\\vspace{1.3ex}\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\\\vspace{1.3ex}\\left(\\rho^{(i)}\\_j\\right)^T&\\left(\\rho^{(i)}\\_j\\right)^{T-1}&\\left(\\rho^{(i)}\\_j\\right)^{T-2}&\\cdots&1\\end{array}\\right],{\\addtocounter{equation}{1}\\tag{\\theequation}}\\label{fullCovSpec}\n}$$ \nand hence the regression coefficients are modelled as a Gaussian copula:\n \n$$F\\left[\\mathbf{b}^{(i)}\\_{:,j}\\right]=\\Phi\\_p\\left[\\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{1,j}]\\right\\},\\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{2,j}]\\right\\},...,\\Phi^{-1}\\left\\{F\\_\\mathcal{L}[b^{(i)}\\_{T,j}]\\right\\};\\boldsymbol{\\Sigma}^{(i)}\\_j\\right].$$ \nThe correlation parameter *ρ**j*(*i*) is assumed to have a reverse-exponential distribution with support [0, 1) and density\n \n*f*rexp[*ρ**j*(*i*)] ∼ *k**e**k**ρ**j*(*i*)/(*e**k* − 1). \n[graphicalModel]\nThe structure of $\\boldsymbol{\\Sigma}^{(i)}\\_j$ is such that transformed model parameters at adjacent points in time, such as Φ− 1{*F*L[*b**t*, *j*(*i*)]} and Φ− 1{*F*L[*b**t* + 1, *j*(*i*)]}, have correlation *ρ**j*(*i*) (Figure [graphicalModel]). Then, the transformed parameters separated by two time-points have correlation (*ρ**j*(*i*))2, etc. Thus, also denoting the sequence of transformed model parameters Φ− 1{*F*L[*b*1, *j*(*i*)]}, Φ− 1{*F*L[*b*2, *j*(*i*)]},...,Φ− 1{*F*L[*b**t*, *j*(*i*)]} forms a Markov chain, meaning that $\\boldsymbol{\\Sigma}^{(i)}\\_j$ is guaranteed to be positive-definite for *ρ**j*(*i*) ∈ [0, 1), and by construction\n \n*b**t* + 1, *j*(*i*) ⊥ *b**t* − 1, *j*(*i*), *b**t* − 2, *j*(*i*), ...∣*b**t*, *j*(*i*). \n[priorTrajSamp]\nSuch a construction for $\\boldsymbol{\\Sigma}^{(i)}\\_j$ discourages differences in the regression coefficients for the same covariate between adjacent time-points, and hence also discourages changes in the network structure over time, resulting in sparsity across time. Then, transforming the Φ− 1{*F*L[*b**t*, *j*(*i*)]} back to *b**t*, *j*(*i*), where the *b**t*, *j*(*i*) are marginally Laplace distributed, achieves sparsity within time by discouraging regression coefficients from taking non-zero values, hence also encouraging discovery of sparse network structures. Figure [priorTrajSamp] shows ten samples from our proposed prior on *b**t*, *j*(*i*) plotted against *t*, and demonstrates the correlation structure enforced by the prior over time. We again note that while *b**t*, *j*(*i*) and *b**t* + 1, *j*(*i*) are correlated, *b**t*, *j*(*i*) and *b**t*, *j*ʹ(*i*) are independent.\nRecent work by that takes a Bayesian approach to generalising the fused lasso could be used similarly to the approach we propose, by modelling the same set of covariates at multiple time-points whilst enforcing smooth changes across time as well as sparsity overall. However, achieve their result by simply multiplying together separate frequentist-inspired priors for smoothness across time and for sparsity. Specifically, they multiply together a Laplace prior to penalise individual non-zero model parameters, with the ultra-sparse negative-exponential-gamma (NEG) prior to penalise non-zero differences in parameters. The Laplace-NEG prior is defined (choosing notation to be consistent with that of our model) as:\n \n*P*(**b**: , *j*(*i*)) ∝ ∏*t* = 1*T*Laplace(*b**t*, *j*(*i*)∣*λ*)∏*t* = 2*T*NEG(*b**t*, *j*(*i*) − *b**t* − 1, *j*(*i*)∣*λ*†, *γ*), \nwhere the Laplace density is defined as $\\frac{\\lambda}{2}e^{-\\lambda|\\cdot|}$, and\n \nNEG( ⋅ ∣*λ*†, *γ*) = ∫0∞∫0∞*f*N( ⋅ ∣0, *τ*2)*f*\\gamma(*τ*2∣1, 1/*ψ*)*f*\\gamma(*ψ*∣*λ*†, 1/*γ*2)*d**τ*2*d**ψ*, \nwhere *f*N and *f*\\gamma are the Normal and Gamma densities, respectively. Sampling from the distribution of equation ([laplNegDist]) is done by simulating exponential and gamma random variables, which are then used to form the precision matrix of a multivariate normal distribution, as specified by. In contrast to the Laplace-NEG prior, the model we propose retains the property that, marginally, each parameter still follows the Bayesian lasso prior. In particular, if we set *ρ**j*(*i*) = 0 (for *j* = 1, 2, ..., *p* − 1), then the model we propose is exactly the same as the Bayesian lasso. This is important because it makes it easier to set priors, including variants of the Bayesian lasso that avoid its well-known shortcomings (see for example the work by and ). Although we will not consider such variants here, they follow directly by mimicking the construction using the Bayesian lasso.\n[priorFig]\nThe novel prior we use on *ρ**j*(*i*) is a ‘reverse exponential prior’ (equation ([revExpPriorDef])). Figure [priorFig]a shows the probability density function of the reverse-exponential prior for different values of hyper-parameter *k*. Figure [priorFig]b then shows heatmaps of the bivariate density distributions of samples from the decoupled-sparsity prior for a parameter *j* over two time-points, i.e., **b**: , *j*(*i*) = [*b*1, *j*(*i*), *b*2, *j*(*i*)]⊤, for a range of values of *λ* and *k* (the corresponding marginal densities are shown in Figures S6 and S7 in the Supplementary Information). For comparison, Figures S3 - S5 in Supplement D show samples from the Laplace-NEG prior as defined in equation ([laplNegDist]), for various values of *λ* (which acts equivalently to *λ* in our model, controlling sparsity of individual model parameters), and various values of *λ*† and *γ* (which both act equivalently to *k* in our model, controlling sparsity of differences between model parameters). The main difference between these priors is that our decoupled-sparsity prior still marginally follows the Bayesian lasso prior, and is hence a direct generalisation of the Bayesian lasso to this setting with time-varying model parameters. In other words, our prior does what the Laplace-NEG does, but with the added benefit that we generalise the Bayesian lasso.\nPosterior inference\n-------------------\nThe order-1 Markovian relations specified by equation ([thetaCondIndepState]) are also computationally attractive, because they result in models with banded precision matrices. From equation ([thetaCondIndepState]), and denoting *θ**t*, *j*(*i*) = Φ− 1{*F*L[*b**t*, *j*(*i*)]} it follows that the partial correlation of *θ**t* + *m*, *j*(*i*) with *θ**t* + *l*, *j*(*i*) will be zero for all ∣*m* − *l*∣ > 1. Hence, all entries of the precision matrix $[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}$ will be zero except the diagonal and the elements immediately adjacent to it (i.e., the sub- and super-diagonals). These relationships allow all the entries of this precision matrix to be found easily in terms of *ρ**j*(*i*) by solving $[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\boldsymbol{\\Sigma}^{(i)}\\_j=\\mathbb{I}$, which gives:\n \n$$\\left(\\left[\\boldsymbol{\\Sigma}^{(i)}\\_j\\right]^{-1}\\right)\\_{t,t^\\prime}=\n\\begin{cases}\n\\vspace\\*{1ex}\n 1/(1-[\\rho^{(i)}\\_j]^2), & \\text{if}\\ t^\\prime=t=1\\ \\text{or}\\ t^\\prime=t=T, \\\\\n\\vspace\\*{1ex}\n (1+[\\rho^{(i)}\\_j]^2)/(1-[\\rho^{(i)}\\_j]^2), & \\text{if}\\ t^\\prime=t>1\\ \\text{and}\\ t^\\prime=t<T, \\\\\n\\vspace{1ex}\n -\\rho^{(i)}\\_j/(1-[\\rho^{(i)}\\_j]^2), & \\text{if}\\ t^\\prime=t+1\\ \\text{or}\\ t^\\prime=t-1, \\\\\n 0, & \\text{otherwise}\\label{precMatSpec}\n\\end{cases}$$ \nwhere $\\left(\\left[\\boldsymbol{\\Sigma}^{(i)}\\_j\\right]^{-1}\\right)\\_{t,t^\\prime}$ represents the (*t*, *t*′) element of the precision matrix $[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}$. A full derivation of equation ([precMatSpec]) is given in Supplement A.\nThe model parameters **B**(*i*) can be sampled directly from multivariate Normal distributions, without needing the intermediate transformation to the marginally Laplace-distributed variables described in Section [priorDef]. This can be achieved with an algebraic manipulation which is an extension from the Bayesian lasso, as follows. The Laplace distribution can be written as an uncountable mixture of zero-mean Normal distributions, with the variances of the mixture components distributed as $\\text{Exp}(\\frac{\\lambda^2}{2})$. Specifically,\n \n$$P(b^{(i)}\\_{t,j}|\\lambda)=\\frac{\\lambda}{2}e^{-\\lambda|b^{(i)}\\_{t,j}|}=\\int\\_0^\\infty P(b^{(i)}\\_{t,j},s^{(i)}\\_j|\\lambda)ds^{(i)}\\_j,$$ \nwhere\n \n$$P(b^{(i)}\\_{t,j},s^{(i)}\\_j|\\lambda)=\\frac{1}{\\sqrt{2\\pi s^{(i)}\\_j}}e^{-[b^{(i)}\\_{t,j}]^2/[2s^{(i)}\\_j]}\\frac{\\lambda^2}{2}e^{-\\lambda^2s^{(i)}\\_j/2},$$ \nfor $s^{(i)}\\_j\\sim\\text{Exp}(\\frac{2}{\\lambda^2})$. This says that we will achieve *b**t*, *j*(*i*) being marginally Laplace distributed by sampling these *s**j*(*i*) from the $\\text{Exp}(\\frac{2}{\\lambda^2})$ prior, and then sampling the *b**t*, *j*(*i*) from zero-mean Normal distributions with variances *s**j*(*i*). Hence\n \n$$P(b^{(i)}\\_{t,j}|s^{(i)}\\_j)=\\frac{1}{\\sqrt{2\\pi s^{(i)}\\_j}}e^{-[b^{(i)}\\_{t,j}]^2/[2s^{(i)}\\_j]},$$ \nand so **b**: , *j*(*i*) has the same Normal distribution as Φ− 1{*F*L[*b**t*, *j*(*i*)]} but with the variances and covariances scaled up by *s**j*(*i*), with $s^{(i)}\\_j\\sim\\text{Exp}(\\frac{2}{\\lambda^2})$. Therefore, also referring back to equation ([fullCovSpec]), it follows that\n \n$$\\begin{aligned}\nP(\\mathbf{b}^{(i)}\\_{:,j},s^{(i)}\\_j|\\rho^{(i)}\\_j,\\lambda)=&\\frac{\\lambda^2}{2}e^{-\\lambda^2s^{(i)}\\_j/2}\\frac{1}{(2\\pi)^{T/2}[s^{(i)}\\_j]^{1/2}|\\boldsymbol{\\Sigma}^{(i)}\\_j|^{1/2}}e^{-{\\mathbf{b}^{(i)}\\_{:,j}}^\\top [s^{(i)}\\_j]^{-1}[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}/2}.\\end{aligned}$$ \nTo make sampling easier, at this stage we let *s**j*(*i*) = [*ν**j*(*i*)]− 1, leading to the density\n \n$$\\begin{aligned}\nP(\\mathbf{b}^{(i)}\\_{:,j},\\nu^{(i)}\\_j|\\rho^{(i)}\\_j,\\lambda)=&\\frac{1}{[\\nu^{(i)}\\_j]^2}\\frac{\\lambda^2}{2}e^{-\\lambda^2/(2\\nu^{(i)}\\_j)}\\frac{[\\nu^{(i)}\\_j]^{1/2}}{(2\\pi)^{T/2}|\\boldsymbol{\\Sigma}^{(i)}\\_j|^{1/2}}e^{-{\\mathbf{b}^{(i)}\\_{:,j}}^\\top \\nu^{(i)}\\_j[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}/2}\\\\\n=&\\frac{\\lambda^2}{2}e^{-\\lambda^2/(2\\nu^{(i)}\\_j)}\\frac{[\\nu^{(i)}\\_j]^{-3/2}}{(2\\pi)^{T/2}|\\boldsymbol{\\Sigma}^{(i)}\\_j|^{1/2}}e^{-{\\mathbf{b}^{(i)}\\_{:,j}}^\\top[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}\\nu^{(i)}\\_j/2},{\\addtocounter{equation}{1}\\tag{\\theequation}}\\label{bNuPriorFinal}\\end{aligned}$$ \nwhere the extra factor of 1/[*ν**j*(*i*)]2 is the factor ∣*d*{[*ν**j*(*i*)]− 1}/*d**ν**j*(*i*)∣ due to the change of variable. Assuming that the model will be fit to data standardised to have unit variance, we set the prior on the intercept as *a* ∼ N(0, 1), and we set the prior on the model precision as *τ**i* ∼ Gamma(1, 1) (which has prior mean 1, with 95% of the prior mass between 0.025 and 3.7, which we believe is reasonable for these data). Now combining equation ([bNuPriorFinal]) with these prior specifications, and $P(\\rho^{(i)}\\_j|k)=\\frac{k}{e^k-1}e^{k\\rho^{(i)}\\_j}$ (for 0 ≤ *ρ**j*(*i*) ≤ 1), as well as with the model likelihood (equation ([dataCondParams])), we get:\n \n$$\\begin{aligned}\nP&(\\mathbf{x}\\_{:,i},\\mathbf{B}^{(i)},\\boldsymbol{\\rho}^{(i)},\\boldsymbol{\\nu}^{(i)},a\\_i,\\tau\\_i|\\mathbf{X}\\_{:,\\fgebackslash i},\\lambda,k)=\\left\\{\\prod\\_{t=1}^T\\prod\\_{k=1}^{n\\_t}\\sqrt{\\frac{\\tau\\_i}{2\\pi}}e^{-\\tau\\_i\\left(x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,:}\\cdot\\mathbf{x}\\_{t,\\fgebackslash i,k}^\\top-a\\_i\\right)^2/2}\\right\\}\\\\\n&\\frac{1}{\\sqrt{2\\pi}}e^{-\\left\\{\\tau\\_i+a\\_i^2/2\\right\\}}\\prod\\_{j=1}^{p-1}\\left\\{\\frac{k}{e^k-1}e^{k\\rho^{(i)}\\_j}\\frac{\\lambda^2}{2}e^{-\\lambda^2/(2\\nu^{(i)}\\_j)}\\frac{[\\nu^{(i)}\\_j]^{-3/2}}{(2\\pi)^{T/2}|\\boldsymbol{\\Sigma}^{(i)}\\_j|^{1/2}}e^{-{\\mathbf{b}^{(i)}\\_{:,j}}^\\top[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}\\nu^{(i)}\\_j/2}\\right\\}.{\\addtocounter{equation}{1}\\tag{\\theequation}}\\label{fullJdist}\\end{aligned}$$ \nFollowing equation ([fullJdist]), posterior sampling for the model described in Sections [mainModel] and [priorDef] can be implemented through a Gibbs sampler with the steps given in Algorithm [gibbsSampEqs]. We note that Algorithm [gibbsSampEqs] has a relatively low computational cost, because each of the steps (with the exception of step 4) involves sampling from a known distribution for which the parameters can be easily calculated. Then for step 4, we can simply use a slice-sampler to sample *ρ**j*(*i*), which has finite support *ρ**j*(*i*) ∈ [0, 1). The full derivations of each step of Algorithm [gibbsSampEqs] appear in Supplement B.\n[gibbsSampEqs] A Gibbs sampler with the following steps:\n1. **Sample:** *a**i* from:\n \nwhere *f*N is the Normal density, *σ**a*− 2 = 1 + *n**τ**i* and $\\mu\\_{a}=\\sigma\\_{a}^2\\tau\\_i\\sum\\_{t=1}^T\\sum\\_{k=1}^{n\\_t}\\{x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,:}\\cdot\\mathbf{x}\\_{t,\\fgebackslash i,k}^\\top\\}$.\n2. **Sample:** *τ**i* from:\n \nwhere *f**γ* is the density of the gamma distribution with $k\\_\\tau=1+\\frac{\\sum\\limits\\_{t=1}^Tn\\_t}{2}$ and\n \n$$\\theta\\_\\tau=1/\\{1+\\sum\\limits\\_{t=1}^T\\sum\\limits\\_{k=1}^{n\\_t}(x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,:}\\cdot\\mathbf{x}\\_{t,\\fgebackslash i,k}^\\top-a\\_i)^2/2\\}.$$\n3. **Sample:** *ν**j*(*i*) from: \nwhere *f**I**G* is the density of the inverse Gaussian distribution with parameters \n*λ**ν* = *λ*2 and $\\mu\\_\\nu=\\lambda\\bigg/\\sqrt{{\\mathbf{b}^{(i)}\\_{:,j}}^\\top[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}\\mathbf{b}^{(i)}\\_{:,j}}$.\n4. **Sample:** *ρ**j*(*i*) from:\n5. **Sample:** **b**: , *j*(*i*) from:\n \nwhere *f*N is the multivariate Normal density, $[\\widetilde{\\boldsymbol{\\Sigma}}^{(i)}\\_j]^{-1}=\\nu^{(i)}\\_j[\\boldsymbol{\\Sigma}^{(i)}\\_j]^{-1}+[\\mathbf{V}^{(i)}\\_j]^{-1}$, \nand $\\widetilde{\\mathbf{m}}^{(i)}\\_j=\\widetilde{\\boldsymbol{\\Sigma}}^{(i)}\\_j[\\mathbf{V}^{(i)}\\_j]^{-1}\\mathbf{m}^{(i)}\\_j$, where the *t*th element of the vector **m***j*(*i*) is\n \n$$m^{(i)}\\_{t,j}=\\sum\\_{t=1}^T\\sum\\_{k=1}^{n\\_t}x\\_{t,j,k}\\left\\{x\\_{t,i,k}-\\mathbf{b}^{(i)}\\_{t,\\fgebackslash j}\\left(\\mathbf{x}\\_{t,\\fgebackslash i,k}\\right)\\_{\\fgebackslash j}^\\top-a\\_i\\right\\}/\\sum\\_{t=1}^T\\sum\\_{k=1}^{n\\_t}x\\_{t,j,k}^2,$$ \nwhere $\\mathbf{b}^{(i)}\\_{t,\\fgebackslash j}$ and $\\left(\\mathbf{x}\\_{t,\\fgebackslash i,k}\\right)\\_{\\fgebackslash j}$ represent **b***t*, :(*i*) and $\\mathbf{x}\\_{t,\\fgebackslash i,k}$ without the $j\\textsuperscript{th}$ elements, respectively, and **V***j*(*i*) is a diagonal matrix, with the *t*th diagonal element equal to 1/{*τ**i*∑*t* = 1*T*∑*k* = 1*n**t**x**t*, *j*, *k*2}.\nSimulation study\n================\nIn this section, we present the results from a simulation study, to test how accurately our model can recover network structure which we know in advance. We generate simulated data with structure that we expect to be typical of real data, and then fit the proposed model to the simulated data. To generate the data, the observations *x**t*, *i* for each node *i* are generated such that they follow a mean time-series of one of four types (illustrated in Figure [simDataEG]), as follows:\n1. Monotonic; decreasing to no signal.\n2. Monotonic; increasing from no signal.\n3. Maximum: increasing from and decreasing to no signal.\n4. Null: random noise.\n[simDataEG]\nTypes (a) and (b) represent node-types of interest to the biological setting, as follows. Type (a) corresponds to genes that are activated (i.e., *x**t*, *i* > 0) early in the time-series before becoming de-activated (as we would expect of genes which are important for stem-like cell identity). Type (b) corresponds to genes which only become activated later in the time-series (as we would expect of genes which are important for the identity of mature cells, such as neurons). Types (c) and (d) make the simulated data closer to what we would expect of the real data, by mixing in nodes with other sorts of signals: type (c) corresponds to genes which are active in the middle of the time-series only, and type (d) are null nodes (with random activation).\nAfter generating each characteristic mean time-series according to fixed types (a)-(d), we then time-stretch the particular characteristic mean time-series chosen for each node *i* by a random amount. We do this to reflect the fact that developmental events (as represented by gene-expression measurements) occur at different times in different cells. We achieve this effect by changing the length of the time-series to have a random but uniformly-distributed period *T*ʹ ∼ U[*T* − 3, *T*], before zero-padding to return the time-series to its original period *T*. This gives a mean profile *x**t*, *i*, *t* = 1, ..., *T* for each node that is distinct from all other nodes of the same type. We then generate observations *x**t*, *i*, *k* for each node according to equation ([mainLinModelEq]) based on these mean profiles, also setting the intercept parameter *a**i* to 0, and sampling via the Markov chain described in Algorithm [simDataGen]:\n[simDataGen] A Markov chain: \n**Loop**: *t*in 1 : *T* \n*X**t*, : , 1 ← 0\n// Initialize Markov chain at 0 \n: *r*in 2 : *R* \n**S** ← **X***t*, : , *r*–1 \n: *i*in 1 : *p* \n: *S**i* ∼ *N*(**S**∖ *i*.**b***t*, :(*i*), *σ*2) \n \n**X***t*, : , *r* ← **s** \n \n**end loop**\nwhere **X** is a *T* × *p* × *R* array containing the sampled data, **S** is a vector of length *p* which temporarily stores intermediate results, and the elements of **b**(*i*) are specified as:\n \n$$b^{(i)}\\_{t,j}=\n\\begin{cases}\n\\vspace\\*{1ex}\n1/{p'}, & \\text{if nodes $i$ and $j$ are of the same type}\\ \\\\\n\\vspace\\*{1ex}\n0, & \\text{otherwise}\\\\\n\\end{cases}$$ \nwhere *p*ʹ is the number of nodes *j* of the same type as *i*. The number of MCMC samples in the Markov chain specified in Algorithm [simDataGen] is given by the variable *R*: we use *R* = 104, and after thinning to take one sample in every 100, we choose the final 25 (thinned) samples to pass forward to the model fitting after adding the mean characteristic profiles. That is, we have 25 samples per time-point, i.e., *n**t* = 25, where autocorrelation analysis and an experimentation with burn-in times show no evidence against them being independently and identically distributed at each time-point group. We note that in the simulation we specify means that vary with time but our model has constant mean. In practice this does not make any difference as the data are always standardised before model fitting, but the user can easily make the intercept time-varying if this is a concern. The procedure for generating the simulated data is also illustrated in Figure [simDataExplain].\n[simDataExplain]\nWe generate each time-series with *T* = 8, *n**t* = 25 (constant for all values of *t*), and *p*ʹ = 10, adding noise with standard deviations *τ**i*− 1/2 = *τ*− 1/2 ∈ {0.1, 0.2, 0.3}. Then, we apply Algorithm [gibbsSampEqs], and calculate each *b̂**t*, *j*(*i*) from the median of the corresponding posterior. We infer an edge between nodes *i* and *j* if *b̂**t*, *j*(*i*) ≠ 0, after thresholding the *b̂**t*, *j*(*i*) to remove trivially small values, i.e., if ∣*b̂**t*, *j*(*i*)∣ ≥ *ϕ*. We generate ROC (receiver-operator characteristic) curves as this threshold *ϕ* is decreased to 0 from max∣*b̂**t*, *j*(*i*)∣ (for *t* ∈ {1, ..., *T*} and all *j*). We generate these curves from the true-positives (TP) and false-positives (FP) which we calculate from the ground-truth network edges *b**t*, *j*(*i*) and estimated network edges *b̂**t*, *j*(*i*) as follows:\n \n$$\\begin{aligned}\n|\\hat{b}^{(i)}\\_{t,j}|>0\\enspace\\text{for}\\enspace|b^{(i)}\\_{t,j}|>0\\implies\\enspace&\\text{TP}\\\\\n\\text{and}\\enspace\\enspace|\\hat{b}^{(i)}\\_{t,j}|>0\\enspace\\text{for}\\enspace|b^{(i)}\\_{t,j}|=0\\implies\\enspace&\\text{FP}\\end{aligned}$$ \nWe generate an average ROC curve over 1000 repetitions of this procedure, and then calculate an AUC (area under curve) statistic for this average ROC curve.\n[simDataNullParam]\n[simDataRes]\nWe assess the performance of our full model using both sparsity within and sparsity across time, compared with the scenarios when one of these priors is excluded from the model. To exclude sparsity across time, we enforce *ρ* = 0, and to exclude sparsity within time, we use *λ* → 0: these results are shown in Figure [simDataNullParam]. With *ρ* = 0, there is no correlation of the model parameters across time, and so we see the effect of inferring the networks separately for each time-point; i.e., sparsity across time is removed: in this case, AUC = 0.5 indicates that none of the intended structure in the data is being detected. Alternatively, as *λ* → 0, the prior becomes flat or uninformative, and so in this case we see the effect of fitting the model without the sparsity within time. We again note that decoupling these types of sparsity is made possible by design with the model structure we propose, unlike alternatives such as Laplace-NEG. Then for the full model (which includes the priors to enforce both the sparsity within time and across time), we repeated the simulation for various values of sparsity parameter *λ*: Figure [simDataRes] shows the results (with hyperparameter *k* = 20, equivalent results with *k* = 10 and *k* = 50 are shown in Figures S10 and S11 in Supplement D). When we include the priors for both sparsity within and across time, we can achieve AUC of 0.9 or more, as long as the sparsity parameter *λ* is large enough. This result demonstrates that our priors are responsible for good detection of network edges with respect to the ground-truth in these simulated data. We also found that these results were not very sensitive to *p*ʹ, the number of covariates included in the simulated data. Figure S12 shows equivalent results to Figure [simDataRes], except with the number of covariates halved to *p*ʹ = 5. In this case we found that the network inference is a bit more accurate, as would be expected with a smaller number of variables to predict; although the difference is minimal as long as the sparsity is great enough.\n[simDataResDrop]\nDropouts, or missing values, are a well known source of technical noise in single-cell transcriptome data. These missing values are replaced by zeros, leading to ‘zero inflation’. A characteristic of this dropout effect is that data-values which are already small are more likely to drop out (i.e., get missed out), than values which are larger in magnitude. This is data missing not-at-random, an effect that can be challenging to model. Dropout rates (i.e., the proportion of data-values missing from the data-set) are often over 60% in typical single-cell transcriptome data-sets that we have seen, such as the one analysed in Section [resSec]. To test the robustness of our method to dropouts, we used a well known and effective model of the dropout effect, published previously by. This model of dropouts specifies the probability of an observed data-value dropping out as\n \n*p**t*, *i*, *k* = exp( − *ω**x̃**t*, *i*, *k*2), \nwhere the parameter *ω* controls the dropout rate (decreasing *ω* increases the number of dropouts), and *x̃**t*, *i*, *k* is the data-value that would have been present without the dropout effect. This is essentially a hurdle model, with *h**t*, *i*, *k* ∼ Bernouilli(*p**t*, *i*, *k*), so that the observed data *x**t*, *i*, *k* (i.e., with dropouts included) is modelled as *x**t*, *i*, *k* = *h**t*, *i*, *k* ⋅ *x̃**t*, *i*, *k*. We found that under this model, *ω* = 2 leads to a dropout rate of around 66% in data-sets generated according to the data-simulation procedure presented earlier in this section. We used this value of *ω* = 2, and repeated our simulation study now with the addition of this dropout effect, carrying out out the same ROC-curve analysis as before. The results of this analysis (again with *λ* = 50 and *k* = 20) are shown in Figure [simDataResDrop]: we found that our method is quite resilient to dropouts, with only a moderate decrease in performance compared to the results shown in Figure [simDataRes]. The time-varying aspect of the model apparently helps to maintain performance when many dropouts are present, because when some values in the time-series are missing, sparsity across time encourages interpolation over the missing values. Interestingly, in very sparse cases, the dropout effect may even be helpful, possibly via a de-noising mechanism, as follows. Referring to the generative model of equation ([dropEq]), it’s clear that the dropouts mostly take place for small values of *x**t*, *i*, *k*. As these are much more likely to correspond to noise than larger values do, this leads to a strong de-noising effect. Finally, we note that to include a hurdle model or dropout effect in a model likelihood such as the one proposed in Section [mainModel] would result in a much more computationally intensive model fitting procedure than the one we propose in our Algorithm [gibbsSampEqs] of Section [postInf].\nSingle-cell gene-expression data\n================================\nIn this section we present an example application of our proposed methodology, to single-cell gene-expression data. These data have been published previously by, and are publicly available from the NCBI database of genotypes and phenotypes (dbGaP), under accession number phs000989.v3 \nIn this context, *x**t*, *i*, *k* represents the log-expression of gene *i*, defined as log(transcript counts + 1), in sample *k* from time *t*. For the pseudo-time assignments for each cell, we use cell-type classifications provided with the data, together with an ordering for these cell-types according to the developmental lineage (for full details see Supplement C). We fitted the model to *n* = 1557 cell samples, and *p* = 22988 genes/nodes, reduced to *p* = 212 for each individual model fit by variable screening. For the fitting we used values of *λ* = 20 and *k* = 1: these values were chosen by grid-search stochastic EM (Figure S13 in Supplement D). Fitting the model as described, we obtained posterior distributions for each model parameter *b**t*, *j*(*i*), and we used the posterior medians as posterior summaries, *b̂**t*, *j*(*i*). To fit each model, we ran the Gibbs’ sampler proposed in Algorithm [gibbsSampEqs] for 1 × 104 samples (after 1 × 103 samples burn-in), which took 1.4 hours for each target-node on one core of a Macbook Pro laptop (mid 2015, 2.8 GHz, 16GB RAM).\n[bestRegs]\nThe model was fitted initially to a panel of 25 genes, as target-nodes: these genes were chosen in an unbiased way by searching the biological sciences literature for genes that are important in this biological setting, and then analysing those that were present in this data-set after quality control. Estimated model parameters *b̂**t*, *j*(*i*) for a selection of these genes are shown in Figure [bestRegs], and the full panel is shown in Figures S14 and S15.\n[convPlotPs]\nWe carried out Geweke and Heidel convergence tests, using the R package CODA, for the sampler outputs for all the parameters *b**t*, *j*(*i*) shown in Figures [bestRegs], S14 and S15. These convergence test results appear in Figure [convPlotPs]. In convergence tests such as these, if an individual *p*-value is significantly small, it can be taken as evidence that the chain has not yet converged. Hence, the uniform distributions of *p*-values shown in Figure [convPlotPs], in which these *p*-values are aggregated over all the test results, indicate that the MCMC sampler has converged for these target-nodes. Then, to give an indication of how ‘stiff’ or ‘sloppy’ these parameters are, we estimated the standard-deviations of these posterior distributions for this panel of genes: these are plotted against the corresponding posterior averages in Figure [postSDs]. These posterior standard deviations are typically much smaller in magnitude than the posterior averages, demonstrating that the posteriors are not ‘sloppy’, and indicating that the estimates from our model are reliable. We also wanted to make sure that our results are not driven by a few outlier cells. So we repeated the inference for this same panel of genes, but now using only a random sample of 50% of the cells originally used, i.e., *n* = 779. The results of this analysis are plotted in Figure S16, for the same genes as are shown in Figure [bestRegs]. The results shown in these figures are clearly very similar, and therefore we conclude that our results here are not driven by outliers.\nFigure [bestRegs]a shows inferred model parameters *b̂**t*, *j*(*i*), for a selection of nodes/genes which are characteristic of stem cells, and of neurons (i.e., mature cells), selected from the full panel of 25 genes. We expect stem cells to predominate at earlier times, and hence we expect to see decreasing time-series for genes which are characteristic of this type of cell. On the other hand, we expect mature cells such as neurons to predominate at later times, and so we expect to see increasing time-series for genes characteristic of this type of cell. As would be expected for stem-cell genes, important model parameters *b**t*, *j*(*i*) tend to decrease in magnitude during the developmental trajectory as cells go from stem-cell to mature cell types (e.g., gene transcript MOXD1). Figure [bestRegs]b then shows, as would be expected, that important model parameters *b̂**t*, *j*(*i*) become non-zero (corresponding to network edges appearing) late in the developmental trajectory, when the cells become neurons and hence their characteristic gene regulatory program is activated (e.g., for SATB2). Equivalent results to Figure [bestRegs]a-b for the full panel of 25 genes analysed then appear in Figures S14 and S15 respectively in Supplement D. In these figures, we also see similar results: for genes that tend to be active in stem-cells, model parameters *b**t*, *j*(*i*) tend to decrease in magnitude during the developmental trajectory as cells go from stem-cell to mature cell types (Figure S14), and *vice-versa* for genes which are important to mature cells such as neurons (Figure S15).\nWe wish to infer a network edge between nodes *i* and *j* if ∣*b̂**t*, *j*(*i*)∣ > 0. We estimate these *b̂**t*, *j*(*i*) from the posterior medians, but because we find that many of these medians are close to, but not exactly zero, we set *b̂**t*, *j*(*i*) to zero in such cases by thresholding. Therefore, we infer ‘no edge’ between nodes *i* and *j* when the posterior median is close to zero. Hence, if (and only if) ∣*b̂**t*, *j*(*i*)∣ > *ϕ*, where *ϕ* is the threshold parameter, we would infer a network edge between nodes *i* and *j* at time *t* (for the model fit around node *i*). We note that the local model fitting (equation ([mainLinModelEq])) does not depend on this network estimation. Hence, this thresholding can take place independently of the computationally-intensive MCMC sampling. Thus, we leave *ϕ* as a tuning parameter, which can be varied by the user in real time to interpret results, equivalently to changing the resolution or granularity in a visualisation. We recommend the user does a full sweep through *ϕ* ∈ [0, ∞] to interpret the results. We also note that if ∣*b̂**t*, *i*(*j*)∣ > *ϕ* (for the independent model fit around node *j* rather than node *i*), we would independently infer an edge between nodes *i* and *j* at time *t*. Thus, some inconsistency may arise, due to these independent model fits around nodes *i* and *j*. To deal with this, we use the ‘min\\_symmetrisation’ scheme of, inferring an edge between nodes *i* and *j* at time *t*, i.e., *Â**i*, *j*, *t* ≠ 0, if and only if ∣*b̂**t*, *j*(*i*)∣ > *ϕ* and ∣*b̂**t*, *i*(*j*)∣ > *ϕ*.\n[postSDs]\nPlots of the inferred network structure around an example of a gene shown in Figure [bestRegs]b, namely SATB2, are shown in Figure [SATB2netPlot], after ‘min\\_symmetrisation’ (Section [modelOverviewSec]) with *ϕ* = 0.05. In addition to the neuronal identity gene SATB2, several of the genes shown in Figure [SATB2netPlot] are already known to be important in neuronal development, including NEUROD1 which initiates the programme of neuronal development and RUNX1T1 which regulates the differentiation of neurons from neural stem cells, as well as the neuronal circuit-formation gene BCL11A. Intriguingly, this network structure also includes MIR133A1HG and LINC00478, which are (respectively) examples of micro-RNA (miRNA) and long non-coding RNA (lncRNA). Non-coding RNA transcripts such as these do not get translated into proteins, as would usually be the case for a transcript from a region of DNA which codes for a gene. Instead, non-coding RNA transcripts are known to play an important role in gene regulation. However, we still only understand a small amount about their function, and gene regulation involving these sorts of non-coding RNA is an important research topic. We note that MIR133A1HG and LINC00478 are promising candidates for for further experimental investigation which have been identified using our proposed methodology.\n[SATB2netPlot]\nNext, we compared our method with alternative network inference method for single-cell transcriptome data. Methods for inferring time-varying network structure in data of this type include alternatives designed for many fewer nodes than our method can handle, such as the work of, which is designed to infer structure in networks with fewer than 100 nodes. However, there is also a static network inference method available for single-cell transcriptome data called ‘SCENIC’, that can be used to infer structure in large networks of 20000 or more nodes, and that is therefore also appropriate for the data-set analysed here. For comparison with our proposed methodology, we ran the SCENIC method on the same single-cell gene-expression data-set already analysed. Equivalently to our proposed method, SCENIC returns fitted model parameters that indicate the strength of the network connection between a pair of nodes, or genes: maintaining equivalent notation, we label these SCENIC model parameters *b**j*′(*i*). Thus, by again choosing a threshold *ϕ*′, it is possible to infer network structure by inferring edges between the pair of nodes *i* and *j* if the corresponding fitted model parameter ∣*b̂**j*′(*i*)∣ > *ϕ*′. We choose *ϕ*′ so as to maintain the same number of connections to each target in the network inferred by the SCENIC method, as compared with our method. Table [SCENICtab] shows the genes inferred in the network structure around the neuronal identity gene SATB2 (summarised from Figure [SATB2netPlot]), together with the genes equivalently found from the SCENIC method (setting *ϕ*′ to maintain the same number of connections).\n| Our method | SCENIC |\n| --- | --- |\n| BCL11A | ARPP21 |\n| DDX1 | CHL1 |\n| LINC00478 | KIAA1598 |\n| MIR133A1HG | MEF2C |\n| NEUROD1 | NFIA |\n| RUNX1T1 | RUNX1T1 |\n[SCENICtab]\nOf the genes shown in Table [SCENICtab], just as with those found by our method, those found by the SCENIC method are mostly already known to be involved in neural development, as follows. ARPP21 is involved with branching of dendrites, CHL1 and KIAA1598 are thought to be involved in neuronal migration and axon formation, and MEF2C and NFIA are known to be important for neural stem and progenitor cell differentiation. However, we note that the SCENIC method is not able to infer time-varying network structure, as our method can: to make the comparison shown Table [SCENICtab], the time-varying aspect of the network structure inferred by our method had to be ‘flattened out’.\nIt is challenging to visualise in a meaningful way the entire structure of a large network, such as the full genome-wide network inferred here, if it is inferred for all 22989 nodes. This challenge becomes even larger when the dimension of time is added. After fitting the\n\n[fullNetPlot]\nmodel to all 22989 target-nodes on a high-performance computing cluster, we inferred the structure of this network, and found the fully connected component (11133 nodes), which is shown in Figure [fullNetPlot]. As a minimum, it can be seen from this figure that the network structure changes gradually rather than suddenly with time, as we would expect from our proposed methodology.\nDiscussion\n==========\nIn this paper, we have proposed a new model to infer time-varying network structure. This model makes use of a novel prior structure we introduce here, which extends the Bayesian lasso to the time-varying case. The novel structure of this prior allows for effective modelling of time-varying network structure even in situations where there are very few time-points, as is typical in cell-biological (i.e., `omics) data. We also found that the model fitting and inference procedure we have proposed works well even in with large networks of over 20000 nodes, which compares very well with alternatives (see for example the work by ).\nWe used simulated data to assess the ability of the proposed model to accurately infer time-varying network structure, and we showed that the model is effective in inferring time-varying genomic network structure from single-cell gene-expression data. However, we note that genomic network structure which is inferred from only gene-expression data (as we do here) is not guaranteed to correspond to true gene regulatory patterns. To strengthen any belief that the inferred genomic network structure corresponds to true gene regulatory patterns rather than simply gene co-expression patterns, evidence from, for example, chromatin binding and epigenomic data could also be incorporated into the model. We intend to incorporate such data as the next stage of the development of this model. Specifically, we will do this by allowing the sparsity parameter *λ* to vary for each pair of nodes *i* and *j*, depending on any prior evidence of a physical interaction between the protein-product of gene *j* with the DNA or surrounding chromatin of gene *i*.\nAnother characteristic of the single-cell transcriptome data analysed here is that the data are zero-inflated. This is a case of data missing-not-at-random, because the dropout events which lead to the extra zeros in the data are more likely to occur when the true transcriptome level is low. As part of the next stage of the development of this model, we intend to account for dropouts as other authors have done, for example by explicitly including the dropout events in the model likelihood. We also note that existing time-inference methods for data such as those presented here are algorithmic, rather than model-based. Hence it is not easy to obtain uncertainties on the inferred times when using these methods. Thus, we would like to develop a model-based time-inference method that will provide such uncertainties, and then feed these uncertainties directly into the time-varying network model we have proposed. We also note that in other contexts, it could complicate the analysis if there is uneven time-sampling. For example, if we expect highly deterministic behaviour with little noise, but have data with time-sampling at known but uneven time-points, the method might need to be adapted. Specifically, in that context we would expect to see larger changes in parameters over larger time-intervals: this structure is not explicitly captured by our model, in"},"data_id":{"kind":"string","value":"arxiv_0002437"}}},{"rowIdx":591,"cells":{"content":{"kind":"string","value":"Quantifying EPR: the resource theory of nonclassicality of common-cause assemblages\n===================================================================================\nEinstein-Podolsky-Rosen (EPR) steering is often (implicitly or explicitly) taken to be evidence for spooky action-at-a-distance. An alternative perspective on steering is that Alice has no causal influence on the physical state of Bob’s system; rather, Alice merely updates her knowledge of the state of Bob’s system by performing a measurement on a system correlated with his. In this work, we elaborate on this perspective (from which the very term ‘steering’ is seen to be inappropriate), and we are led to a resource-theoretic treatment of correlations in EPR scenarios. For both bipartite and multipartite scenarios, we develop the resulting resource theory, wherein the free operations are local operations and shared randomness (LOSR). We show that resource conversion under free operations in this paradigm can be evaluated with a single instance of a semidefinite program, making the problem numerically tractable. Moreover, we find that the structure of the pre-order of resources features interesting properties, such as infinite families of incomparable resources. In showing this, we derive new EPR resource monotones. We also discuss advantages of our approach over a pre-existing proposal for a resource theory of ‘steering’, and discuss how our approach sheds light on basic questions, such as which multipartite assemblages are classically explainable.\nIntroduction\n============\nThe notion of Einstein-Podolsky-Rosen (EPR) ‘steering’ refers to a form of nonclassical correlations that arise when one considers measurements performed on half of a bipartite system prepared on an entangled state. Such correlations have multiple applications in quantum information ; for example, they allow for the certification of entanglement under relaxed assumptions , and constitute an information-theoretic resource for various cryptographic tasks . For these reasons, many previous works have begun the process of characterizing the resourcefulness of such correlations.\nIn this paper, we develop a novel resource-theoretic approach to quantifying the nonclassicality of a given correlation in an EPR scenario. Our approach is conceptually motivated by a particular perspective on EPR correlations arising from the language of causal modelling . In this approach, it is assumed that the relevant causal structure contains no directed causal influences between the parties that share the quantum system, which implies that ‘steering’ is not a form of action-at-a-distance, but rather is a form of inference through a nonclassical common cause. This view leads to a ‘resource theory of steering’ which is distinct from that proposed in previous works . Our approach follows the recent developments of resource theories for studying nonclassicality of common-cause processes wherein the free operations are local operations and shared randomness (LOSR) .\nWe now motivate our approach and compare it to prior approaches. Readers already familiar with the conceptual framework of Refs. may wish to skip to Section [sec:intro-summary], where we summarize our main technical results.\nReimagining (and renaming) EPR ‘steering’\n-----------------------------------------\nConsider a scenario wherein two parties (Alice and Bob) share a bipartite quantum system prepared in an entangled state, and where Alice then chooses a measurement, performs it on her share of the system, and obtains an outcome. Conditioned on the outcome of the measurement, Alice updates her description of the quantum state that corresponds to Bob’s system. Consequently, each of Alice’s possible measurements leads to an ensemble of potential updated states of Bob’s system together with their associated probabilities of arising. The collection of these ensembles – an ‘ensemble of ensembles’ – is termed an *assemblage* . Depending on *which* measurement she chooses to carry out, then, Alice chooses from which of these ensembles the quantum state of Bob’s system will be drawn from.\nSchrödinger considered this dependence of the wavefunction describing Bob’s system on Alice’s measurement choice to be ‘magic’ , and termed this phenomena *quantum steering*. This terminology is motivated by the idea that Alice’s choice of measurement exerts a *causal influence* on Bob’s system, even when the two are at an arbitrary distance. If one instead considers the quantum state to be merely a representation of one’s *information* about a system, then Alice’s ability to update this information conditioned on her measurement outcome is not in itself surprising. When Alice *learns* something about the true state of Bob’s system, by virtue of her measurement on a system correlated with it, it is only natural that she would adjust her description of it (as a wavefunction), accordingly.\nIn this work we do not view ‘quantum steering’ as evidence for nonlocal causal influences. Since the term ‘steering’ has an intrinsic bias towards action-at-a-distance, in this manuscript we forgo its use in favor of more neutral terminology.[1](#fn1) For example, we will refer to a ‘steering scenario’ as an *EPR scenario*. This attitude towards terminology follows the lines of Ref. , which avoided the use of the term ‘nonlocality’ since it similarly suggests the existence of superluminal causal influences. We will only use the term ‘steering’ when referring to prior work, and even then we will write it between quotations marks.\nWith this in mind, without loss of generality, we recast the EPR scenario as follows. Firstly, the fundamental causal structure underpinning an EPR scenario is that depicted in Fig. [fig:bipartite-scenario-entangled], wherein Alice and Bob are connected only by a *common cause*, with no causal influence between them. This is exactly the causal structure suggested by relativity theory. Second, we assume Alice has access to a classical input system and a classical output system, while Bob has access to a quantum output system [2](#fn2). A process with these two features defines an assemblage, and can be depicted as in Fig. [fig:bipartite-scenario-entangled].\n(-60,0) (-20,-20)(*a*) \n(-60,0) (-20,-20)(*b*) \nA fundamental question about assemblages in an EPR scenario is which assemblages admit a classical description, that is, one in terms of an underlying classical system that generates the observed correlations and that always takes a definite value. In the next section, we discuss that there exist assemblages which cannot be understood as being prepared with a shared classical random variable while setting up the relevant resource theory of nonclassicality in EPR scenarios.\nThe resource theory of nonclassicality in EPR scenarios\n-------------------------------------------------------\nWe follow the standard approach to quantifying the resourcefulness of given processes, namely, the framework of resource theories . In this approach, one determines two defining features of the resource theory: the *enveloping theory*, which is the set of all possible resources one can produce in the setup; and the *free subtheory* – the set of operations on the resources that are considered to be freely available, as dictated by the physical constraints in the scenario under study. From this, one can directly determine the relative value of any pair of resources: if resource R can be converted to resource S, then R is at least as valuable as S, since (in the context of these free operations) R can clearly be used for any purpose that S can be used for. There are by now many useful resource theories, including those for entanglement (relative to local operations and classical communication or relative to local operations and shared randomness ), for nonclassicality in Bell scenarios and other common-cause processes , for post-quantumness , for coherence , for athermality , and so on.\nHistorically, the EPR scenario has been viewed as a means of indirectly studying the properties of entangled states . More recently, researchers have begun to study assemblages as processes in and of themselves, since these contain all the relevant information for characterizing an EPR scenario. Here, we will follow this latter approach[3](#fn3), and so the resources in our resource theory are taken to be assemblages. We will only consider assemblages which can be realized within quantum theory (as opposed to those which can be realized with post-quantum resources ), and so we take our enveloping theory to be given by assemblages that can be generated by quantum common causes as in Fig. [fig:bipartite-scenario-entangled].\nThe critical step in defining any resource theory is to determine the relevant set of free operations. It has previously been argued that the relevant free operations for studying nonclassicality in common-cause scenarios are given by the set of local operations and shared randomness . In this work, we also adopt this approach. In other words, free resources and free transformations on resources are those which can be generated by classical common causes. In doing so, our approach unifies the study of ‘steering’ resources with resources of ‘nonlocality’ and (LOSR-)entanglement, showing that these are all simply different manifestations of nonclassicality of common-cause processes.\nWe now briefly reiterate the basic motivations for taking LOSR as free operations, while specializing these arguments to the case of EPR scenarios. Our choice of free operations is guided by our assumptions about the causal structure, depicted in Fig. [fig:bipartite-scenario-entangled]. Firstly, we do not allow Alice or Bob to freely exert causal influences on one another, thus ruling out both classical and quantum communication. We place no limitations on the local quantum operations that each can carry out on the systems in their lab. Finally, as we wish to quantify the *non*classical properties of assemblages, we restrict the set of common causes which are taken to be free to be the *classical* common causes, so that a resource may be valuable only by virtue of having a nonclassical common cause. In other words, we take the set of free operations to include local operations and classical common causes. Since a more common term synonymous to ‘classical common cause’ is ‘shared randomness’, we refer to the set of free operations as *local operations and shared randomness*.\nFrom the perspective endorsed in this manuscript, then, the notion of resourcefulness embodied in a useful assemblage is *nonclassicality of its common-cause*. The free (classically explainable) assemblages are all and only those which can be generated by classical common causes, i.e., those of the form shown in Fig. [fig:bipartite-scenario-classical]. As we will see, this class of assemblages coincides exactly with the set of “unsteerable” assemblages—those that admit of so-called “hidden-state models”, i.e., classical common-cause explanations. For such assemblages, Bob’s system can always be viewed as locally prepared in a state that depends on a classical random variable which always takes a definite classical value, and Alice’s choice of measurement can then be viewed as simply providing her information about this classical value, which in turn allows her to refine her description of the state of Bob’s quantum system.\nIn contrast, the common cause generating any nonfree assemblages is necessarily nonclassical (i.e., cannot be viewed as a shared classical random variable), and so Alice’s refinements of her knowledge about Bob’s system are *not* compatible in this sense. However, there is currently no agreed-upon framework for formalizing Alice’s refinements of her knowledge in this sense. Indeed, to fully understand how inferences are made through such a nonclassical common cause would presumably require a nonclassical generalization of the classical theory of inference. While some first attempts at such a program have been made, arguably with some success when applied specifically to EPR scenarios, this ambitious problem remains very much unsolved. A convenient feature of our resource theoretic approach is that it does not require us to resolve these difficult issues. This is because the structure of the resource theory is entirely determined by the free set of *classical* common-cause processes, which are well-understood even without such a novel theory of inference.\nComparison to prior work\n------------------------\nA resource theory of ‘steering’ in bipartite scenarios, distinct from ours, was defined a few years ago . Therein, the free operations were taken to be (stochastic) local operations together with classical communication from Bob to Alice. This set of free operations was motivated by one particular application for which assemblages are known to serve as resources – namely, this set is the most general one that does not compromise the security of one-sided device-independent quantum key distribution protocols. In contrast, our resource theory is developed based on the natural physical limitations arising in any common-cause scenario. As such, our approach follows a recent body of work which demonstrates the importance of studying nonclassicality in common-cause scenarios (like the Bell scenario) within a resource-theoretic perspective underpinned by LOSR operations.\nWe compare these two approaches further in Section [sec:relwork-comparison]. In particular, we note that our approach has the technical advantage that its set of free transformations is simpler to characterize and study. More importantly, our resource theory has the conceptual advantage that it allows for the unification of every type of nonclassical correlation in Bell-like scenarios, since all of these (including entanglement, ‘steering’, and ‘nonlocality’) can be viewed as resources of nonclassicality of common-cause processes. We expect this unification to be useful for better understanding the relationships between entanglement, EPR nonclassicality, and Bell nonclassicality, as well as for understanding the possibilities for interconversion between these (and many other ) forms of nonclassicality in common-cause scenarios. Furthermore, the principled approach we follow here to distill the essence of nonclassicality allows us to directly and uniquely generalize our framework to the case of multipartite EPR scenarios (which we provide a resource theory for in this work) and to Bob-with-input EPR scenarios and channel EPR scenarios [4](#fn4) (which we provide a resource theory for in a follow-up work).\nSummary of main results\n-----------------------\nIn this paper, we construct a resource theory of bipartite and multipartite ‘steering’ under LOSR operations. This is the first time a resource-theoretic approach has been applied to the latter scenario. Even within the traditional bipartite scenario, our approach differs from previous approaches to resource theories of ‘steering’, and clarifies some issues which arose within them.\nThis paper is divided into three main sections, where we discuss the definition and implications of the LOSR resource-theoretic approach to bipartite EPR scenarios (Section [sec:bipartite]), multipartite EPR scenarios (Section [sec:multi]), and the relation of this resource theory to previous work (Section [sec:relwork]). In Sections [sec:bipartite] and [sec:multi], we first recall the definition of the scenario of interest, and explicitly specify its most general LOSR processing of the corresponding EPR assemblage. Next, we show that resource conversion under free operations in the corresponding scenario can be evaluated with a single instance of a semidefinite program. We conclude each of these sections by highlighting properties of the pre-order of resources. In Section [sec:relwork], we discuss the advantages of our LOSR approach in relation to the problem of choosing the set of free operations in a resource theory (Section [sec:relwork-comparison]) and consistently defining the set of free resources (Sections [sec:relwork-multi] and [sec:relwork-exposure]).\nThe bipartite EPR scenario\n==========================\nIn this section, we define the resource theory of nonclassicality of bipartite assemblages under LOSR operations. We begin by formalizing the set of free (LOSR) operations. Then, we explicitly show that a resource conversion in this paradigm can be decided with a single instance of a semidefinite program. This is the first tool that allows one to systematically determine the relative nonclassicality of assemblages, that is, the convertibility relations that hold among them. We use this semidefinite program to study the possible conversions among a family of infinite assemblages. Moreover, we define new EPR measures and study the properties of the pre-order analytically.\nDefinition of the scenario and free assemblages\n-----------------------------------------------\nThe bipartite EPR scenario (see Fig. [fig:bipartite-scenario]) consists of two distant parties, Alice and Bob, that share a physical system and perform local actions on it. Alice performs (possibly incompatible) measurements on her subsystem: upon choosing a measurement setting denoted by *x*, she obtains a classical outcome *a* with probability *p*(*a*∣*x*). We denote by X the set of classical labels that denote Alice’s choice of measurement, and by A the set of labels for her measurement outcomes. Without loss of generality, we assume all the measurements to have the same outcome cardinality. By measuring her system, Alice refines her knowledge of Bob’s system, which is now described by a conditional marginal state *ρ**a*∣*x*. The relevant object of study is the ensemble of ensembles of quantum states, dubbed an *assemblage*, that contains all the information characterizing an EPR scenario. It is defined as ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}=\\{ \\{\\sigma\\_{a|x}\\}\\_{a\\in {\\mathbb{A}}} \\}\\_{ x \\in {\\mathbb{X}}}$, where each unnormalised state *σ**a*∣*x* is given by *σ**a*∣*x* : = *p*(*a*∣*x*)*ρ**a*∣*x*. That is, the probability that an updated state arises is given by the normalization factor for the corresponding state in the assemblage.\nOne can depict an assemblage as in Fig. [fig:bipartite-scenario], where the classical and quantum systems are depicted with single and double lines, respectively. It is worth noticing that, even though we have chosen our enveloping theory to consist only of quantumly-realizable assemblages, the results derived in this section also apply to broader enveloping theories including those generated by arbitrary (e.g. post-quantum) common-causes. In fact, in the bipartite case, there is no difference between the assemblages[5](#fn5) that can be realized by quantum common causes and those that can be realized by arbitrary common causes. This follows from the GHJW theorem, proven by Gisin and Hughston, Jozsa, and Wootters, which (in our causal language) shows that *all* bipartite common-cause assemblages are quantumly-realizable.\n(-60,40) (-45,75)*x* (-44,125)*a* (25,125)*ρ**a*∣*x* (-20,0)(*a*) \n(-85,0) (-55,75)*x* (-55,120)*a* (-95,100)*c* (-21,5)*λ* (15,120)*ρ**a*∣*x* (-72,10)*x*′ (-60,165)*a*′ (26,165)*ρ**a*′∣*x*′ (-25,-20)(*b*) \nIn quantum theory, the state of the shared system can be described by a density matrix *ρ* and Alice implements generalised measurements, i.e., positive operator-valued measures (POVMs), which we denote by {{*M**a*∣*x*}*a* ∈ A}*x* ∈ X. In this case, the unnormalised states admit a quantum realisation of the form *σ**a*∣*x* = tr*A*{(*M**a*∣*x* ⊗ I)*ρ*}. This is formalised as follows:\n **Quantumly-realizable assemblage.** \nAn assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}=\\{ \\{\\sigma\\_{a|x}\\}\\_{a\\in {\\mathbb{A}}} \\}\\_{ x \\in {\\mathbb{X}}}$ has a quantum realisation iff there exists a Hilbert space H*A*, a state *ρ* in H*A* ⊗ H*B*, and POVMs {{*M**a*∣*x*}*a* ∈ A}*x* ∈ X on H*A* such that\n \n$$\\begin{aligned}\n\\label{eq:quantum-bipartite}\n\\sigma\\_{a|x} = \\mathrm{tr}\\_A\\{(M\\_{a|x} \\otimes {\\mathbb{I}}) \\rho\\}\\end{aligned}$$ \nfor all *x* ∈ X and *a* ∈ A. [def:Q-bipartite]\nFinally, we consider the question of which assemblages admit a classical explanation: that is, which assemblages can be understood as a classical common-cause process. As argued in the introduction, these are all and only those that can be constructed freely from LOSR operations. Assemblages of this form are depicted in Fig. [fig:bipartite-scenario-classical]. Now, one can associate:\na probability distribution *p*(*λ*) with the box representing the state of the common cause (where we imagine that *λ* is sampled according to this distribution, copied, and shared with both parties),\na conditional probability distribution *p*(*a*∣*x*, *λ*) with the box representing the process by which Alice’s outcome is generated (depending on her setting and the value of *λ*),\na normalized quantum state *ρ**λ* with the process by which Bob’s quantum state is locally generated (depending on *λ*).\nHence, the elements of an LOSR-free assemblage can be written as *σ**a*∣*x* = ∑*λ**p*(*λ*)*p*(*a*∣*x*, *λ*)*ρ**λ*, with *p*(*a*∣*x*, *λ*) being valid conditional probability distributions for all values of *λ*, which is the classical variable corresponding to the shared classical common cause. Traditionally, these LOSR-free assemblages have been referred to as `unsteerable’ assemblages and are mathematically expressed in terms of the so-called ‘local hidden states’ *σ**λ* : = *p*(*λ*) *ρ**λ*[6](#fn6).\nLOSR transformations between bipartite assemblages\n--------------------------------------------------\nThe most general LOSR transformation of an assemblage is illustrated in Fig. [fig:bipartite-scenario-LOSR]. It consists of a comb (which locally pre- and post-processes the classical variables in Alice’s wing) and a completely positive trace preserving (CPTP) map (which post-processes Bob’s quantum system). Both operations are correlated by a classical variable *λ*. Formally, a generic LOSR transformation illustrated in Fig. [fig:bipartite-scenario-LOSR] transforms one assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ into a new assemblage ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$ as follows:\n \n$$\\begin{aligned}\n\\label{eq:LOSRtrans-bipartite} \n\\sigma'\\_{a^\\prime|x^\\prime} = \\sum\\_{\\lambda,c} \\sum\\_{a,x} \\, p(\\lambda) \\, p(c,x|x^{\\prime},\\lambda) \np(a|x) p(a^{\\prime}|a,c) \\mathcal{E}\\_{\\lambda}(\\rho\\_{a|x})\\,,\\end{aligned}$$ \nwhere\n*p*(*c*, *x*∣*x*′, *λ*) encodes the classical pre-processing of Alice’s input *x* as a function of *x*′ and the shared classical randomness *λ*. Here *c* denotes the variable to be transmitted through Alice’s classical side channel toward the post-processing stage.\n*p*(*a*′∣*a*, *c*) encodes the classical post-processing of Alice’s output *a*, as a function of the classical information *c* kept from the pre-processing stage. The output of the process is Alice’s new outcome *a*′.\nE*λ*[ ⋅ ] is the CPTP map corresponding to Bob’s local post-processing of his quantum system, as a function of the shared classical randomness *λ*.\nNotice that if ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ is free, then ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$ is free as well – that is, the set of free assemblages is closed under LOSR operations. This highlights the basic property of a resource theory: applying a free transformation on a free resource cannot create a resourceful object. Any assemblage that cannot be realised by local operations and shared randomness is nonfree and constitutes a resource of LOSR nonclassicality.\nThe local pre- and post-pocessings of Alice’s classical variables are represented by the product *p*(*c*, *x*∣*x*′, *λ*) *p*(*a*′∣*a*, *c*) in Eq. . Notice, however, that the classical variable *c* is a function only of the classical values *x*′ and *λ*, and that we have no constraints on the dimension of the classical variable *c*. Therefore, Eq. can be written as\n \n$$\\begin{aligned}\n\\label{eq:LOSRtrans-bipartite2} \n\\sigma'\\_{a^\\prime|x^\\prime} = \\sum\\_{\\lambda} \\sum\\_{a,x} \\, p(\\lambda) \\, p(a^{\\prime}|a,x^{\\prime}, \\lambda)\\, p(x|x^{\\prime}, \\lambda)\\, p(a|x)\\, \\mathcal{E}\\_{\\lambda}(\\rho\\_{a|x})\\,.\\end{aligned}$$ \nNotice that this expression satisfies the condition of no-retrocausation – the variable *a* cannot influence the value of the variable *x*.\nA final remark pertains to a particular way to express a generic LOSR transformation, based on Fine’s argument and discussed in Ref. . In the central point of this expression is the fact that the set of LOSR operations is convex and its extremal elements are deterministic. This implies that Alice’s pre- and post-processing can be decomposed as a convex combination of deterministic operations:\n \n$$\\begin{aligned}\n\\label{eq:det-bipartite}\n p(a^{\\prime}|a,x^{\\prime}, \\lambda)\\, p(x|x^{\\prime}, \\lambda)= \\sum\\_{\\tilde{\\lambda}} p(\\tilde{\\lambda}|\\lambda) D(a^{\\prime}|a,x^{\\prime},\\tilde{\\lambda})\\,D(x|x^{\\prime},\\tilde{\\lambda}).\\end{aligned}$$ \nHere, *D*(*a*′∣*a*, *x*′, *λ̃*) assigns a fixed outcome *a*′ for each possible choice of *a*, *x*′, and *λ̃*, i.e., *D*(*a*′∣*a*, *x*′, *λ̃*) = *δ**a*′, *f**λ̃*(*a*, *x*′). Similarly, *D*(*x*∣*x*′, *λ̃*) assigns a fixed outcome *x* for each measurement *x*′ and value of *λ̃*, i.e., *D*(*x*∣*x*′, *λ̃*) = *δ**x*, *g**λ̃*(*x*′). Let us define a new completely positive and trace non-increasing (CPTNI) map\n \n$$\\begin{aligned}\n\\tilde{\\mathcal{E}}\\_{\\tilde{\\lambda}}(\\sigma\\_{a|x})= \\sum\\_{\\lambda} p(\\lambda) p(\\tilde{\\lambda}|\\lambda) \\mathcal{E}\\_{\\lambda}(p(a|x)\\rho\\_{a|x}).\\end{aligned}$$ \nNotice that $\\sum\\_{\\tilde{\\lambda}} \\tilde{\\mathcal{E}}\\_{\\tilde{\\lambda}}(\\sigma\\_{a|x})$ forms a CPTP map. We are now in the position to rewrite Eq. as follows\n \n$$\\begin{aligned}\n\\label{eq:det-bipartite2}\n \\sigma'\\_{a^{\\prime}|x^{\\prime}}= \\sum\\_{\\tilde{\\lambda}}\\sum\\_{a,x} D(a^{\\prime}|a,x^{\\prime},\\tilde{\\lambda})\\,D(x|x^{\\prime},\\tilde{\\lambda}) \\tilde{\\mathcal{E}}\\_{\\tilde{\\lambda}}(\\sigma\\_{a|x}),\\end{aligned}$$ \nwhich yields the simplified characterisation of a generic LOSR transformation that we will use throughout.\nA semidefinite test for deciding resource conversions\n-----------------------------------------------------\nAn assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ can be converted into a different assemblage ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$ under LOSR operations if and only if there exist a collection of CPTNI maps $\\tilde{\\mathcal{E}}\\_{\\lambda}$ such that ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$ can be decomposed as in Eq. . Therefore, deciding whether ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ can be converted into ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$ under LOSR operations is equivalent to checking whether this decomposition is possible. We will now show that this can be decided with a single instance of a semidefinite program (SDP).\nRecall that due to Choi-Jamiołkowski isomorphism every CPTP map E : H*B* → H*B*ʹ can be associated with an operator *W* on H*B* ⊗ H*B*ʹ such that E(*ρ**B*) = *d**B* tr*B*{*W* ( I*B*′ ⊗ *ρ**B**T*)}, where *d**B* is the dimension of the system *ρ**B*. Conversely, the operator *W* can be written as $W=(\\mathcal{E}\\otimes{\\mathbb{I}}\\_{B'})\\ket{\\Omega}\\bra{\\Omega}$, with $\\ket{\\Omega}=\\frac{1}{\\sqrt{d\\_B}}\\sum\\_{i=1}^{d\\_B} \\ket{ii}$. This isomorphism is crucial for reformulating our problem as an SDP. For Eq. to hold, each *σ*ʹ*a*′∣*x*′ must admit the following decomposition\n \n$$\\begin{aligned}\n\\label{eq:sigSDP-bipartite}\n \\sigma'\\_{a^{\\prime}|x^{\\prime}}= \\sum\\_{\\lambda}\\sum\\_{a,x}D(a^{\\prime}|a,x^{\\prime},\\lambda)\\,D(x|x^{\\prime},\\lambda)\\,d\\_B\\, {\\mathrm{tr}\\_{B} \\left\\{ W\\_{\\lambda}\\,(\\,{\\mathbb{I}}\\_{B^{\\prime}} \\otimes \\sigma\\_{a|x}^{T}) \\right\\}}\\,.\\end{aligned}$$ \nHere, the map $\\tilde{\\mathcal{E}}\\_{\\lambda}(\\sigma\\_{a|x})$ is written in the operator form with *W**λ* being the Choi state, i.e., $\\tilde{\\mathcal{E}}\\_{\\lambda}(\\sigma\\_{a|x})=d\\_B\\, {\\mathrm{tr}\\_{B} \\left\\{ W\\_{\\lambda}\\,(\\,{\\mathbb{I}}\\_{B^{\\prime}} \\otimes \\sigma\\_{a|x}^{T}) \\right\\}}$, with *W**λ* being a (*d**B* × *d**B*’) by (*d**B* × *d**B*’) matrix. Notice also that Eq. involves only finite sums, since the variable *λ* enumerates the finitely-many deterministic distributions *D*(*a*′∣*a*, *x*′, *λ*) and *D*(*x*∣*x*′, *λ*). The total number of the deterministic strategies encoded in *λ* is equal to ∣A′∣∣A∣ × ∣Xʹ∣ × ∣X∣∣Xʹ∣. This correspondence between elements *σ*ʹ*a*′∣*x*′ and *σ**a*∣*x* enables us to construct an SDP that checks whether ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ can be converted into ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$ under LOSR operations:\n[SDP-bipartite]**${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}} \\, \\overset{\\text{LOSR}}{\\longrightarrow} \\, {\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$.** \nThe assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ can be converted into the assemblage ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$ under LOSR operations, denoted by ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}} \\, \\overset{\\text{LOSR}}{\\longrightarrow} \\, {\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$, if and only if the following SDP is feasible:\n \n$$\\begin{aligned}\n\\begin{split}\n\\textrm{given} \\;\\;\\;& \\{ \\{\\sigma\\_{a|x}\\}\\_{a} \\}\\_{ x}\\,,\\; \\{ \\{ \\sigma'\\_{a^{\\prime}|x^{\\prime}}\\}\\_{a^{\\prime}}\\}\\_{x^{\\prime}}\\,,\\; \\{D(a^{\\prime}|a,x^{\\prime},\\lambda)\\}\\_{\\lambda,a^{\\prime},a,x^{\\prime}} \\,, \\; \\{D(x|x^{\\prime},\\lambda)\\}\\_{\\lambda,x,x^{\\prime}} \\\\\n \\textrm{find} \\;\\;\\;& \\{(W\\_{\\lambda})\\_{BB'} \\}\\_{\\lambda} \\\\\n \\textrm{s.t.} \\;\\;\\;& \\begin{cases} W\\_{\\lambda} \\geq 0\\,,\\\\\n {\\mathrm{tr}\\_{B^{\\prime}} \\left\\{ W\\_{\\lambda} \\right\\}} \\propto \\frac{1}{d}\\,{\\mathbb{I}}\\_B \\;\\;\\;\\; \\forall \\lambda\\,, \\\\\n \\sum\\_{\\lambda} {\\mathrm{tr}\\_{B^{\\prime}} \\left\\{ W\\_{\\lambda} \\right\\}} = \\frac{1}{d}\\, {\\mathbb{I}}\\_B\\,,\\\\\n \\sigma'\\_{a^{\\prime}|x^{\\prime}}= \\sum\\_{\\lambda}\\sum\\_{a,x} D(a^{\\prime}|a,x^{\\prime},\\lambda)\\, D(x|x^{\\prime},\\lambda)\\,d\\_B\\, {\\mathrm{tr}\\_{B} \\left\\{ W\\_{\\lambda}\\,(\\,{\\mathbb{I}}\\_{B^{\\prime}} \\otimes \\sigma\\_{a|x}^{T}) \\right\\}}\\,.\n \\end{cases}\n \\end{split}\\end{aligned}$$ \nWhen the conversion is not possible, we denote it by ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}} \\, \\overset{\\text{LOSR}}{\\not\\longrightarrow} \\, {\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$.\nSDP [SDP-bipartite] is a feasibility problem, i.e., it checks if the feasible set is equal to the empty set. If this is the case, the primal optimal value is equal to − ∞, which means that there exists no set {*W**λ*}*λ* that satisfies the constraints specified by Eq. . If the feasible set is not equal to the empty set, the optimal value is equal to zero, and the problem is feasible. Therefore, checking whether ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ can be converted into ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}^{\\prime}|{\\mathbb{X}}^{\\prime}}$ under LOSR operations requires a single instance of an SDP.\nProperties of the pre-order\n---------------------------\nWhen one assemblage can be freely converted into another, then the former is said to be *at least as nonclassical* as the latter. Therefore, the pre-order of assemblages gives information about their relative nonclassicality. In this section, we study the nonclassicality among members of an infinite family of assemblages defined below.\nConsider an EPR scenario where A = X = {0, 1}, and Bob’s dimension is 2. Imagine Alice and Bob share an entangled state of the form $\\ket{\\theta}=\\cos{\\theta} \\ket{00} + \\sin{\\theta} \\ket{11}$, where $\\theta \\in \\left(0,\\sfrac{\\pi}{4}\\right]$. The measurements that Alice performs on her share of the entangled state are given by $\\widetilde{M}\\_{a|0} = \\frac{1}{2}\\{{\\mathbb{I}}+ (-1)^a \\sigma\\_z\\}\\,$ and $\\widetilde{M}\\_{a|1} = \\frac{1}{2}\\{{\\mathbb{I}}+ (-1)^a \\sigma\\_x\\}$, with *σ**z* and *σ**x* being Pauli matrices. Then, the assemblage elements can be written as\n \n$$\\begin{aligned}\n\\label{eq:ass\\_elem}\n\\sigma^\\theta\\_{a|x} = {\\mathrm{tr}\\_{\\mathrm{A}} \\left\\{ \\widetilde{M}\\_{a|x} \\otimes {\\mathbb{I}}\\ket{\\theta}\\bra{\\theta} \\right\\}}.\\end{aligned}$$ \nThis infinite family of assemblages is indexed by one parameter – the angle {*θ*}. We will now introduce one more parameter to this family. The new parameter {*p*} is responsible for mixing the elements *σ**a*∣*x**θ* with noise. Let us define a family of assemblages S as:\n \n$$\\begin{aligned}\n\\label{eq:thefam}\n{\\mathsf{S}}&= \\left\\{ {\\boldsymbol{\\Sigma}}^{\\theta,p}\\_{{\\mathbb{A}}|{\\mathbb{X}}} \\, \\Big\\vert \\, \\theta \\in \\left(0,\\sfrac{\\pi}{4}\\right], p \\in [0,1] \\right\\}\\,,\\\\ \\nonumber\n\\text{where} \\quad {\\boldsymbol{\\Sigma}}^{\\theta,p}\\_{{\\mathbb{A}}|{\\mathbb{X}}} &= \\left\\{ \\left\\{p \\, \\sigma^\\theta\\_{a|x} + (1-p) \\, \\frac{{\\mathbb{I}}}{4} \\right\\}\\_{a\\in {\\mathbb{A}}}\\right\\}\\_{ x \\in {\\mathbb{X}}} \\,.\\end{aligned}$$ \nThis family of assemblages has an infinite number of elements. Each element is indexed by two parameters – the angle and the probability, {*θ*, *p*} respectively. We will sometimes focus on a subset of S with a fixed value of *p*. In such cases, for $p=\\varrho$, we define ${\\mathsf{S}}\\_\\varrho=\\left\\{ {\\boldsymbol{\\Sigma}}^{\\theta,\\varrho}\\_{{\\mathbb{A}}|{\\mathbb{X}}} \\, \\Big\\vert \\, \\theta \\in \\left(0,\\sfrac{\\pi}{4}\\right], p=\\varrho \\right\\}$. For example, for *p* = 1, we have the following family of assemblages:\n \n$$\\begin{aligned}\n\\label{eq:thefam1}\n{\\mathsf{S}}\\_1 &= \\left\\{ {\\boldsymbol{\\Sigma}}^{\\theta,1}\\_{{\\mathbb{A}}|{\\mathbb{X}}} \\, \\Big\\vert \\, \\theta \\in \\left(0,\\sfrac{\\pi}{4}\\right], p =1 \\right\\}\\,,\\\\ \\nonumber\n\\text{where} \\quad {\\boldsymbol{\\Sigma}}^{\\theta,1}\\_{{\\mathbb{A}}|{\\mathbb{X}}} &= \\left\\{ \\left\\{ \\sigma^\\theta\\_{a|x} \\right\\}\\_{a\\in {\\mathbb{A}}} \\right\\} \\_{ x \\in {\\mathbb{X}}} \\,.\\end{aligned}$$ \nFor simplicity, we denote ${\\boldsymbol{\\Sigma}}^{\\theta,p}\\_{{\\mathbb{A}}|{\\mathbb{X}}}={\\boldsymbol{\\Sigma}}^{\\theta,p}$ in this section.\nWe now study the properties of the pre-order among the resources in S. First, we run the SDP [SDP-bipartite] and test which conversions between resources parametrized by different values of {*θ*, *p*} are possible. Second, in Section [sec:new-monotones], we focus on S1 and confirm the results obtained from the SDP analytically.\nTo test whether two assemblages in S can be converted into each other, we check if they satisfy the constraints specified by Eq. . The solutions to the SDP (computed in Matlab, using the software CVX, the solver SDPT3 and the toolbox QETLAB ; see the code at ) are illustrated in Fig. [fig:SDP-bipartite]. In this figure, each dot represents one assemblage ${\\boldsymbol{\\Sigma}}^{\\theta,p}$. For example, the point indexed by {*θ* = *π*/6, *p* = 0.9} corresponds to ${\\boldsymbol{\\Sigma}}^{\\pi/6,0.9} = \\left\\{\\left\\{0.9 \\, \\sigma^{\\pi/6}\\_{a|x} + 0.1 \\, \\frac{{\\mathbb{I}}}{4} \\right\\}\\_{a\\in {\\mathbb{A}}}\\right\\}\\_{x \\in {\\mathbb{X}}}$. The black dots correspond to nonfree assemblages, and the grey dot represents a free assemblage. Note that the assemblage ${\\boldsymbol{\\Sigma}}^{\\pi/4,1}$ is one obtained when Alice and Bob share a quantum system prepared in a ‘maximally entangled’[7](#fn7) state, and Alice performs two suitable Pauli measurements in her share of the system. The arrows represent possible conversions, e.g., the arrow pointing from ${\\boldsymbol{\\Sigma}}^{\\pi/4,1}$ to ${\\boldsymbol{\\Sigma}}^{\\pi/6,0.8}$ means that ${\\boldsymbol{\\Sigma}}^{\\pi/4,1}$ can be converted into ${\\boldsymbol{\\Sigma}}^{\\pi/6,0.8}$ under LOSR operations. The grey, dashed lines represent trivial conversions. For the sake of simplicity, Fig. [fig:SDP-bipartite] represents only nine assemblages, which is already sufficient to illustrate some interesting features of the pre-order among assemblages in S.\n[fig:SDP-bipartite]\nFig. [fig:SDP-bipartite] displays some elements of S1 that are unordered in our LOSR resource theory of assemblages; e.g., notice that no conversions are possible among assemblages where *p* = 1. In contrast, some conversions *are* possible among assemblages for which *p* = 0.9, and among assemblages for which *p* = 0.8. Moreover, this figure is not symmetric: conversions of assemblages with higher value of *θ* to those with lower value of *θ* are more common. For example, ${\\boldsymbol{\\Sigma}}^{\\pi/4,1}$ can be converted into ${\\boldsymbol{\\Sigma}}^{\\pi/12,0.9}$, but ${\\boldsymbol{\\Sigma}}^{\\pi/12,1}$ cannot be converted into ${\\boldsymbol{\\Sigma}}^{\\pi/4,0.9}$. Finally, all arrows point in one direction only, suggesting that there are no equivalent assemblages in this family.\nNotice that we used the SDP [SDP-bipartite] to study the pre-order of assemblages in S, where A = X = {0, 1}, and Bob’s dimension is 2. However, the SDP is not constrained to such simple examples. In principle, resource conversion can be evaluated via the SDP [SDP-bipartite] for families of assemblages with an arbitrary cardinality of A and X, and arbitrary Bob’s dimension.\n### EPR monotones\nA common approach to unraveling the pre-order of resources in a resource theory is by employing so-called *resource monotones*. Formally, a resource monotone is a function which is monotonic under free operations. Resource monotones enable comparison of resources and identification of conversion relations and equivalence classes. That is, if the real numbers that resource monotone *M* assigns to the pair of resources *R*1 and *R*2 satisfy *M*(*R*1) < *M*(*R*2), then one can conclude that the monotone *M* witnesses that $R\\_1 \\not\\longrightarrow R\\_2$. Strictly speaking, hence, a resource monotone usually gives only partial information about the pre-order of resources, and one generically may need a collection of resource monotones (called a *complete set* of them) to have the complete information to fully specify the pre-order of resources based on them. Nevertheless, resource monotones are friendly ways to explore the pre-order of resources, and the real numbers in their co-domain are interpreted as a (possibly incomplete) *quantification* of the nonfreeness of the resources in its domain.\nIn the particular case of our resource theory, an EPR monotone is a function from the space of assemblages into real numbers, whose value does not increase under LOSR operations. Among the existing measures of bipartite assemblages, the *‘steerable’ weight*, *robustness of ’steering’*, and *relative entropy* were shown to be EPR monotones in a resource theory of ‘steering’ with stochastic local operations assisted by one-way classical communication (S-1W-LOCC) being the set of free operations. The set of S-1W-LOCC operations allows, in particular, for one party to generate randomness and share it with the second party by classical communication; hence, the set of S-1W-LOCC operations strictly includes the set of LOSR operations. It follows that all EPR monotones defined for S-1W-LOCC are therefore also monotones for LOSR. It has moreover been shown that the first two of these monotones – ‘steerable’ weight and robustness of ‘steering’ – can be reformulated in a type-independent form, i.e., they can be used to compare the LOSR-nonclassicality of resources of arbitrary types (where an EPR assemblage is one specific type of a resource). Let us now recall their formal definitions for the case of study in this section.\n[‘Steerable’ weight ] The ‘steerable’ weight of an assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ is given by the minimum *μ* such that ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ can be decomposed as\n \n$$\\begin{aligned}\n {\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}} = \\mu \\, {\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{S} + (1-\\mu) \\, {\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{\\mathrm{free}},\\end{aligned}$$ \nwhere ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{S}$ is an arbitrary nonfree assemblage and ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{\\mathrm{free}}$ is an arbitrary free assemblage. [def:monotoneweight]\nWe provide the following intuition for this definition. Imagine Alice wants to prepare a given assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ using the minimal amount of a nonfree resource. To achieve her goal, she prepares a free assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{\\mathrm{free}}$ most of the time (that is, a 1 − *μ* fraction of the rounds), and sometimes (a *μ* fraction of the rounds), she prepares a nonfree assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{S}$. On average, hence she prepares ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$. The ‘steerable’ weight quantifies the minimal amount of ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{S}$ needed to generate the desired assemblage in this way.\n[Robustness of ‘steering’ ] The ‘steering’ robustness of an assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ is given by the minimum *ν* such that the assemblage\n \n$$\\begin{aligned}\n {\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{\\mathrm{free}} = \\frac{1}{1+\\nu} \\, {\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}} + \\frac{\\nu}{1+\\nu} \\, {\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{\\prime}\\end{aligned}$$ \nis free, with ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{\\prime}$ being an arbitrary assemblage. [def:monotonerobustness]\nRobustness of ‘steering’ can be understood as follows. Imagine Alice holds an assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ and she wants to make it free by mixing it with some other assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{\\prime}$. Robustness of ‘steering’ quantifies the minimal amount of mixing needed to make ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ a free assemblage. If one optimises ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}^{\\prime}$ over all assemblages, this leads to mixing ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ with worst-case noise. If one restricts the type of noise that can be added to the original assemblage, this measure is referred to as generalized robustness or random robustness, depending on the type of noise added.\nWe conjecture that relative entropy could be defined in a type-independent way as well, as similar measures exist for states and Bell scenarios.\n### New EPR monotones\nWe now develop a method for obtaining EPR monotones from Bell inequalities and use it to construct a family of monotones that certify the incomparability of elements of S1. We only focus on S1 in this section, hence we drop the index *p* = 1 and denote ${\\boldsymbol{\\Sigma}}^{\\theta,1}={\\boldsymbol{\\Sigma}}^{\\theta}$.\nTo prove that the elements of S1 are unordered as per the LOSR resource theory of assemblages, it suffices to find a set of EPR LOSR monotones M = {M*j*} such that, for every pair (*θ*1, *θ*2) there exists a pair (M*θ*1, M*θ*2) with the following properties:\n \n$$\\begin{aligned}\n\\label{eq:monotones\\_condition}\n\\begin{cases}\n{\\mathrm{M}}\\_{\\theta\\_1}({\\boldsymbol{\\Sigma}}^{\\theta\\_1}) > {\\mathrm{M}}\\_{\\theta\\_1}({\\boldsymbol{\\Sigma}}^{\\theta\\_2}) \\quad \\text{-- which implies } {\\boldsymbol{\\Sigma}}^{\\theta\\_2} \\not\\xrightarrow{\\mathrm{LOSR}} {\\boldsymbol{\\Sigma}}^{\\theta\\_1} \\,,\\\\\n{\\mathrm{M}}\\_{\\theta\\_2}({\\boldsymbol{\\Sigma}}^{\\theta\\_1}) < {\\mathrm{M}}\\_{\\theta\\_2}({\\boldsymbol{\\Sigma}}^{\\theta\\_2}) \\quad \\text{-- which implies } {\\boldsymbol{\\Sigma}}^{\\theta\\_1} \\not\\xrightarrow{\\mathrm{LOSR}} {\\boldsymbol{\\Sigma}}^{\\theta\\_2} \\,.\n\\end{cases}\\end{aligned}$$ \nTo construct such a family of monotones, we will make use of the EPR inequalities constructed from the Bell inequalities presented in Ref. . These Bell inequalities, which we recall in the Appendix [app:bipartite-Bell], are uniquely maximized by the states $\\ket{\\theta}$ (via a self-testing result ). In the Appendix [app:steeringin-bipartite], we use these inequalities to construct an EPR functional, which we denote $S\\_\\eta[{\\boldsymbol{\\Sigma}}]$. This functional defines a family of EPR inequalities that are uniquely maximized by assemblages living in S1. For a formalization of this statement and definition of $S\\_\\eta[{\\boldsymbol{\\Sigma}}]$, see Appendix [app:steeringin-bipartite] and Definition [def:SFT] therein.\nWe construct the monotone M*η* from the EPR functional $S\\_\\eta[{\\boldsymbol{\\Sigma}}]$ following a yield-based construction:\n[def:Meta] The resource monotone M*η*, for $\\eta \\in (0,\\sfrac{\\pi}{4}]$, is defined as\n \n$$\\begin{aligned}\n{\\mathrm{M}}\\_\\eta[{\\boldsymbol{\\Sigma}}] := \\max\\_{\\widetilde{{\\boldsymbol{\\Sigma}}}} \\{ S\\_\\eta[\\widetilde{{\\boldsymbol{\\Sigma}}}]: {\\boldsymbol{\\Sigma}}\\xrightarrow{\\mathrm{LOSR}} \\widetilde{{\\boldsymbol{\\Sigma}}} \\}.\\end{aligned}$$ \nWe will now show that each of the monotones ${\\mathrm{M}}\\_\\eta[{\\boldsymbol{\\Sigma}}]$, when evaluated on assemblages in S1, is uniquely maximized by ${\\boldsymbol{\\Sigma}}^{\\eta}$. First, note that\n \n$$\\begin{aligned}\n\\label{eq:property1}\n{\\mathrm{M}}\\_\\eta[{\\boldsymbol{\\Sigma}}] \\leq S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\eta}] \\quad \\forall \\, \\eta \\in \\left(0,\\sfrac{\\pi}{4}\\right]\\,,\\end{aligned}$$ \nsince all assemblages in this traditional bipartite scenario admit a quantum realisation. Hence, $S\\_\\eta[\\widetilde{{\\boldsymbol{\\Sigma}}}]$ is upperbounded by its maximum quantum violation, which is given by $S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\eta}]$. Moreover,\n \n$$\\begin{aligned}\n \\label{eq:property2}\n{\\mathrm{M}}\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\eta}] = S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\eta}] \\,.\\end{aligned}$$ \nWe now show that that equality in Eq. only holds when M*η* and *S**η* are evaluated on the same assemblage.\n[thm:4] Let M*η* be an EPR monotone from Definition [def:Meta] and *S**η* be an EPR functional given in Definition [def:SFT]. Then, if *θ* ≠ *η*, ${\\mathrm{M}}\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\theta}] < S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\eta}]$.\nLet us prove this by contradiction. Our starting assumption is that there exists a pair (*θ* , *η*) with *θ* ≠ *η*, such that ${\\mathrm{M}}\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\theta}] = S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\eta}]$. Then, one of the two should happen:\n${\\mathrm{M}}\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\theta}] = S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\theta}]$. \nIn this case, the solution to the maximisation problem in the computation of M*η* is achieved by ${\\boldsymbol{\\Sigma}}^{\\theta}$ itself. \nOur starting assumption then tells us that $S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\theta}] = S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\eta}]=S\\_\\eta^{\\max}$. \nFrom Remark [rem:iff] (see Appendix [app:steeringin-bipartite]) it follows that necessarily *θ* = *η*, which contradicts our initial condition.\n${\\mathrm{M}}\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\theta}] = S\\_\\eta[\\widetilde{{\\boldsymbol{\\Sigma}}}]$, with ${\\boldsymbol{\\Sigma}}^{\\theta} \\xrightarrow{\\mathrm{LOSR}} \\widetilde{{\\boldsymbol{\\Sigma}}}$. \nIn this case, the solution to the maximisation problem in the computation of M*η* is achieved by an LOSR processing of ${\\boldsymbol{\\Sigma}}^{\\theta}$. \nOur starting assumption then tells us that $S\\_\\eta[\\widetilde{{\\boldsymbol{\\Sigma}}}] = S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\eta}]=S\\_\\eta^{\\max}$. \nLet *ρ* and {{*M**a*∣*x*}*a* ∈ *A*}*x* ∈ *X* be any quantum state and measurements that realise the quantum assemblage $\\widetilde{{\\boldsymbol{\\Sigma}}}$. From Remark [rem:thelast] (see Appendix [app:steeringin-bipartite]), we know that *ρ* is equivalent to $\\ket{\\eta}$ up to local isometries. But since local isometries are free LOSR operations, this means that $\\ket{\\theta}$ is more LOSR-entangled[8](#fn8) than $\\ket{\\eta}$, which contradicts the result of Ref. that all two-qubit pure entangled states are LOSR-inequivalent.\nWe can now prove that S1 is composed of pairwise unordered resources by identifying a pair of monotones that satisfies Eqs. . As we see next, this is achieved by choosing M*j* = M*θ**j*.\n[thm:M-bipartite] For every pair *θ*1 ≠ *θ*2, the monotones *M**θ*1 and *M**θ*2 given by Definition [def:Meta] satisfy\n \n$$\\begin{aligned}\n \\label{eq:6}\n{\\mathrm{M}}\\_{\\theta\\_1}({\\boldsymbol{\\Sigma}}^{\\theta\\_1}) &> {\\mathrm{M}}\\_{\\theta\\_1}({\\boldsymbol{\\Sigma}}^{\\theta\\_2}) \\,, \\\\\n{\\mathrm{M}}\\_{\\theta\\_2}({\\boldsymbol{\\Sigma}}^{\\theta\\_1}) &< {\\mathrm{M}}\\_{\\theta\\_2}({\\boldsymbol{\\Sigma}}^{\\theta\\_2}) \\,. \\label{eq:7}\\end{aligned}$$ \nLet us first prove Eq. . On the one hand, ${\\mathrm{M}}\\_{\\theta\\_1}({\\boldsymbol{\\Sigma}}^{\\theta\\_1}) = S\\_{\\theta\\_1}[{\\boldsymbol{\\Sigma}}^{\\theta\\_1}]$. On the other hand, since *θ*1 ≠ *θ*2, Theorem [thm:4] implies that ${\\mathrm{M}}\\_{\\theta\\_1}({\\boldsymbol{\\Sigma}}^{\\theta\\_2}) < S\\_{\\theta\\_1}[{\\boldsymbol{\\Sigma}}^{\\theta\\_1}]$. Therefore, Eq. follows. \nThe proof of Eq. follows similarly.\n[cor:bip] The infinite family of EPR monotones ${\\mathsf{M}}= \\{{\\mathrm{M}}\\_\\eta \\, \\vert \\, \\eta \\in (0,\\sfrac{\\pi}{4}] \\}$ certifies that the infinite family of assemblages S1 is composed of pairwise unordered resources.\nWe defined new measures of EPR assemblages and showed that they certify that the elements of S1 are unordered in the resource theory of assemblages under LOSR operations. As we showed using the SDP [SDP-bipartite], this property of the pre-order appears to be unique to assemblages ${\\boldsymbol{\\Sigma}}^{\\theta,p}$ with *p* = 1.\nThe multipartite EPR scenario\n=============================\nWe now develop a resource theory for multipartite EPR scenarios under LOSR operations. This is the first time multipartite EPR scenarios have been studied in a resource-theoretic framework. Similarly to the bipartite case, we show that resource conversion can be decided with a single instance of an SDP in this paradigm. Moreover, we define new measures of multipartite nonclassicality and use them to analytically study the properties of the pre-order. Our results show that multipartite scenarios are easier and more natural to understand from an LOSR perspective rather than an LOCC one; we elaborate on this in Section [sec:relwork-comparison].\nIn this paper, we focus on one particular type of multipartite EPR scenario with multiple Alices and one Bob. However, the number of quantum parties (Bobs) can also be increased. Indeed, one possible multipartite setup consists of a single Alice and multiple Bobs. In such a scenario, results from bipartite scenarios may sometimes directly generalise. However, this generalisation might not be straight-forward and a detailed analysis of this multipartite scenario is beyond the scope of this work. For a description of a scenario with multiple Bobs, we refer the reader to Refs. . Focusing on scenarios with more-than-one Alice is however necessary when one wants to allow for post-quantum assemblages, which is crucial when exploring ways to single out quantum phenomena from basic principles – a.k.a. studying quantum ‘from the outside’.\nDefinition of the scenario and free assemblages\n-----------------------------------------------\nIn the multipartite EPR scenario of interest to us, *k* + 1 separated parties share a physical system. In analogy to the bipartite scenario, we consider *k* parties called Alices that hold measurement devices. Each Alice, labeled by *A**i* ∈ {1...*k*}, decides on a classical input *x**i* from the set *x**i* ∈ {1, ..., *m**A*} = : X and generates a classical outcome *a**i* from the set *a**i* ∈ {0, ..., *o**A* − 1} = : A with probability *p**i*(*a**i*∣*x**i*). Without loss of generality, we will take the sets X and A to be the same for all the Alices. When Alices perform the measurements on their subsystems, they update their knowledge about Bob’s subsystem which is now described by a conditional marginal state *ρ**a*1...*a**k*∣*x*1...*x**k*, which depends on all Alices inputs and outputs. In this scenario, the elements of the assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1\\ldots{\\mathbb{A}}\\_k|{\\mathbb{X}}\\_1\\ldots{\\mathbb{X}}\\_k} = \\{\\{\\sigma\\_{a\\_1...a\\_k|x\\_1...x\\_k}\\}\\_{a\\_1...a\\_k}\\}\\_{ x\\_1...x\\_k}$ are given by *σ**a*1...*a**k*∣*x*1...*x**k* : = *p*(*a*1...*a**k*∣*x*1...*x**k*)*ρ**a*1...*a**k*∣*x*1...*x**k*. The multipartite EPR scenario is illustrated in Fig. [fig:multipartite-scenario] for *k* = 2 (two Alices, one Bob).\nIn this paper, we are interested in studying the common-cause processes between multiple parties, hence we focus on the global object given by the assemblage, and not on the individual agents. Nevertheless, we would like to note that it is always possible to recover the individual parties by tracing out the irrelevant subsystems. It would be interesting to study what inferences each Alice can make about the other parties. However, this problem is related to the quantum marginal problem and is beyond the scope of our framework.\n(-70,70) (-58,105)*x*1 (-53,152)*a*1 (5,105)*x*2 (10,152)*a*2 (70,152)*ρ**a*1*a*2∣*x*1*x*2 (0,0)(*a*) \n(-130,0) (-100,71)*x*1 (-33,71)*x*2 (-100,118)*a*1 (-33,118)*a*2 (-64,90)*c*2 (-134,90)*c*1 (-5,5)*λ* (32,118)*ρ**a*1*a*2∣*x*1*x*2 (-117,5)*x*1′ (-47,5)*x*2′ (-37,160)*a*2′ (-107,160)*a*1′ (40,160)*ρ**a*1′*a*2′∣*x*1′*x*2′ (-47,-30)(*b*) \nIn a quantum common-cause multipartite EPR scenario, then, Bob and the Alices share a quantum system, and for each *i* ∈ {1, …, *k*}, the *i*-th Alice is allowed to perform on her share of the system generalised measurements represented by the set {{*M**a**i*∣*x**i*}*a**i*}*x**i*. Then, Bob’s unnormalised states can be realised as *σ**a*1...*a**k*∣*x*1...*x**k* = tr*A*1...*A**k*{(*M**a*1∣*x*1 ⊗ ... ⊗ *M**a**k*∣*x**k* ⊗ I)*ρ*}, where *ρ* is the quantum state of the shared system. These quantumly-realised assemblages are formalised as follows:\n**Multipartite quantum assemblage.** \nAn assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1\\ldots{\\mathbb{A}}\\_k|{\\mathbb{X}}\\_1\\ldots{\\mathbb{X}}\\_k} $ has a quantum realisation iff there exist Hilbert spaces H*A**i* with *i* ∈ {1, …, *k*}, a state *ρ* in H*A*1 ⊗ … ⊗ H*A**k* ⊗ H*B*, and a POVM {*M**a**i*∣*x**i*}*a**i* ∈ A on H*A**i* for each *x**i* ∈ X and *i* ∈ {1, …, *k*}, such that\n \n$$\\begin{aligned}\n\\label{eq:quantum-multi}\n\\sigma\\_{a\\_1...a\\_k|x\\_1...x\\_k}=\\text{tr}\\_{A\\_1...A\\_k} \\{ (M\\_{a\\_1|x\\_1} \\otimes... \\otimes M\\_{a\\_k|x\\_k} \\otimes {\\mathbb{I}}) \\rho\\}\\end{aligned}$$ \nfor all *a*1, …, *a**k* ∈ A and *x*1, …, *x**k* ∈ X. [def:Q-multipartite]\nIn analogy to the bipartite case, a multipartite assemblage is LOSR-free if the parties (Alices and Bob) can generate it using local operations (classical and quantum) and classical shared randomness. The elements of an LOSR-free assemblage can be decomposed as *σ**a*1…*a**k*∣*x*1…*x**k* = ∑*λ**p*(*λ*)*p*1(*a*1∣*x*1, *λ*)…*p**k*(*a**k*∣*x**k*, *λ*) *σ**λ*, where *p**j*(*a**j*∣*x**j*, *λ*) is a conditional probability distribution for the *j*-th Alice for every *λ*, and *σ**λ* are unnormalised quantum states which, similarly to the bipartite case, satisfy tr{∑*λ**σ**λ*} = 1. Notice that this definition of a free multipartite assemblage coincides with the definition of an “unsteerable multipartite assemblage” in Refs. , in the sense of “fully unsteerable”[9](#fn9). Therefore our LOSR underpinning of the resource-theoretic understanding of assemblages coincides with what people in the literature understand as a ``totally-useless assemblage\". Notice, however, that unlike in Ref. , we do not render unsteerable assemblages where the correlations observed among the Alices’ measurement outputs are non-classical. In Section [sec:relwork-multi], we review different definitions of a multipartite free assemblage and compare them to our approach.\nLOSR transformations between multipartite assemblages\n-----------------------------------------------------\nThe most general LOSR transformation of a multipartite assemblage consists of a comb for each Alice (which locally pre- and post-processes the variables on each Alice’s wing), and a CPTP map (which post-processes Bob’s quantum system), with all these actions being coordinated by a classical random variable *λ*. Formally, a generic LOSR operation is illustrated in Fig. [fig:multipartite-scenario-LOSR], and transforms an assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1\\ldots {\\mathbb{A}}\\_k|{\\mathbb{X}}\\_1\\ldots {\\mathbb{X}}\\_k}$ into a new one as follows:\n \n$$\\begin{aligned}\n\\label{eq:LOSRtrans-multi} \n\\sigma'\\_{a'\\_1\\ldots a'\\_k|x'\\_1\\ldots x'\\_k} = \\sum\\_{\\substack{\\lambda \\\\ c\\_1 \\ldots c\\_k}} \\sum\\_{\\substack{a\\_1\\ldots a\\_k \\\\ x\\_1\\ldots x\\_k }} \\quad p(c\\_1,x\\_1|&x\\_1^{\\prime}, \\lambda) \\ldots p(c\\_k,x\\_k|x\\_k^{\\prime}, \\lambda) \n\\quad p(a\\_1^{\\prime}|a\\_1,c\\_1) \\ldots p(a\\_k^{\\prime}|a\\_k,c\\_k) \\nonumber \\\\\n& p(\\lambda) \\quad p(a\\_1, \\ldots, a\\_k|x\\_1, \\ldots, x\\_k) \\quad\n\\mathcal{E}\\_{\\lambda}(\\rho\\_{a\\_1\\ldots a\\_k|x\\_1\\ldots x\\_k})\\,,\\end{aligned}$$ \nwhere, similarly to the bipartite case,\n*p*(*c**i*, *x**i*∣*x**i*′, *λ*) encodes the classical pre-processing of the *i*-th Alice’s input *x**i* as a function of *x**i*′ and the shared classical randomness *λ*. Here, *c**i* denotes the variable to be transmitted through the *i*-th Alice’s classical side channel towards her post-processing stage.\n*p*(*a**i*′∣*a**i*, *c**i*) encodes the classical post-processing of the *i*-th Alice’s output *a**i*, as a function of the classical information *c**i* kept from the pre-processing stage. The output of the process is the *i*-th Alice’s new outcome *a**i*′.\nE*λ*[ ⋅ ] is the CPTP map corresponding to Bob’s local post-processing of his quantum system, as a function of the shared classical randomness *λ*.\nNow recall that, similarly to the bipartite case, an LOSR-free assemblage is one that can be created from local operations and shared randomness. Hence, classical assemblages are all and only the ones that can be generated through the free operations of choice, consistent with the unifying assessment of ‘free of cost’ that this resource-theoretic underpinning brings. Moreover, if ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1\\ldots {\\mathbb{A}}\\_k|{\\mathbb{X}}\\_1\\ldots {\\mathbb{X}}\\_k}$ is free, ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}\\_1^{\\prime}\\ldots {\\mathbb{A}}\\_k^{\\prime}|{\\mathbb{X}}\\_1^{\\prime}\\ldots {\\mathbb{X}}\\_k^{\\prime}}$ is free as well, hence the set of free multipartite assemblages (our free resources) is closed under LOSR operations, as it should be.\nA final remark pertains to a particular way to express a generic LOSR transformation, similarly to the discussion in the bipartite scenario. Let us represent each Alice’s local comb with a single probability distribution *p*(*a**i*′, *x**i*∣*a**i*, *x**i*′, *λ*). By Fine’s argument and discussion in Ref. , such local combs can be decomposed as a convex combination of deterministic combs as follows\n \n$$\\begin{aligned}\n p(a\\_{i}^{\\prime}, x\\_i|a\\_i,x\\_{i}^{\\prime}, \\lambda) = \\sum\\_{\\tilde{\\lambda}} p(\\tilde{\\lambda}|\\lambda) D(x\\_i|x\\_i^{\\prime}, \\tilde{\\lambda})\\,D(a\\_i^{\\prime}|a\\_i,x\\_i^{\\prime}, \\tilde{\\lambda}).\\end{aligned}$$ \nHere, each deterministic probability distribution assigns a fixed outcome *a**i*′ (resp. *x**i*) for each possible choice of *a**i*, *x**i*′, and *λ̃* (resp. *x**i*′ and *λ̃*). Let us now use this observation to rewrite Eq. ; for clarity in the presentation let us focus on the tripartite case (two Alices, one Bob). Define the CPTNI map $\\tilde{\\mathcal{E}}\\_{\\tilde{\\lambda}}$ as:\n \n$$\\begin{aligned}\n\\tilde{\\mathcal{E}}\\_{\\tilde{\\lambda}}(\\sigma\\_{a\\_1 a\\_2|x\\_1 x\\_2})= \\sum\\_{\\lambda} p(\\lambda) p(\\tilde{\\lambda}|\\lambda) \\mathcal{E}\\_{\\lambda}(p(a\\_1 a\\_2|x\\_1 x\\_2)\\rho\\_{a\\_1 a\\_2|x\\_1 x\\_2}).\\end{aligned}$$ \nA generic LOSR operation transforms an assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1 {\\mathbb{A}}\\_2|{\\mathbb{X}}\\_1 {\\mathbb{X}}\\_2}$ into a new one as follows:\n \n$$\\begin{aligned}\n\\label{eq:LOSRtrans-multi2} \n\\sigma'\\_{a'\\_1 a'\\_2|x'\\_1 x'\\_2} = \\sum\\_{\\tilde{\\lambda}} \\sum\\_{\\substack{a\\_1 a\\_2 \\\\ x\\_1 x\\_2 }} \\, D(x\\_1|x\\_1^{\\prime}, \\tilde{\\lambda})\\,D(a\\_1^{\\prime}|a\\_1,x\\_1^{\\prime}, \\tilde{\\lambda})\\, D(x\\_2|x\\_2^{\\prime}, \\tilde{\\lambda})\\,D(a\\_2^{\\prime}|a\\_2,x\\_2^{\\prime}, \\tilde{\\lambda})\\,\n\\tilde{\\mathcal{E}}\\_{\\tilde{\\lambda}}(\\sigma\\_{a\\_1 a\\_2|x\\_1 x\\_2}).\\end{aligned}$$ \nEq. provides the simplified characterisation of a generic multipartite LOSR transformation that we will use throughout.\nResource conversion as a semidefinite test\n------------------------------------------\nFor clarity in the presentation, we will still focus on the specific multipartite scenario, illustrated in Fig. [fig:multipartite-scenario]. Our method, however, extends to scenarios with an arbitrary number of Alices. Given two assemblages generated in this setup, ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1 {\\mathbb{A}}\\_2|{\\mathbb{X}}\\_1 {\\mathbb{X}}\\_2}$ and ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}\\_1^{\\prime}{\\mathbb{A}}\\_2^{\\prime}|{\\mathbb{X}}\\_1^{\\prime}{\\mathbb{X}}\\_2^{\\prime}}$, testing whether ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1 {\\mathbb{A}}\\_2|{\\mathbb{X}}\\_1 {\\mathbb{X}}\\_2}$ can be converted into ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}\\_1^{\\prime}{\\mathbb{A}}\\_2^{\\prime}|{\\mathbb{X}}\\_1^{\\prime}{\\mathbb{X}}\\_2^{\\prime}}$ under LOSR operations amounts to checking if ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}\\_1^{\\prime}{\\mathbb{A}}\\_2^{\\prime}|{\\mathbb{X}}\\_1^{\\prime}{\\mathbb{X}}\\_2^{\\prime}}$ admits a decomposition as per Eq. for the case of two Alices. Similarly to the bipartite scenario, the possibility of the conversion can be decided with a single instance of an SDP.\nFirst, notice that the map $\\tilde{\\mathcal{E}}\\_{\\lambda}(\\sigma\\_{a\\_1a\\_2|x\\_1x\\_2})$ can be represented in the operator form in terms of its Choi state *W**λ* as follows:\n \n$$\\begin{aligned}\n\\tilde{\\mathcal{E}}\\_{\\lambda}(\\sigma\\_{a\\_1a\\_2|x\\_1x\\_2})= d\\_B\\, {\\mathrm{tr}\\_{B} \\left\\{ W\\_{\\lambda}\\,(\\,{\\mathbb{I}}\\_{B^{\\prime}} \\otimes \\sigma\\_{a\\_1a\\_2|x\\_1x\\_2}^{T}) \\right\\}}\\,,\\end{aligned}$$ \nwhere *d**B* is the dimension of Bob’s Hilbert space. Therefore, for ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}\\_1^{\\prime}{\\mathbb{A}}\\_2^{\\prime}|{\\mathbb{X}}\\_1^{\\prime}{\\mathbb{X}}\\_2^{\\prime}}$ to decompose as in Eq. , each *σ*ʹ*a*1′*a*2′∣*x*1′*x*2′ must admit the following decomposition:\n \n$$\\begin{aligned}\n\\label{eq:sigSDP-multi}\n \\sigma'\\_{a\\_1^{\\prime}a\\_2^{\\prime}|x\\_1^{\\prime}x\\_2^{\\prime}}= \\sum\\_{\\substack{\\lambda \\\\ c\\_1,c\\_2}}\\sum\\_{\\substack{a\\_1a\\_2\\\\x\\_1x\\_2}} D(x\\_1|x\\_1^{\\prime}, {\\lambda})\\,D(a\\_1^{\\prime}|a\\_1,x\\_1^{\\prime}, {\\lambda})\\, D(x\\_2|x\\_2^{\\prime}, {\\lambda})\\,&D(a\\_2^{\\prime}|a\\_2,x\\_2^{\\prime}, {\\lambda})\\, \\nonumber \\\\\n &d\\_B\\, {\\mathrm{tr}\\_{B} \\left\\{ W\\_{\\lambda}\\,(\\,{\\mathbb{I}}\\_{B^{\\prime}} \\otimes \\sigma\\_{a\\_1a\\_2|x\\_1x\\_2}^{T}) \\right\\}}\\,.\\end{aligned}$$ \nThis relation between elements *σ*ʹ*a*1′*a*2′∣*x*1′*x*2′ and *σ**a*1*a*2∣*x*1*x*2 can be tested via an SDP. The SDP that tests whether ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1{\\mathbb{A}}\\_2|{\\mathbb{X}}\\_1{\\mathbb{X}}\\_2}$ can be converted into ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}\\_1^{\\prime}{\\mathbb{A}}\\_2^{\\prime}|{\\mathbb{X}}\\_1^{\\prime}{\\mathbb{X}}\\_2^{\\prime}}$ under LOSR operations reads as follows.\n[SDP-multi]**${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1{\\mathbb{A}}\\_2|{\\mathbb{X}}\\_1{\\mathbb{X}}\\_2}\\, \\overset{\\text{LOSR}}{\\longrightarrow} \\, {\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}\\_1^{\\prime}{\\mathbb{A}}\\_2^{\\prime}|{\\mathbb{X}}\\_1^{\\prime}{\\mathbb{X}}\\_2^{\\prime}}$.** \nThe assemblage ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1{\\mathbb{A}}\\_2|{\\mathbb{X}}\\_1{\\mathbb{X}}\\_2}$ can be converted into the assemblage ${\\boldsymbol{\\Sigma}}'\\_{{\\mathbb{A}}\\_1^{\\prime}{\\mathbb{A}}\\_2^{\\prime}|{\\mathbb{X}}\\_1^{\\prime}{\\mathbb{X}}\\_2^{\\prime}}$ under LOSR operations if and only if the following SDP is feasible\n \n$$\\begin{aligned}\n\\begin{split}\n\\textrm{given} \\;\\;\\;& \\{ \\{\\sigma\\_{a\\_1a\\_2|x\\_1x\\_2}\\}\\_{a\\_1,a\\_2}\\}\\_{x\\_1,x\\_2}\\,,\\; \\{\\{ \\sigma'\\_{a\\_1^{\\prime}a\\_2^{\\prime}|x\\_1^{\\prime}x\\_2^{\\prime}}\\}\\_{a\\_1^{\\prime},a\\_2^{\\prime}}\\}\\_{x\\_1^{\\prime},x\\_2^{\\prime}}\\,,\\\\ \n& \\{D(a\\_1^{\\prime}|a\\_1,x\\_1^{\\prime},\\lambda)\\}\\_{\\lambda,a\\_1^{\\prime},a\\_1,x\\_1^{\\prime}}\\,,\\, D(x\\_1|x\\_1^{\\prime},\\lambda)\\}\\_{\\lambda,x\\_1,x\\_1^{\\prime}} \\,, \\\\\n& \\{D(a\\_2^{\\prime}|a\\_2,x\\_2^{\\prime},\\lambda)\\}\\_{\\lambda,a\\_2^{\\prime},a\\_2,x\\_2^{\\prime}}\\,,\\, D(x\\_2|x\\_2^{\\prime},\\lambda)\\}\\_{\\lambda,x\\_2,x\\_2^{\\prime}} \\\\\n \\textrm{find} \\;\\;\\;& \\{(W\\_{\\lambda})\\_{BB'}\\}\\_{\\lambda} \\\\\n \\textrm{s.t.} \\;\\;\\;& \\begin{cases} W\\_{\\lambda} \\geq 0\\,,\\\\\n {\\mathrm{tr}\\_{B^{\\prime}} \\left\\{ W\\_{\\lambda} \\right\\}} \\propto \\frac{1}{d}\\,{\\mathbb{I}}\\_B \\;\\;\\;\\; \\forall \\lambda\\,, \\\\\n \\sum\\_{\\lambda} {\\mathrm{tr}\\_{B^{\\prime}} \\left\\{ W\\_{\\lambda} \\right\\}} = \\frac{1}{d}\\, {\\mathbb{I}}\\_B\\,,\\\\\n \\sigma'\\_{a\\_1^{\\prime}a\\_2^{\\prime}|x\\_1^{\\prime}x\\_2^{\\prime}}= \\sum\\_{\\lambda}\\sum\\_{\\substack{a\\_1a\\_2\\\\x\\_1x\\_2}} & D(a\\_1^{\\prime}|a\\_1,x\\_1^{\\prime},\\lambda)\\, D(x\\_1|x\\_1^{\\prime},\\lambda) \\, D(a\\_2^{\\prime}|a\\_2,x\\_2^{\\prime},\\lambda) \\\\\n &D(x\\_2|x\\_2^{\\prime},\\lambda) \\, d\\_B\\, {\\mathrm{tr}\\_{B} \\left\\{ W\\_{\\lambda}\\,(\\,{\\mathbb{I}}\\_{B^{\\prime}} \\otimes \\sigma\\_{a\\_1a\\_2|x\\_1x\\_2}^{T}) \\right\\}}\\,.\n \\end{cases}\n \\end{split}\\end{aligned}$$ \nSimilarly to SDP [SDP-bipartite], SDP [SDP-multi] is a feasibility problem.\nProperties of the pre-order\n---------------------------\nThe properties of the pre-order of multipartite assemblages can be studied numerically, with the SDP [SDP-multi], or analytically, with EPR LOSR monotones. The SDP [SDP-multi] can be used to test conversions between multipartite assemblages, and a plot similar to that in Fig. [fig:SDP-bipartite] can be made to illustrate the pre-order of any multipartite family. Hence, we will not repeat this analysis for the multipartite case. In this section, we study the pre-order analytically, but the results can be easily verified with SDP [SDP-multi]. Although our methods apply to general assemblages, here we focus on quantumly-realisable assemblages, and study the properties of the pre-order for a particular family of resources therein.\nConsider an EPR scenario with *N* = *k* + 1 parties, where A*i* = X*i* = {0, 1} for *i* ∈ {1, ..., *N* − 1}. Assume all parties share the state *ρ**N**θ* defined as\n \n$$\\begin{aligned}\n\\label{eq:Nass}\n\\rho\\_N^{\\theta}&=\\ket{GHZ\\_N^{\\theta}}\\bra{GHZ\\_N^{\\theta}} \\,, \\\\\n\\text{with} \\quad \\ket{GHZ\\_N^{\\theta}}&=\\cos{\\theta} \\ket{0}^{\\otimes N} + \\sin{\\theta} \\ket{1}^{\\otimes N} \\,, \\nonumber \\end{aligned}$$ \nand the measurements that Alices perform upon an input *x**i* ∈ X are given by\n \n$$\\begin{aligned}\n\\label{eq:Nmeas}\n\\widetilde{M}\\_{a\\_i|0} &= \\frac{{\\mathbb{I}}+ (-1)^{a\\_i} \\sigma\\_z}{2}\\,,\\quad \\widetilde{M}\\_{a\\_i|1} = \\frac{{\\mathbb{I}}+ (-1)^{a\\_i} \\sigma\\_x}{2}\\,. \\end{aligned}$$ \nLet us define a family of assemblages S(*N*) as:\n \n$$\\begin{aligned}\n\\label{eq:thefamN}\n{\\mathsf{S}}^{(N)} &= \\left\\{ {\\boldsymbol{\\Sigma}}^\\theta\\_{{\\mathbb{A}}\\_1\\ldots {\\mathbb{A}}\\_{N-1}|{\\mathbb{X}}\\_1\\ldots {\\mathbb{X}}\\_{N-1}} \\, \\Big\\vert \\, \\theta \\in \\left(0,\\sfrac{\\pi}{4}\\right] \\right\\}\\,,\\\\ \\nonumber\n\\text{where} \\quad{\\boldsymbol{\\Sigma}}^\\theta\\_{{\\mathbb{A}}\\_1\\ldots {\\mathbb{A}}\\_{N-1}|{\\mathbb{X}}\\_1\\ldots {\\mathbb{X}}\\_{N-1}} &= \\left\\{\\left\\{\\sigma^\\theta\\_{a\\_1...a\\_{N-1}|x\\_1...x\\_{\\_{N-1}}}\\right\\}\\_{a\\_i\\in{\\mathbb{A}}\\_i}\\right\\}\\_{ x\\_i \\in {\\mathbb{X}}\\_i} \\,,\\\\\n\\text{with} \\quad \\sigma^\\theta\\_{a\\_1...a\\_{N-1}|x\\_1...x\\_{\\_{N-1}}} &= {\\mathrm{tr}\\_{A\\_1...A\\_{N-1}} \\left\\{ \\widetilde{M}\\_{a\\_1|x\\_1} \\otimes... \\otimes \\widetilde{M}\\_{a\\_{N-1}|x\\_{N-1}} \\otimes {\\mathbb{I}}\\, \\rho\\_N^{\\theta} \\right\\}} \\,. \\nonumber\\end{aligned}$$ \nFor simplicity in the notation, we here denote ${\\boldsymbol{\\Sigma}}^\\theta\\_N := {\\boldsymbol{\\Sigma}}^\\theta\\_{{\\mathbb{A}}\\_1\\ldots {\\mathbb{A}}\\_{N-1}|{\\mathbb{X}}\\_1\\ldots {\\mathbb{X}}\\_{N-1}}$ and ${\\boldsymbol{\\Sigma}}\\_N := {\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}\\_1\\ldots {\\mathbb{A}}\\_{N-1}|{\\mathbb{X}}\\_1\\ldots {\\mathbb{X}}\\_{N-1}}$.\nThis family S(*N*), for fixed *N*, has an infinite number of elements. We claim that the elements of S(*N*) are unordered as per the LOSR resource theory of assemblages. To show this, we define a set of EPR monotones {*S**η*(*N*)}*η* using the Bell inequalities derived in Ref. . This procedure is presented in the Appendix [app:steeringin-multi], and the EPR functional is given in Definition [def:NSFT-multi]. For each value of *η*, we construct the monotone M*η*(*N*) from the EPR functional *S**η*(*N*) following a yield-based construction:\n[def:MetaN] The EPR monotone M*η*(*N*), for $\\eta \\in (0,\\sfrac{\\pi}{4}]$, is defined as\n \n$$\\begin{aligned}\n{\\mathrm{M}}\\_{\\eta}^{(N)}[{\\boldsymbol{\\Sigma}}\\_N] := \\max\\_{\\widetilde{{\\boldsymbol{\\Sigma}}}\\_N} \\{ S\\_{\\eta}^{(N)}[\\widetilde{{\\boldsymbol{\\Sigma}}}\\_N]: {\\boldsymbol{\\Sigma}}\\_N \\xrightarrow{\\mathrm{LOSR}} \\widetilde{{\\boldsymbol{\\Sigma}}}\\_N \\}.\\end{aligned}$$ \nIn Appendix [app:steeringin-multi], we show that the monotones ${\\mathrm{M}}\\_{\\eta}^{(N)}[{\\boldsymbol{\\Sigma}}\\_N]$ satisfy properties analogous to these given in Eqs. and, and Theorems [thm:4] and [thm:M-bipartite], for the monotones ${\\mathrm{M}}\\_{\\eta}[{\\boldsymbol{\\Sigma}}]$. It follows that:\n[cor:multi] The infinite family of EPR monotones ${\\mathsf{M}}^{(N)} = \\{{\\mathrm{M}}\\_\\eta^{(N)} \\, \\vert \\, \\eta \\in (0,\\sfrac{\\pi}{4}] \\}$ certifies that the infinite family of assemblages S(*N*) is composed of pairwise unordered resources.\nThis result shows how, when LOSR are considered to be free operations in a resource theory, methods used in the bipartite scenarios can be leveraged to the multipartite case.\nRelated Work\n============\nAlthough the causal approach we take in this paper is not the standard description of the EPR scenario, the objects that we study, i.e., assemblages, have been widely investigated for their information-theoretic and foundational relevance. In particular, their role in one-side device-independent quantum key distribution (1S-DI-QKD) protocols motivated a formulation of the resource theory of ‘steering’, where the set of free operations is deemed to be stochastic local operations assisted by one-way classical communication (S-1W-LOCC). This set of operations – just like LOSR – is a valid set of free operations in a resource theory of assemblages, as it maps free resources to free resources. However, the conceptual underpinnings of our resource theory of nonclassicality of assemblages and of the resource theory of ‘steering’ under S-1W-LOCC operations are very different. In this section, we discuss some conceptual and technical differences between these two resource theories. Moreover, we show that the choice of free operations in a resource theory has significant consequences for the definition of a free multipartite assemblage.\nS-1W-LOCC as the set of free operations\n---------------------------------------\n### Formalisation of S-1W-LOCC operations\nTo start with, let us recall the formal definition of a S-1W-LOCC operation which is illustrated in Fig. [fig:1W-LOCC]. Consider a bipartite EPR scenario. First, Bob performs an instrument on his subsystem that produces both a classical system represented by the variable *ω*, and a quantum system. This action corresponds to a completely-positive, trace-non-increasing map E*ω* on Bob’s quantum system. Next, he communicates a classical message that depends on *ω* to Alice; without loss of generality, we can take the communicated message to be *ω* itself. In the second laboratory, Alice at first generates the classical input variable *x*ʹ. Once she has received *ω*, she generates the new input variable *x* which can depend on both *x*ʹ and *ω*. Then, she performs the measurement labelled by *x* on her share of the system; she will then obtain a measurement outcome *a*. Finally, Alice classically processes the variables *x*ʹ, *ω* and *a*, to produce the classical output *a*ʹ of her measurement process. In the S-1W-LOCC operations, the instrument does not necessary need to be complete, i.e., if the probability for each outcome *ω* is *P**ω*, then the only guarantee on the sum of these probabilities is 0 < ∑*ω**P**ω* ≤ 1. It means that the total S-1W-LOCC operation does not need to happen with certainty, but with some non-zero probability. It was shown in Ref. that these bipartite S-1W-LOCC operations map free assemblages to free assemblages in bipartite EPR scenarios; hence, they are formally a valid set of free operations in a resource theory of assemblages therein.\n[fig:1W-LOCC]\n### Comparing S-1W-LOCC and LOSR\nThe resource theory of ‘steering’ under S-1W-LOCC and our resource theory of nonclassicality of assemblages are both formally valid resource theories that quantify nonclassicality in the bipartite EPR scenario. In this subsection, we will argue that the causal view on assemblages has several advantages over the S-1W-LOCC approach.\nThe set of free operations in any resource theory should be motivated by natural physical constraints in the scenario of interest. The set of S-1W-LOCC operations is hence a meaningful choice in scenarios wherein one-way classical communication is possible. Indeed, the choice of S-1W-LOCC as the set of free operations was motivated by the fact that it is the most general set that does not compromise the security of 1S-DI-QKD protocols . In contrast, the motivation behind our LOSR approach is fundamentally different. Rather than using a methodology which singles out one particular task for which assemblages are known to be useful, we take a principled approach that applies to any scenario with a common-cause causal structure. In particular, we follow a general construction that can be applied to study nonclassicality in arbitrary causal networks, as first introduced in Ref. (see in particular Appendix A3 therein). Indeed, our resource theory of EPR assemblages is simply a type-specific instance of the type-independent resource theory introduced in Ref. . This is the most significant conceptual advantage of our LOSR approach – it unifies the study of nonclassicality of assemblages with the study of nonclassicality of arbitrary processes in common-cause (e.g., Bell) scenarios. This unified view leads to a deeper understanding of nonclassical correlations, as well as to new technical tools for studying them .\nIt is also worth noting that, at a technical level, LOSR conversions of assemblages are much easier to characterize and study than the S-1W-LOCC conversions.\nThe fact that we have taken a principled approach to constructing our resource theory also implies a final advantage: our framework extends naturally and uniquely to more general scenarios, including multipartite EPR scenarios, as well as EPR scenarios where the processes about which Alice is learning are dynamic (e.g., general channels or Bob-with-input processes[10](#fn10)). It is not clear how prior approaches generalize to these cases; indeed, in past work there is no consensus even on the simplest question of how to define free assemblages in multipartite EPR scenarios (we discuss this point further in the following sections).\nFinally, for completeness, we note that the LOSR and S-1W-LOCC resource theories do in fact lead to different pre-orders, and hence give different answers to questions about the relative value of assemblages.\n[prop:pre-order] The pre-order of assemblages under LOSR operations is different than the pre-order of assemblages under S-1W-LOCC operations.\nThe proof of this proposition is given in Appendix [app:proof-prop-preorder]. The main idea is to find a conversion that is possible with S-1W-LOCC operations and is impossible with LOSR operations. In the proof we focus on the family ${\\boldsymbol{\\Sigma}}^{\\theta,p}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ defined in Eq. . We show that for S-1W-LOCC operations, the assemblage ${\\boldsymbol{\\Sigma}}^{\\pi/4,1}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ is above all others in the family; whereas in LOSR, all assemblages ${\\boldsymbol{\\Sigma}}^{\\theta,1}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ are incomparable for $\\theta \\in \\left(0,\\sfrac{\\pi}{4}\\right]$. In Appendix [app:proof-prop-preorder] we provide an S-1W-LOCC map that maps ${\\boldsymbol{\\Sigma}}^{\\pi/4,1}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ to ${\\boldsymbol{\\Sigma}}^{\\theta,1}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ for $\\theta \\in \\left(0,\\sfrac{\\pi}{4}\\right)$. This result is analogous to the difference in the pre-order in the resource theories of LOCC-entanglement and LOSR-entanglement [11](#fn11).\nIn proving Proposition [prop:pre-order], we note a deficiency in Theorem 5 in Ref. . The theorem claims that there is no universal assemblage in a particular setting from which we obtain any other assemblage in the same setting with S-1W-LOCC maps. The proof applies if one assumes that the universal assemblage is not ${\\boldsymbol{\\Sigma}}^{\\pi/4,1}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$, thus limiting the applicability of the theorem. Therefore, the proof given in Ref. is incomplete.\nMultipartite free assemblages\n-----------------------------\nHistorically, a bipartite assemblage would be considered nonfree or nonclassical[12](#fn12) if Alice and Bob need to share a quantum system prepared in an entangled state in order to produce it. If the assemblage could be generated by performing measurements on a system prepared on a separable state, then the assemblage is free.\n*Both* for bipartite *and* multipartite scenarios, this is exactly the distinction between free and nonfree in our approach: an assemblage is free if and only if it can be generated by measurements on a fully-separable state. Although the definition of a free multipartite assemblage coincides with our approach in some past literature , alternative approaches to studying multipartite EPR scenarios also exist. We now discuss some of the different sets of assemblages that have been proposed as free sets in the multipartite case, and contrast them with our proposal.\nFor simplicity, we will focus on the case of a tripartite EPR scenario with two Alices and one Bob, although the three definitions easily generalise to EPR scenarios with more than two Alices. In this setup, one proposal is given in Ref. , where an assemblage is called free if it can be decomposed as\n \n$$\\begin{aligned}\n\\label{eq:uola\\_unsteer}\n \\sigma\\_{a\\_1a\\_2|x\\_1x\\_2}=\\sum\\_\\lambda p(\\lambda) p(a\\_1,a\\_2|x\\_1,x\\_2,\\lambda) \\rho\\_{\\lambda},\\end{aligned}$$ \nwhere *p*(*λ*) is a normalised probability distribution on *λ*, *ρ**λ* is a normalised quantum state for each *λ*, and *p*(*a*1, *a*2∣*x*1, *x*2, *λ*) is only required to be a valid conditional probability distribution. Indeed, here *p*(*a*1, *a*2∣*x*1, *x*2, *λ*) is allowed to even be a signalling correlation between the Alices, for each *λ*, as long as this signalling is averaged out in Eq. to give rise to a valid quantum assemblage {{*σ**a*1*a*2∣*x*1*x*2}*a*1, *a*2 ∈ A}*x*1, *x*2 ∈ X. This condition for {*p*(*a*1, *a*2∣*x*1, *x*2, *λ*)} is a form of fine-tuning , and is conceptually problematic. Furthermore, it is clear that any assemblage that is nontrivially fine-tuned in this manner is not consistent with our hypothesis about the causal structure of an EPR scenario. Note also that, by this definition, there exist assemblages that are free and yet for which the correlations shared by the Alices (when one marginalizes over Bob’s system) are highly nonclassical– e.g., a Popescu-Rohrlich box correlations. As such, this approach is not suited to studying nonclassicality in general, but could only be suitable for studying nonclassicality *across the Alices-vs-Bob* partition. A different proposal for the definition of a free assemblage is given in Ref. , where an assemblage is deemed free if it can be expressed as\n \n$$\\begin{aligned}\n\\label{eq:TO-LHS}\n\\begin{split}\n \\sigma\\_{a\\_1a\\_2|x\\_1x\\_2}&=\\sum\\_{\\lambda} p(\\lambda) p^{A\\_1 \\rightarrow A\\_2}(a\\_1,a\\_2|x\\_1,x\\_2,\\lambda) \\rho\\_{\\lambda} \\\\\n &=\\sum\\_{\\lambda} p^{\\prime}(\\lambda) p^{A\\_2 \\rightarrow A\\_1}(a\\_1,a\\_2|x\\_1,x\\_2,\\lambda) \\rho^{\\prime}\\_{\\lambda},\n \\end{split}\\end{aligned}$$ \nwhere for each value of *λ*, the probability distribution *p**A*1 → *A*2(*a*1, *a*2∣*x*1, *x*2, *λ*) is non-signalling from Alice2 to Alice1, and the probability distribution *p**A*2 → *A*1(*a*1, *a*2∣*x*1, *x*2, *λ*) is non-signalling from Alice1 to Alice2. Such a decomposition is called the time-ordered local hidden state model (TO-LHS) of an assemblage. This set has been motivated by the fact that it is the largest set of free assemblages for which an anomalous phenomenon termed ‘exposure of steering’ does not occur (under particular assumptions about the set of free operations, distinct from those we have advocated for), and we elaborate further on this point in Section [sec:relwork-exposure]. In particular, we note that in any standard resource theory (i.e., in the sense of Ref. ), both the free resources and the free transformations on them should be derivable from a single set of free operations. What this implies here is that it should be the case that the assemblages which admit of TO-LHS models constitute *all and only* those assemblages which can be constructively generated using S-1W-LOCC operations. This is, however, not the case, since the set of TO-LHS assemblages includes some post-quantum tripartite assemblages (e.g., where the Alices share a Popescu-Rohrlich box; see the next subsection), and these cannot be generated by a S-1W-LOCC protocol.\nThese two definitions of free assemblage contrast with our LOSR resource theory of nonclassicality of assemblages, where an assemblage is LOSR-free if it can be written as\n \n$$\\begin{aligned}\n\\label{eq:unsteer\\_multi}\n \\sigma\\_{a\\_1a\\_2|x\\_1x\\_2}=\\sum\\_\\lambda p(\\lambda) p(a\\_1|x\\_1,\\lambda) p(a\\_2|x\\_2,\\lambda) \\sigma\\_{\\lambda}.\\end{aligned}$$ \nClassicality in our resource-theoretic approach is captured by the structure of the network, and imposes that the assemblage is free, i.e., classical, if the parties – Alices and Bob – are related solely by a shared classical common cause. This definition of a free assemblage has indeed appeared in the literature before in the context of EPR scenarios, and not from a resource-theoretic perspective: in Refs. , a free assemblage is defined as one that arises when all parties share a quantum system prepared in a fully separable state, and the Alices perform local measurements on their subsystems – this leads to all and only assemblages of the form of Eq. .\nFinally, it is worth noting that in this discussion we have focused on multipartite EPR scenarios with multiple Alices and one Bob. However, a possible multipartite EPR scenario comprises also multiple Bobs. Such a multipartite scenario is beyond the scope of this work, however, it would be interesting to see which other approaches to defining free assemblages it gives rise to[13](#fn13).\nExposure of ‘steering’\n----------------------\n*Exposure of ‘steering’* refers to a process in which a resourceful assemblage in an EPR scenario can be created from a free assemblage in a multipartie EPR scenario, by performing a global operation on the Alices. Notice that such an operation may change the type of scenario under study; e.g., it may transform a tripartite assemblage into a bipartite one.\nA type-changing transformation of this nature is considered in Ref. , where Taddei *et al.* focus on a tripartite EPR scenario with two Alices and one Bob. As a case study, they take a free tripartite assemblage to be defined as per Eq. , when no extra constraints are imposed on *p*(*a*1, *a*2∣*x*1, *x*2, *λ*) other than the implicit fine tuning condition. The global operation on the two Alices is a wiring operation – the measurement outcome of one Alice is taken as the choice of measurement for the other Alice. Taddei *et al.* argue that this global operation should not increase the Alices’ capability to remotely ‘steer’ Bob’s system, since no global action is performed across the Bob-vs-Alices divide, i.e., this global operation should be a free operation (at least relative to the Bob-vs-Alices divide). Despite this intuition, they find that a wiring between the Alices in a tripartite free assemblage may give rise to a new bipartite assemblage that is nonfree.\nThis then raises important conceptual questions, since a well-defined *compositional* resource theory should not allow one to create nonfree resources by using free operations on free resources, *even if* the free transformation is type-changing. The set of free resources in any resource theory must be closed under free operations . The spectre of ‘steering’ exposure emphasizes the importance of choosing one’s set of free resources and free operations wisely and consistently.\nAs Taddei *et al.* observe, then, ‘steering’ exposure implies an inconsistency taking Eq. to define the set of free resources (together with taking wirings-and-S-1W-LOCC as free operations). They then derive the largest set of assemblages that do not allow for ‘steering’ exposure under these free operations, and show this set to be those assemblages that admit a TO-LHS model. Hence, they argue that the set of free assemblages should be taken to be defined by Eq. .\nOne awkwardness of this proposal is that the set of TO-LHS assemblages includes assemblages that are not quantumly-realizable; i.e., that cannot be produced by Bob and the Alices sharing a multipartite quantum system and performing local measurements on it. An example of this arises when *p**A*1 → *A*2(*a*1, *a*2∣*x*1, *x*2, *λ*) = *p**A*2 → *A*1(*a*1, *a*2∣*x*1, *x*2, *λ*) is equal to ‘a Popescu-Rohrlich box’ for all *λ*. This sort of definition, then, is clearly not suitable for studying generic nonclassicality in multipartite EPR scenarios, but rather could *only* be suitable as a way of studying *nonclassicality across the Alices-vs-Bob partition*.\nAnother problem with Taddei *et al.*’s approach is that it comes at the expense of introducing an arbitrary divide between free resources and free processes. That is, as argued in the previous section, the set of free resources defined by Eq. does not constitute all and only those that can be constructively built out of S-1W-LOCC operations, as the standard approach to resource theories requires. Indeed, any TO-LHS assemblage for which the marginal correlations shared between the Alices is post-quantum will be a counterexample, since the classical communication from Bob to the Alices is not sufficient to set up such correlations.\nNote that Taddei *et al.*’s approach is only *one* way to avoid exposure phenomena. A second way to avoid it is to consider a different set of free operations. Indeed, in our approach, neither the free operations *nor* the free resources are taken to be the sets that Taddei *et al.* considered. Furthermore, our approach does not allow for any exposure phenomena, nor does it face the issues we just outlined which arise in Taddei *et al.*’s TO-LHS approach. Note first that in our approach, wirings are not free LOSR operations, since our approach does not only aim to characterize nonclassicality between the Alices-vs-Bob partition, but rather *any* nonclassicality of common cause, shared between *any* of the parties. As such, wirings among black-box parties can of course increase the nonclassicality of a given common-cause process. In our approach, then, the question of exposure is whether nonfree transformations that act globally on the Alices—but do not act on Bob—are capable of transforming free resources to resources which require nonclassical common causes *shared across the Alices-vs-Bob divide*. But such exposure phenomena are not possible in our approach. Indeed, the set of LOSR-free assemblages is strictly contained within the set of TO-LHS assemblages, and hence (by Taddei *et al.*’s result) they cannot yield exposure.\nFor completeness, we further note that in our resource theory, it is indeed the case that the free resources are all and only those that can be constructed out of free (i.e., LOSR) operations, so that the consistency condition mentioned above is satisfied.\nConclusions and outlook\n=======================\nIn this paper, we developed a resource theory of nonclassicality for assemblages in EPR scenarios, with its free operations defined as local operations and shared randomness. This choice of free processes is motivated by the causal structure of the EPR scenario, together with a viewpoint on EPR scenarios wherein Alice’s actions allow her to make *inferences about* the physical state of Bob’s system rather than to *remotely steer* (i.e., influence) it. This is the first resource theory for assemblages in multipartite EPR scenarios.\nWe proved that resource conversions in our resource theory can be tested using a single instance of a semidefinite program, in both bipartite and multipartite EPR scenarios. These semidefinite programs – which we give – are the first tools that allow one to explore systematically free conversions of assemblages. We also proved that the pre-order of (both bipartite and multipartite) assemblages contain an infinite number of incomparable assemblages. To prove this, we constructed new EPR monotones, which may be useful in their own right for other tasks.\nThe causal approach that we have endorsed here brings a new perspective to fundamental questions regarding EPR scenarios. Firstly, we have motivated a principled approach to defining the set of classically-explainable assemblages in multipartite EPR scenarios from a resource-theoretic perspective. In addition, our approach (in contrast to the approach of Ref. ) ensures that the free set of resources and the free set of operations are consistent, in the sense that the free resources are all and only those that can be constructed out of free operations. Finally, our approach ensures that there are no free type-changing operations that do not preserve the free set (i.e., that there is no ‘steering exposure’), and sheds light on previous approaches to ensuring this property. As far as we know, this is the first resource theory defined for EPR assemblages that satisfies these (in our opinion) important properties, highlighting the value of taking our principled LOSR approach.\nThe resource theory of nonclassicality of assemblages is an instance of a general construction for studying the nonclassicality of common-cause processes. This principled approach was previously used to study LOSR-entanglement , nonclassical correlations in Bell scenarios, and indeed arbitrary types of common-cause processes . Our work continues the development of this program in the specific context of EPR scenarios, and in doing so provides a perspective on EPR ‘steering’ which unifies it with other pertinent notions of nonclassicality. A natural continuation of this work would be to study other EPR-like scenarios, wherein the process of Bob about which Alice is learning is not merely a quantum output system, but rather a dynamical process. Indeed, in our forthcoming paper we develop a resource theory for Bob-with-input scenarios and channel EPR scenarios.\nAnother relevant research avenue would be to extend the enveloping theory to include post-quantum common-cause processes (e.g., described by processes in arbitrary generalized probabilistic theories ). Scenarios such as the multipartite EPR scenario and the Bob-with-Input scenario are of particular importance for these questions because – unlike the bipartite EPR scenario – they allow for post-quantum assemblages. In order to quantify the nonclassicality of such assemblages, one would again take the free set to be defined by LOSR operations, and apply all the tools and monotones developed herein. Alternatively, one could study not the nonclassicality, but rather the *post-quantumness* of these resources, by taking the free set to be defined by *local operations and shared entanglement* (LOSE operations), as argued in Ref. . In this latter resource theory, one would be studying not the possibilities allowed by quantum theory, but rather the limitations imposed by quantum theory.\nGiven that both entanglement and incompatibility are necessary resources for generating nonclassical assemblages, another interesting line of work would be to explore how our resource theory of EPR assemblages emerges from the resource theories of incompatibility and entanglement .\nFinally, an interesting question pertains to the notion of self-testing of quantum states, which is relevant for the certification of quantum devices in communication and information processing protocols. Ref. shows that self-testing of states can indeed be defined in a resource-theoretic framework. It would be interesting to see how self-testing of quantum assemblages (cf. Refs. ) may be understood from the tools we developed in our work, such as the resource-conversion SDP tests.\nAcknowledgments\n===============\nBZ, DS, and ABS acknowledge support by the Foundation for Polish Science (IRAP project, ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth Operational Programme). MJH and ABS acknowledge the FQXi large grant “The Emergence of Agents from Causal Order” (FQXi FFF Grant number FQXi-RFP-1803B). BZ acknowledges partial support by the National Science Centre, Poland 2021/41/N/ST2/02242. This research was supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. The diagrams within this manuscript were prepared using TikZ and Inkscape.\nThe bipartite tilted Bell inequalities\n======================================\nConsider a bipartite Bell scenario with two dichotomic measurements per party. The Bell inequality studied in Refs. for this scenario, defined in, reads:\n \n$$\\begin{aligned}\n\\label{eq:app-tiltedBell}\nI\\_\\alpha = \\alpha {\\langle A\\_0 \\rangle} + {\\langle A\\_0B\\_0 \\rangle} + {\\langle A\\_0B\\_1 \\rangle} + {\\langle A\\_1B\\_0 \\rangle} - {\\langle A\\_1B\\_1 \\rangle}\\,,\\end{aligned}$$ \nwhere *α* ∈ [0, 2] is a real parameter. For *α* = 0, *I*0 correspond to the CHSH inequality.\nThe set I = {*I**α* | *α* ∈ [0, 2]} defines a family of Bell inequalities for the bipartite Bell scenario, indexed by the parameter *α*. It was proved in Ref. that the maximum quantum violation of *I**α* is given by $I\\_\\alpha^{\\max} = \\sqrt{8+2\\alpha^2}$, and the classical bound of *I**α* is given by *I**α*C = 2 + *α*. The value *I**α*max is achieved by measuring the observables\n \n$$\\begin{aligned}\n\\label{eq:app-observables}\nA\\_0 &= \\sigma\\_z\\,,\\quad & B\\_0= \\cos(\\mu) \\, \\sigma\\_z + \\sin(\\mu) \\, \\sigma\\_x\\,,\\\\ \\nonumber\nA\\_1 &=\\sigma\\_x \\,,\\quad &B\\_1= \\cos(\\mu) \\, \\sigma\\_z - \\sin(\\mu) \\, \\sigma\\_x \\,,\\end{aligned}$$ \non the quantum state\n \n$$\\begin{aligned}\n\\label{eq:app-thetastate}\n\\ket{\\theta}=\\cos{\\theta} \\ket{00} + \\sin{\\theta} \\ket{11} \\,, \\end{aligned}$$ \nwith the relation between the parameters *α*, *θ*, and *μ* being the following:\n \n$$\\begin{aligned}\n\\alpha = \\frac{2}{\\sqrt{1+2\\tan^2(2\\theta) }}\\,,\\quad \\tan(\\mu) = \\sin(2\\theta)\\,.\\end{aligned}$$ \nIn Ref., it was proved that the Bell inequalities specified by Eq. provide a robust self-test for the reference states given in Eq. .\n[rem:STarg] Let $\\widetilde{A}\\_x$, $\\widetilde{B}\\_y$, and $\\ket{\\widetilde{\\psi}}$ be physical observables and a physical quantum state that achieve the value *I**α*max for the Bell inequality *I**α*. Then, these are equal to (up to local isometries) the state and observables given in Eqs. and.\nConstructing bipartite EPR inequalities from Bell inequalities\n==============================================================\nWe now show how to use the tilted Bell inequalities from the set I, defined in Appendix [app:bipartite-Bell], to construct a family of EPR inequalities . Let us begin by transforming a Bell functional *I**α* ∈ I into an EPR functional *S*. A generic EPR functional is defined as:\n \n$$\\begin{aligned}\nS[{\\boldsymbol{\\Sigma}}] = {\\mathrm{tr}\\left\\{ \\sum\\_{a\\in {\\mathbb{A}}, \\, x\\in {\\mathbb{X}}} F\\_{a,x} \\, \\sigma\\_{a|x} \\right\\}}\\,.\\end{aligned}$$ \nThe Bell expression *I**α* can be expressed in terms of the elements of ${\\boldsymbol{\\Sigma}}$ as follows:\n \n$$\\begin{aligned}\nI\\_\\alpha &= \\alpha {\\langle A\\_0 \\rangle} + {\\langle A\\_0B\\_0 \\rangle} + {\\langle A\\_0B\\_1 \\rangle} + {\\langle A\\_1B\\_0 \\rangle} - {\\langle A\\_1B\\_1 \\rangle} \\nonumber \\\\\n&= \\alpha {\\mathrm{tr}\\left\\{(M\\_{0|0} - M\\_{1|0})\\otimes {\\mathbb{I}}\\_B \\, \\rho \\right\\}} + {\\mathrm{tr}\\left\\{(M\\_{0|0} - M\\_{1|0})\\otimes B\\_0 \\, \\rho \\right\\}} + {\\mathrm{tr}\\left\\{(M\\_{0|0} - M\\_{1|0})\\otimes B\\_1 \\, \\rho \\right\\}} \\nonumber \\\\\n&\\quad + {\\mathrm{tr}\\left\\{(M\\_{0|1} - M\\_{1|1})\\otimes B\\_0 \\, \\rho \\right\\}} - {\\mathrm{tr}\\left\\{(M\\_{0|1} - M\\_{1|1})\\otimes B\\_1 \\, \\rho \\right\\}} \\nonumber \\\\\n&= {\\mathrm{tr}\\left\\{\\alpha \\, \\sigma\\_{0|0} - \\alpha \\, \\sigma\\_{1|0} + B\\_0 \\, \\sigma\\_{0|0} - B\\_0 \\, \\sigma\\_{1|0} + B\\_1 \\, \\sigma\\_{0|0} - B\\_1 \\, \\sigma\\_{1|0} + B\\_0 \\, \\sigma\\_{0|1} - B\\_0 \\, \\sigma\\_{1|1} \\right\\}} \\nonumber \\\\\n&\\quad +{\\mathrm{tr}\\left\\{- B\\_1 \\, \\sigma\\_{0|1} + B\\_1 \\, \\sigma\\_{1|1} \\right\\}} \\nonumber\\end{aligned}$$ \nFrom the above expansion of *I**α*, we then define an EPR functional $S[{\\boldsymbol{\\Sigma}}]$ by setting:\n \n$$\\begin{aligned}\n\\label{eq:theF}\nF\\_{0,0} = \\alpha {\\mathbb{I}}+ B\\_0 + B\\_1 = - F\\_{1,0} = \\,,\\, \nF\\_{0,1} = B\\_0 - B\\_1 = - F\\_{1,1} \\,.\\end{aligned}$$ \nIn the expression for $S[{\\boldsymbol{\\Sigma}}]$ given by the operators of Eq. there are plenty of things that still need to be specified: the value of the parameter *α*, as well as the operators *B*0 and *B*1. This is the freedom we will leverage to construct a family of EPR inequalities.\n[def:SFT] **EPR functional $S\\_\\eta[{\\boldsymbol{\\Sigma}}]$** \nWe define the EPR functional $S\\_\\eta[{\\boldsymbol{\\Sigma}}]$ via the operators of Eq. by taking:\n \n$$\\begin{aligned}\n\\label{eq:theStpar}\n\\alpha &= \\frac{2}{\\sqrt{1+2\\tan^2(2\\eta) }}\\,, \\\\ \\nonumber\nB\\_0 &= \\cos(\\mu) \\, \\sigma\\_z + \\sin(\\mu) \\, \\sigma\\_x \\,,\\\\ \\nonumber\nB\\_1 &= \\cos(\\mu) \\, \\sigma\\_z - \\sin(\\mu) \\, \\sigma\\_x \\,, \\\\ \\nonumber\n\\tan(\\mu) &= \\sin(2\\eta)\\,.\\end{aligned}$$ \nA few remarks regarding the properties of $S\\_\\eta[{\\boldsymbol{\\Sigma}}]$ are in order.\nThe maximum quantum value *S**η*max of $S\\_\\eta[{\\boldsymbol{\\Sigma}}]$ is given by\n \n$$\\begin{aligned}\n\\label{eq:Smax}\nS\\_\\eta^{\\max} = 2\\sqrt{2} \\, \\sqrt{1 + \\frac{1}{1+2\\tan^2(2\\eta)}}\\,.\\end{aligned}$$ \nFirst notice that ‘optimising $S\\_\\eta[{\\boldsymbol{\\Sigma}}]$ over quantum assemblages’ is equivalent to ‘optimising *I**α* over quantum correlations constrained on Bob’s observables being those from Eq. , and on *α* and *η* being related as in Eq. ’.\nSince the maximum quantum violation *I**α*max can indeed be achieved by the operators for Bob from Eq. , then *S**η*max = *I**α*max. Using the relation between *α* and *η* we obtain Eq. .\n[rem:iff] The quantum assemblage ${\\boldsymbol{\\Sigma}}^{\\theta}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$, defined in Eq. , satisfies $S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\theta}\\_{{\\mathbb{A}}|{\\mathbb{X}}}] = S\\_\\eta^{\\max}$ if and only if *η* = *θ*.\nThe ‘if’ direction is straightforward to prove, by noticing that the state and measurements used to prepare ${\\boldsymbol{\\Sigma}}^{\\theta}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$, together with Bob’s observables from Eq. , produce correlations that achieve *I**α*max in the corresponding Bell experiment, where *α* and *η* are related as in Eq. .\nThe ‘only if’ direction follows from a proof by contradiction. Assume that there exists ${\\boldsymbol{\\Sigma}}^{\\theta}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ with *θ* ≠ *η*, with $S\\_\\eta[{\\boldsymbol{\\Sigma}}^{\\theta}\\_{{\\mathbb{A}}|{\\mathbb{X}}}] = S\\_\\eta^{\\max}$. Then, the state and measurements used to prepare ${\\boldsymbol{\\Sigma}}^{\\theta}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$, together with Bob’s observables from Eq. , become a quantum realisation of correlations that achieve *I**α*max for the Bell functional *I**α*, where $\\alpha = \\frac{2}{\\sqrt{1+2\\tan^2(2\\eta) }}$.\nBy Remark [rem:STarg] then one concludes that there exists a local isometry in Alice’s lab that takes $\\ket{\\theta}$ to $\\ket{\\eta}$. This local isometry, however, can never exist, since $\\ket{\\theta}$ cannot be transformed into $\\ket{\\eta}$ by such local operations. Hence, it must be the case that *θ* = *η*.\n[rem:thelast] Let ${\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}$ be a quantum assemblage, and let *ρ* and {{*M**a*∣*x*}*a* ∈ A}*x* ∈ X be a quantum state and measurements that realise it. Then, $S\\_\\eta[{\\boldsymbol{\\Sigma}}\\_{{\\mathbb{A}}|{\\mathbb{X}}}] = S\\_\\eta^{\\max}$ if and only if: \n- The observables *A*0′ = *M*0∣0 − *M*1∣0 and *A*1′ = *M*0∣1 − *M*1∣1 are equivalent (up to local isometries) to *A*0 and *A*1, respectively, from Eq. , \n- The quantum state *ρ* is equivalent, up to local isometries, to $\\ket{\\eta}$.\nThe ‘if’ direction follows trivially from Remark [rem:iff]. \nTo prove the ‘only if’ direction, notice that *ρ*, {*A**j*′}*j* = 0, 1, and Bob’s observables {*B**j*}*j* = 0, 1 from Eq. , provide a quantum realisation of correlations that achieve the maximum value of the Bell functional *I**α*, with *α* and *η* related as in Eq. . From Remark [rem:STarg] it follows that *ρ* and {*A**j*′}*j* = 0, 1 must be equal to (up to local isometries) the state and observables of Eqs. and, which proves our claim.\nThe multipartite Bell inequalities\n==================================\nConsider *N*-partite Bell scenario with two dichotomic measurements per party. The Bell inequality studied in Ref. for this scenario, defined in Ref. , reads:\n \n$$\\begin{aligned}\n \\label{eq:app-multi-Bell}\n\\begin{split}\nI\\_\\alpha^{(N)} &= (N-1) {\\langle (B\\_0+B\\_1)A\\_0^{(1)}...A\\_0^{(N-1)} \\rangle} \n+ (N-1) \\frac{\\cos{2\\alpha}}{\\sqrt{1-\\cos^2{2\\alpha}}} ({\\langle B\\_0 \\rangle}-{\\langle B\\_1 \\rangle}) \\\\\n&+ \\frac{1}{\\sqrt{1-\\cos^2{2\\alpha}}} \\sum\\_{i=1}^{N-1} {\\langle (B\\_0-B\\_1)A\\_1^{(i)} \\rangle}\\,,\n\\end{split}\\end{aligned}$$ \nwhere *α* ∈ (0, *π*/4].\nThe set I(*N*) = {*I**α*(*N*) | *α* ∈ (0, *π*/4]} defines, for fixed *N*, a family of Bell inequalities for the multipartite scenario, indexed by the parameter *α*. It was shown in Ref. that the maximum quantum violation of *I**α*(*N*) is given by $I\\_\\alpha^{(N) \\max} = 2\\sqrt{2}(N-1)$, and the classical bound of *I**α*(*N*) is given by $I\\_\\alpha^{(N)\\,\\mathrm{C}} = (N-1) \\frac{1-\\cos{2\\alpha}}{\\sqrt{1-\\cos{2\\alpha}}}$. The value *I**α*(*N*)max is achieved by measuring the observables\n \n$$\\begin{aligned}\n\\label{eq:app-multi-observables}\nA\\_0^{(i)} &= \\sigma\\_x\\,,\\quad & B\\_0= \\cos(\\mu) \\, \\sigma\\_z + \\sin(\\mu) \\, \\sigma\\_x\\,,\\\\\nA\\_1^{(i)} &=\\sigma\\_z \\,,\\quad &B\\_1= - \\cos(\\mu) \\, \\sigma\\_z + \\sin(\\mu) \\, \\sigma\\_x \\,, \\label{eq:app-multi-observables2}\\end{aligned}$$ \nfor *i* ∈ {1...N-1}, on the quantum state\n \n$$\\begin{aligned}\n\\label{eq:app-multi-state}\n \\ket{GHZ\\_N^{\\theta}}&=\\cos{\\theta} \\ket{0}^{\\otimes N} + \\sin{\\theta} \\ket{1}^{\\otimes N} \\,, \\end{aligned}$$ \nwith the relation between the parameters *α*, *θ*, and *μ* being the following:\n \n$$\\begin{aligned}\n\\alpha = \\theta\\,,\\quad 2 \\sin^2(\\mu) = \\sin^2(2\\theta)\\,.\\end{aligned}$$ \nIt was also proved in Ref. that the inequality can be used to self-test the the state in Eq. for any *θ* ∈ (0, *π*/4].\n[rem:STarg-multi] Let $\\widetilde{A}\\_x^{(i)}$, $\\widetilde{B}\\_y$, and $\\ket{\\widetilde{\\psi}}$ be physical observables and a physical quantum state that achieve the value *I**α*(*N*)max for the Bell inequality *I**α*(*N*). Then, these are equal to (up to local isometries) the state and observables of Eqs. ,, and.\nConstructing multipartite EPR inequalities from Bell inequalities\n=================================================================\nWe now define a family of multipartite EPR functionals *S**α*(*N*) from the multipartite Bell inequalities *I**α*(*N*) ∈ I(*N*). A generic *N*-partite EPR functional is defined as:\n \n$$\\begin{aligned}\nS^{(N)}[{\\boldsymbol{\\Sigma}}\\_N] = {\\mathrm{tr}\\left\\{ \\sum\\_{\\substack{a\\_i\\in {\\mathbb{A}}, \\\\ x\\_i\\in {\\mathbb{X}}}} F\\_{a\\_1\\ldots a\\_{N-1},x\\_1\\ldots x\\_{N-1}} \\, \\sigma\\_{a\\_1\\ldots a\\_{N-1}|x\\_1\\ldots x\\_{N-1}} \\right\\}}\\,.\\end{aligned}$$ \nTo transform the Bell functional to the EPR functional, notice the following relations. First, for any choice of *x*1…*x**N* − 1, we can write:\n \n$$\\begin{aligned}\n {\\langle B \\rangle}={\\mathrm{tr}\\left\\{B \\, \\rho\\_B \\right\\}}=\\sum\\_{a\\_1\\ldots a\\_{N-1}} {\\mathrm{tr}\\left\\{B \\, \\sigma\\_{a\\_1\\ldots a\\_{N-1}|x\\_1\\ldots x\\_{N-1}} \\right\\}} \\,.\\end{aligned}$$ \nMoreover,\n \n$$\\begin{aligned}\n\\begin{split}\n {\\langle (B\\_0-B\\_1)A\\_1^{(i)} \\rangle}&={\\mathrm{tr}\\left\\{{\\mathbb{I}}^{(1)} \\otimes \\ldots \\otimes (\\widetilde{M}\\_{0|1}-\\widetilde{M}\\_{1|1})^{(i)} \\otimes \\ldots \\otimes {\\mathbb{I}}^{(N-1)} \\otimes (B\\_0-B\\_1) \\, \\rho \\right\\}} \\\\\n &= {\\mathrm{tr}\\left\\{(B\\_0-B\\_1) \\, (\\sigma\\_{0|1}^{(i)}-\\sigma\\_{1|1}^{(i)}) \\right\\}} \\,,\n\\end{split}\\end{aligned}$$ \nwhere *σ**a*∣*x*(*i*) = ∑*a**j*, *j* ≠ *i**σ**a*1…(*a**i* = *a*)…*a**N* − 1∣*x*1…(*x**i* = *x*)…*x**N* − 1.\nLastly,\n \n$$\\begin{aligned}\n\\begin{split}\n {\\langle (B\\_0+B\\_1)A\\_0^{(1)}\\ldots A\\_0^{(N-1)} \\rangle}&={\\mathrm{tr}\\left\\{(\\widetilde{M}\\_{0|0}-\\widetilde{M}\\_{1|0})^{(1)} \\otimes \\ldots \\otimes (\\widetilde{M}\\_{0|0}-\\widetilde{M}\\_{1|0})^{(N-1)} \\otimes (B\\_0+B\\_1) \\, \\rho \\right\\}} \\\\\n &= \\sum\\_{a\\_1,\\ldots,a\\_{N-1}} (-1)^{a\\_1+\\ldots+a\\_{N-1}} {\\mathrm{tr}\\left\\{\\widetilde{M}\\_{a\\_1|0}^{(1)} \\otimes \\ldots \\otimes \\widetilde{M}\\_{a\\_{N-1}|0}^{(N-1)} \\otimes (B\\_0+B\\_1) \\, \\rho \\right\\}} \\\\\n &= \\sum\\_{a\\_, \\ldots, a\\_{N-1}} (-1)^{a\\_1+\\ldots+a\\_{N-1}} {\\mathrm{tr}\\left\\{ (B\\_0+B\\_1) \\, \\sigma\\_{a\\_1\\ldots a\\_{N-1}|0\\ldots 0} \\right\\}}.\n\\end{split}\\end{aligned}$$ \nUsing the above relations, we define an EPR functional $S^{(N)}\\_{\\alpha}[{\\boldsymbol{\\Sigma}}\\_N]$ by setting:\n \n$$\\begin{aligned}\n\\label{eq:theF-multi}\nF\\_{a\\_1\\ldots a\\_{N-1},0\\ldots0} = (N-1) (-1)^{a\\_1+\\ldots+a\\_{N-1}} (B\\_0 + B\\_1) +& (N-1) \\frac{\\cos{2\\alpha}}{\\sqrt{1-\\cos^2{2\\alpha}}} (B\\_0 - B\\_1), \\\\ \nF\\_{a\\_1\\ldots (a\\_i=a) \\ldots x\\_{N-1},x\\_1\\ldots (x\\_i=1)\\ldots x\\_{N-1}} = (-1)^{a} \\frac{1}{\\sqrt{1-\\cos^2{2\\alpha}}} &(B\\_0 - B\\_1). \\nonumber\\end{aligned}$$ \nFrom the operators of Eq. , we construct a family of EPR functionals as follows.\n[def:NSFT-multi] **EPR functional $S\\_\\eta^{(N)}[{\\boldsymbol{\\Sigma}}\\_N]$.** \nWe define the EPR functional $S\\_\\eta^{(N)}[{\\boldsymbol{\\Sigma}}\\_N]$ via the operators of Eq. by taking:\n \n$$\\begin{aligned}\n\\label{eq:theStpar-multi}\n\\alpha &= \\eta \\,, \\\\ \\nonumber\nB\\_0 &= \\cos(\\mu) \\, \\sigma\\_z + \\sin(\\mu) \\, \\sigma\\_x \\,,\\\\ \\nonumber\nB\\_1 &=- \\cos(\\mu) \\, \\sigma\\_z + \\sin(\\mu) \\, \\sigma\\_x \\,, \\\\ \\nonumber\n2 \\sin^2(\\mu) &= \\sin^2(2\\eta)\\,.\\end{aligned}$$ \nThe properties of the multipartite EPR functional from Definition [def:NSFT-multi] can be studied in a similar manner to the properties of the bipartite EPR functional from Definition [def:SFT]. We now make a few remarks about these properties.\nThe maximum quantum value *S**η*(*N*)max of $S\\_\\eta^{(N)}[{\\boldsymbol{\\Sigma}}\\_N]$ is given by\n \n$$\\begin{aligned}\n\\label{eq:SmaxN}\nS\\_\\eta^{(N)\\max} = 2\\sqrt{2} \\, (N-1)\\,.\\end{aligned}$$ \nThe problem of optimising $S\\_\\eta^{(N)}[{\\boldsymbol{\\Sigma}}\\_N]$ over quantum assemblages is equivalent to the problem optimising *I**α*(*N*) over quantum correlations constrained on Bob’s"},"data_id":{"kind":"string","value":"arxiv_0002439"}}},{"rowIdx":592,"cells":{"content":{"kind":"string","value":"In this paper, we consider the evaluation of the effective action for photons coupled to charged scalar fields in the framework of a (2 + 1)-dimensional noncommutative spacetime. In order to determine the noncommutative Maxwell Lagrangian density, we follow a perturbative approach, by integrating out the charged scalar fields, to compute the respective graphs for the vev’s ⟨*A**A*⟩, ⟨*A**A**A*⟩ and ⟨*A**A**A**A*⟩. Surprisingly, it is shown that these contributions are planar and that, in the highly noncommutative limit, correspond to the Maxwell effective action and its higher-derivative corrections. It is explicitly verified that the one-loop effective action is gauge invariant, as well as under discrete symmetries: parity, time reversal, and charge conjugation. Moreover, a comparison of the main results with the noncommutative QED3 is established. In particular, the main difference is the absence of parity violating terms in the photon’s effective action coming from integrating out the charged scalar fields.\nIntroduction\n============\nIn recent years a great amount of attention has been paid in the analysis and calculation of covariant effective action for different types of quantum fields, exploring the diversity of new interactions that mainly depend on the spin of the fields involved as well as the spacetime dimensionality. One may say that the canonical example of a complete analysis is the Euler-Heisenberg effective action, where quantum effects from QED are responsible to induce nonlinear interactions among photons. Moreover, the effective action framework has served as an important tool to explore different point of views about the quantum gravitational theory, where the Einstein-Hilbert action is augmented by metric and/or torsion fields higher-order terms.\nNaturally, since the framework of effective action is a powerful tool, there is a great expectation that this approach can be used to make contact with modern phenomenology of physics beyond the standard model. The main idea behind this formulation is that at energies below some cutoff scale *μ*, [5](#fn5) all the effects of the massive degrees of freedom above *μ* can be encoded as new interactions among the fields remaining active below *μ*. The effective action approach has been extensively used to the study of Lorentz violating field theories, where the energy scale *μ* is related to the Planck energy scale $E\\_{\\rm Pl}$ (or length $\\ell\\_{\\rm Pl}$) where our notion of smooth geometry is expected to break. In this case, the current understanding is that the low energy Lorentz violating terms come as quantum corrections from heavy modes.\nAlthough the majority of studies of Lorentz violating field theories is developed in a four-dimensional spacetime, there are considerable interests in the description of three-dimensional (3D) ones. Besides the algebraic richness of odd dimensional spacetimes, one may say that the most appealing aspect of 3D field theories is the UV finiteness in some models. This feature might provide an ambiguity free description of Lorentz violation, allowing thus a close contact of violating effects with physical planar phenomena. In particular, it is worth recall the example of the description of quantum hall fluids in terms of noncommutative geometry.\nOver the past two decades, field theories defined in a noncommutative (NC) geometry have been considered as one of the most prominent candidates presenting Lorentz violation to make contact with quantum gravity phenomenology. Within this description, the noncommutativity measurement parameter is related to a length scale $\\ell\\_{\\rm nc} \\sim \\sqrt{\\theta}$. In one hand, this length scale can be seen as a manifestation of the discreteness of the spacetime, presenting a smooth profile in the UV region. On the other hand, this same scale is responsible for introducing instabilities in the dispersion relations of the fields, the so-called UV/IR mixing.\nNC field theories have been studied through the effective action approach, where the behavior of the new couplings was deeply analyzed, where the presence of UV/IR mixing in the 1PI functions signals that applying the usual Wilsonian field theory notions and techniques to NC QFT’s one should be careful. This type of analysis was also developed to two and three-dimensional NC models. These studies of effective action in 3D models were exclusive to the coupling of gauge and fermion fields, no much attention has been paid to the case involving scalar fields, in particular the case of spinless charged fields interacting with photons.\nIn one hand, it is of physical significance to study scalars in 3D field theories independently of fermions in condensed matter systems, as in Quantum Hall systems, since we have scalar quasiparticle excitations. On the other hand, recently 3D versions of fermionization/bosonization have also been introduced. In these studies, it was discussed the duality between nonspin Chern-Simons theory and a spin Chern-Simons theory, exploring precisely the spin structure of the given models. In this sense, the present work could be the first step in extending such analysis to the NC case. Motivated by these facts, we will analyze throughout the paper to what extent the spin of the matter fields can change the effective action when charged scalar and fermion fields are considered in the presence of the spacetime noncommutativity. A straightforward result is that in the case of the 3D scalar quantum electrodynamics (scalar QED3), it is not possible to generate the parity odd Chern-Simons terms, showing thus that the dynamics of the 3D gauge field is significantly different in the presence of either charged scalar or fermion fields. It is well known that the presence of the degree of freedom associated with the spin changes in most of the cases only the magnitude of physical quantities, e.g. the beta function and electron’s magnetic moment.\nIn this paper we discuss the effective action for the photon in the scalar QED in the noncommutative three-dimensional spacetime. In Sec. [sec2] we present an overview of the scalar QED, where the charged scalar fields are minimally coupled with the photons. There we define the main aspects regarding the Moyal product used in our analysis,[6](#fn6) we also discuss the content of discrete symmetries in the NC 3D spacetime. In addition, all the Feynman rules are presented for the propagators and 1PI vertices. Section [sec4] is focused in the perturbative computation of the relevant graphs corresponding to the one-loop effective action for the photon gauge field. It is also discussed the generation of higher-derivative terms, similarly to the Alekseev-Arbuzov-Baikov effective Lagrangian for non-Abelian fields. In Sec. [sec5] we establish a comparison of the obtained results for the effective action in the NC-scalar QED to those of ordinary NC-QED, exploring the part played by the spin in these cases. We present our final remarks in Sec. [conc].\nThe model\n=========\nIn this section, we introduce the model and fix our notation. The noncommutative extension of the bosonic electrodynamics is described by the following action\n \n$$S=\\int d^{3}x\\Big[\\left(D\\_{\\mu}\\phi\\right)^{\\dagger}\\star D^{\\mu}\\phi-m^{2}\\phi^{\\dagger}\\star \\phi\\Big],\n\\label{eq:a1}$$ \nthis functional action consists of the interaction of charged scalar fields minimally coupled with an external gauge field. We consider the covariant derivative form in the fundamental representation *D**μ**ϕ* = ∂*μ**ϕ* + *i**e**A**μ* ⋆ *ϕ*. This action is invariant under the infinitesimal gauge transformation\n \n*δ**A**μ* = ∂*μ**λ* + *i**e*[*A**μ*, *λ*]⋆ , *δ**ϕ* = *i**e**λ* ⋆ *ϕ*, \nwhere [ , ]⋆ is the Moyal bracket. Moreover, the Moyal star product between the functions *f* and *g* is defined as\n \n$$f\\left( x\\right)\\star g\\left(x\\right) = f\\left( x\\right) \\exp \\left(\\frac{i}{2}\\theta ^{\\mu \\nu}\n\\overleftarrow{\\partial\\_\\mu}\n \\overrightarrow{\\partial\\_\\nu}\\right) g\\left( x\\right),\n \\label{eq:a3}$$ \nwhere *θ**μ**ν* = − *θ**ν**μ* are constant parameters that measure the noncommutative structure of the space-time. In order to avoid unitarity violation, we assume that *θ*0*i* = 0, hence we have only one nonzero independent component *θ*12 in our model.\nIt is worth mentioning that although the couplings are simply modified by the presence of a nonplanar phase due to the Moyal product, the noncommutativity of spacetime coordinates shows its importance in the computation of the one-loop effective action for the gauge field, where nonlinear self-couplings are present solely due to the NC framework. The one-loop effective action for the gauge field can be readily obtained by integrating out the charged scalar fields of\n \n$$e^{i\\Gamma\\_{\\rm eff}[A]}=\\int D\\phi^{\\dagger} D\\phi~e^{-i\\int d^{3}x~\\phi^{\\dagger}\\star(D^{2}+m^{2})\\star\\phi}.$$ \nUsing the Gaussian functional integration formulas for the case of interacting charged scalar fields, we can write the noncommutative 1PI effective action as below\n \n$$i\\Gamma\\_{\\rm eff}[A]={\\rm Tr}\\ln\\left[\\frac{(i\\partial\\_{\\mu}-eA\\_{\\mu})\\star(i\\partial^{\\mu}-eA^{\\mu})\\star-m^{2}}{-\\partial^{2}-m^{2}}\\right],$$ \nwhere ${\\rm Tr}$ is a sum over eigenvalues of the operator inside the bracket which can also be evaluated in momentum space. Similarly to the description of one-loop effective action for the gauge field in the case of NC-QED, one can show that $\\Gamma\\_{\\rm eff}[A]$ has a convergent series expansion in coupling constant *e*. From a diagrammatic point of view, it includes the one-loop graphs contributing to the gauge field *n*-point functions which is considered as\n \n$$\\Gamma\\_{\\rm eff}[A]={\\cal{S}}\\_{\\rm eff}[AA]+{\\cal{S}}\\_{\\rm eff}[AAA]+{\\cal{S}}\\_{\\rm eff}[AAAA]+\\cdots.\n\\label{eq:aa}$$ \nHowever, the functional $\\Gamma\\_{\\rm eff}[A]$, in comparison to the NC-QED case, has more graphs due to the presence of an additional interacting vertex.\nMoreover, it is important to emphasize that as we will show in our model, similarly to the case of NC-QED, the one-loop effective action for the photons is completely planar. Explicitly, in the evaluation of the one-loop diagrams with an arbitrary number of external legs of photons, for energies below the mass scale *m*, only planar diagrams contribute. This means the absence of IR/UV mixing.\nDiscrete symmetries\n-------------------\nSince we are interested in computing the one-loop effective action for the photon, it is useful to analyze the behavior of the original action under discrete symmetries: parity, charge conjugation and time reversal. This study will allow us to determine which of them may be anomalous in the obtained results for the one-loop order effective action.\n* *Parity*\nParity transformation in *d* = 2 + 1 is defined as *x*1 → − *x*1 and *x*2 → *x*2, in this case we have that the field *ϕ* is even under parity, and the components of the gauge field *A**μ* behave as *A*0 → *A*0, *A*1 → − *A*1 and *A*2 → *A*2. Moreover, we observe from the NC algebra that the *θ* parameter changes under this transformation as *θ*12 → − *θ*12. With these considerations, it is easy to show that the whole of the action is parity invariant.\n* *Time Reversal*\nUnder time reversal, we have that *x*0 → − *x*0. In this case, the components of the gauge field behave as (*A*0, *A**i*) → (*A*0, − *A**i*). By demanding that the scalar field does not change *ϕ* → *ϕ*, and that necessarily the NC parameter transforms as *θ*12 → − *θ*12 under time reversal, we are left with a *T*-invariant action.\n* *Charge Conjugation*\nAs we know, the behavior of the gauge field under charge conjugation is given by *A**μ* → − *A**μ* for any space-time dimensionality. Taking the scalar field to be unchanged under *C*, and the transformation for the NC parameter *θ* → − *θ*, we conclude that the action is *C*-invariant.\nPropagators and vertex functions\n--------------------------------\nIn order to discuss the computation of the perturbative effective action, we must determine the basic propagators and 1PI vertex functions. From the functional action described in, we can obtain the bosonic propagator\n \n$${\\cal{D}}(p) = \\frac{i}{{p^{2}-m^{2}}},\n \\label{eq:b1}$$ \nthe cubic vertex ⟨*A* *ϕ* *ϕ*†⟩\n \n$$\\begin{aligned}\n\\Gamma^\\mu(p,q)= -ie\\left(p+q\\right)^{\\mu}\\exp\\Big(\\frac{i}{2} p \\wedge q\\Big),\n \\label{eq:b2}\\end{aligned}$$ \nand the quartic vertex ⟨*A**A* *ϕ* *ϕ*†⟩\n \n$$\\begin{aligned}\n\\Lambda^{\\mu\\nu}(p,q,s)=2ie^{2}\\eta^{\\mu\\nu}\\exp{\\Big(\\frac{i}{2}k \\wedge s\\Big)}\\cos\\Big(\\frac{p \\wedge q}{2}\\Big),\n \\label{eq:b3}\\end{aligned}$$ \nwhere we have introduced the notation *p* ∧ *q* = *p**μ**θ**μ**ν**q**ν*. A straightforward difference of the scalar and fermionic electrodynamics is the presence of the quartic vertex ⟨*A**A**ϕ**ϕ*†⟩, which increases significantly the number of the one-loop graphs. Moreover, the scalar vertices are rather simpler due to the absence of the Dirac *γ* matrices, resulting in a much simpler algebraic analysis.\nPerturbative Effective Action\n=============================\nNow that we have determined the basic Feynman rules for the 1PI functions, we shall proceed to the computation of the one-loop diagrams related to the effective action for the gauge field. For this purpose, we shall compute along this section the respective contributions: the free part of the effective action ⟨*A**A*⟩, and the interacting parts for the cubic vertex ⟨*A**A**A*⟩ and quartic vertex ⟨*A**A**A**A*⟩. In general, the final results of our analysis related to the graphs contributing to ⟨*A**A*⟩, ⟨*A**A**A*⟩ and ⟨*A**A**A**A*⟩ vertices shall be a function of *e*, *p̃**μ* = *θ**μ**ν**p**ν* and *p*2/*m*2.\nTo highlight the effects of noncommutativity in the low energy effective action and the photon two, three and four-point functions, while we take the external momenta *p* such that *p*2/*m*2 ≪ 1, we also take the highly noncommutative limit, i.e., the low-energy regime *p*2/*m*2 → 0 while *p̃* is kept finite. In this limit the noncommutative (planar) phase factors, which are a function of *p̃*, remain finite. Moreover, we shall focus our attention on those terms of order *m*− 1. We present by complementarity, at the next to leading order, the terms of order *m*− 3 that correspond to higher-derivative corrections.\nOne-loop ⟨*A**A*⟩ part\n----------------------\nFrom the Feynman rules we can compute the one-loop contribution to the *A**A*-term corresponding to the free part of the photon effective action. The two diagrams contributing at this order are depicted in Fig. [oneloop1], which the respective expressions have the form\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu}\\_{(a)}(p) &= e^{2} \\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(p+2k)^{\\nu}}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]}, \\nonumber \\\\\n\\Pi^{\\mu\\nu}\\_{(b)}(p) &= -e^{2} \\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{2\\eta^{\\mu\\nu}[(p+k)^{2}-m^{2}]}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]},\n \\label{eq:c1}\\end{aligned}$$ \nso that the full contribution is written as\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu}(p) &= e^{2}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(p+2k)^{\\nu}-2\\eta^{\\mu\\nu}[(p+k)^{2}-m^{2}]}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]}.\n \\label{eq:c2}\\end{aligned}$$ \nA first comment is that this piece is completely planar, carrying no noncommutative effects. The explicit computation is straightforward using dimensional regularization. After some algebraic calculation, we can consider the low-energy limit, *p*2/*m*2 → 0, resulting into\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu}(p) =\\frac{ie^{2}}{48\\pi m}\\left(p^{\\mu}p^{\\nu}-\\eta^{\\mu\\nu}p^{2}\\right).\n \\label{eq:c3}\\end{aligned}$$ \nMoreover, for the next to leading order contribution, ${\\cal{O}}(m^{-3})$, we find that\n \n$$\\Pi^{\\mu\\nu}\\_{\\rm hd}(p)=\\frac{ie^{2}}{960\\pi m^{3}}\\left(p^{\\mu}p^{\\nu}-\\eta^{\\mu\\nu}p^{2}\\right)p^{2}.\n \\label{eq:c4}$$ \nThese two terms Eqs. and satisfy straightforwardly the Ward identity, *p**μ*Π*μ**ν* = 0, as we expected. We can determine the respective contribution to the effective action by means of\n \n$$i{\\cal{S}}\\_{\\rm eff}[AA]=\\int\\int d^{3}x\\_{1}d^{3}x\\_{2}~A\\_{\\mu}(x\\_{1})\\Gamma^{\\mu\\nu}(x\\_{1},x\\_{2})A\\_{\\nu}(x\\_{2}),\n \\label{eq:c5}$$ \nwhere ${\\cal{S}}\\_{\\rm eff}[AA]$ is the quadratic part of the effective action Γ[*A*] in. Here, we have defined by simplicity\n \n$$\\Gamma^{\\mu\\nu}(x\\_{1},x\\_{2})=\\int\\frac{d^{3}p}{(2\\pi)^{3}} e^{-i p\\cdot (x\\_{1}-x\\_{2})}\\Pi^{\\mu\\nu}(p ).\n \\label{eq:c6}$$ \n [oneloop1]\nAfter some algebra, the quadratic part of the induced effective action for the photon, considering and, is given by\n \n$$\\begin{aligned}\ni{\\cal{S}}\\_{\\rm eff}[AA]&=-\\frac{ie^{2}}{48\\pi m}\\int d^{3}x\\Big(\\partial\\_{\\mu}A\\_{\\nu}\\partial^{\\mu}A^{\\nu}-\\partial^{\\mu}A\\_{\\nu}\\partial^{\\nu}A\\_{\\mu}\\Big) \\nonumber \\\\\n&+\\frac{ie^{2}}{960\\pi m^{3}}\\int d^{3}x\\Big(\\partial\\_{\\mu}A\\_{\\nu}\\Box\\partial^{\\mu}A^{\\nu}-\\partial^{\\mu}A\\_{\\nu}\\Box\\partial^{\\nu}A\\_{\\mu}\\Big).\n \\label{eq:c7}\\end{aligned}$$ \nAs we have previously mentioned, the first term of the expression corresponds to the kinetic part of the noncommutative Maxwell action, ${\\cal O}(m^{-1})$, while the second term is the higher-derivative correction to the kinetic term, of order ${\\cal O}(m^{-3})$. Moreover, the obtained result does not contain any noncommutativity effect, since the produced phase factors cancel for *n* = 2. It is worth noticing the absence of the parity odd Chern-Simons term in the scalar QED3, which in turn is generated in the fermionic electrodynamics due to the algebraic structure of the two-dimensional realization of *γ* matrices.\nOne-loop ⟨*A**A**A*⟩ vertex\n---------------------------\nThe relevant graphs for the ⟨*A**A**A*⟩ part of the effective action are shown in Fig. [oneloop2]. However, in order to determine correctly the full contribution to the effective action, it is necessary to consider all different permutations of the external bosonic lines of the given graphs. It is easy to see that the diagram (a) has an additional contribution (b), corresponding to a permutation of the external photon legs, which has an equivalent structure but with a reversed momentum flow, which comes exactly from the S-matrix expansion at the order of *e*3. With help of the Feynman rules, we can easily write the relevant expression for the sum of the graphs (a) and (b)\n \n$$\\Pi^{\\mu\\nu\\rho}\\_{(a+b)}(p,q)=2i e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(2p+2k+q)^{\\nu}(p+q+2k)^{\\rho}}{[(p+k)^{2}-m^{2}][(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]} \\sin\\big(\\frac{p \\wedge q}{2}\\big),\n \\label{eq:c8}$$ \nwhich is a planar quantity, we can see that its integrand is independent of the noncommutativity. The contribution from the graph (c) also has a simple planar structure, which is given by the expression\n \n$$\\Pi^{\\mu\\nu\\rho}\\_{(c)}(p,q)= -e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}(p+q+2k)^{\\rho}}{[(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]} \\cos\\big(\\frac{p \\wedge q}{2}\\big).\n \\label{eq:c9}$$ \nSince the graph (c) is planar, one can perform straightforward manipulations to show that this contribution is identically zero, i.e. Π(*c*)*μ**ν**ρ* = 0, for any value of the external momenta. Hence the full contribution for the ⟨*A**A**A*⟩ vertex reads\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho}(p,q)=2i e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(2k+p)^{\\mu}(2k+p+s)^{\\nu}(s+2k)^{\\rho}}{[(p+k)^{2}-m^{2}][(s+k)^{2}-m^{2}][k^{2}-m^{2}]} \\sin\\big(\\frac{p \\wedge q}{2}\\big).\n \\label{eq:c10}\\end{aligned}$$ \n [oneloop2]\nThe computation of the loop integral is lengthy but straightforward using dimensional regularization, and in the low-energy limit *p*2, *q*2 ≪ *m*2, we find that\n \n$$\\Pi^{\\mu\\nu\\rho}(p,q)=\\frac{e^{3}}{12\\pi m}\\bigg[\\big(p-q\\big)^{\\rho}\\eta^{\\mu\\nu}-\\big(2p+ q\\big)^{\\nu}\\eta^{\\mu\\rho} +\\big(p+2 q\\big)^{\\mu}\\eta^{\\nu\\rho}\\bigg] \\sin\\big(\\frac{p \\wedge q}{2}\\big).\n \\label{eq:c11}$$ \nHere, we notice that the Eq. corresponds exactly to the standard Feynman vertex of the 3-photon interaction term in the NC spacetime. Moreover, in the next to leading order, ${\\cal{O}}(m^{-3})$, we have the contribution from the higher-derivative terms\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho}\\_{\\rm hd}(p,q)=-\\frac{e^3}{240\\pi m^3}\n\\bigg\\{&\\eta^{\\mu\\nu}\\Big[ p^2(2q-p)^{\\rho}+q^2(q-2p)^{\\rho}+(p.q)(q-p)^{\\rho}\\Big]\\nonumber\\\\\n+&\\eta^{\\mu\\rho}\\Big[p^2(4p+2q)^{\\nu}+q^2(3p+q)^{\\nu}+(p.q)(4p+q)^{\\nu}\\Big]\\nonumber\\\\\n-& \\eta^{\\nu\\rho}\\Big[p^2(p+3q)^{\\mu}+q^2(2p+4q)^{\\mu}+(p.q)(p+4q)^{\\mu}\\Big]\\nonumber\\\\\n+&p^{\\mu}q^{\\rho}(q-p)^{\\nu}+p^{\\rho}q^{\\mu}(q-p)^{\\nu}-p^{\\mu}\np^{\\rho}(2p+q)^{\\nu}+q^{\\mu}q^{\\rho}(p+2q)^{\\nu}\\bigg\\}\\sin\\big(\\frac{p\\wedge q}{2}\\big).\n \\label{eq:c12}\\end{aligned}$$ \nThis expression corresponds to the higher-derivative correction to the 3-photon vertex.\nIt is important to observe that in the commutative limit, the graphs (a) and (b) cancel each other, so that the induced 3-photon vertex is completely removed in the scalar QED. We can understand this result from the charge conjugation invariance of the scalar QED in any space-time dimension, known as Furry’s theorem, that forbids the presence of an odd number of photon lines in the case of commutative theory. Another important aspect from our analysis is the absence of the Chern-Simons self-coupling *ε**μ**ν**λ**A**μ* ⋆ *A**ν* ⋆ *A**λ* for the noncommutative scalar QED3 effective action, which is only generated in the case of fermionic electrodynamics.\nOne-loop ⟨*A**A**A**A*⟩ vertex\n------------------------------\nThe full contribution to the ⟨*A**A**A**A*⟩ part is determined by considering three different types of diagrams that are depicted in Fig. [oneloop3]. Since all of these graphs have 4 external bosonic legs, 24 different permutations for each graph must be considered in order to obtain the fully symmetrized contribution. Hence, the full contribution can be formally written as\n \n$$\\begin{aligned}\n \\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}= \\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a)} +\n \\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(b)} +\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(c)} =\\sum\\_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a,i)}+\n \\sum\\_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(b,i)}+\n \\sum\\_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(c,i)}.\n \\label{eq:c17}\\end{aligned}$$ \nWe shall present next the explicit discussion for the first contribution of each graph, whereas the remaining graphs are obtained by a direct permutation of momenta and spacetime indices.\n [oneloop3]\nThe box diagram contribution represented in graph (a) has the following expression\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho\\sigma}\\_{(a,1)}&=e^{4}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(2p+q+2k)^{\\nu}(2p+2q+s+2k)^{\\rho}(p+q+s+2k)^{\\sigma}}{[(p+q+s+k)^{2} -m^{2}][(p+q+k)^{2}-m^{2}][(p+k)^{2}-m^{2}][k^{2}-m^{2}]}\ne^{\\frac{i}{2}p \\wedge q}e^{\\frac{i}{2}(p+q)\\wedge s},\n \\label{eq:c13}\\end{aligned}$$ \nwhere we have labeled the momenta (*p*, *q*, *s*, *r*) accordingly to the spacetime indices of the external legs (*μ*, *ν*, *ρ*, *σ*). Moreover we have adopted the notation, in order to satisfy the energy-momentum conservation, where the momenta flow satisfies the relation *r* = *p* + *q* + *s*. The remaining contributions from the other 23 box diagrams, coming from the S-matrix expansion, can easily be obtained from the equation by considering the respective permutation. Next, we have the contribution from the bubble diagram represented in (b), which is given by\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho\\sigma}\\_{(b,1)} =e^{4} \\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}}{[(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]}\\cos\\big(\\frac{p \\wedge q}{2}\\big)\\cos\\big(\\frac{s\\wedge (p+q)}{2}\\big).\n \\label{eq:c14}\\end{aligned}$$ \nAt last, the triangle contribution shown in graph (c) is written as\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho\\sigma}\\_{(c,1)} =-e^{4}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}(2p+2q+s+2k)^{\\rho}(p+q+s+2k)^{\\sigma}}{[(p+q+s+k)^{2}-m^{2}][(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]}\\cos\\big( \\frac{p \\wedge q}{2}\\big) e^{\\frac{i}{2} (p+q) \\wedge s}.\n \\label{eq:c15}\\end{aligned}$$ \nThe Feynman expressions of the graphs (a), (b) and (c) show that all of them are planar, making the evaluation of the momentum integration easier by dimensional regularization. Hence the resulting expressions from the contributions to, evaluated in the highly noncommutative limit, where *p*2, *q*2, *s*2 ≪ *m*2, are written as follows\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a,1)}&=\\frac{1}{4}\\times \\frac{ie^{4}}{12\\pi m}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n~e^{\\frac{i}{2}p\\wedge q}e^{\\frac{i}{2}r\\wedge s},\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(b,1)}&= \\frac{1}{2}\\times\\frac{ie^{4}}{8\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\n\\cos\\big(\\frac{p\\wedge q}{2}\\big)\\cos\\big(\\frac{r\\wedge s}{2}\\big),\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(c,1)}&=1\\times \\frac{-ie^{4}}{8\\pi m} ~\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(\\frac{p\\wedge q}{2}\\big)\\cos\\big(\\frac{r\\wedge s}{2}\\big),\n \\label{eq:c16}\\end{aligned}$$ \nwhere the coefficients $\\frac{1}{4}$, $\\frac{1}{2}$ and 1 are the symmetry factors for the graphs (a), (b) and (c) of Fig. [oneloop3], respectively. We then apply to the results all the 24 permutations, necessary to evaluate, yielding\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a)}&=\\frac{ie^{4}}{6\\pi m}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n\\Big[\\cos\\big(12\\big)\\cos\\big(34\\big)+\\cos\\big(13\\big)\\cos\\big(24\\big)+\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(b)}&=\\frac{ie^{4}}{2\\pi m}\n\\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(12\\big)\\cos\\big(34\\big)+\\eta^{\\mu\\rho}\n\\eta^{\\nu\\sigma}\\cos\\big(13\\big)\\cos\\big(24\\big)+\n\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(c)}&=- \\frac{ie^{4}}{\\pi m}\n\\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(12\\big)\\cos\\big(34\\big)+\\eta^{\\mu\\rho}\n\\eta^{\\nu\\sigma}\\cos\\big(13\\big)\\cos\\big(24\\big)+\n\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n \\label{eq:c18}\\end{aligned}$$ \nwhere we have introduced, by simplicity of the upcoming analysis, the following notation for the NC momenta product: $\\left(12 \\right) \\equiv \\left(\\frac{p\\wedge q}{2}\\right)$, $\\left(13 \\right) \\equiv \\left(\\frac{p\\wedge s}{2}\\right)$, $\\left(14 \\right) \\equiv \\left(\\frac{p\\wedge r}{2}\\right)$, $\\left(23 \\right) \\equiv \\left(\\frac{q\\wedge s}{2}\\right)$, $\\left(24 \\right) \\equiv \\left(\\frac{q\\wedge r}{2}\\right)$, and $ \\left(34\\right) \\equiv \\left(\\frac{s\\wedge r}{2}\\right)$. Finally, we substitute the results into the Eq. to obtain the total one-loop contribution expression corresponding to the photon 4-point function\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}=\\frac{ie^{4}}{\\pi m}\\Bigg\\{\n&\\frac{1}{6}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big) \\Big[\\cos(12)\\cos(34)+\\cos(14)\\cos(23)+\\cos(13)\\cos(24)\\Big]\\nonumber\\\\\n-&\\frac{1}{2} \\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos(12)\n\\cos(34)+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\cos(13)\\cos(24)+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos(14)\\cos(23)\\Big]\\Bigg\\}.\n \\label{eq:c19}\\end{aligned}$$ \nWe can verify whether the quartic vertex satisfy the Ward identity in scalar QED. First, we consider the commutative limit, i.e. *θ* → 0, so that the one-loop contribution is reduced to\n \n$$\\lim\\_{\\theta\\rightarrow 0}\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}=\\frac{ie^{4}}{\\pi m}\n\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big) \\Big(\\frac{1}{2}+\\frac{1}{2}-1\\Big)=0,\n \\label{eq:c20}$$ \nshowing that the photon quartic self-coupling in the order ${\\cal{O}}(m^{-1})$ is absent in the Abelian theory; however higher order contributions could be nonvanishing, corresponding to nonlinear Euler-Heisenberg-like terms. Moreover, in the case of the violation of the Ward identity, a nonvanishing result for the contribution would generate a four photon interaction term in the effective action of the type\n \n$$\\begin{aligned}\n\\lim\\_{\\theta\\rightarrow 0}{\\cal{S}}\\_{\\rm eff}[AAAA]\\sim\\int d^{3}x~\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)A\\_{\\mu}(x) A\\_{\\nu}(x) A\\_{\\rho}(x)A\\_{\\sigma}(x),\n \\label{eq:c21}\\end{aligned}$$ \nthat is not explicitly gauge invariant. Hence, with the result we conclude that the gauge invariance is satisfied in our analysis of ${\\cal{O}}(m^{-1})$ terms at the one-loop approximation.\nOn the other hand, in the noncommutative case, as it is well known, we expect to find the relevant Feynman rule corresponding to the 4-photon interaction term at this order. To accomplish that, we start working separately each one of the tensor terms present in the function $\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}$ Eq. . We shall illustrate the analysis for the terms proportional to *η**μ**ν**η**ρ**σ*, the remaining terms can be evaluated in the same fashion. Hence, by picking the pieces that are proportional to *η**μ**ν**η**ρ**σ* in, we have that\n \n$$\\mathcal{I}^{\\mu\\nu,\\rho\\sigma}\\equiv\n\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big[-2\\cos(12)\\cos(34)+ \\cos(14)\\cos(23)+\\cos(13)\\cos(24)\\Big]. \\label{eq:c22}$$ \nThe major work here consist in simplifying the trigonometric part of this function by making use of the energy-momentum conservation *r* = *p* + *q* + *s*, which in the new notation is renamed as 4 = 1 + 2 + 3, together with the manipulation of some trigonometric identities, e.g. cos*α*cos*β* = cos(*α* + *β*) + sin*α*sin*β*. After some laborious but straightforward calculation, we arrive at the desired expression\n \n$$\\mathcal{I}^{\\mu\\nu,\\rho\\sigma}=\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big[\\sin(14)\\sin(23)+\\sin(13)\\sin(24)\\Big].\n \\label{eq:c24}$$ \nSimilarly, we can apply the same process in order to simplify the remaining terms, proportional to *η**μ**ρ**η**ν**σ* and *η**μ**σ**η**ν**ρ*, so that it yields to\n \n$$\\begin{aligned}\n\\mathcal{I}^{\\mu\\rho,\\nu\\sigma}&=\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big[\\sin(12)\\sin(34)-\\sin(14)\\sin(23)\\Big],\n\\nonumber\\\\\n\\mathcal{I}^{\\mu\\sigma,\\nu\\rho}&=- \\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big[\\sin(12)\\sin(34)+\\sin(13)\\sin(24)\\Big].\n \\label{eq:c25}\\end{aligned}$$ \nHence, by considering the results from our manipulations, Eqs. and, we can rewrite in a convenient form as the following\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}=\\frac{ie^{4}}{6\\pi m}\\bigg[&\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin\\big(\\frac{p\\wedge s}{2}\\big)\\sin\\big(\\frac{q\\wedge r}{2}\\big)\\nonumber\\\\\n+&\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big)\\sin\\big(\\frac{p\\wedge r}{2}\\big)\\sin\\big(\\frac{q\\wedge s}{2}\\big)\\nonumber\\\\\n+&\\Big(\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin\\big(\\frac{p\\wedge q}{2}\\big)\n\\sin\\big(\\frac{s\\wedge r}{2}\\big)\\bigg],\n \\label{eq:c27}\\end{aligned}$$ \nwhere we have reintroduced the notation in terms of the external momenta *p*, *q*, *s*, *r*. As we can observe, the expression inside the bracket corresponds exactly to the Feynamn vertex of the 4-photon interaction within the noncommutative *U*⋆(1) gauge theory.\nFinally, we can gather the leading ${\\cal{O}}(m^{-1})$ contributions from the one-loop order parts related to the two, three and four-point functions, Eqs. , and, respectively, so that we can write the complete expression of the NC Maxwell action as\n \n$$i{\\cal{S}}\\_{\\rm eff}\\bigg|\\_{{\\cal{O}}(m^{-1})}=-\\frac{ie^{2}}{96\\pi m}\\int~d^{3}x~F\\_{\\mu\\nu}\\star F^{\\mu\\nu},\n\\label{eq:c28}$$ \nin which the field strength tensor in the NC framework is defined as *F**μ**ν* = ∂*μ**A**ν* − ∂*ν**A**μ* + *i**e*[*A**μ*, *A**ν*]⋆. As we have previously discussed, this action is manifestly *U*⋆(1) gauge invariant under the transformation *U* = *e*⋆*i**e**λ*, where the field strength has the following transformation law *F**μ**ν* → *U* ⋆ *F**μ**ν* ⋆ *U*− 1.\nRegarding the higher-derivative corrections to the 4-photon vertex, corresponding to the next to leading order ${\\cal{O}}(m^{-3})$ terms, we arrive at a result involving a long expression which can be found in the Appendix [sec-appA]. This ${\\cal{O}}(m^{-3})$ result can be seen as the 3D version of the Euler-Heisenberg Lagrangian. We notice that in the commutative limit, the gauge invariant field strength is defined as *f**μ**ν* = ∂*μ**A**ν* − ∂*ν**A**μ*, so that the commutative version of the effective action only receives contribution from the one-loop ⟨*A**A*⟩ part, the remaining contributions are vanishing. Thus, the commutative one-loop effective action in the presence of the higher-derivative term, at the next to leading order, is described as\n \n$$\\lim\\limits\\_{\\theta\\rightarrow 0}i{\\cal{S}}\\_{\\rm eff}\\bigg|\\_{{\\cal{O}}(m^{-3})}=\n -\\frac{ie^{2}}{96\\pi m}\\int d^{3}x~f\\_{\\mu\\nu}f^{\\mu\\nu}+\\frac{ie^{2}}{1920\\pi m^{3}}\\int d^{3}x~\nf\\_{\\mu\\nu}\\Box f^{\\mu\\nu}$$ \nSome comments about the result are now in place. Regarding discrete symmetries, following the aforementioned discussion in Sec. [sec2.1], it is easy to show that the above one-loop effective action is also invariant under all of the discrete symmetries and therefore we have not faced any anomalous symmetry at this order.\nSince the effective action, arising from the 2, 3 and 4-point functions at the order ${\\cal{O}}(m^{-1})$, is exactly gauge invariant, it is possible to conclude that no further ${\\cal{O}}(m^{-1})$ terms are generated from higher-order graphs with *n* > 4 external photon legs. According to this reasoning, we can also discuss the gauge invariance of the higher-derivative terms generated by considering the next to leading order ${\\cal{O}}(m^{-3})$ of our expansion. Actually, it is possible to make use of a dimensional analysis, based on arguments of gauge invariance, to establish the perturbative generation of all possible gauge invariant higher-derivative terms in the one-loop effective action. As an example, if we consider all of the ${\\cal{O}}(m^{-3})$ contributions up to the diagrams with *n* = 6 external photon legs, we can generate the following effective higher-derivative Lagrangian\n \n$$\\mathcal{L}\\_{\\rm hd} = \\frac{1}{6\\mu^2} \\nabla \\_\\mu F^{\\mu \\nu }\\star \\nabla^\\lambda F\\_{\\lambda \\nu }\n+\\frac{1}{6\\mu^2} \\nabla \\_\\lambda F^{\\mu \\nu }\\star \\nabla^\\lambda F\\_{\\mu \\nu }\n- \\frac{e}{18\\mu^2} F^{\\mu \\nu }\\star F\\_{\\nu \\lambda}\\star F^{\\lambda } \\_{\\;\\;\\; \\mu},$$ \nwhere ∇*μ* = ∂*μ* + *i**e*[*A**μ*, ]⋆ is the covariant derivative in the adjoint representation and *μ* ∼ *m* is the mass scale of the theory. This expression can been seen as a noncommutative extension of the Alekseev-Arbuzov-Baikov effective Lagrangian.\nOne last comment about the photon effective action is in regard of some of the nonlinear contributions. It is well known that either fermionic or scalar electrodynamics generate nonlinear corrections of quantum character to the photon dynamics. In the case of a (2 + 1) spacetime the effective Euler-Heisenberg Lagrangian density has the appearance of fractional powers of the field strength \n \n$$\\mathcal{L}\\_{\\rm EH} \\sim \\left( e \\sqrt{B^2-E^2} \\right)^{\\frac{3}{2}}.$$ \nHence, it is reasonable to expect that the coordinates noncommutativity would also present corrections to this nonlinear coupling term.\nComparison with NC-QED3\n=======================\nIn this section, we shall present a comparative discussion of the 2,3 and 4-point functions in the case of fermionic and bosonic matter fields coupled to the photon. It is notable that the presence of the trace of *γ* matrices in fermionic QED3 leads to two sectors characterized as: odd and even in regard to parity symmetry. The former is related to the induced Chern-Simons (CS) terms, appearing with odd powers of the fermion mass *m**e*, while the latter sector contributes to the induced Maxwell (M) terms, with even powers of *m**e*. Now in the scalar framework, odd parity terms are absent, and only parity preserving terms are present in the induced effective action, which can be understood as the main difference between these two matter fields. These results can be briefly described in terms of a mass expansion as the following:\n|c|c|c|c|c|c|c| & Induced action & NC-QED3& NC-scalar QED3 \n${\\cal{O}}(m^{0})$ & ordinary NC-CS & $\\checkmark$ & × \n ${\\cal{O}}(m^{-1})$ & ordinary NC-M & $\\checkmark$ & $\\checkmark$ \n ${\\cal{O}}(m^{-2})$& higher-derivative NC-CS & $\\checkmark$ & × \n ${\\cal{O}}(m^{-3})$& higher-derivative NC-M & $\\checkmark$ & $\\checkmark$ \n& & & \n ${\\cal{O}}(m^{-2\\ell})$& higher-derivative NC-CS & $\\checkmark$ & × \n ${\\cal{O}}(m^{-2\\ell-1})$& higher-derivative NC-M & $\\checkmark$ & $\\checkmark$ \nHere some important comments are in order. In NC-QED3, the structure of the kinetic part, coming from the *n* = 2 photon external legs analysis, has the contribution of two types of terms: a CS-type parity violating term *e*2*A*∂□ℓ*A*, at the order ${\\cal{O}}(m^{-2\\ell})$, and a M-type parity preserving term *e*2*A*∂∂□ℓ*A*, at the order ${\\cal{O}}(m^{-2\\ell-1})$. We can observe that the mass dimension of the CS and M-type terms is given by 3 + 2ℓ and 4 + 2ℓ, respectively. Thus, we can conclude that in order to have a gauge invariant CS-type action at the order ${\\cal{O}}(m^{-2\\ell})$, it is necessary to consider all contributions originating from the graphs with *n* = 2, 3, …, 3 + 2ℓ photon legs. On the other hand, to obtain a gauge invariant M-type action at the order ${\\cal{O}}(m^{-2\\ell-1})$, it is necessary to consider all contributions arising from the relevant graphs with *n* = 2, 3, …, 4 + 2ℓ photon legs.\nFurthermore, in (2 + 1) dimensions, the number of degrees of freedom of the charged boson and Dirac fermion (in the 2 dimensional representation) is equal and hence it is easy to see that the numerical coefficient appearing in the 2-point function would be the same as in the fermionic QED3. Now for the *n* = 3 graphs, in the case of NC-QED3, there are only the contribution of two triangle graphs in the one-loop order. These contributions are the same as in the NC-scalar QED3 because the additional graph is identically zero. Thus, it is easy to realize that the final result in the parity preserving sector for both cases is the same. The last type of diagrams is for *n* = 4 legs, that for the fermionic case we have the contribution of the box diagram only, i.e. type (a). The expression for this diagram at the leading order of ${\\cal{O}}(m^{-1})$ is given by\n \n$$\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a,1)}\\bigg|\\_{\\tiny\\mbox{NC-QED}}=-\\frac{ie^{4}}{3\\pi m}\\Big(\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}-2\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+ \\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big)e^{\\frac{i}{2}p\\wedge q}e^{\\frac{i}{2}r\\wedge s}.$$ \nBy considering all of the 24 permutations and some manipulations we obtain the following\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}\\bigg|\\_{\\tiny\\mbox{NC-QED}}=\\frac{4ie^4}{3\\pi m}\\bigg[ \\Big(&\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big)\\sin{\\big(\\frac{p\\wedge r}{2}\\big)}\\sin{\\big(\\frac{q\\wedge s}{2}\\Big)}\\nonumber\\\\\n+\\Big(&\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin{\\big(\\frac{q\\wedge r}{2}\\big)}\\sin{\\big(\\frac{p\\wedge s}{2}\\big)}\\nonumber\\\\\n+\\Big(&\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin{\\big(\\frac{p\\wedge q}{2}\\big)}\\sin{\\big(\\frac{s\\wedge r}{2}\\big)}\\bigg],\\end{aligned}$$ \nwhich has the same tensor and momenta structure as the standard 4-photon vertex, but with a different numerical coefficient. In this case, we can conclude that the commutative limit of this function is also satisfied, and have a vanishing result as we would expect.\nFinal remarks\n=============\nIn this paper we have considered the perturbative evaluation of the effective action for photon in the context of scalar QED in the (2 + 1) noncommutative spacetime. Our main interest was to determine to what extent the spin of the matter fields can change the effective action in the presence of the spacetime noncommutativity. Since the number of degrees of freedom of the charged boson and Dirac fermion (in the 2 dimensional representation) is equal, the main difference between these fields is solely to the well known presence of different couplings in the case of scalar QED, implying in new types of graphs. An important drawback from the scalar QED in terms of the induced effective action for the photon is the absence of parity violating terms, showing that no Chern-Simons terms are generated when charged scalar fields are considered.\nThe perturbative analysis followed the computation of the ⟨*A**A*⟩, ⟨*A**A**A*⟩ and ⟨*A**A**A**A*⟩ vertex functions. A more detailed and careful analysis was necessary to the computation of the 4 point vertex, where the process of symmetrization is rather intricate. The evaluation of the noncommutative Maxwell action ∫*F**μ**ν* ⋆ *F**μ**ν* was done by considering the highly noncommutative limit of these 1*P**I* functions at order ${\\cal{O}}(m^{-1})$.\nIn addition, we have considered the generation of ${\\cal{O}}(m^{-3})$ terms, which are higher-derivative terms for the photon fields, and can be seen as the noncommutative generalization of the phenomenological Alekseev-Arbuzov-Baikov effective Lagrangian. Another possible terms to be present in the gauge invariant photon’s effective action are those nonlinear couplings, e.g. analogous to the Euler-Heisenberg action, thus within our study of NC-scalar QED, we would find noncommutative corrections to the fractional powers of the field strength that appear in the (2 + 1) dimensional Euler-Heisenberg action.\nAcknowledgements\n----------------\nWe are grateful to M.M. Sheikh-Jabbari for fruitful discussions and comments on the manuscript. R.B. acknowledges partial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq Projects No. 304241/2016-4 and 421886/2018-8 ) and Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG Project No. APQ-01142-17).\nHigher-derivative corrections to the 4-photon vertex\n====================================================\nThe higher-derivative contributions of the graphs, depicted in Fig. [oneloop3], to the 4-photon vertex at the next to leading order, ${\\cal{O}}(m^{-3})$, are described as the following:\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}}=\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(a)}\n+\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(b)}\n+\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(c)}\\end{aligned}$$ \nFor simplicity, we define $\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(a)}=\\frac{ie^4}{240 \\pi m^3 }\\widetilde{\\Gamma}^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(a)}$ and $\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(b+c)}=-\\frac{ie^{4}}{24\\pi m^{3}}\\widetilde{\\Gamma}^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(b+c)}$ in which\n \n$$\\begin{aligned}\n\\widetilde{\\Gamma}^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(a)}=\\Bigg\\{&\\eta^{\\mu \\nu} \\Big[ - 2 p^{\\sigma} p^{\\rho} + 4 q^{\\sigma} p^{\\rho} -\n6 s^{\\sigma} p^{\\rho} - q^{\\rho} p^{\\sigma} - s^{\\rho} p^{\\sigma} + 2\nq^{\\sigma} s^{\\rho} + q^{\\rho} q^{\\sigma} - 3 q^{\\rho} s^{\\sigma} - 4\ns^{\\rho} s^{\\sigma}\\Big]\n\\nonumber\\\\\n +& \\eta^{\\mu \\rho} \\Big[ - 2 \\left(p^{\\sigma} + 3\nq^{\\sigma} - 2 s^{\\sigma} \\right) p^{\\nu} - q^{\\nu} p^{\\sigma} -\ns^{\\nu} p^{\\sigma} - 4 q^{\\nu} q^{\\sigma} - 3 s^{\\nu} q^{\\sigma} + 2\nq^{\\nu} s^{\\sigma} + s^{\\nu} s^{\\sigma}\\Big]\n\\nonumber\\\\\n+& \\eta^{\\mu \\sigma} \\Big[4 \\left(2\np^{\\rho} + q^{\\rho} + s^{\\rho} \\right) p^{\\nu} + 4 q^{\\nu} p^{\\rho} +\n4 s^{\\nu} p^{\\rho} + 2 q^{\\rho} s^{\\nu} + q^{\\nu} q^{\\rho} + 2 q^{\\nu}\ns^{\\rho} + s^{\\nu} s^{\\rho}\\Big]\n\\nonumber\\\\\n+& \\eta^{\\nu \\rho} \\Big[ - 2 p^{\\sigma} p^{\\mu} -\nq^{\\sigma} p^{\\mu} - s^{\\sigma} p^{\\mu} - q^{\\mu} p^{\\sigma} - s^{\\mu}\np^{\\sigma} + q^{\\mu} q^{\\sigma} - 3 s^{\\mu} q^{\\sigma} + s^{\\mu}\ns^{\\sigma} - 3 q^{\\mu} s^{\\sigma}\\Big]\n\\nonumber\\\\\n+& \\eta^{\\nu \\sigma} \\Big[ - 2 p^{\\rho}\np^{\\mu} - q^{\\rho} p^{\\mu} - s^{\\rho} p^{\\mu} + 4 s^{\\mu} p^{\\rho} - 6\nq^{\\mu} p^{\\rho} + 2 q^{\\rho} s^{\\mu} - 4 q^{\\mu} q^{\\rho} + s^{\\mu}\ns^{\\rho} - 3 q^{\\mu} s^{\\rho}\\Big]\n\\nonumber\\\\\n +& \\eta^{\\rho \\sigma} \\Big[ - 2 p^{\\nu} p^{\\mu}\n- q^{\\nu} p^{\\mu} - s^{\\nu} p^{\\mu} + 4 q^{\\mu} p^{\\nu} - 6 s^{\\mu}\np^{\\nu} + q^{\\mu} q^{\\nu} + 2 q^{\\mu} s^{\\nu} - 3 s^{\\mu} q^{\\nu} - 4\ns^{\\mu} s^{\\nu}\\Big]\n\\nonumber\\\\\n+&\n\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n\\Big[4p^{2}+3q^{2}+3s^{2}+4(p.q)+4(p.s)+2(q.s)\\Big]\\Bigg\\}\\cos (\\frac{r\\wedge s+q\\wedge p}{2})\\nonumber\\\\\\end{aligned}$$ \n \n$$\\begin{aligned}\n+\\Bigg\\{&\\eta^{\\mu \\nu}\n\\Big[p^{\\rho} p^{\\sigma} - p^{\\rho} q^{\\sigma} - 3 p^{\\rho} s^{\\sigma} +\n4 p^{\\sigma} q^{\\rho} + 2 p^{\\sigma} s^{\\rho} - 2 q^{\\rho} q^{\\sigma}\n- q^{\\sigma} s^{\\rho} - 6 q^{\\rho} s^{\\sigma} - 4 s^{\\rho} s^{\\sigma}\\Big]\n\\nonumber\\\\\n+&\\eta^{\\mu \\rho}\\Big[(p^{\\sigma} - q^{\\sigma} - 3 s^{\\sigma})\np^{\\nu} - p^{\\sigma} q^{\\nu} - 3 p^{\\sigma} s^{\\nu} - 2 q^{\\nu}\nq^{\\sigma} - q^{\\sigma} s^{\\nu} - q^{\\nu} s^{\\sigma} + s^{\\nu}\ns^{\\sigma}\\Big]\n\\nonumber\\\\\n+&\\eta^{\\mu \\sigma} \\Big[- (4 p^{\\rho} + 6 q^{\\rho} + 3\n s^{\\rho}) p^{\\nu} - p^{\\rho} q^{\\nu} + 2 p^{\\rho} s^{\\nu} + 4\nq^{\\rho} s^{\\nu} - 2 q^{\\nu} q^{\\rho} - q^{\\nu} s^{\\rho} + s^{\\nu}\ns^{\\rho})\n\\nonumber\\\\\n+&\\eta^{\\nu \\rho}\\Big[ - 4 p^{\\mu} p^{\\sigma} - p^{\\mu}\nq^{\\sigma} + 2 p^{\\mu} s^{\\sigma} - 6 p^{\\sigma} q^{\\mu} - 3\np^{\\sigma} s^{\\mu} - 2 q^{\\mu} q^{\\sigma} - q^{\\sigma} s^{\\mu} + 4\nq^{\\mu} s^{\\sigma} + s^{\\mu} s^{\\sigma}\\Big]\n\\nonumber\\\\\n +&\\eta^{\\nu \\sigma}\\Big[p^{\\mu}\np^{\\rho} + 4 p^{\\mu} q^{\\rho} + 2 p^{\\mu} s^{\\rho} + 4 p^{\\rho}\nq^{\\mu} + 2 p^{\\rho} s^{\\mu} + 4 q^{\\rho} s^{\\mu} + 8 q^{\\mu} q^{\\rho}\n+ 4 q^{\\mu} s^{\\rho} + s^{\\mu} s^{\\rho}\\Big]\n\\nonumber\\\\\n+&\\eta^{\\rho \\sigma}\\Big[p^{\\mu}\np^{\\nu} - p^{\\mu} q^{\\nu} - 3 p^{\\mu} s^{\\nu} + 4 p^{\\nu} q^{\\mu} + 2\np^{\\nu} s^{\\mu} - 2 q^{\\mu} q^{\\nu} - q^{\\nu} s^{\\mu} - 6 q^{\\mu}\ns^{\\nu} - 4 s^{\\mu} s^{\\nu}\\Big]\n\\nonumber\\\\\n+ &\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n\\Big[3 p^{2} + 4 q^{2} + 3 s^{2} + 4(p.q) + 2(p.s) + 4(q.s)\\Big] \\Bigg\\}\\cos (\\frac{p\\wedge r+q\\wedge s}{2})\\nonumber\\\\\n+\\Bigg\\{&\n\\eta^{\\mu \\nu}\\Big[p^{\\rho} p^{\\sigma} - 3 p^{\\rho} q^{\\sigma} - p^{\\rho} s^{\\sigma} -\n3 p^{\\sigma} q^{\\rho} - p^{\\sigma} s^{\\rho} + q^{\\rho} q^{\\sigma} -\nq^{\\sigma} s^{\\rho} - q^{\\rho} s^{\\sigma} - 2 s^{\\rho} s^{\\sigma}\\Big]\n\\nonumber\\\\\n+& \\eta^{\\mu \\rho}\\Big[(p^{\\sigma} - 3 q^{\\sigma} - s^{\\sigma})\np^{\\nu} + 2 p^{\\sigma} q^{\\nu} + 4 p^{\\sigma} s^{\\nu} - 4 q^{\\nu}\nq^{\\sigma} - 6 q^{\\sigma} s^{\\nu} - q^{\\nu} s^{\\sigma} - 2 s^{\\nu}\ns^{\\sigma}\\Big]\n\\nonumber\\\\\n +& \\eta^{\\mu \\sigma}\\Big[ -(4 p^{\\rho} - 2 q^{\\rho} +\ns^{\\rho}) p^{\\nu} - 3 p^{\\rho} q^{\\nu} - 6 p^{\\rho} s^{\\nu} + 4\nq^{\\rho} s^{\\nu} + q^{\\nu} q^{\\rho} - q^{\\nu} s^{\\rho} - 2 s^{\\nu}\ns^{\\rho}\\Big]\n\\nonumber\\\\\n+& \\eta^{\\nu \\rho}\\Big[ - 4 p^{\\mu} p^{\\sigma} + 2 p^{\\mu}\nq^{\\sigma} - p^{\\mu} s^{\\sigma} - 3 p^{\\sigma} q^{\\mu} - 6 p^{\\sigma}\ns^{\\mu} + 4 q^{\\sigma} s^{\\mu} + q^{\\mu} q^{\\sigma} - q^{\\mu}\ns^{\\sigma} - 2 s^{\\mu} s^{\\sigma}\\Big]\n\\nonumber\\\\\n+&\\eta^{\\nu \\sigma}\\Big[p^{\\mu} p^{\\rho} -\n3 p^{\\mu} q^{\\rho} - p^{\\mu} s^{\\rho} + 2 p^{\\rho} q^{\\mu} + 4\np^{\\rho} s^{\\mu} - 4 q^{\\mu} q^{\\rho} - 6 q^{\\rho} s^{\\mu} - q^{\\mu}\ns^{\\rho} - 2 s^{\\mu} s^{\\rho}\\Big]\n\\nonumber\\\\\n+& \\eta^{\\rho \\sigma}\\Big[p^{\\mu} p^{\\nu} + 2\np^{\\mu} q^{\\nu} + 4 p^{\\mu} s^{\\nu} + 2 p^{\\nu} q^{\\mu} + 4 p^{\\nu}\ns^{\\mu} + 4 q^{\\nu} s^{\\mu} + q^{\\mu} q^{\\nu} + 4 q^{\\mu} s^{\\nu} + 8\ns^{\\mu} s^{\\nu}\\Big]\n\\nonumber\\\\\n +& \\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n \\Big[3p^{2} + 3 q^{2} + 4 s^{2} + 2(p.q)+4(p.s)+4(q.s)\\Big]\\Bigg\\}\\cos(\\frac{p\\wedge s+r\\wedge q}{2})\\nonumber\\\\\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\widetilde{\\Gamma}^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(b+c)}=\n\\Bigg\\{&\\Big[\\big(p^2+q^2+p.q+s.r\\big)\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+p^{\\nu}q^{\\mu}\\eta^{\\rho\\sigma}\n-r^{\\rho}s^{\\sigma}\\eta^{\\mu\\nu}\\Big]\\cos(\\frac{p\\wedge q}{2})\\cos(\\frac{s\\wedge r}{2})\n\\nonumber\\\\\n+&\n\\Big[\\big(p^2+s^2+p.s+q.r\\big)\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+p^{\\rho}s^{\\mu}\\eta^{\\nu\\sigma}\n-r^{\\nu}q^{\\sigma}\\eta^{\\mu\\rho}\\Big]\\cos(\\frac{p\\wedge s}{2})\\cos(\\frac{q\\wedge r}{2})\n\\nonumber\\\\\n+&\n\\Big[\\big(q^2+s^2+q.s+p.r\\big)\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}+q^{\\rho}s^{\\nu}\\eta^{\\mu\\sigma}\n-r^{\\mu}p^{\\sigma}\\eta^{\\nu\\rho}\\Big]\\cos(\\frac{p\\wedge r}{2})\\cos(\\frac{q\\wedge s}{2})\n\\Bigg\\}\\end{aligned}$$ \n#1#2[#2](http://eudml.org/#1) #1#2[#2](http://dx.doi.org/#1) #1#2[arXiv:#1 [#2]](http://arxiv.org/abs/#1) #1[arXiv:#1](http://arxiv.org/abs/#1)\n99\nL. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari, B. Lima de Souza and T. Štemberga, “*One-loop effective actions and higher spins*,”,. L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari and T. Štemberga, “*One-loop effective actions and higher spins. Part II*,”,. J. Quevillon, C. Smith and S. Touati, “*Effective action for gauge bosons*,”,. W. Heisenberg and H. Euler, “*Consequences of Dirac’s theory of positrons*,”,. I. L. Buchbinder, S. D. Odintsov and I. L. Shapiro, “*Effective action in quantum gravity*,” Bristol, UK: IOP (1992) 413 p M. Chaichian, M. Oksanen and A. Tureanu, “*Sakharov’s induced gravity and the Poincaré gauge theory*,” in Jacob Bekenstein Memorial Volume, World Scientific,. D. Mattingly, “*Modern tests of Lorentz invariance*,”,. R. Bluhm, “*Overview of the SME: Implications and phenomenology of Lorentz violation*,”,. L. H. C. Borges, A. G. Dias, A. F. Ferrari, J. R. Nascimento and A. Y. Petrov, “*Generation of axionlike couplings via quantum corrections in a Lorentz-violating background*,”,. L. H. C. Borges, A. G. Dias, A. F. Ferrari, J. R. Nascimento and A. Y. Petrov, “*Generation of higher derivatives operators and electromagnetic wave propagation in a Lorentz-violation scenario*,”,. B. Charneski, M. Gomes, T. Mariz, J. R. Nascimento and A. J. da Silva, “*Dynamical Lorentz symmetry breaking in 3D and charge fractionalization*,”,. J. R. Nascimento, A. Y. Petrov, C. Wotzasek and C. A. D. Zarro, “*Three-dimensional Lorentz-violating action*,”,. R. Casana, C. F. Farias and M. M. Ferreira, “*Topological self-dual configurations in a Lorentz-violating gauged O(3) sigma model*,”,. L. Susskind, “*The Quantum Hall fluid and noncommutative Chern-Simons theory*,”. M. R. Douglas and N. A. Nekrasov, “*Noncommutative field theory*,”,. G. Amelino-Camelia, “*Quantum-Spacetime Phenomenology*,”,. M. Arzano and J. Kowalski-Glikman, “*Non-commutative fields and the short-scale structure of spacetime*,”,. A. Matusis, L. Susskind and N. Toumbas, “*The IR/UV connection in the noncommutative gauge theories*,”,. D. V. Vassilevich, “*Heat kernel, effective action and anomalies in noncommutative theories*,”,. M. Ghasemkhani, “*Study of the photon’s pole structure in the noncommutative Schwinger model*,”,. M. Ghasemkhani, A. A. Varshovi and R. Bufalo, “*Perturbative effective action for the photon in noncommutative QED2 and exactness of the Schwinger mass*,”. C. S. Chu, “*Induced Chern-Simons and WZW action in noncommutative space-time*,”,. R. Banerjee, T. Shreecharan and S. Ghosh, “*Three dimensional noncommutative bosonization*,”,. R. Bufalo and M. Ghasemkhani, “*Higher derivative Chern-Simons extension in the noncommutative QED3*,”,. P. S. Hsin and N. Seiberg, “*Level/rank Duality and Chern-Simons-Matter Theories*,”,. F. Benini, P. S. Hsin and N. Seiberg, “*Comments on global symmetries, anomalies, and duality in (2 + 1)d*,”,. M. Ghasemkhani, R. Bufalo, V. Rahmanpour and E. Nouri, “*One-loop **\\beta** function of noncommutative scalar *Q**E**D*4*,”,. P. K. Panigrahi and T. Shreecharan, “*Induced magnetic moment in noncommutative Chern-Simons scalar QED*,”,. A. I. Alekseev, B. A. Arbuzov and V. A. Baikov, “*Infrared Asymptotics of Gluon Green’s Functions in Quantum Chromodynamics*,”. A. N. Redlich, “*Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions*,”.\n---\n1. <!--\nh='ipm.ir';a='@';n='ghasemkhani';e=n+a+h;\ndocument.write('<a h'+'ref'+'=\"ma'+'ilto'+':'+e+'\">'+e+'<\\/'+'a'+'>');\n// -->\nghasemkhani at ipm dot ir[↩](#fnref1)\n2. <!--\nh='ufla.br';a='@';n='rodrigo.bufalo';e=n+a+h;\ndocument.write('<a h'+'ref'+'=\"ma'+'ilto'+':'+e+'\">'+e+'<\\/'+'a'+'>');\n// -->\nrodrigo.bufalo at ufla dot br[↩](#fnref2)\n3. <!--\nh='mail.sbu.ac.ir';a='@';n='v.rahmanpour';e=n+a+h;\ndocument.write('<a h'+'ref'+'=\"ma'+'ilto'+':'+e+'\">'+e+'<\\/'+'a'+'>');\n// -->\nv.rahmanpour at mail dot sbu dot ac dot ir[↩](#fnref3)\n4. <!--\nh='yahoo.com';a='@';n='moj.alipour';e=n+a+h;\ndocument.write('<a h'+'ref'+'=\"ma'+'ilto'+':'+e+'\">'+e+'<\\/'+'a'+'>');\n// -->\nmoj.alipour at yahoo dot com[↩](#fnref4)\n5. That may signal symmetry violation, for instance Lorentz symmetry violation.[↩](#fnref5)\n6. The noncommutativity we will be using in the paper is defined by the algebra [*x̂**μ*, *x̂**ν*] = *i**θ**μ**ν*. So in order to construct a noncommutative field theory, using the Weyl-Moyal (symbol) correspondence, the ordinary product is replaced by the Moyal star product as defined below.[↩](#fnref6)\n\nIn this paper, we consider the evaluation of the effective action for photons coupled to charged scalar fields in the framework of a (2 + 1)-dimensional noncommutative spacetime. In order to determine the noncommutative Maxwell Lagrangian density, we follow a perturbative approach, by integrating out the charged scalar fields, to compute the respective graphs for the vev’s ⟨*A**A*⟩, ⟨*A**A**A*⟩ and ⟨*A**A**A**A*⟩. Surprisingly, it is shown that these contributions are planar and that, in the highly noncommutative limit, correspond to the Maxwell effective action and its higher-derivative corrections. It is explicitly verified that the one-loop effective action is gauge invariant, as well as under discrete symmetries: parity, time reversal, and charge conjugation. Moreover, a comparison of the main results with the noncommutative QED3 is established. In particular, the main difference is the absence of parity violating terms in the photon’s effective action coming from integrating out the charged scalar fields.\nIntroduction\n============\nIn recent years a great amount of attention has been paid in the analysis and calculation of covariant effective action for different types of quantum fields, exploring the diversity of new interactions that mainly depend on the spin of the fields involved as well as the spacetime dimensionality. One may say that the canonical example of a complete analysis is the Euler-Heisenberg effective action, where quantum effects from QED are responsible to induce nonlinear interactions among photons. Moreover, the effective action framework has served as an important tool to explore different point of views about the quantum gravitational theory, where the Einstein-Hilbert action is augmented by metric and/or torsion fields higher-order terms.\nNaturally, since the framework of effective action is a powerful tool, there is a great expectation that this approach can be used to make contact with modern phenomenology of physics beyond the standard model. The main idea behind this formulation is that at energies below some cutoff scale *μ*, [5](#fn5) all the effects of the massive degrees of freedom above *μ* can be encoded as new interactions among the fields remaining active below *μ*. The effective action approach has been extensively used to the study of Lorentz violating field theories, where the energy scale *μ* is related to the Planck energy scale $E\\_{\\rm Pl}$ (or length $\\ell\\_{\\rm Pl}$) where our notion of smooth geometry is expected to break. In this case, the current understanding is that the low energy Lorentz violating terms come as quantum corrections from heavy modes.\nAlthough the majority of studies of Lorentz violating field theories is developed in a four-dimensional spacetime, there are considerable interests in the description of three-dimensional (3D) ones. Besides the algebraic richness of odd dimensional spacetimes, one may say that the most appealing aspect of 3D field theories is the UV finiteness in some models. This feature might provide an ambiguity free description of Lorentz violation, allowing thus a close contact of violating effects with physical planar phenomena. In particular, it is worth recall the example of the description of quantum hall fluids in terms of noncommutative geometry.\nOver the past two decades, field theories defined in a noncommutative (NC) geometry have been considered as one of the most prominent candidates presenting Lorentz violation to make contact with quantum gravity phenomenology. Within this description, the noncommutativity measurement parameter is related to a length scale $\\ell\\_{\\rm nc} \\sim \\sqrt{\\theta}$. In one hand, this length scale can be seen as a manifestation of the discreteness of the spacetime, presenting a smooth profile in the UV region. On the other hand, this same scale is responsible for introducing instabilities in the dispersion relations of the fields, the so-called UV/IR mixing.\nNC field theories have been studied through the effective action approach, where the behavior of the new couplings was deeply analyzed, where the presence of UV/IR mixing in the 1PI functions signals that applying the usual Wilsonian field theory notions and techniques to NC QFT’s one should be careful. This type of analysis was also developed to two and three-dimensional NC models. These studies of effective action in 3D models were exclusive to the coupling of gauge and fermion fields, no much attention has been paid to the case involving scalar fields, in particular the case of spinless charged fields interacting with photons.\nIn one hand, it is of physical significance to study scalars in 3D field theories independently of fermions in condensed matter systems, as in Quantum Hall systems, since we have scalar quasiparticle excitations. On the other hand, recently 3D versions of fermionization/bosonization have also been introduced. In these studies, it was discussed the duality between nonspin Chern-Simons theory and a spin Chern-Simons theory, exploring precisely the spin structure of the given models. In this sense, the present work could be the first step in extending such analysis to the NC case. Motivated by these facts, we will analyze throughout the paper to what extent the spin of the matter fields can change the effective action when charged scalar and fermion fields are considered in the presence of the spacetime noncommutativity. A straightforward result is that in the case of the 3D scalar quantum electrodynamics (scalar QED3), it is not possible to generate the parity odd Chern-Simons terms, showing thus that the dynamics of the 3D gauge field is significantly different in the presence of either charged scalar or fermion fields. It is well known that the presence of the degree of freedom associated with the spin changes in most of the cases only the magnitude of physical quantities, e.g. the beta function and electron’s magnetic moment.\nIn this paper we discuss the effective action for the photon in the scalar QED in the noncommutative three-dimensional spacetime. In Sec. [sec2] we present an overview of the scalar QED, where the charged scalar fields are minimally coupled with the photons. There we define the main aspects regarding the Moyal product used in our analysis,[6](#fn6) we also discuss the content of discrete symmetries in the NC 3D spacetime. In addition, all the Feynman rules are presented for the propagators and 1PI vertices. Section [sec4] is focused in the perturbative computation of the relevant graphs corresponding to the one-loop effective action for the photon gauge field. It is also discussed the generation of higher-derivative terms, similarly to the Alekseev-Arbuzov-Baikov effective Lagrangian for non-Abelian fields. In Sec. [sec5] we establish a comparison of the obtained results for the effective action in the NC-scalar QED to those of ordinary NC-QED, exploring the part played by the spin in these cases. We present our final remarks in Sec. [conc].\nThe model\n=========\nIn this section, we introduce the model and fix our notation. The noncommutative extension of the bosonic electrodynamics is described by the following action\n \n$$S=\\int d^{3}x\\Big[\\left(D\\_{\\mu}\\phi\\right)^{\\dagger}\\star D^{\\mu}\\phi-m^{2}\\phi^{\\dagger}\\star \\phi\\Big],\n\\label{eq:a1}$$ \nthis functional action consists of the interaction of charged scalar fields minimally coupled with an external gauge field. We consider the covariant derivative form in the fundamental representation *D**μ**ϕ* = ∂*μ**ϕ* + *i**e**A**μ* ⋆ *ϕ*. This action is invariant under the infinitesimal gauge transformation\n \n*δ**A**μ* = ∂*μ**λ* + *i**e*[*A**μ*, *λ*]⋆ , *δ**ϕ* = *i**e**λ* ⋆ *ϕ*, \nwhere [ , ]⋆ is the Moyal bracket. Moreover, the Moyal star product between the functions *f* and *g* is defined as\n \n$$f\\left( x\\right)\\star g\\left(x\\right) = f\\left( x\\right) \\exp \\left(\\frac{i}{2}\\theta ^{\\mu \\nu}\n\\overleftarrow{\\partial\\_\\mu}\n \\overrightarrow{\\partial\\_\\nu}\\right) g\\left( x\\right),\n \\label{eq:a3}$$ \nwhere *θ**μ**ν* = − *θ**ν**μ* are constant parameters that measure the noncommutative structure of the space-time. In order to avoid unitarity violation, we assume that *θ*0*i* = 0, hence we have only one nonzero independent component *θ*12 in our model.\nIt is worth mentioning that although the couplings are simply modified by the presence of a nonplanar phase due to the Moyal product, the noncommutativity of spacetime coordinates shows its importance in the computation of the one-loop effective action for the gauge field, where nonlinear self-couplings are present solely due to the NC framework. The one-loop effective action for the gauge field can be readily obtained by integrating out the charged scalar fields of\n \n$$e^{i\\Gamma\\_{\\rm eff}[A]}=\\int D\\phi^{\\dagger} D\\phi~e^{-i\\int d^{3}x~\\phi^{\\dagger}\\star(D^{2}+m^{2})\\star\\phi}.$$ \nUsing the Gaussian functional integration formulas for the case of interacting charged scalar fields, we can write the noncommutative 1PI effective action as below\n \n$$i\\Gamma\\_{\\rm eff}[A]={\\rm Tr}\\ln\\left[\\frac{(i\\partial\\_{\\mu}-eA\\_{\\mu})\\star(i\\partial^{\\mu}-eA^{\\mu})\\star-m^{2}}{-\\partial^{2}-m^{2}}\\right],$$ \nwhere ${\\rm Tr}$ is a sum over eigenvalues of the operator inside the bracket which can also be evaluated in momentum space. Similarly to the description of one-loop effective action for the gauge field in the case of NC-QED, one can show that $\\Gamma\\_{\\rm eff}[A]$ has a convergent series expansion in coupling constant *e*. From a diagrammatic point of view, it includes the one-loop graphs contributing to the gauge field *n*-point functions which is considered as\n \n$$\\Gamma\\_{\\rm eff}[A]={\\cal{S}}\\_{\\rm eff}[AA]+{\\cal{S}}\\_{\\rm eff}[AAA]+{\\cal{S}}\\_{\\rm eff}[AAAA]+\\cdots.\n\\label{eq:aa}$$ \nHowever, the functional $\\Gamma\\_{\\rm eff}[A]$, in comparison to the NC-QED case, has more graphs due to the presence of an additional interacting vertex.\nMoreover, it is important to emphasize that as we will show in our model, similarly to the case of NC-QED, the one-loop effective action for the photons is completely planar. Explicitly, in the evaluation of the one-loop diagrams with an arbitrary number of external legs of photons, for energies below the mass scale *m*, only planar diagrams contribute. This means the absence of IR/UV mixing.\nDiscrete symmetries\n-------------------\nSince we are interested in computing the one-loop effective action for the photon, it is useful to analyze the behavior of the original action under discrete symmetries: parity, charge conjugation and time reversal. This study will allow us to determine which of them may be anomalous in the obtained results for the one-loop order effective action.\n* *Parity*\nParity transformation in *d* = 2 + 1 is defined as *x*1 → − *x*1 and *x*2 → *x*2, in this case we have that the field *ϕ* is even under parity, and the components of the gauge field *A**μ* behave as *A*0 → *A*0, *A*1 → − *A*1 and *A*2 → *A*2. Moreover, we observe from the NC algebra that the *θ* parameter changes under this transformation as *θ*12 → − *θ*12. With these considerations, it is easy to show that the whole of the action is parity invariant.\n* *Time Reversal*\nUnder time reversal, we have that *x*0 → − *x*0. In this case, the components of the gauge field behave as (*A*0, *A**i*) → (*A*0, − *A**i*). By demanding that the scalar field does not change *ϕ* → *ϕ*, and that necessarily the NC parameter transforms as *θ*12 → − *θ*12 under time reversal, we are left with a *T*-invariant action.\n* *Charge Conjugation*\nAs we know, the behavior of the gauge field under charge conjugation is given by *A**μ* → − *A**μ* for any space-time dimensionality. Taking the scalar field to be unchanged under *C*, and the transformation for the NC parameter *θ* → − *θ*, we conclude that the action is *C*-invariant.\nPropagators and vertex functions\n--------------------------------\nIn order to discuss the computation of the perturbative effective action, we must determine the basic propagators and 1PI vertex functions. From the functional action described in, we can obtain the bosonic propagator\n \n$${\\cal{D}}(p) = \\frac{i}{{p^{2}-m^{2}}},\n \\label{eq:b1}$$ \nthe cubic vertex ⟨*A* *ϕ* *ϕ*†⟩\n \n$$\\begin{aligned}\n\\Gamma^\\mu(p,q)= -ie\\left(p+q\\right)^{\\mu}\\exp\\Big(\\frac{i}{2} p \\wedge q\\Big),\n \\label{eq:b2}\\end{aligned}$$ \nand the quartic vertex ⟨*A**A* *ϕ* *ϕ*†⟩\n \n$$\\begin{aligned}\n\\Lambda^{\\mu\\nu}(p,q,s)=2ie^{2}\\eta^{\\mu\\nu}\\exp{\\Big(\\frac{i}{2}k \\wedge s\\Big)}\\cos\\Big(\\frac{p \\wedge q}{2}\\Big),\n \\label{eq:b3}\\end{aligned}$$ \nwhere we have introduced the notation *p* ∧ *q* = *p**μ**θ**μ**ν**q**ν*. A straightforward difference of the scalar and fermionic electrodynamics is the presence of the quartic vertex ⟨*A**A**ϕ**ϕ*†⟩, which increases significantly the number of the one-loop graphs. Moreover, the scalar vertices are rather simpler due to the absence of the Dirac *γ* matrices, resulting in a much simpler algebraic analysis.\nPerturbative Effective Action\n=============================\nNow that we have determined the basic Feynman rules for the 1PI functions, we shall proceed to the computation of the one-loop diagrams related to the effective action for the gauge field. For this purpose, we shall compute along this section the respective contributions: the free part of the effective action ⟨*A**A*⟩, and the interacting parts for the cubic vertex ⟨*A**A**A*⟩ and quartic vertex ⟨*A**A**A**A*⟩. In general, the final results of our analysis related to the graphs contributing to ⟨*A**A*⟩, ⟨*A**A**A*⟩ and ⟨*A**A**A**A*⟩ vertices shall be a function of *e*, *p̃**μ* = *θ**μ**ν**p**ν* and *p*2/*m*2.\nTo highlight the effects of noncommutativity in the low energy effective action and the photon two, three and four-point functions, while we take the external momenta *p* such that *p*2/*m*2 ≪ 1, we also take the highly noncommutative limit, i.e., the low-energy regime *p*2/*m*2 → 0 while *p̃* is kept finite. In this limit the noncommutative (planar) phase factors, which are a function of *p̃*, remain finite. Moreover, we shall focus our attention on those terms of order *m*− 1. We present by complementarity, at the next to leading order, the terms of order *m*− 3 that correspond to higher-derivative corrections.\nOne-loop ⟨*A**A*⟩ part\n----------------------\nFrom the Feynman rules we can compute the one-loop contribution to the *A**A*-term corresponding to the free part of the photon effective action. The two diagrams contributing at this order are depicted in Fig. [oneloop1], which the respective expressions have the form\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu}\\_{(a)}(p) &= e^{2} \\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(p+2k)^{\\nu}}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]}, \\nonumber \\\\\n\\Pi^{\\mu\\nu}\\_{(b)}(p) &= -e^{2} \\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{2\\eta^{\\mu\\nu}[(p+k)^{2}-m^{2}]}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]},\n \\label{eq:c1}\\end{aligned}$$ \nso that the full contribution is written as\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu}(p) &= e^{2}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(p+2k)^{\\nu}-2\\eta^{\\mu\\nu}[(p+k)^{2}-m^{2}]}{[(p+k)^{2}-m^{2}][k^{2}-m^{2}]}.\n \\label{eq:c2}\\end{aligned}$$ \nA first comment is that this piece is completely planar, carrying no noncommutative effects. The explicit computation is straightforward using dimensional regularization. After some algebraic calculation, we can consider the low-energy limit, *p*2/*m*2 → 0, resulting into\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu}(p) =\\frac{ie^{2}}{48\\pi m}\\left(p^{\\mu}p^{\\nu}-\\eta^{\\mu\\nu}p^{2}\\right).\n \\label{eq:c3}\\end{aligned}$$ \nMoreover, for the next to leading order contribution, ${\\cal{O}}(m^{-3})$, we find that\n \n$$\\Pi^{\\mu\\nu}\\_{\\rm hd}(p)=\\frac{ie^{2}}{960\\pi m^{3}}\\left(p^{\\mu}p^{\\nu}-\\eta^{\\mu\\nu}p^{2}\\right)p^{2}.\n \\label{eq:c4}$$ \nThese two terms Eqs. and satisfy straightforwardly the Ward identity, *p**μ*Π*μ**ν* = 0, as we expected. We can determine the respective contribution to the effective action by means of\n \n$$i{\\cal{S}}\\_{\\rm eff}[AA]=\\int\\int d^{3}x\\_{1}d^{3}x\\_{2}~A\\_{\\mu}(x\\_{1})\\Gamma^{\\mu\\nu}(x\\_{1},x\\_{2})A\\_{\\nu}(x\\_{2}),\n \\label{eq:c5}$$ \nwhere ${\\cal{S}}\\_{\\rm eff}[AA]$ is the quadratic part of the effective action Γ[*A*] in. Here, we have defined by simplicity\n \n$$\\Gamma^{\\mu\\nu}(x\\_{1},x\\_{2})=\\int\\frac{d^{3}p}{(2\\pi)^{3}} e^{-i p\\cdot (x\\_{1}-x\\_{2})}\\Pi^{\\mu\\nu}(p ).\n \\label{eq:c6}$$ \n [oneloop1]\nAfter some algebra, the quadratic part of the induced effective action for the photon, considering and, is given by\n \n$$\\begin{aligned}\ni{\\cal{S}}\\_{\\rm eff}[AA]&=-\\frac{ie^{2}}{48\\pi m}\\int d^{3}x\\Big(\\partial\\_{\\mu}A\\_{\\nu}\\partial^{\\mu}A^{\\nu}-\\partial^{\\mu}A\\_{\\nu}\\partial^{\\nu}A\\_{\\mu}\\Big) \\nonumber \\\\\n&+\\frac{ie^{2}}{960\\pi m^{3}}\\int d^{3}x\\Big(\\partial\\_{\\mu}A\\_{\\nu}\\Box\\partial^{\\mu}A^{\\nu}-\\partial^{\\mu}A\\_{\\nu}\\Box\\partial^{\\nu}A\\_{\\mu}\\Big).\n \\label{eq:c7}\\end{aligned}$$ \nAs we have previously mentioned, the first term of the expression corresponds to the kinetic part of the noncommutative Maxwell action, ${\\cal O}(m^{-1})$, while the second term is the higher-derivative correction to the kinetic term, of order ${\\cal O}(m^{-3})$. Moreover, the obtained result does not contain any noncommutativity effect, since the produced phase factors cancel for *n* = 2. It is worth noticing the absence of the parity odd Chern-Simons term in the scalar QED3, which in turn is generated in the fermionic electrodynamics due to the algebraic structure of the two-dimensional realization of *γ* matrices.\nOne-loop ⟨*A**A**A*⟩ vertex\n---------------------------\nThe relevant graphs for the ⟨*A**A**A*⟩ part of the effective action are shown in Fig. [oneloop2]. However, in order to determine correctly the full contribution to the effective action, it is necessary to consider all different permutations of the external bosonic lines of the given graphs. It is easy to see that the diagram (a) has an additional contribution (b), corresponding to a permutation of the external photon legs, which has an equivalent structure but with a reversed momentum flow, which comes exactly from the S-matrix expansion at the order of *e*3. With help of the Feynman rules, we can easily write the relevant expression for the sum of the graphs (a) and (b)\n \n$$\\Pi^{\\mu\\nu\\rho}\\_{(a+b)}(p,q)=2i e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(2p+2k+q)^{\\nu}(p+q+2k)^{\\rho}}{[(p+k)^{2}-m^{2}][(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]} \\sin\\big(\\frac{p \\wedge q}{2}\\big),\n \\label{eq:c8}$$ \nwhich is a planar quantity, we can see that its integrand is independent of the noncommutativity. The contribution from the graph (c) also has a simple planar structure, which is given by the expression\n \n$$\\Pi^{\\mu\\nu\\rho}\\_{(c)}(p,q)= -e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}(p+q+2k)^{\\rho}}{[(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]} \\cos\\big(\\frac{p \\wedge q}{2}\\big).\n \\label{eq:c9}$$ \nSince the graph (c) is planar, one can perform straightforward manipulations to show that this contribution is identically zero, i.e. Π(*c*)*μ**ν**ρ* = 0, for any value of the external momenta. Hence the full contribution for the ⟨*A**A**A*⟩ vertex reads\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho}(p,q)=2i e^{3}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(2k+p)^{\\mu}(2k+p+s)^{\\nu}(s+2k)^{\\rho}}{[(p+k)^{2}-m^{2}][(s+k)^{2}-m^{2}][k^{2}-m^{2}]} \\sin\\big(\\frac{p \\wedge q}{2}\\big).\n \\label{eq:c10}\\end{aligned}$$ \n [oneloop2]\nThe computation of the loop integral is lengthy but straightforward using dimensional regularization, and in the low-energy limit *p*2, *q*2 ≪ *m*2, we find that\n \n$$\\Pi^{\\mu\\nu\\rho}(p,q)=\\frac{e^{3}}{12\\pi m}\\bigg[\\big(p-q\\big)^{\\rho}\\eta^{\\mu\\nu}-\\big(2p+ q\\big)^{\\nu}\\eta^{\\mu\\rho} +\\big(p+2 q\\big)^{\\mu}\\eta^{\\nu\\rho}\\bigg] \\sin\\big(\\frac{p \\wedge q}{2}\\big).\n \\label{eq:c11}$$ \nHere, we notice that the Eq. corresponds exactly to the standard Feynman vertex of the 3-photon interaction term in the NC spacetime. Moreover, in the next to leading order, ${\\cal{O}}(m^{-3})$, we have the contribution from the higher-derivative terms\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho}\\_{\\rm hd}(p,q)=-\\frac{e^3}{240\\pi m^3}\n\\bigg\\{&\\eta^{\\mu\\nu}\\Big[ p^2(2q-p)^{\\rho}+q^2(q-2p)^{\\rho}+(p.q)(q-p)^{\\rho}\\Big]\\nonumber\\\\\n+&\\eta^{\\mu\\rho}\\Big[p^2(4p+2q)^{\\nu}+q^2(3p+q)^{\\nu}+(p.q)(4p+q)^{\\nu}\\Big]\\nonumber\\\\\n-& \\eta^{\\nu\\rho}\\Big[p^2(p+3q)^{\\mu}+q^2(2p+4q)^{\\mu}+(p.q)(p+4q)^{\\mu}\\Big]\\nonumber\\\\\n+&p^{\\mu}q^{\\rho}(q-p)^{\\nu}+p^{\\rho}q^{\\mu}(q-p)^{\\nu}-p^{\\mu}\np^{\\rho}(2p+q)^{\\nu}+q^{\\mu}q^{\\rho}(p+2q)^{\\nu}\\bigg\\}\\sin\\big(\\frac{p\\wedge q}{2}\\big).\n \\label{eq:c12}\\end{aligned}$$ \nThis expression corresponds to the higher-derivative correction to the 3-photon vertex.\nIt is important to observe that in the commutative limit, the graphs (a) and (b) cancel each other, so that the induced 3-photon vertex is completely removed in the scalar QED. We can understand this result from the charge conjugation invariance of the scalar QED in any space-time dimension, known as Furry’s theorem, that forbids the presence of an odd number of photon lines in the case of commutative theory. Another important aspect from our analysis is the absence of the Chern-Simons self-coupling *ε**μ**ν**λ**A**μ* ⋆ *A**ν* ⋆ *A**λ* for the noncommutative scalar QED3 effective action, which is only generated in the case of fermionic electrodynamics.\nOne-loop ⟨*A**A**A**A*⟩ vertex\n------------------------------\nThe full contribution to the ⟨*A**A**A**A*⟩ part is determined by considering three different types of diagrams that are depicted in Fig. [oneloop3]. Since all of these graphs have 4 external bosonic legs, 24 different permutations for each graph must be considered in order to obtain the fully symmetrized contribution. Hence, the full contribution can be formally written as\n \n$$\\begin{aligned}\n \\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}= \\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a)} +\n \\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(b)} +\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(c)} =\\sum\\_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a,i)}+\n \\sum\\_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(b,i)}+\n \\sum\\_{i=1}^{24}\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(c,i)}.\n \\label{eq:c17}\\end{aligned}$$ \nWe shall present next the explicit discussion for the first contribution of each graph, whereas the remaining graphs are obtained by a direct permutation of momenta and spacetime indices.\n [oneloop3]\nThe box diagram contribution represented in graph (a) has the following expression\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho\\sigma}\\_{(a,1)}&=e^{4}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{(p+2k)^{\\mu}(2p+q+2k)^{\\nu}(2p+2q+s+2k)^{\\rho}(p+q+s+2k)^{\\sigma}}{[(p+q+s+k)^{2} -m^{2}][(p+q+k)^{2}-m^{2}][(p+k)^{2}-m^{2}][k^{2}-m^{2}]}\ne^{\\frac{i}{2}p \\wedge q}e^{\\frac{i}{2}(p+q)\\wedge s},\n \\label{eq:c13}\\end{aligned}$$ \nwhere we have labeled the momenta (*p*, *q*, *s*, *r*) accordingly to the spacetime indices of the external legs (*μ*, *ν*, *ρ*, *σ*). Moreover we have adopted the notation, in order to satisfy the energy-momentum conservation, where the momenta flow satisfies the relation *r* = *p* + *q* + *s*. The remaining contributions from the other 23 box diagrams, coming from the S-matrix expansion, can easily be obtained from the equation by considering the respective permutation. Next, we have the contribution from the bubble diagram represented in (b), which is given by\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho\\sigma}\\_{(b,1)} =e^{4} \\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}}{[(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]}\\cos\\big(\\frac{p \\wedge q}{2}\\big)\\cos\\big(\\frac{s\\wedge (p+q)}{2}\\big).\n \\label{eq:c14}\\end{aligned}$$ \nAt last, the triangle contribution shown in graph (c) is written as\n \n$$\\begin{aligned}\n\\Pi^{\\mu\\nu\\rho\\sigma}\\_{(c,1)} =-e^{4}\\int \\frac{d^{d}k}{(2\\pi)^{d}} \\frac{\\eta^{\\mu\\nu}(2p+2q+s+2k)^{\\rho}(p+q+s+2k)^{\\sigma}}{[(p+q+s+k)^{2}-m^{2}][(p+q+k)^{2}-m^{2}][k^{2}-m^{2}]}\\cos\\big( \\frac{p \\wedge q}{2}\\big) e^{\\frac{i}{2} (p+q) \\wedge s}.\n \\label{eq:c15}\\end{aligned}$$ \nThe Feynman expressions of the graphs (a), (b) and (c) show that all of them are planar, making the evaluation of the momentum integration easier by dimensional regularization. Hence the resulting expressions from the contributions to, evaluated in the highly noncommutative limit, where *p*2, *q*2, *s*2 ≪ *m*2, are written as follows\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a,1)}&=\\frac{1}{4}\\times \\frac{ie^{4}}{12\\pi m}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n~e^{\\frac{i}{2}p\\wedge q}e^{\\frac{i}{2}r\\wedge s},\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(b,1)}&= \\frac{1}{2}\\times\\frac{ie^{4}}{8\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\n\\cos\\big(\\frac{p\\wedge q}{2}\\big)\\cos\\big(\\frac{r\\wedge s}{2}\\big),\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(c,1)}&=1\\times \\frac{-ie^{4}}{8\\pi m} ~\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(\\frac{p\\wedge q}{2}\\big)\\cos\\big(\\frac{r\\wedge s}{2}\\big),\n \\label{eq:c16}\\end{aligned}$$ \nwhere the coefficients $\\frac{1}{4}$, $\\frac{1}{2}$ and 1 are the symmetry factors for the graphs (a), (b) and (c) of Fig. [oneloop3], respectively. We then apply to the results all the 24 permutations, necessary to evaluate, yielding\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a)}&=\\frac{ie^{4}}{6\\pi m}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\n\\Big[\\cos\\big(12\\big)\\cos\\big(34\\big)+\\cos\\big(13\\big)\\cos\\big(24\\big)+\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(b)}&=\\frac{ie^{4}}{2\\pi m}\n\\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(12\\big)\\cos\\big(34\\big)+\\eta^{\\mu\\rho}\n\\eta^{\\nu\\sigma}\\cos\\big(13\\big)\\cos\\big(24\\big)+\n\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n\\nonumber\\\\\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(c)}&=- \\frac{ie^{4}}{\\pi m}\n\\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos\\big(12\\big)\\cos\\big(34\\big)+\\eta^{\\mu\\rho}\n\\eta^{\\nu\\sigma}\\cos\\big(13\\big)\\cos\\big(24\\big)+\n\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos\\big(14\\big)\\cos\\big(23\\big)\\Big],\n \\label{eq:c18}\\end{aligned}$$ \nwhere we have introduced, by simplicity of the upcoming analysis, the following notation for the NC momenta product: $\\left(12 \\right) \\equiv \\left(\\frac{p\\wedge q}{2}\\right)$, $\\left(13 \\right) \\equiv \\left(\\frac{p\\wedge s}{2}\\right)$, $\\left(14 \\right) \\equiv \\left(\\frac{p\\wedge r}{2}\\right)$, $\\left(23 \\right) \\equiv \\left(\\frac{q\\wedge s}{2}\\right)$, $\\left(24 \\right) \\equiv \\left(\\frac{q\\wedge r}{2}\\right)$, and $ \\left(34\\right) \\equiv \\left(\\frac{s\\wedge r}{2}\\right)$. Finally, we substitute the results into the Eq. to obtain the total one-loop contribution expression corresponding to the photon 4-point function\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}=\\frac{ie^{4}}{\\pi m}\\Bigg\\{\n&\\frac{1}{6}\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big) \\Big[\\cos(12)\\cos(34)+\\cos(14)\\cos(23)+\\cos(13)\\cos(24)\\Big]\\nonumber\\\\\n-&\\frac{1}{2} \\Big[\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\cos(12)\n\\cos(34)+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\cos(13)\\cos(24)+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\cos(14)\\cos(23)\\Big]\\Bigg\\}.\n \\label{eq:c19}\\end{aligned}$$ \nWe can verify whether the quartic vertex satisfy the Ward identity in scalar QED. First, we consider the commutative limit, i.e. *θ* → 0, so that the one-loop contribution is reduced to\n \n$$\\lim\\_{\\theta\\rightarrow 0}\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}=\\frac{ie^{4}}{\\pi m}\n\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big) \\Big(\\frac{1}{2}+\\frac{1}{2}-1\\Big)=0,\n \\label{eq:c20}$$ \nshowing that the photon quartic self-coupling in the order ${\\cal{O}}(m^{-1})$ is absent in the Abelian theory; however higher order contributions could be nonvanishing, corresponding to nonlinear Euler-Heisenberg-like terms. Moreover, in the case of the violation of the Ward identity, a nonvanishing result for the contribution would generate a four photon interaction term in the effective action of the type\n \n$$\\begin{aligned}\n\\lim\\_{\\theta\\rightarrow 0}{\\cal{S}}\\_{\\rm eff}[AAAA]\\sim\\int d^{3}x~\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}+\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)A\\_{\\mu}(x) A\\_{\\nu}(x) A\\_{\\rho}(x)A\\_{\\sigma}(x),\n \\label{eq:c21}\\end{aligned}$$ \nthat is not explicitly gauge invariant. Hence, with the result we conclude that the gauge invariance is satisfied in our analysis of ${\\cal{O}}(m^{-1})$ terms at the one-loop approximation.\nOn the other hand, in the noncommutative case, as it is well known, we expect to find the relevant Feynman rule corresponding to the 4-photon interaction term at this order. To accomplish that, we start working separately each one of the tensor terms present in the function $\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}$ Eq. . We shall illustrate the analysis for the terms proportional to *η**μ**ν**η**ρ**σ*, the remaining terms can be evaluated in the same fashion. Hence, by picking the pieces that are proportional to *η**μ**ν**η**ρ**σ* in, we have that\n \n$$\\mathcal{I}^{\\mu\\nu,\\rho\\sigma}\\equiv\n\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big[-2\\cos(12)\\cos(34)+ \\cos(14)\\cos(23)+\\cos(13)\\cos(24)\\Big]. \\label{eq:c22}$$ \nThe major work here consist in simplifying the trigonometric part of this function by making use of the energy-momentum conservation *r* = *p* + *q* + *s*, which in the new notation is renamed as 4 = 1 + 2 + 3, together with the manipulation of some trigonometric identities, e.g. cos*α*cos*β* = cos(*α* + *β*) + sin*α*sin*β*. After some laborious but straightforward calculation, we arrive at the desired expression\n \n$$\\mathcal{I}^{\\mu\\nu,\\rho\\sigma}=\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big[\\sin(14)\\sin(23)+\\sin(13)\\sin(24)\\Big].\n \\label{eq:c24}$$ \nSimilarly, we can apply the same process in order to simplify the remaining terms, proportional to *η**μ**ρ**η**ν**σ* and *η**μ**σ**η**ν**ρ*, so that it yields to\n \n$$\\begin{aligned}\n\\mathcal{I}^{\\mu\\rho,\\nu\\sigma}&=\\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big[\\sin(12)\\sin(34)-\\sin(14)\\sin(23)\\Big],\n\\nonumber\\\\\n\\mathcal{I}^{\\mu\\sigma,\\nu\\rho}&=- \\frac{ie^{4}}{6\\pi m}\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big[\\sin(12)\\sin(34)+\\sin(13)\\sin(24)\\Big].\n \\label{eq:c25}\\end{aligned}$$ \nHence, by considering the results from our manipulations, Eqs. and, we can rewrite in a convenient form as the following\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}=\\frac{ie^{4}}{6\\pi m}\\bigg[&\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin\\big(\\frac{p\\wedge s}{2}\\big)\\sin\\big(\\frac{q\\wedge r}{2}\\big)\\nonumber\\\\\n+&\\Big(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big)\\sin\\big(\\frac{p\\wedge r}{2}\\big)\\sin\\big(\\frac{q\\wedge s}{2}\\big)\\nonumber\\\\\n+&\\Big(\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin\\big(\\frac{p\\wedge q}{2}\\big)\n\\sin\\big(\\frac{s\\wedge r}{2}\\big)\\bigg],\n \\label{eq:c27}\\end{aligned}$$ \nwhere we have reintroduced the notation in terms of the external momenta *p*, *q*, *s*, *r*. As we can observe, the expression inside the bracket corresponds exactly to the Feynamn vertex of the 4-photon interaction within the noncommutative *U*⋆(1) gauge theory.\nFinally, we can gather the leading ${\\cal{O}}(m^{-1})$ contributions from the one-loop order parts related to the two, three and four-point functions, Eqs. , and, respectively, so that we can write the complete expression of the NC Maxwell action as\n \n$$i{\\cal{S}}\\_{\\rm eff}\\bigg|\\_{{\\cal{O}}(m^{-1})}=-\\frac{ie^{2}}{96\\pi m}\\int~d^{3}x~F\\_{\\mu\\nu}\\star F^{\\mu\\nu},\n\\label{eq:c28}$$ \nin which the field strength tensor in the NC framework is defined as *F**μ**ν* = ∂*μ**A**ν* − ∂*ν**A**μ* + *i**e*[*A**μ*, *A**ν*]⋆. As we have previously discussed, this action is manifestly *U*⋆(1) gauge invariant under the transformation *U* = *e*⋆*i**e**λ*, where the field strength has the following transformation law *F**μ**ν* → *U* ⋆ *F**μ**ν* ⋆ *U*− 1.\nRegarding the higher-derivative corrections to the 4-photon vertex, corresponding to the next to leading order ${\\cal{O}}(m^{-3})$ terms, we arrive at a result involving a long expression which can be found in the Appendix [sec-appA]. This ${\\cal{O}}(m^{-3})$ result can be seen as the 3D version of the Euler-Heisenberg Lagrangian. We notice that in the commutative limit, the gauge invariant field strength is defined as *f**μ**ν* = ∂*μ**A**ν* − ∂*ν**A**μ*, so that the commutative version of the effective action only receives contribution from the one-loop ⟨*A**A*⟩ part, the remaining contributions are vanishing. Thus, the commutative one-loop effective action in the presence of the higher-derivative term, at the next to leading order, is described as\n \n$$\\lim\\limits\\_{\\theta\\rightarrow 0}i{\\cal{S}}\\_{\\rm eff}\\bigg|\\_{{\\cal{O}}(m^{-3})}=\n -\\frac{ie^{2}}{96\\pi m}\\int d^{3}x~f\\_{\\mu\\nu}f^{\\mu\\nu}+\\frac{ie^{2}}{1920\\pi m^{3}}\\int d^{3}x~\nf\\_{\\mu\\nu}\\Box f^{\\mu\\nu}$$ \nSome comments about the result are now in place. Regarding discrete symmetries, following the aforementioned discussion in Sec. [sec2.1], it is easy to show that the above one-loop effective action is also invariant under all of the discrete symmetries and therefore we have not faced any anomalous symmetry at this order.\nSince the effective action, arising from the 2, 3 and 4-point functions at the order ${\\cal{O}}(m^{-1})$, is exactly gauge invariant, it is possible to conclude that no further ${\\cal{O}}(m^{-1})$ terms are generated from higher-order graphs with *n* > 4 external photon legs. According to this reasoning, we can also discuss the gauge invariance of the higher-derivative terms generated by considering the next to leading order ${\\cal{O}}(m^{-3})$ of our expansion. Actually, it is possible to make use of a dimensional analysis, based on arguments of gauge invariance, to establish the perturbative generation of all possible gauge invariant higher-derivative terms in the one-loop effective action. As an example, if we consider all of the ${\\cal{O}}(m^{-3})$ contributions up to the diagrams with *n* = 6 external photon legs, we can generate the following effective higher-derivative Lagrangian\n \n$$\\mathcal{L}\\_{\\rm hd} = \\frac{1}{6\\mu^2} \\nabla \\_\\mu F^{\\mu \\nu }\\star \\nabla^\\lambda F\\_{\\lambda \\nu }\n+\\frac{1}{6\\mu^2} \\nabla \\_\\lambda F^{\\mu \\nu }\\star \\nabla^\\lambda F\\_{\\mu \\nu }\n- \\frac{e}{18\\mu^2} F^{\\mu \\nu }\\star F\\_{\\nu \\lambda}\\star F^{\\lambda } \\_{\\;\\;\\; \\mu},$$ \nwhere ∇*μ* = ∂*μ* + *i**e*[*A**μ*, ]⋆ is the covariant derivative in the adjoint representation and *μ* ∼ *m* is the mass scale of the theory. This expression can been seen as a noncommutative extension of the Alekseev-Arbuzov-Baikov effective Lagrangian.\nOne last comment about the photon effective action is in regard of some of the nonlinear contributions. It is well known that either fermionic or scalar electrodynamics generate nonlinear corrections of quantum character to the photon dynamics. In the case of a (2 + 1) spacetime the effective Euler-Heisenberg Lagrangian density has the appearance of fractional powers of the field strength \n \n$$\\mathcal{L}\\_{\\rm EH} \\sim \\left( e \\sqrt{B^2-E^2} \\right)^{\\frac{3}{2}}.$$ \nHence, it is reasonable to expect that the coordinates noncommutativity would also present corrections to this nonlinear coupling term.\nComparison with NC-QED3\n=======================\nIn this section, we shall present a comparative discussion of the 2,3 and 4-point functions in the case of fermionic and bosonic matter fields coupled to the photon. It is notable that the presence of the trace of *γ* matrices in fermionic QED3 leads to two sectors characterized as: odd and even in regard to parity symmetry. The former is related to the induced Chern-Simons (CS) terms, appearing with odd powers of the fermion mass *m**e*, while the latter sector contributes to the induced Maxwell (M) terms, with even powers of *m**e*. Now in the scalar framework, odd parity terms are absent, and only parity preserving terms are present in the induced effective action, which can be understood as the main difference between these two matter fields. These results can be briefly described in terms of a mass expansion as the following:\n|c|c|c|c|c|c|c| & Induced action & NC-QED3& NC-scalar QED3 \n${\\cal{O}}(m^{0})$ & ordinary NC-CS & $\\checkmark$ & × \n ${\\cal{O}}(m^{-1})$ & ordinary NC-M & $\\checkmark$ & $\\checkmark$ \n ${\\cal{O}}(m^{-2})$& higher-derivative NC-CS & $\\checkmark$ & × \n ${\\cal{O}}(m^{-3})$& higher-derivative NC-M & $\\checkmark$ & $\\checkmark$ \n& & & \n ${\\cal{O}}(m^{-2\\ell})$& higher-derivative NC-CS & $\\checkmark$ & × \n ${\\cal{O}}(m^{-2\\ell-1})$& higher-derivative NC-M & $\\checkmark$ & $\\checkmark$ \nHere some important comments are in order. In NC-QED3, the structure of the kinetic part, coming from the *n* = 2 photon external legs analysis, has the contribution of two types of terms: a CS-type parity violating term *e*2*A*∂□ℓ*A*, at the order ${\\cal{O}}(m^{-2\\ell})$, and a M-type parity preserving term *e*2*A*∂∂□ℓ*A*, at the order ${\\cal{O}}(m^{-2\\ell-1})$. We can observe that the mass dimension of the CS and M-type terms is given by 3 + 2ℓ and 4 + 2ℓ, respectively. Thus, we can conclude that in order to have a gauge invariant CS-type action at the order ${\\cal{O}}(m^{-2\\ell})$, it is necessary to consider all contributions originating from the graphs with *n* = 2, 3, …, 3 + 2ℓ photon legs. On the other hand, to obtain a gauge invariant M-type action at the order ${\\cal{O}}(m^{-2\\ell-1})$, it is necessary to consider all contributions arising from the relevant graphs with *n* = 2, 3, …, 4 + 2ℓ photon legs.\nFurthermore, in (2 + 1) dimensions, the number of degrees of freedom of the charged boson and Dirac fermion (in the 2 dimensional representation) is equal and hence it is easy to see that the numerical coefficient appearing in the 2-point function would be the same as in the fermionic QED3. Now for the *n* = 3 graphs, in the case of NC-QED3, there are only the contribution of two triangle graphs in the one-loop order. These contributions are the same as in the NC-scalar QED3 because the additional graph is identically zero. Thus, it is easy to realize that the final result in the parity preserving sector for both cases is the same. The last type of diagrams is for *n* = 4 legs, that for the fermionic case we have the contribution of the box diagram only, i.e. type (a). The expression for this diagram at the leading order of ${\\cal{O}}(m^{-1})$ is given by\n \n$$\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{(a,1)}\\bigg|\\_{\\tiny\\mbox{NC-QED}}=-\\frac{ie^{4}}{3\\pi m}\\Big(\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}-2\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+ \\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}\\Big)e^{\\frac{i}{2}p\\wedge q}e^{\\frac{i}{2}r\\wedge s}.$$ \nBy considering all of the 24 permutations and some manipulations we obtain the following\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\rm total}\\bigg|\\_{\\tiny\\mbox{NC-QED}}=\\frac{4ie^4}{3\\pi m}\\bigg[ \\Big(&\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}\\Big)\\sin{\\big(\\frac{p\\wedge r}{2}\\big)}\\sin{\\big(\\frac{q\\wedge s}{2}\\Big)}\\nonumber\\\\\n+\\Big(&\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin{\\big(\\frac{q\\wedge r}{2}\\big)}\\sin{\\big(\\frac{p\\wedge s}{2}\\big)}\\nonumber\\\\\n+\\Big(&\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}-\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho}\\Big)\\sin{\\big(\\frac{p\\wedge q}{2}\\big)}\\sin{\\big(\\frac{s\\wedge r}{2}\\big)}\\bigg],\\end{aligned}$$ \nwhich has the same tensor and momenta structure as the standard 4-photon vertex, but with a different numerical coefficient. In this case, we can conclude that the commutative limit of this function is also satisfied, and have a vanishing result as we would expect.\nFinal remarks\n=============\nIn this paper we have considered the perturbative evaluation of the effective action for photon in the context of scalar QED in the (2 + 1) noncommutative spacetime. Our main interest was to determine to what extent the spin of the matter fields can change the effective action in the presence of the spacetime noncommutativity. Since the number of degrees of freedom of the charged boson and Dirac fermion (in the 2 dimensional representation) is equal, the main difference between these fields is solely to the well known presence of different couplings in the case of scalar QED, implying in new types of graphs. An important drawback from the scalar QED in terms of the induced effective action for the photon is the absence of parity violating terms, showing that no Chern-Simons terms are generated when charged scalar fields are considered.\nThe perturbative analysis followed the computation of the ⟨*A**A*⟩, ⟨*A**A**A*⟩ and ⟨*A**A**A**A*⟩ vertex functions. A more detailed and careful analysis was necessary to the computation of the 4 point vertex, where the process of symmetrization is rather intricate. The evaluation of the noncommutative Maxwell action ∫*F**μ**ν* ⋆ *F**μ**ν* was done by considering the highly noncommutative limit of these 1*P**I* functions at order ${\\cal{O}}(m^{-1})$.\nIn addition, we have considered the generation of ${\\cal{O}}(m^{-3})$ terms, which are higher-derivative terms for the photon fields, and can be seen as the noncommutative generalization of the phenomenological Alekseev-Arbuzov-Baikov effective Lagrangian. Another possible terms to be present in the gauge invariant photon’s effective action are those nonlinear couplings, e.g. analogous to the Euler-Heisenberg action, thus within our study of NC-scalar QED, we would find noncommutative corrections to the fractional powers of the field strength that appear in the (2 + 1) dimensional Euler-Heisenberg action.\nAcknowledgements\n----------------\nWe are grateful to M.M. Sheikh-Jabbari for fruitful discussions and comments on the manuscript. R.B. acknowledges partial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq Projects No. 304241/2016-4 and 421886/2018-8 ) and Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG Project No. APQ-01142-17).\nHigher-derivative corrections to the 4-photon vertex\n====================================================\nThe higher-derivative contributions of the graphs, depicted in Fig. [oneloop3], to the 4-photon vertex at the next to leading order, ${\\cal{O}}(m^{-3})$, are described as the following:\n \n$$\\begin{aligned}\n\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}}=\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(a)}\n+\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(b)}\n+\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(c)}\\end{aligned}$$ \nFor simplicity, we define $\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(a)}=\\frac{ie^4}{240 \\pi m^3 }\\widetilde{\\Gamma}^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(a)}$ and $\\Gamma^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(b+c)}=-\\frac{ie^{4}}{24\\pi m^{3}}\\widetilde{\\Gamma}^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(b+c)}$ in which\n \n$$\\begin{aligned}\n\\widetilde{\\Gamma}^{\\mu\\nu\\rho\\sigma}\\_{\\tiny\\mbox{hd}~(a)}=\\Bigg\\{&\\eta^{\\mu \\nu} \\Big[ - 2 p^{\\sigma} p^{\\rho} + 4 q^{\\sigma} p^{\\rho} -\n6 s^{\\sigma} p^{\\rho} - q^{\\rho} p^{\\sigma} - s^{\\rho} p^{\\sigma} + 2\nq^{\\sigma} s^{\\rho} + q^{\\rho} q^{\\sigma} - 3 q^{\\rho} s^{\\sigma} - 4\ns^{\\rho} s^{\\sigma}\\Big]\n\\nonumber\\\\\n +& \\eta^{\\mu \\rho} \\Big[ - 2 \\left(p^{\\sigma} + 3\nq^{\\sigma} - 2 s^{\\sigma} \\right) p^{\\nu} - q^{\\nu} p^{\\sigma} -\ns^{\\nu} p^{\\sigma} - 4 q^{\\nu} q^{\\sigma} - 3 s^{\\nu} q^{\\sigma} + 2\nq^{\\nu} s^{\\sigma} + s^{\\nu} s^{\\sigma}\\Big]\n\\nonumber\\\\\n+& \\eta^{\\mu \\sigma} \\Big[4 \\left(2\np^{\\rho} + q^{\\rho} + s^{\\rho} \\right) p^{\\nu} + 4 q^{\\nu} p^{\\rho} +\n4 s^{\\nu} p^{\\rho} + 2 q^{\\rho} s^{\\nu} + q^{\\nu} q^{\\rho} + 2 q^{\\nu}\ns^{\\rho} + s^{\\nu} s^{\\rho}\\Big]\n\\nonumber\\\\\n+& \\eta^{\\nu \\"},"data_id":{"kind":"string","value":"arxiv_0002442"}}},{"rowIdx":593,"cells":{"content":{"kind":"string","value":"Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order\n=======================================================================================================\nWe study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the *H*1(Ω)-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate 1/*t*1/*β* for some *β* > 0. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in *H*1(Ω).\nMSC (2010): 76A05, 76A20, 35B40, 35Q35, 35K35, 35K65\nKeywords: non-Newtonian fluid, power-law fluid, Ellis fluid, degenerate parabolic equation, weak solution, long-time asymptotics, thin-film equation\nIntroduction\n============\nAim of the paper\n----------------\nThe present paper is concerned with the asymptotic behaviour of positive weak solutions to fourth-order quasilinear (doubly) degenerate parabolic problems as they arise in the modelling of non-Newtonian thin-film flows. It turns out that, for large times, fluids with a shear-rate dependent viscosity exhibit a specific asymptotic behaviour, depending on their shear-thickening or shear-thinning nature, respectively.\nWe consider a thin layer of a viscous, non-Newtonian and incompressible fluid on an impermeable flat bottom, as sketched in Figure [fig:thin-film].\n[domain=0:3\\*pi, xscale=1.2, yscale=0.8] plot (,0.3\\*cos(r)+2); plot[domain=0:3\\*pi] (,0) – plot[domain=3\\*pi:0] (,0.3\\*cos(r)+2); (3\\*pi+0.4,0) node[right] *x* – (0,0) – (0,3.5) node[above] *z*; (0,0) – (-0.75,-0.75) node[below] *y*; (0,-0.3) – (0.3, 0); (0.5,-0.3) – +(0.3, 0.3); (1,-0.3) – +(0.3, 0.3); (1.5,-0.3) – +(0.3, 0.3); (2,-0.3) – +(0.3, 0.3); (2.5,-0.3) – +(0.3, 0.3); (3,-0.3) – +(0.3, 0.3); (3.5,-0.3) – +(0.3, 0.3); (4,-0.3) – +(0.3, 0.3); (4.5,-0.3) – +(0.3, 0.3); (5,-0.3) – +(0.3, 0.3); (5.5,-0.3) – +(0.3, 0.3); (6,-0.3) – +(0.3, 0.3); (6.5,-0.3) – +(0.3, 0.3); (7,-0.3) – +(0.3, 0.3); (7.5,-0.3) – +(0.3, 0.3); (8,-0.3) – +(0.3, 0.3); (8.5,-0.3) – +(0.3, 0.3); (9,-0.3) – +(0.3, 0.3);\n[fig:thin-film]\nIn addition to the non-Newtonian fluid rheology, the following modelling assumptions are crucial for the analysis of the resulting partial differential equations. First, the fluid flow is assumed to be uniform in one horizontal direction (in *y*-direction in Figure [fig:thin-film]), such that we obtain a (spatially) one-dimensional problem. Moreover, we assume that the characteristic height of the fluid layer is rather thin compared to its characteristic length and consider the asymptotic limit of a vanishing aspect ratio. Based on a non-Newtonian Navier–Stokes system, we use the so-called lubrication approximation in order to derive an evolution equation for the height *u* = *u*(*t*, *x*) ≥ 0 of the fluid film at time *t* > 0 and spatial position *x* ∈ Ω, where Ω ⊂ R is a bounded interval. We neglect gravitational effects and assume that the dynamics of the flow is driven by capillarity only. Finally, we prescribe a no-slip condition on the lower boundary of the fluid film. However, the mathematical analysis of the present paper does also apply to the case of Navier-slip conditions.\nAs constitutive laws for the non-Newtonian shear-dependent fluid we consider so-called **power-law fluids**, also called **Ostwald-de Waele fluids**, and so-called **Ellis-fluids**; see below for more details on these material laws. In the case of power-law fluids, when prescribing a no-slip condition on the lower boundary, the resulting evolution problem reads\n \n$$\\label{eq:PDE\\_power-law}\n \\begin{cases}\n u\\_t + \\bigl(u^{\\alpha+2} |u\\_{xxx}|^{\\alpha-1} u\\_{xxx}\\bigr)\\_x = 0, &\n t > 0,\\ x \\in \\Omega,\n \\\\\n u\\_x(t,x) = u\\_{xxx}(t,x) = 0,\n &\n t > 0,\\ x \\in \\partial\\Omega,\n \\\\\n u(0,x) = u\\_0(x), \n &\n x \\in \\Omega.\n \\end{cases}$$ \nNote that $\\eqref{eq:PDE\\_power-law}\\_1$ is a fourth-order quasilinear parabolic equation that is doubly degenerate in the sense that the degeneracy occurs both with respect to the unknown *u* and with respect to its third spatial derivative *u**x**x**x*. The Neumann-type boundary conditions *u**x* = *u**x**x**x* = 0 on ∂Ω reflect the zero-contact angle condition and the no-flux condition at the lateral boundary, respectively. Finally, *u*0 > 0 denotes the given positive initial film height. Note that for 0 < *α* < 1, the coefficients of the highest-order term depend only Hölder continuously on the unknown and lower-order derivatives.\nIn the case of Ellis-fluids, we obtain the evolution equation\n \n$$\\label{eq:PDE\\_Ellis}\n \\begin{cases}\n u\\_t + a\\bigl(u^3 \\bigl[1+b|uu\\_{xxx}|^{\\alpha-1}\\bigr] u\\_{xxx}\\bigr)\\_x = 0, &\n t > 0,\\ x \\in \\Omega,\n \\\\\n u\\_x(t,x) = u\\_{xxx}(t,x) = 0,\n &\n t > 0,\\ x \\in \\partial\\Omega,\n \\\\\n u(0,x) = u\\_0(x), \n &\n x \\in \\Omega.\n \\end{cases}$$ \nHere, *a*, *b* > 0 are positive physical parameters, depending on the constant surface tension, the flow-behaviour exponent *α* and the characteristic viscosity of the fluid. However, for clarity of presentation, we drop these parameters in our analysis since they do not affect our arguments. This equation has for instance been studied in in the context of self-similar solutions and local strong solutions, respectively.\nThe main difference in the classification of and is that is doubly degenerate in the sense that we loose parabolicity if either the unknown *u* or its third spatial derivative *u**x**x**x* become zero. In contrast, is degenerate only in the unknown *u* itself.\nFor *α* = 1 we recover in both equations and the well-known Newtonian thin-film equation\n \n$$\\label{eq:PDE\\_Newtonian}\n u\\_t + \\bigl(u^3 u\\_{xxx}\\bigr)\\_x = 0, \n \\quad\n t > 0,\\ x \\in \\Omega.$$ \nThis equation is studied extensively in the mathematical literature. For results concerning existence, uniqueness and stability of weak solutions to we refer the reader for instance to the works.\nMain results of the paper – Stability of steady states and long-time behaviour of positive weak solutions\n---------------------------------------------------------------------------------------------------------\nIn the present paper we study the behaviour of positive weak solutions to and, respectively, for large times. Note that we consider only the case of strictly positive initial values *u*0 > 0 since these allow us to find a positive time up to which solutions remain strictly positive.\nThe main results of the paper are the following: We prove local existence of positive weak solutions to the power-law thin-film equation for all flow-behaviour exponents *α* > 0, see Theorem [thm:LocalExPL] below. In the case *α* > 1 of shear-thinning power-law fluids, even global existence of non-negative weak solutions has been established in, using a two-step regularisation scheme, Galerkin approximation and energy/entropy methods. Since the present paper is concerned with stability of positive steady states, we are only interested in positive weak solutions. Therefore, we use a simpler regularisation method that allows us (only) to construct local positive weak solutions, but for all flow-behaviour exponents *α* > 0. These solutions can then be extended to global weak solutions as long as they are close to steady states.\nMoreover, again for all *α* > 0, we can characterise positive steady states of the power-law thin-film equation by positive constants, cf. Theorem [thm:charsteadystates] below. As already mentioned, the long-time behaviour of solutions that are initially close to a steady state $\\bar{u}\\_0 = \\fint\\_\\Omega u\\_0\\, dx$ depends strongly on the choice of the flow-behaviour exponent *α*, i.e., on the shear-thinning, respectively shear-thickening nature of the fluid. The main result concerning global existence and stability properties of steady states is the following:\nFix *α* > 0. Then there exists an *ɛ* > 0 such that, for all positive initial values *u*0 ∈ *H*1(Ω) with ∥*u*0 − *ū*0∥*H*1(Ω) ≤ *ɛ*, problem possesses at least one global positive weak solution\n \n$$u \\in C\\bigl([0,\\infty);H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1,\\text{loc}}\\bigl((0,\\infty);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\quad \\text{with} \\quad\n u\\_t \\in L\\_{\\frac{\\alpha+1}{\\alpha},\\text{loc}}\\bigl((0,\\infty);(W^1\\_{\\alpha+1,B}(\\Omega))^\\prime\\bigr),$$ \nsatisfying the boundary condition *u**x* = 0 on ∂Ω pointwise for almost every *t* ≥ 0. Moreover, this global solution has the following asymptotic behaviour:\n* In the shear-thickening case 0 < *α* < 1, there exists a positive but finite time 0 < *t*\\* < ∞ such that\n \n*u*(*t*, ⋅ ) → *ū*0in *H*1(Ω), as *t* → *t*\\*, and *u*(*t*, *x*) = *ū*0, *t* ≥ *t*\\*, *x* ∈ Ω.\n* In the shear-thinning case 1 < *α* < ∞, there exists a constant *C* > 0 such that\n \n$$\\|u(t) - \\bar{u}\\_0\\|\\_{H^1(\\Omega)} \\leq\n \\frac{C {\\varepsilon}}{\\bigl(1 + C {\\varepsilon}^{\\alpha-1} t\\bigr)^\\frac{1}{\\alpha-1}},\n \\quad\n 0 \\leq t < \\infty.$$\n* In the Newtonian case *α* = 1, there exist positive constants *C*, *γ* > 0 such that\n \n∥*u*(*t*) − *ū*0∥*H*1(Ω) ≤ *C**e*− *γ**t*, 0 ≤ *t* < ∞.\nNote that statement (iii) of this theorem is already well-known and can even be proved in ‘better’ function spaces with standard theory, see for instance the text books. Moreover, in the shear-thinning case (ii), convergence to steady states has already been proved in but without rate of convergence. In the cylindrical Taylor–Couette setting, statement (iii) has first been shown in in the framework of stable center manifolds. Similarly, the results in (i) and (ii) have been obtained in and, also in the cylindrical Taylor–Couette geometry.\nFinally, we prove global existence of positive weak solutions to the Ellis-law thin-film equation and provide a description of their asymptotic behaviour. For *α* ≥ 2, stability and exponential decay to equilibrium can again be obtained by standard techniques. However, for 1 < *α* < 2, these techniques are not applicable since the coefficients of the differential operator are merely Hölder continuous. For this range of flow-behaviour exponents we use energy methods to prove exponential asymptotic stability of steady states in *H*1(Ω).\nShear-Dependent non-Newtonian Fluids\n------------------------------------\nMany common liquids and gases, such as water and air, may reasonably be considered Newtonian. However, there is still a multitude of real fluids which are in fact non-Newtonian. Newtonian fluids are characterised by a perfectly linear dependence of the shear stress *σ*(*ε*) on the local strain rate *ε*, the constant fluid viscosity *μ* > 0 being the factor of proportionality. In contrast to that, shear-dependent non-Newtonian fluids feature a non-linear relation between the shear-rate and the viscous stress, *σ*(*ε*) = *μ*(∣*ε*∣)*ε*, where *μ*(∣*ε*∣) is the shear-dependent viscosity. That is, these fluids become more solid or more liquid under shear force. In the case in which the fluid viscosity increases with increasing shear rate, the corresponding fluids are called **shear-thickening**. On the contrary, fluids are called **shear-thinning** if their viscosity decreases with increasing shear-rate. In this paper, we are concerned with two classes of non-Newtonian fluids, so-called **power-law fluids** or **Ostwald–de Waele fluids** and **Ellis-fluids**.\n**Power-Law Fluids.** For **power-law fluids** or **Ostwald–de Waele fluids** the constitutive law for the effective fluid viscosity reads\n \n$$\\label{eq:mu\\_power-law}\n \\mu(|\\epsilon|) = \\mu\\_0 |\\epsilon|^{\\frac{1}{\\alpha}-1},$$ \nwith a characteristic viscosity *μ*0 > 0 and a flow-behaviour exponent *α* > 0. For these fluids, the relation between the local strain and the viscous stress is\n \n$$\\sigma(\\epsilon) = \\mu\\_0 |\\epsilon|^{\\frac{1}{\\alpha}-1} \\epsilon.$$ \nNote that the corresponding fluid is shear-thickening for flow-behaviour exponents 0 < *α* < 1, while it is shear-thinning for *α* > 1. In the case *α* = 1, we recover the Newtonian regime *μ*(∣*ε*∣) ≡ *μ*0 > 0 of a constant viscosity.\nHowever, it is observed in real-world applications (e.g. in polymeric systems) that, at ‘intermediate’ shear rates, fluids behave according to, while the at rather low and/or rather high shear rates, the viscosity approaches a Newtonian plateau. This is obviously not reflected by.\n**Ellis fluids.** As a second class of shear-dependent non-Newtonian fluids we consider fluids the rheology of which is described by the so-called **Ellis constitutive law** \n \n$$\\label{eq:mu\\_Ellis}\n \\frac{1}{\\mu(|\\epsilon|)}\n =\n \\frac{1}{\\mu\\_0} \\left(1 + \\left|\\frac{\\sigma(\\epsilon)}{\\sigma\\_{1/2}}\\right|^{\\alpha-1}\\right),\n \\quad\n \\alpha \\geq 1,\\\n 0 < \\sigma\\_{1/2} < \\infty,$$ \nwhere *σ*(*ε*) = *μ*(∣*ε*∣)*ε* is the viscous shear stress. Here, *μ*0 > 0 denotes the viscosity at zero shear stress and *σ*1/2 > 0 is the viscous shear stress at which the viscosity is reduced to *μ*0/2. Thus, for *α* > 1 and 0 < *σ*1/2 < ∞ the Ellis constitutive law describes a shear-thinning behaviour, i.e., the fluid viscosity decreases with increasing shear rate. For *α* = 1 or for *σ*1/2*α* − 1 → ∞, we recover a Newtonian behaviour. As an advantage over, the Ellis law has the ability to describe a shear-thinning behaviour for ‘moderate’ shear rates and a Newtonian plateau for rather low shear stresses, since for all *σ* ∈ R,\n \n$$\\frac{1}{\\mu(|\\epsilon|)}\n =\n \\frac{1}{\\mu\\_0} \\left(1 + \\left|\\frac{\\sigma(\\epsilon)}{\\sigma\\_{1/2}}\\right|^{\\alpha-1}\\right)\n \\longrightarrow\n \\frac{1}{\\mu\\_0},\n \\quad\n \\text{as}\n \\quad\n \\sigma\\_{1/2}^{\\alpha-1} \\to \\infty.$$ \nFor the majority of polymers and polymer solutions the flow-behaviour exponent *α* in varies in a range between 1 and 2, see e.g..\nA plot of the different constitutive laws for the fluid viscosity (Newtonian fluids, shear-thickening and shear-thinning power-law fluids and Ellis fluids) is offered in Figure [fig:constitutivelaws].\n[yscale=1.5, xscale=1.5] (0,0)–(5,0) node[right] shear rate ∣*ε*∣; (0,0)– (0,2.5) node[above] viscosity *μ*(∣*ε*∣); (0,1.5)–(4,1.5) node[right]; plot (,(1.5-0.3\\*abs())2 ); plot (,(0.4\\*abs())2); plot (, (1.5/pi)\\*(pi/2 - rad(atan(-8+(-0.2)3))) ); plot (, (sqrt((1.5/pi)\\*(pi/2 - rad(atan(-8+1.83))))-0.3\\*abs(-2.0))2 ); at (3.8,2.4) power-law, *α* < 1; at (1.3,2.1) power-law, *α* > 1; at (4.32,0.4) Ellis;\n[fig:constitutivelaws]\nOutline of the paper\n--------------------\nThe structure of the paper is as follows: In Section [sec:FunctionalSetting] we introduce the functional setting we will work in.\nIn Section [sec:LocalExistence] we prove local existence of positive weak solutions to the power-law thin-film equation and characterise positive steady states by positive constants.\nIn Section [sec:regularity-estimates] we derive regularity estimates for weak solutions that are valid as long as the solution stays bounded away from zero. More precisely, we prove a Łojasiewicz-Simon-type inequality that estimates the dissipation functional in terms of powers of the energy functional. Moreover, we provide a local *L*1-in-time estimate for the dissipation functional in terms of the energy at a slightly earlier time.\nSection [sec:shear-thickening] is concerned with the dynamic behaviour of solutions to the shear-thickening power-law problem. First, we prove global existence of positive weak solutions for initial film heights that are initially close to a constant in *H*1(Ω). Moreover, we show that these solutions converge to a positive constant in finite time and stay constant for all later times.\nSection [sec:shear-thinning] is concerned with the stability properties of solutions to the shear-thinning power-law thin-film equation. As in the shear-thickening case, it is shown that weak solutions exist globally time and stay positive if they are initially close to a steady state. Moreover, these positive global weak solutions are polynomially stable in *H*1(Ω) in the sense that they converge to a steady state (positive constant) at rate 1/*t*1/(*α* − 1), as time tends to infinity.\nIn Section [sec:Ellis] we study the non-Newtonian thin-film equation that arises when the constitutive law for the fluid viscosity is the Ellis-law. The corresponding Ellis fluids have a Newtonian plateau for small shear rates and behave like a shear-thinning power-law fluid for high shear rates. We observe exponential asymptotic stability of steady states in the *H*1(Ω)-norm.\nFunctional Framework\n====================\nIn this section we provide the functional setting that will be needed for the study of both the power-law and Ellis-law thin-film equations.\nThroughout this paper, we assume that Ω ⊂ R is a bounded interval. For *k* ∈ N and *p* ∈ [1, ∞) we denote by *W**p**k*(Ω) the usual Sobolev spaces with norm\n \n∥*v*∥*W**p**k*(Ω) = (∑*j* = 0*k*∥∂*j**v*∥*L**p*(Ω)*p*)1/*p*. \nWe then define the seminorm\n \n$$[v]\\_{W^s\\_p(\\Omega)}\n=\n\\int\\_{\\Omega} \\int\\_{\\Omega} \\frac{|v(x) - v(z)|^p}{|x-z|^{1+sp}}\\, dx\\, dz,\n\\quad\n1 \\leq p < \\infty,\\ 0 < s < 1,$$ \nand introduce the **fractional Sobolev spaces** by\n \n*W**p**s*(Ω) = {*v* ∈ *W**p*[*s*](Ω); ∥*v*∥*W**p**s*(Ω) < ∞}, 1 ≤ *p* < ∞, *s* ∈ R+ \\ N, \nwhere\n \n∥*v*∥*W**p**s*(Ω) = (∥*v*∥*W**p*[*s*](Ω)*p* + [∂[*s*]*v*]*W**p**s* − [*s*](Ω)*p*)1/*p*, 1 ≤ *p* < ∞, *s* ∈ R+ \\ N, \nwith [*s*] denoting the largest integer such that [*s*] ≤ *s*.\nWe now recall some important properties of these spaces. It is well-known (see, for instance, ) that, for 0 ≤ *s*0 < *s*1 < ∞, 1 < *p* < ∞, and 0 < *ρ* < 1, the space *W**p**s*(Ω) with *s* = (1 − *ρ*)*s*0 + *ρ**s*1, is the complex interpolation space between *W**p**s*1(Ω) and *W**p**s*0(Ω), in symbols\n \n*W**p**s*(Ω) = [*W**p**s*0(Ω), *W**p**s*1(Ω)]*ρ*. \nIn order to take the Neumann-type boundary conditions into account, we further introduce the Banach spaces\n \n$$W^{4\\rho}\\_{p,B}(\\Omega)\n=\n\\begin{cases}\n\\bigl\\{v \\in W^{4\\rho}\\_p(\\Omega); \nv\\_x = v\\_{xxx} = 0 \\text{ on } \\partial\\Omega\\bigr\\}, \n& 3 + \\frac{1}{p} < 4\\rho \\leq 4,\n\\\\[1ex]\n\\bigl\\{v \\in W^{4\\rho}\\_p(\\Omega); \nv\\_x = 0 \\text{ on } \\partial\\Omega\\bigr\\}, \n& 1+ \\frac{1}{p} < 4\\rho \\leq 3 + \\frac{1}{p},\n\\\\[1ex] \nW^{4\\rho}\\_p(\\Omega), & 0 \\leq 4\\rho \\leq 1+ \\frac{1}{p}.\n\\end{cases}$$ \nFor 4*ρ* ∈ (0, 4) \\ {1 + 1/*p*, 3 + 1/*p*}, the spaces *W**p*, *B*4*ρ*(Ω) are closed linear subspaces of *W**p*4*ρ*(Ω) and satisfy the interpolation property \n \n$$W^{4\\rho}\\_{p,B}(\\Omega) = \\bigl(L\\_p(\\Omega),W^4\\_{p,B}(\\Omega)\\bigr)\\_{\\rho,p},\n\\quad\n1 < p < \\infty.$$ \nLastly, we use *W**p*, 01(Ω) to denote the space of functions belonging to *W**p*1(Ω) with zero boundary condition.\nLocal Existence for the Power-Law Thin-Film Equation\n====================================================\nIn this section we prove local existence of positive weak solutions to the evolution problem\n \n$$\\label{eq:PDE}\n \\begin{cases}\n u\\_t + \\bigl(u^{\\alpha+2} |u\\_{xxx}|^{\\alpha-1} u\\_{xxx}\\bigr)\\_x\n =\n 0,\n & t>0,\\ x \\in \\Omega,\n \\\\\n u\\_x(t,x) = u\\_{xxx}(t,x) = 0, \n &\n t > 0,\\ x \\in \\partial\\Omega,\n \\\\\n u(0,x) = u\\_0(x),\n &\n x \\in \\Omega,\n \\end{cases}$$ \nfor flow-behaviour exponents *α* > 0, i.e., for both shear-thinning (*α* > 1) and shear-thickening (*α* < 1) power-law fluids. Moreover, we characterise the positive steady states of by positive constants (flat films of positive height).\nOur analysis strongly relies on an energy-dissipation estimate for the **energy functional**\n \n$$E[u] = \\frac{1}{2} \\int\\_\\Omega |u\\_x|^2\\,dx.$$ \nFormally testing the equation with the second derivative *u**x**x*, one finds that *E*[*u*](*t*) decreases along solutions to. More precisely, solutions *u* to satisfy\n \n$$\\frac{d}{dt} E[u](t) = -D[u](t) = -\\int\\_\\Omega u^{\\alpha+2} |u\\_{xxx}|^{\\alpha+1}\\, dx.$$ \nWe call *D*[ ⋅ ] the **dissipation functional**.\nFor the purpose of local existence, we introduce in Section [ssec:ExistencePLreg] a regularised version of that removes the degeneracy in the third derivative *u**x**x**x*. For the regularised problem we apply standard parabolic theory in order to prove existence of positive strong solutions, emanating from positive initial values. In Section [ssec:ExistencePL] we provide uniform a-priori bounds for the solutions to the regularised problem and pass to the limit of a vanishing regularisation parameter in order to obtain local existence of positive weak solutions to the original problem.\nNote that for *α* > 1 (shear-thinning fluids) existence of global non-negative weak solutions is already proved in, where the authors use a more involved regularisation scheme. However, in the present paper we are only interested in positive solutions, but for all flow-behaviour exponents *α* > 0.\nIn order to simplify notation, we introduce, for a fixed *α* > 0, the function\n \n*ψ*: R → R, *s* ↦ *ψ*(*s*) = ∣*s*∣*α* − 1*s*, \nand rewrite the partial differential equation $\\eqref{eq:PDE}\\_1$ as\n \n$$u\\_t + \\bigl(u^{\\alpha+2} \\psi(u\\_{xxx})\\bigr)\\_x = 0,\n \\quad\n t > 0,\\ x \\in \\Omega.$$ \nNote that if *α* ≥ 1, then *ψ* ∈ *C*1(R) with *ψ*′(*s*) = *α*∣*s*∣*α* − 1. For *α* < 1 the function *ψ* is only *α*-Hölder-continuous.\n[def:weaksol] For a given *T* > 0 and initial value *u*0 ∈ *H*1(Ω), a weak solution to is defined as a function\n \n$$u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n \\cap\n L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\quad\n \\text{with}\n \\quad\n u\\_t \\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$$ \nthat has the following properties:\n* (Weak formulation) *u* satisfies the differential equation $\\eqref{eq:PDE}\\_1$ in the weak sense, i.e.,\n \n∫0*T*⟨*u**t*, *φ*⟩*W**α* + 11(Ω) *d**t* = ∫0*T*∫Ω*u**α* + 2*ψ*(*u**x**x**x*) *φ**x* *d**x* *d**t* \nfor all test functions ${\\varphi}\\in L\\_{\\alpha+1}\\bigl((0,T);W^1\\_{\\alpha+1,B}(\\Omega)\\bigr)$.\n* (Initial and boundary values) *u* satisfies the contact angle condition *u**x* = 0 on ∂Ω and the initial condition $\\eqref{eq:PDE}\\_3$ pointwise.\nThe following theorem contains the main result of this section.\n[Local existence of positive weak solutions] [thm:LocalExPL] Given a positive initial value *u*0 ∈ *W**α* + 1, *B*4*ρ*(Ω), 4*ρ* > 3 + 1/(*α* + 1), with $u\\_0(x) > 0,\\ x \\in \\bar{\\Omega}$, there exists a time *T* > 0 such that problem admits at least one positive weak solution\n \n$$u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n \\cap\n L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\quad \\text{with} \\quad\n u\\_t \\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$$ \non (0, *T*) in the sense of Definition [def:weaksol]. Moreover, such a solution has the following properties:\n* (Positivity) *u* is bounded away from zero\n \n$$0\n < C\\_T \\leq\n u(t,x),\n \\quad\n 0 \\leq t \\leq T,\\ x \\in \\bar{\\Omega}.$$\n* (Conservation of mass) *u* conserves its mass in the sense that\n \n∥*u*(*t*)∥*L*1(Ω) = ∥*u*0∥*L*1(Ω), 0 ≤ *t* ≤ *T*.\n* (Energy-dissipation identity) Energy is dissipated along solutions\n \n*E*[*u*](*t*) + ∫0*t**D*[*u*](*s*) *d**s* = *E*[*u*0] \nfor almost every *t* ∈ [0, *T*].\nObserve that due to the positivity of a solution *u* to we have\n \n∫Ω*u*(*t*, *x*) *d**x* = ∥*u*(*t*)∥*L*1(Ω) = ∥*u*0∥*L*1(Ω), 0 ≤ *t* ≤ *T*. \n[Rem:H1-initial] In fact, the above theorem holds true for initial values *u*0 ∈ *H*1(Ω). We choose *u*0 in the smaller space *W**α* + 14*ρ*(Ω) since the solutions *u* to the original problem are constructed as accumulation points of **strong** solutions *u**σ* to a regularised problem, not only as functions satisfying a suitable weak formulation. In order to apply semigroup theory, we require the initial value to satisfy *u*0 ∈ *W**α* + 14*ρ*(Ω). That *u*0 ∈ *H*1(Ω) is enough can be seen by replacing *u*0 by *u*0*σ* ∈ *W**α* + 14*ρ*(Ω) with\n \n$$u\\_0^\\sigma(x) > 0,\\ x \\in \\bar{\\Omega},\n \\quad \n \\bar{u}\\_0^\\sigma = \\bar{u}\\_0 = \\fint\\_\\Omega u\\_0\\, dx\n \\quad \\text{and} \\quad \n u\\_0^\\sigma \\longrightarrow u\\_0 \\quad \\text{strongly in } H^1(\\Omega), \\text{ as } \\sigma \\searrow 0.$$ \nThis can for instance be obtained by a symmetric extension of the initial value *u*0 ∈ *H*1(Ω) at the lateral boundaries and mollification.\n[Rem:extension] Given a positive weak solution $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n\\cap L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ to as obtained in Theorem [thm:LocalExPL], we may extend it beyond time *T* by restarting the equation with initial datum *u*(*T*) and using that *u*(*T*, *x*) > 0 for all $x\\in \\bar{\\Omega}$ and Remark [Rem:H1-initial]. In fact, in this way we can construct a weak solution to in the sense of Definition [def:weaksol] up to a time *T*\\* > 0 at which *u*(*T*\\*, *x*) = 0 for some $x\\in \\bar{\\Omega}$. Note though, that the solutions in Theorem [thm:LocalExPL] are not unique, so that the ‘maximal’ time *T*\\* of existence of positive solutions is not unique.\n**Positive Steady states of.** We are interested in the stability properties of steady-state solutions to, i.e., in functions *u*\\* ∈ *W**α* + 1, *B*3(Ω) that solve the ordinary differential equation\n \n*U**α* + 2∣*U*′′′∣*α* − 1*U*′′′ = 0, *x* ∈ Ω. \nIn physical parlance, says that there is no flux of the fluid through the boundaries of the interval. Positive steady states of may be easily characterised by the following theorem.\n[Characterisation of positive steady states] [thm:charsteadystates] A function *u* ∈ *W**α* + 1, *B*3(Ω) is a positive steady-state solution of if and only if *u* ≡ *u*\\* ∈ R> 0 is given by a positive constant.\n**(i)** Let *u* ≡ *u*\\* ∈ R> 0. Then *u*\\* ∈ *W**α* + 1, *B*3(Ω) clearly satisfies the ODE.\n**(ii)** Let *u* = *u*\\* ∈ *W**α* + 1, *B*3(Ω) be an arbitrary positive steady-state solution of, i.e., a solution to the ODE. Then *u*\\* satisfies\n \n$$0 \n =\n \\frac{d}{dt} E[u^\\ast]\n =\n -D[u^\\ast]\n =\n -\\int\\_\\Omega |u^\\ast|^{\\alpha+2} |u^\\ast\\_{xxx}|^{\\alpha+1}\\, dx.$$ \nSince the integrand on right-hand side of this equation is non-negative and $u^\\ast(x) > 0,\\, x \\in \\bar{\\Omega}$, it follows that *u**x**x**x*\\* ≡ 0 on $\\bar{\\Omega}$. Consequently, *u**x**x*\\* is constant and this in turn implies that *u**x*\\* is linear. Taking the Neumann boundary conditions into account, we find that *u*\\* must be constant.\nLocal Existence of Positive Solutions to the Regularised Problem and Uniform A-Priori Bounds\n--------------------------------------------------------------------------------------------\nIn order to handle the difficulties caused by the doubly nonlinear and doubly degenerate nature of the evolution problem, we introduce, for a fixed regularisation parameter *σ* ∈ (0, 1) and all *s* ∈ R, the smooth function\n \n$$\\psi\\_\\sigma(s) = (s^2 + \\sigma^2)^\\frac{\\alpha-1}{2} s,\n \\quad \n s \\in {\\mathbb{R}},$$ \nand substitute the nonlinear term *ψ*(*u**x**x**x*) in accordingly. The **regularised problem** corresponding to then reads\n \n$$\\label{eq:PDE\\_reg}\\tag{$P\\_\\sigma$}\n \\begin{cases}\n u^\\sigma\\_t + \\bigl((u^\\sigma)^{\\alpha+2} \\psi\\_\\sigma(u^\\sigma\\_{xxx})\\bigr)\\_x\n =\n 0,\n &\n t > 0,\\, x \\in \\Omega,\n \\\\\n u^\\sigma\\_x(t,x) = u^\\sigma\\_{xxx}(t,x) = 0,\n &\n t > 0,\\, x \\in \\partial\\Omega,\n \\\\\n u^\\sigma(0,x)\n =\n u\\_0(x), &\n x \\in \\Omega.\n \\end{cases}$$ \nIt follows from standard parabolic theory that the regularised problem possesses, for each fixed *σ* ∈ (0, 1) and suitable initial data, a unique maximal strong solution *u**σ*. This is the content of Theorem [thm:LocalExPLreg] below. Moreover, in Lemma [lem:uniformbounds] below, we provide a-priori bounds for the strong solution that are uniform in the regularisation parameter *σ* > 0. First, though, we define what we mean by a maximal strong solution to.\nFix *α* > 0 and *σ* ∈ (0, 1). Let 1 < *p* < ∞. Given a positive initial value *u*0 ∈ *L**p*(Ω), we call a function *u*: [0, *T**u*) → *L**p*(Ω) with *u*(*t*, *x*) > 0 for *t* ∈ [0, *T**u*) and $x \\in \\bar{\\Omega}$ a **maximal positive strong solution** to on [0, *T**u*) in *L**p*(Ω) if the following conditions are satisfied: \n* $u \\in C\\bigl([0,T\\_u);L\\_p(\\Omega)\\bigr) \\cap C^1\\bigl((0,T\\_u);L\\_p(\\Omega)\\bigr)$;\n* *u*(0) = *u*0 ∈ *L**p*(Ω) and *u*(*t*) ∈ *W**p*, *B*4(Ω) for all *t* ∈ (0, *T**u*);\n* (Positivity) *u*(*t*, *x*) > 0 for *t* ∈ [0, *T**u*) and $x \\in \\bar{\\Omega}$;\n* *u* satisfies the differential equation $\\eqref{eq:PDE\\_reg}\\_1$ pointwise;\n* (Maximality) There is no other solution *v* on [0, *T**v*) with *T**u* < *T**v*.\nClearly, solutions to, as obtained in the following theorem, do also dissipate energy. We therefore introduce the notation\n \n*D**σ*[*u**σ*](*t*) = ∫Ω(*u**σ*)*α* + 2∣*u**x**x**x**σ*∣*α* + 1 *d**x* \nfor the dissipation functional corresponding to the energy functional *E*[ ⋅ ] and the regularised equation.\n[Local existence for ] [thm:LocalExPLreg] Fix *α* > 0 and *σ* ∈ (0, 1). Let 1/(*α* + 1) < *s* < *r* < 1. Moreover, let $\\theta = \\frac{3+s}{4}$ and $\\rho = \\frac{3+r}{4}$. Then, given an initial film height *u*0 ∈ *W**α* + 1, *B*4*ρ*(Ω) such that *u*0(*x*) > 0 for all $x \\in \\bar{\\Omega}$, problem possesses a unique maximal solution\n \n$$u^\\sigma \\in C\\bigl([0,T\\_\\sigma);W^{4\\rho}\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\cap\n C^{\\rho}\\bigl([0,T\\_\\sigma);L\\_{\\alpha+1}(\\Omega)\\bigr)\n \\cap\n C\\bigl((0,T\\_\\sigma);W^4\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\cap\n C^1\\bigl((0,T\\_\\sigma);L\\_{\\alpha+1}(\\Omega)\\bigr).$$ \nMoreover, the solution enjoys the following properties.\n* (Positivity) *u**σ* is positive\n \n$$u^\\sigma(t,x) > 0,\n \\quad\n 0 \\leq t < T\\_\\sigma,\\, x\\in \\bar{\\Omega}.$$\n* (Conservation of mass) *u**σ* conserves its mass in the sense that\n \n∥*u**σ*(*t*)∥*L*1(Ω) = ∥*u*0∥*L*1(Ω), 0 ≤ *t* < *T**σ*.\n* (Energy-dissipation identity) *u**σ* satisfies the energy-dissipation identity\n \n*E*[*u**σ*](*t*) + ∫0*t**D**σ*[*u**σ*](*s*) = *E*[*u*0], 0 ≤ *t* < *T**σ*.\n* (Maximal time of existence) Suppose that *T**σ* < ∞. Then\n \n$$\\liminf\\_{t \\nearrow T\\_\\sigma} \\frac{1}{\\min\\_{x \\in \\bar{\\Omega}} u^\\sigma(t)} \n + \n \\|u^\\sigma(t)\\|\\_{W^{4\\gamma}\\_{\\alpha+1,B}(\\Omega)} \n = \n \\infty$$ \nfor all *γ* ∈ (*θ*, 1].\n**(i) Local existence, uniqueness and positivity.** In order to prove local existence and uniqueness of a strong solution we apply. To this end, we verify that fits into the corresponding abstract functional setting. Moreover, after rewriting in non-divergence form, we define for *v*(*t*) ∈ *W**α* + 1, *B*4*θ*(Ω) with *θ* = (3 + *s*)/4 such that $v(x) > 0,\\ x \\in \\bar{\\Omega}$, the linear differential operator ${\\mathcal{A}}(v(t)) \\in {\\mathcal{L}}\\bigl(W^4\\_{\\alpha+1,B}(\\Omega);L\\_{\\alpha+1}(\\Omega)\\bigr)$ of fourth order by\n \nA(*v*(*t*))*u**σ* = *A*(*v*(*t*))∂*x*4*u**σ* with *A*(*v*(*t*)) = *v**α* + 2*ψ**σ*′(*v**x**x**x*), \nwhere\n \n$$\\psi\\_\\sigma^\\prime(s) \n = \n (\\alpha-1) (s^2 + \\sigma^2)^\\frac{\\alpha-3}{2} s^2 + (s^2 + \\sigma^2)^\\frac{\\alpha-1}{2}\n =\n \\alpha (s^2 + \\sigma^2)^\\frac{\\alpha-1}{2}\n -\n \\sigma^2 (\\alpha-1) (s^2 + \\sigma^2)^\\frac{\\alpha-3}{2},\n \\quad s \\in {\\mathbb{R}}.$$ \nNote that for positive *σ* ∈ (0, 1) we have *ψ**σ*′(*s*) > *C**σ*, *α* > 0 for all *s* ∈ R and all fixed *α* > 0. Moreover, we introduce the right-hand side\n \nF(*v*(*t*)) = − (*α* + 2)*v**α* + 1*v**x* *ψ**σ*(*v**x**x**x*) \nand perceive as an abstract quasilinear Cauchy problem\n \n$$\\begin{cases}\n \\dot{u^\\sigma} + {\\mathcal{A}}(u^\\sigma)u^\\sigma = {\\mathcal{F}}(u^\\sigma), \\quad t > 0,\n \\\\\n u^\\sigma(0) = u\\_0.\n \\end{cases}$$ \nNote that the Neumann-type boundary conditions $\\eqref{eq:PDE\\_reg}\\_2$ are incorporated in the domain *W**α* + 1, *B*4(Ω) of the operator A(*v*(*t*)). Due to the smoothness of *ψ**σ* the maps\n \n$${\\mathcal{A}}\\colon W^{3+s}\\_{\\alpha+1,B}(\\Omega) \\longrightarrow\n {\\mathcal{L}}\\bigl(W^4\\_{\\alpha+1,B}(\\Omega);L\\_{\\alpha+1}(\\Omega)\\bigr)\n \\quad \\text{and} \\quad\n {\\mathcal{F}}\\colon W^{3+s}\\_{\\alpha+1,B}(\\Omega) \\longrightarrow L\\_{\\alpha+1}(\\Omega)$$ \nare, for all *α* > 0, locally Lipschitz continuous. In order to guarantee parabolicity, we extend the differential operator A to the differential operator\n \n$$\\bar{{\\mathcal{A}}}\\_{\\varepsilon}(v(t)) \\in {\\mathcal{L}}\\bigl(W^4\\_{\\alpha+1,B}(\\Omega);L\\_{\\alpha+1}(\\Omega)\\bigr),\n \\quad\n \\bar{{\\mathcal{A}}}\\_{\\varepsilon}(v(t))u^\\sigma = \\bar{A}\\_{{\\varepsilon}}(v(t)) \\partial\\_x^4 u^\\sigma,$$ \nwhere\n \n*Ā**ɛ*(*v*(*t*)) = max{*v*+*α* + 2*ψ**σ*′(*v**x**x**x*), *ɛ*/2} \nand *v*+ = max{*v*, 0}. Following the lines of, we study the extended parabolic problem with $\\bar{{\\mathcal{A}}}\\_{\\varepsilon}$ instead of A and show that the corresponding local positive solution *u**σ* = *u**σ*(*ɛ*) also solves the non-extended problem for a short but strictly positive time. More precisely, the extended regularised problem is, for each fixed *σ*, *ɛ* ∈ (0, 1), parabolic in the sense that $\\bar{{\\mathcal{A}}}\\_{{\\varepsilon}}(v(t))$ generates an analytic semigroup on *L**α* + 1(Ω). Indeed, due to the embedding $W^{3+s}\\_{\\alpha+1,B}(\\Omega) \\hookrightarrow C^3(\\bar{\\Omega})$ and the positivity of *σ*, *ɛ* > 0, we have that $\\bar{A}\\_{\\varepsilon}(v(t,\\cdot)) \\in C(\\bar{\\Omega})$. Moreover, the principal symbol *a**ɛ*(*x*, *ξ*) satisfies\n \n$$\\text{Re}(a\\_{\\varepsilon}(x,\\xi)\\eta | \\eta)\n \\geq\n C\\_{\\sigma,\\alpha,{\\varepsilon}}(i\\xi)^4 \\eta^2 > 0,\n \\quad\n (x,\\xi) \\in \\bar{\\Omega}\\times \\{-1,1\\},\\, \\eta \\in {\\mathbb{R}}\\setminus\\{0\\},$$ \nfor a positive constant *C**σ*, *α*, *ɛ* > 0. Consequently, $\\bar{{\\mathcal{A}}}\\_{\\varepsilon}(v(t))$, together with the Neumann-type boundary conditions, is normally elliptic in the sense of and we can apply to conclude that $\\bar{{\\mathcal{A}}}\\_{\\varepsilon}(v(t))$ generates an analytic semigroup on *L**α* + 1(Ω). Thus, we are in the abstract setting of which yields existence and uniqueness of a local positive strong solution to the extended problem in *L**α* + 1(Ω). On a potentially smaller time interval, this solution *u**σ* = *u**σ*(*ɛ*) is, for *ɛ* small enough, also a local positive strong solution to, see step (iii) in the proof of.\n**(ii) Conservation of mass.** This follows by testing the regularised partial differential equation $\\eqref{eq:PDE\\_reg}\\_1$ with the constant function *φ* ≡ 1, integration by parts and using the Neumann boundary conditions $\\eqref{eq:PDE\\_reg}\\_2$.\n**(iii) Energy-dissipation identity.** Since the solution obtained in step (i) enjoys the regularity\n \n$$u\\_x^\\sigma \\in C\\bigl((0,T);W^1\\_{\\alpha+1,0}(\\Omega)\\bigr) \\cap C^1\\bigl((0,T);(W^1\\_{\\alpha+1,0}(\\Omega))^\\prime\\bigr),$$ \nwe may apply in order to guarantee that the expression\n \n$$\\frac{d}{dt} E[u^\\sigma](t)\n =\n \\int\\_\\Omega u^\\sigma\\_{xt} u^\\sigma\\_x\\, dx\n =\n -\\int\\_\\Omega |u^\\sigma|^{\\alpha+2} \\bigl(|u^\\sigma\\_{xxx}|^2 + \\sigma^2\\bigr)^\\frac{\\alpha-1}{2} |u^\\sigma\\_{xxx}|^2\\, dx\n =\n -D\\_\\sigma[u^\\sigma](t)$$ \nis well-defined for all *t* ∈ (0, *T*). Integrating with respect to time gives the energy-dissipation identity.\n**(iv) Maximal time of existence.** Using the notation introduced in step (i), this result is a minor adaptation of.\nIn order to prove the local-existence result for the original problem (Theorem [thm:LocalExPL]), we need suitable uniform (in *σ*) a-priori estimates for the solution to as given in the following lemma.\n[Uniform bounds] [lem:uniformbounds] Let *u**σ* be the maximal solution to for a fixed *σ* ∈ (0, 1) and an initial value *u*0 ∈ *W**α* + 1, *B*4*ρ*(Ω) such that *u*0(*x*) > 0 for all $x \\in \\bar{\\Omega}$. Then the following holds true. There is *T* > 0 such that the family (*u**σ*)*σ* has the following properties: \n* (*u**σ*)*σ* is uniformly bounded in $L\\_\\infty\\bigl((0,T);H^1(\\Omega)\\bigr)$;\n* $\\bigl(\\left|u^\\sigma\\right|^{\\alpha+2} \\psi\\_\\sigma(u^\\sigma\\_{xxx})\\bigr)\\_\\sigma$ is uniformly bounded in $L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T)\\times \\Omega\\bigr)$;\n* (*u**t**σ*)*σ* is uniformly bounded in $L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$;\n* (*u**x**x**x**σ*)*σ* is uniformly bounded in $L\\_{\\alpha+1}\\bigl((0,T)\\times \\Omega\\bigr)$;\n* (*u**σ*)*σ* is uniformly bounded in $L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$;\n* ((*u**x**σ*)*t*)*σ* is uniformly bounded in $L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);\\bigl(W^1\\_{\\alpha+1,0}(\\Omega)\\cap W^2\\_{\\alpha+1}(\\Omega)\\bigr)'\\bigr)$.\nNote that once we have proved items (i) and (iii), the Aubin–Lions–Simon Theorem implies that the family (*u**σ*)*σ* is equicontinuous. Hence, we may choose *T* > 0 such that *u**σ* is bounded away uniformly from zero on the interval [0, *T*].\nWithin this proof, *C* > 0 denotes a positive constant, possibly depending on *α*, Ω, and ∥*u*0∥*W**α* + 14*ρ*(Ω), but independent of *σ*.\n**(i)** Since\n \n*D**σ*[*u**σ*](*t*) = ∫Ω(*u**σ*)*α* + 2*ψ**σ*(*u**x**x**x**σ*) *u**x**x**x**σ* *d**x* ≥ 0, *t* ∈ [0, *T**σ*), \nwe have\n \n$$\\label{eq:unifen}\n E[u^{\\sigma}](t) = \\frac{1}{2} \\|u\\_x^{\\sigma}( t) \\|\\_{L\\_2(\\Omega)}^2 \\leq E[u\\_0], \\quad t \\in [0,T\\_{\\sigma}).$$ \nUsing Poincaré’s inequality and, we obtain for *t* ∈ [0, *T**σ*)\n \n∥*u**σ*(*t*)∥*L*2(Ω) ≤ ∥*u**σ*(*t*) − *ū**σ*(*t*)∥*L*2(Ω + ∥*ū**σ*(*t*)∥*L*2(Ω) ≤ *C*∥*u**x**σ*(*t*)∥*L*2(Ω) + ∥*ū*0∥*L*2(Ω), \nwhich, together with, yields\n \n$$\\sup \\limits\\_{0\\leq t\\leq T\\_{\\sigma}} \\|u^{\\sigma}(t)\\|\\_{H^1(\\Omega)} \\leq C\\bigl( \\|\\bar{u}\\_0\\|\\_{L\\_2(\\Omega)} + E[u\\_0]^{1/2} \\bigr).$$ \nHence, (*u**σ*)*σ* is uniformly bounded in $L\\_{\\infty}\\bigl((0,T\\_{\\sigma});H^1(\\Omega)\\bigr)$.\n**(ii)** First we consider the case 0 < *α* < 1. Observe that\n \n$$\\begin{aligned}\n \\bigl\\| |u^{\\sigma}|^{\\alpha+2}\\psi\\_{\\sigma}(u^{\\sigma}\\_{xxx}) \\bigr\\|\\_{L\\_{\\frac{\\alpha+1}{\\alpha}}((0,T\\_{\\sigma})\\times \\Omega)}^{\\frac{\\alpha+1}{\\alpha}} & = \\int\\_0^{T\\_{\\sigma}} \\int\\_{\\Omega} |u^{\\sigma}|^{(\\alpha+2)\\frac{\\alpha+1}{\\alpha}} \\bigl(|u^{\\sigma}\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{\\frac{\\alpha+1}{\\alpha}} \\, dx \\, dt \\\\\n & = \\int\\_0^{T\\_{\\sigma}} \\int\\_{\\Omega} |u^{\\sigma}|^{(\\alpha+2)\\frac{\\alpha+1}{\\alpha}} \\bigl(|u^{\\sigma}\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{\\frac{1-\\alpha}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{2} \\, dx \\, dt.\n \\end{aligned}$$ \nUsing that $\\frac{1-\\alpha}{\\alpha} >0$, we get the pointwise estimate $|u^{\\sigma}\\_{xxx}|^{\\frac{1-\\alpha}{\\alpha}} \\leq \\bigl( |u^{\\sigma}\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{1-\\alpha}{2\\alpha}}$. Furthermore, in view of (i) and $H^1(\\Omega) \\hookrightarrow L\\_\\infty (\\Omega)$, we find that (*u**σ*)*σ* is uniformly bounded in $L\\_{\\infty}\\bigl((0,T\\_{\\sigma}); L\\_\\infty(\\Omega)\\bigr)$. Combining this, we obtain the estimate\n \n$$\\begin{aligned}\n \\int\\_0^{T\\_{\\sigma}} \\int\\_{\\Omega} & |u^{\\sigma}|^{(\\alpha+2)\\frac{\\alpha+1}{\\alpha}} \\bigl(|u^{\\sigma}\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{\\frac{1-\\alpha}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{2} \\, dx \\, dt \\\\\n \\leq & \\int\\_0^{T\\_{\\sigma}} \\int\\_{\\Omega} |u^{\\sigma}|^{(\\alpha+2)\\frac{\\alpha+1}{\\alpha}} \\bigl(|u^{\\sigma}\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}}|u^{\\sigma}\\_{xxx}|^{2} \\, dx \\, dt \\\\\n \\leq & C \\int\\_0^{T\\_\\sigma} D\\_\\sigma[u^{\\sigma}](t) \\, dt \\\\\n \\leq & C E[u\\_0],\n \\end{aligned}$$ \nwhere the last step is due to. In the case 1 < *α* < ∞, we have to use a different argument. Note that by (i) and, we have\n \n$$\\begin{aligned}\n & \\bigl\\| |u^{\\sigma}|^{\\alpha+2}\\psi\\_{\\sigma}(u^{\\sigma}\\_{xxx}) \\bigr\\|\\_{L\\_{\\frac{\\alpha+1}{\\alpha}}((0,T\\_{\\sigma})\\times \\Omega)}^{\\frac{\\alpha+1}{\\alpha}}\\\\\n = & \\int\\_0^{T\\_{\\sigma}} \\int\\_{\\Omega} |u^{\\sigma}|^{(\\alpha+2)\\frac{\\alpha+1}{\\alpha}} \\bigl(|u^{\\sigma}\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{\\frac{\\alpha+1}{\\alpha}} \\, dx \\, dt \\\\\n \\leq & C \\int\\_{\\{|u^{\\sigma}\\_{xxx}| \\leq \\sigma\\}} \\bigl(|u^{\\sigma}\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{\\frac{1-\\alpha}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{2} \\, dx \\, dt \\\\\n & \\quad + \\int\\_{\\{|u^{\\sigma}\\_{xxx}| > \\sigma\\}} |u^{\\sigma}|^{(\\alpha+2)\\frac{\\alpha+1}{\\alpha}} \\bigl(|u^{\\sigma}\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}}|u^{\\sigma}\\_{xxx}|^{\\frac{\\alpha+1}{\\alpha}} \\, dx \\, dt \\\\\n \\leq & C T\\_{\\sigma} \\sigma^{\\alpha+1} + C\\int\\_0^{T\\_{\\sigma}} D\\_\\sigma[u^{\\sigma}](t) \\, dt\\\\\n \\leq & C\\bigl(T\\_{\\sigma} \\sigma^{\\alpha+1} + E[u\\_0] \\bigr).\n \\end{aligned}$$ \n**(iii)** Since *u**σ* is a weak solution to, we have\n \n∫0*T**σ*⟨*u**t**σ*, *φ*⟩*W**α* + 11(Ω) *d**t* = ∫0*T**σ*∫Ω(*u**σ*)*α* + 2 *ψ**σ*(*u**x**x**x**σ*) *φ**x* *d**x* *d**t* \nfor all $\\varphi \\in L\\_{\\alpha+1}\\bigl((0,T\\_\\sigma); \n W^1\\_{\\alpha +1,B}(\\Omega)\\bigr)$. Applying Hölder’s inequality and (i), we obtain\n \n$$\\begin{aligned}\n &\n \\left|\\int\\_0^{T\\_\\sigma} \\langle u^\\sigma\\_t, \\varphi \\rangle\\_{W^1\\_{\\alpha+1}(\\Omega)}\\, dt\n \\right|\n \\leq\n \\int\\_0^{T\\_\\sigma} \\int\\_{\\Omega} |u^\\sigma|^{\\alpha+2} \\left|\\psi\\_\\sigma(u^\\sigma\\_{xxx})\\right|\\, |\\varphi\\_x|\\, dx\\, dt\n \\\\\n &\\quad \n \\leq\n \\left(\\int\\_0^{T\\_\\sigma} \\int\\_{\\Omega} |u^\\sigma|^{\\alpha+2} |\\varphi\\_x|^{\\alpha+1}\\, dx\\, dt\n \\right)^\\frac{1}{\\alpha+1} \\cdot\n \\\\\n &\\quad\\quad \\cdot\n \\left(\\int\\_0^{T\\_\\sigma} \\int\\_{\\Omega} |u^\\sigma|^{\\alpha+2} \\bigl(|u^\\sigma\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}} |u^\\sigma\\_{xxx}|^\\frac{\\alpha+1}{\\alpha}\\, dx\\, dt\n \\right)^\\frac{\\alpha}{\\alpha+1}\n \\\\\n &\\quad \n \\leq\n C \\|\\varphi\\|\\_{L\\_{\\alpha+1}((0,T\\_{\\sigma});W^1\\_{\\alpha+1}(\\Omega))}\n \\left(\\int\\_0^{T\\_\\sigma} \\int\\_{\\Omega} |u^\\sigma|^{\\alpha+2} \\bigl(|u^\\sigma\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}} |u^\\sigma\\_{xxx}|^\\frac{\\alpha+1}{\\alpha}\\, dx\\, dt\n \\right)^\\frac{\\alpha}{\\alpha+1}.\\end{aligned}$$ \nFor 0 < *α* < 1, we obtain similar as in step (ii) that\n \n$$\\int\\_0^{T\\_\\sigma} \\int\\_{\\Omega} |u^\\sigma|^{\\alpha+2} \\bigl(|u^\\sigma\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}} |u^\\sigma\\_{xxx}|^\\frac{\\alpha+1}{\\alpha}\\, dx\\, dt \n \\leq \\int\\_0^{T\\_\\sigma} D\\_\\sigma[u^\\sigma](t) \\, dt \n \\leq E[u\\_0].$$ \nFor 1 < *α* < ∞, we get, similarly as in step (ii),\n \n$$\\begin{aligned}\n \\int\\_0^{T\\_\\sigma} \\int\\_{\\Omega} |u^\\sigma|^{\\alpha+2} \\bigl(|u^\\sigma\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}\\frac{\\alpha+1}{\\alpha}} |u^\\sigma\\_{xxx}|^\\frac{\\alpha+1}{\\alpha}\\, dx\\, dt \n &\\leq C T\\_\\sigma \\sigma^{\\alpha+1} + C \\int\\_0^{T\\_\\sigma} D\\_\\sigma[u^\\sigma](t) \\, dt \n \\\\\n & \\leq C\\left(T\\_\\sigma \\sigma^{\\alpha+1} + E[u\\_0]\\right). \\end{aligned}$$ \n**(iv)** We prove that (*u**x**x**x**σ*)*σ* is uniformly bounded in $L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,T\\_\\sigma)\\times \\Omega\\bigr)$. Note that by definition of *T**σ* and continuity of *u**σ*, we have *u**σ*(*t*, *x*) > *c**δ* > 0 for all (*t*, *x*) ∈ [0, *T**σ* − *δ*) × Ω, for every *δ* > 0.\nIn the case 1 < *α* < ∞, we get\n \n$$\\begin{aligned}\n \\int\\_0^{T\\_\\sigma -\\delta} \\int\\_{\\Omega} | u\\_{xxx}^\\sigma|^{\\alpha+1} \\, dx\\, dt \n &\\leq \\int\\_0^{T\\_\\sigma-\\delta} \\int\\_{\\Omega} \\bigl(|u^\\sigma\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}} | u\\_{xxx}^\\sigma|^2 \\, dx\\, dt\n \\\\\n &\\leq C \\int\\_0^{T\\_\\sigma -\\delta}D\\_\\sigma[u^\\sigma](t)\\, dt\n \\\\\n &\\leq C E[u\\_0], \n \\end{aligned}$$ \nwhere the constant *C* depends also on *δ*, and where in the last step we used.\nNow we consider the case 0 < *α* < 1. We have\n \n$$\\begin{aligned}\n \\int\\_0^{T\\_\\sigma-\\delta} \\int\\_{\\Omega} | u\\_{xxx}^\\sigma|^{\\alpha+1} \\, dx\\, dt \n & = \\int\\_{\\{|u^{\\sigma}\\_{xxx}| \\leq \\sigma\\}} \n |u\\_{xxx}^\\sigma|^{\\alpha+1} \\, dx\\, dt + \n \\int\\_{\\{|u^{\\sigma}\\_{xxx}| > \\sigma\\}} \n |u\\_{xxx}^\\sigma|^{\\alpha+1} \\, dx\\, dt \n \\\\\n &\\leq C (T\\_\\sigma-\\delta) \\sigma^{\\alpha+1} + \\int\\_{\\{|u^{\\sigma}\\_{xxx}| > \\sigma\\}} \n |u\\_{xxx}^\\sigma|^{\\alpha+1} \\, dx\\, dt. \n \\end{aligned}$$ \nUsing the inequality\n \n$$\\begin{aligned}\n |x|^{\\alpha+1} = \\left( \\tfrac{1}{2} |x|^2 + \\tfrac{1}{2} |x|^2\n \\right)^{\\frac{\\alpha-1}{2}} |x|^2 \n \\leq \\left(\\tfrac{1}{2} \\right)^{\\frac{\\alpha-1}{2}} \n \\left( |x|^2 + \\sigma^2\\right)^{\\frac{\\alpha-1}{2}} |x|^2, \n \\qquad |x|>\\sigma, \\; x \\in {\\mathbb{R}}, \n \\end{aligned}$$ \nwe obtain\n \n$$\\begin{aligned}\n \\int\\_0^{T\\_\\sigma -\\delta} \\int\\_{\\Omega} | u\\_{xxx}^\\sigma|^{\\alpha+1} \\, dx\\, dt \n &\\leq C (T\\_\\sigma-\\delta) \\sigma^{\\alpha+1} \n + C \\int\\_0^{T\\_\\sigma -\\delta} \\int\\_{\\Omega} \n \\bigl(|u^\\sigma\\_{xxx}|^2 + \\sigma^2\\bigr)^{\\frac{\\alpha-1}{2}} | u\\_{xxx}^\\sigma|^2 \\, dx\\, dt\n \\\\\n &\\leq C (T\\_\\sigma-\\delta) \\sigma^{\\alpha+1} \n + C \\int\\_0^{T\\_{\\sigma}-\\delta} D\\_\\sigma[u^\\sigma](t) \\, dt\n \\\\\n &\\leq C \\left((T\\_\\sigma-\\delta) \\sigma^{\\alpha+1} \n + E[u\\_0]\\right) \n \\end{aligned}$$ \nwith *C* depending also on *δ*. In the last step we used again.\n**(v)** As observed in (i), *u**σ* is uniformly bounded in $L\\_{\\infty}\\bigl((0,T\\_{\\sigma});L\\_{\\infty}(\\Omega)\\bigr)$, and hence also in $L\\_{\\alpha+1}\\bigl((0,T\\_{\\sigma})\\times \\Omega\\bigr)$. From (iv), we also know that *u**x**x**x**σ* is uniformly bounded in $L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,T\\_{\\sigma})\\times \\Omega\\bigr)$. Combining this, we find that *u**σ* is uniformly bounded in $L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,T\\_{\\sigma});W\\_{\\alpha+1,B}^3(\\Omega)\\bigr)$ by interpolation.\n**(vi)** This follows as in (iii) using a duality argument.\nProof of Theorem [thm:LocalExPL]: Local Existence of Positive Weak Solutions to the Original Problem\n----------------------------------------------------------------------------------------------------\nIn this section we pass to the limit of a vanishing regularisation parameter $\\sigma \\searrow 0$. Using the uniform bounds provided in Lemma [lem:uniformbounds], we show that the family (*u**σ*)*σ* admits an accumulation point that is a positive weak solution to the original problem. As usual, we use Minty’s trick in order to identify the (nonlinear) limit flux.\n[Convergence of approximations] [lem:convergence] Let *u**σ* be the maximal solution to for a fixed *σ* ∈ (0, 1) and a positive initial value *u*0 ∈ *W**p*, *B*4*ρ*(Ω) such that *u*0(*x*) > 0 for all $x \\in \\bar{\\Omega}$. Then the following holds true. There are a positive time *T* > 0 and a subsequence (*u**σ*)*σ* (not relabelled) such that, as $\\sigma \\searrow 0$, we have convergence in the following sense: \n* *u**σ* → *u* strongly in $C\\bigl([0,T];C^{\\rho}(\\bar{\\Omega})\\bigr)$;\n* $\\left|u^\\sigma\\right|^{\\alpha+2} \\psi\\_\\sigma(u^\\sigma\\_{xxx}) \\rightharpoonup \\chi$ weakly in $L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T)\\times \\Omega\\bigr)$ for some limit function *χ*;\n* $u^\\sigma\\_t \\rightharpoonup u\\_t$ weakly in $L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T); (W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$;\n* $u^\\sigma\\_{xxx} \\rightharpoonup u\\_{xxx}$ weakly in $L\\_{\\alpha+1}\\bigl((0,T)\\times \\Omega\\bigr)$;\n* $(u^\\sigma\\_x)\\_t \\rightharpoonup u\\_{xt}$ weakly in $L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);\\bigl(W^1\\_{\\alpha+1,0}(\\Omega)\\cap W^2\\_{\\alpha+1}(\\Omega)\\bigr)'\\bigr)$.\nSince the proof of this lemma differs only very slightly from that in, we shift it to the appendix.\nWe are left to prove the convergence of the nonlinear flux term $\\bigl(\\left|u^\\sigma\\right|^{\\alpha+2} \\psi\\_\\sigma(u^\\sigma\\_{xxx}\\bigr)\\bigr) \\rightharpoonup \\bigl(\\left|u\\right|^{\\alpha+2} \\psi(u\\_{xxx})\\bigr)$ in $L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T)\\times \\Omega\\bigr)$. This is done in the next lemma the proof of which is based on the monotonicity of the regularisation and Minty’s trick.\n[lem:limitflux] Given *σ* ∈ (0, 1), let *u**σ* be the maximal solution to, corresponding to an initial value *u*0 ∈ *W**α* + 1, *B*4*ρ*(Ω). Then there exists a subsequence (*u**σ*)*σ* (not relabelled) such that\n \n$$\\left|u^\\sigma\\right|^{\\alpha+2} \\psi\\_\\sigma(u^\\sigma\\_{xxx})\\ \n \\rightharpoonup\n |u|^{\\alpha+2} \\psi(u\\_{xxx}) \n \\quad\n \\text{weakly in }\n L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T)\\times \\Omega\\bigr)$$ \nas $\\sigma \\searrow 0$.\nThe proof of the above stated lemma uses the same arguments as the one in. For the sake of completeness, we include it in the appendix.\n[Rem:contH1] Note that the limit *u* is bounded in $C\\bigl([0,T];H^1(\\Omega)\\bigr)$. Indeed, from Lemma [lem:convergence] (i) we already know that\n \n$$u \\in C\\bigl([0,T];C^\\rho(\\bar{\\Omega})\\bigr) \n \\hookrightarrow\n C\\bigl([0,T];L\\_2(\\Omega)\\bigr).$$ \nFurthermore,\n \n$$u\\_x \\in L\\_{\\alpha+1}\\bigl((0,T);W^1\\_{\\alpha+1,0}(\\Omega) \\cap W^2\\_{\\alpha+1}(\\Omega)\\bigr) \\quad \\text{and} \\quad\n u\\_{xt}\\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);\\bigl(W^1\\_{\\alpha+1,0}(\\Omega)\\cap W^2\\_{\\alpha+1}(\\Omega)\\bigr)'\\bigr)$$ \ndue to Lemma [lem:convergence] (iv) and (v) and lower semicontinuity of the norm. Using, this yields that $u\\_x \\in C\\bigl([0,T];L\\_2(\\Omega)\\bigr)$. Therefore, $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr)$.\n[**Proof of Theorem [thm:LocalExPL]**] **(i)** We first show that the limit *u* is bounded away from zero on $[0,T] \\times \\bar{\\Omega}$. This follows immediately from the positivity of *u**σ* on $[0,T\\_\\sigma)\\times \\bar{\\Omega}$ and the convergence in Lemma (i).\n**(ii)** Thanks to Lemma [lem:uniformbounds] (iii) and (iv) and Remark [Rem:contH1] above, we obtain the regularity properties \n$$u \\in C\\bigl([0,T]; H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1}\\bigl((0,T); W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\quad \\text{and} \\quad u\\_t \\in L\\_{\\frac{\\alpha+1}{\\alpha}}\\bigl(\n (0,T); (W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr).$$ \n**(iii)** We now prove that *u* satisfies the weak integral formulation in Definition [def:weaksol]. To do so, note that for solutions to the regularised problem we have that \n∫0*T*⟨*u**t**σ*, *φ*⟩*W**α* + 11(Ω) *d**t* = ∫0*T*∫Ω∣*u**σ*∣*α* + 2*ψ**σ*(*u**x**x**x**σ*) *φ**x* *d**x* *d**t* \n for all test functions ${\\varphi}\\in L\\_{\\alpha+1}\\bigl((0,T); W^1\\_{\\alpha+1,B}(\\Omega)\\bigr)$. On the one hand, since ${\\varphi}\\_x \\in L\\_{\\alpha+1}\\bigl((0,T)\\times \\Omega)\\bigr)$, it follows from Lemma [lem:limitflux] that \n∫0*T*⟨*u**t**σ*, *φ*⟩*W**α* + 11(Ω) *d**t* → ∫0*T*∫Ω∣*u*∣*α* + 2*ψ*(*u**x**x**x*) *φ**x* *d**x* *d**t*. \n On the other hand, Lemma [lem:uniformbounds] (iii) gives \n∫0*T*⟨*u**t**σ*, *φ*⟩*W**α* + 11(Ω) *d**t* → ∫0*T*⟨*u**t*, *φ*⟩*W**α* + 11(Ω) *d**t*. \n Combining both, we then find that *u* satisfies the desired integral identity \n∫0*T*⟨*u**t*, *φ*⟩*W**α* + 11(Ω) *d**t* = ∫0*T*∫Ω*u**α* + 2*ψ*(*u**x**x**x**σ*) *φ**x* *d**x* *d**t* \n for all ${\\varphi}\\in L\\_{\\alpha+1}\\bigl((0,T); W^1\\_{\\alpha+1,B}(\\Omega)\\bigr)$.\n**(iv)** By Lemma [lem:convergence] (i) the initial condition is satisfied in the limit. That the first boundary condition in $\\eqref{eq:PDE}\\_2$ is fulfilled by *u* follows from Lemma [lem:convergence] (v).\n**(v)** This follows from the conservation of mass property \n∫Ω*u**σ*(*t*) *d**x* = ∫Ω*u*0 *d**x*, *t* ∈ [0, *T**σ*), \n for the approximation *u**σ* (see Theorem [thm:LocalExPLreg] (ii)) and the convergence in Lemma [lem:convergence] (i).\n**(vi)** In Lemma [lem:limitflux] we have already shown that the solution *u* to the original problem satisfies the energy-dissipation identity for almost every *t* ∈ [0, *T*].\nDifferential Inequality for the Energy and Regularity Estimates\n===============================================================\nThe content of this section is twofold. First, we derive a differential inequality of Łojasiewicz–Simon type for the energy functional *E* which is valid as long as the weak solution to remains bounded away from zero. Then, we derive *L*1-in-time regularity estimates for the weak solution to. The results are the same as in the cylindrical Taylor–Couette setting in. However, since the present paper deals with the flat case, the proofs cannot rely on Fourier analysis.\n[prop:EDestimate] Fix *α* > 0 and a positive initial value *u*0 ∈ *H*1(Ω) with *u*0(*x*) > 0 for $x\\in \\bar{\\Omega}$. Let $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n \\cap\n L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ with $u\\_t \\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$ be a weak solution to with initial value *u*0, as obtained in Theorem [thm:LocalExPL]. Let $m = \\min\\_{(t,x)\\in [0,T] \\times\\bar{\\Omega}}u(t,x) > 0$. Then there is a constant *C* = *C**α*, Ω, *m* > 0 such that\n \n$$\\frac{d}{dt} E[u](t) = - D[u](t) \\leq -C \\bigl(E[u](t)\\bigr)^{\\frac{\\alpha+1}{2}}$$ \nfor almost every *t* ∈ [0, *T*].\nThe proof of Proposition [prop:EDestimate] is based on the following crucial Poincaré estimate. It is worthwhile to emphasise that this estimate is valid in both the shear-thinning case and the shear-thickening case.\n[Lem:Poincare] Fix *α* > 0 and let *v* ∈ *H*1(Ω) ∩ *W**α* + 1, *B*3(Ω) with *v̄* = 0 and *v**x*(*x*) = 0 for *x* ∈ ∂Ω. Then there exists a constant *C* = *C**α*, Ω > 0 such that \n*E*[*v*] ≤ *C*∥*v**x**x**x*∥*L**α* + 1(Ω)2. \nWe distinguish the cases *α* = 1, *α* > 1 and *α* < 1.\n**The case *α* = 1.** This is just a direct application of Poincaré’s inequality.\n**The case *α* > 1.** Define *w* = *v**x**x* ∈ *W**α* + 11(Ω) ⊂ *L*2(Ω). Observe that *v* is a weak solution to the Neumann boundary-value problem given by\n \n$$\\begin{cases}\n v\\_{xx}\n =\n w,\n &\n x\\in \\Omega,\n \\\\\n v\\_x \n =\n 0,\n &\n x\\in \\partial\\Omega.\n \\end{cases}$$ \nHence, we obtain the estimate ∥*v**x*∥*L*2(Ω) ≤ *C*∥*w*∥*L*2(Ω). Furthermore, note that *w̄* = 0. Using this, applying Poincaré’s inequality and then Jensen’s inequality for the concave function *s* ↦ *s*2/(*α* + 1), *s* ∈ (0, ∞), we find that\n \n$$\\begin{aligned}\n E[v] = \\frac{1}{2} \\|v\\_x\\|\\_{L\\_2(\\Omega)}^2 \\leq C\\|w\\|\\_{L\\_2(\\Omega)}^2 \\leq C\\|w\\_x\\|\\_{L\\_2(\\Omega)}^2 = C \\int\\_{\\Omega} |v\\_{xxx}|^{(\\alpha+1)\\frac{2}{\\alpha+1}} \\, dx \\leq C\\left(\\int\\_{\\Omega} |v\\_{xxx}|^{\\alpha+1}\\, dx\\right)^{\\frac{2}{\\alpha+1}}.\n \\end{aligned}$$ \n**The case *α* < 1.** In this case we have 2/(*α* + 1) > 1 and we cannot use Jensen’s inequality anymore. Instead, we rely on the Sobolev embedding and a-priori estimates for the Bi-Laplace equation. Define *w* = *v**x**x**x* ∈ *L**α* + 1(Ω). Then *v* is a weak solution to\n \n$$\\begin{cases}\n v\\_{xxxx}\n =\n w\\_x,\n &\n x\\in \\Omega,\n \\\\\n v\\_x = v\\_{xxx}\n =\n 0,\n &\n x\\in \\partial\\Omega,\n \\end{cases}$$ \nin the sense that \n$$\\int\\_{\\Omega} v\\_{xx} \\varphi\\_{xx} \\, dx = - \\int\\_{\\Omega} w \\varphi\\_x \\, dx \\quad \\text{ for all } \\varphi \\in W^2\\_{\\frac{\\alpha+1}{\\alpha},B}(\\Omega).$$ \n Since $v \\in C^2(\\bar{\\Omega})$ by the Sobolev embedding, we may use $v \\in W^2\\_{\\frac{\\alpha+1}{\\alpha},B}(\\Omega)$ as a test function and find that \n$$\\|v\\_{xx}\\|\\_{L\\_2(\\Omega)}^2 \\leq \\int\\_{\\Omega} |w| |v\\_x| \\, dx \\leq \\|w\\|\\_{L\\_{\\alpha+1}(\\Omega)}\\|v\\_x\\|\\_{L\\_{\\frac{\\alpha}{\\alpha+1}}(\\Omega)} \\leq C \\|w\\|\\_{L\\_{\\alpha+1}(\\Omega)}\\|v\\_{xx}\\|\\_{L\\_{2}(\\Omega)}.$$ \n Dividing by ∥*v**x**x*∥*L*2(Ω), we conclude that ∥*v**x**x*∥*L*2(Ω)2 ≤ *C*∥*w*∥*L**α* + 1(Ω)2. Finally, the desired estimate\n \n$$\\begin{aligned}\n E[v] = \\frac{1}{2} \\|v\\_x\\|\\_{L\\_2(\\Omega)}^2 \\leq C\\|v\\_{xx}\\|\\_{L\\_2(\\Omega)}^2 \\leq C\\|w\\|\\_{L\\_{\\alpha+1}(\\Omega)}^2 = C\\|v\\_{xxx}\\|\\_{L\\_{\\alpha+1}(\\Omega)}^2\n \\end{aligned}$$ \nfollows by Poincaré’s inequality.\n[**Proof of Proposition [prop:EDestimate]**] From Theorem [thm:LocalExPL] we know that weak solutions to satisfy the energy-dissipation identity. Taking the derivative in time, we find that\n \n$$\\frac{d}{dt} E[u](t) + D[u](t) = 0$$ \nfor almost every *t* ∈ [0, *T*]. Furthermore, since $m = \\min\\_{(t,x)\\in [0,T] \\times\\bar{\\Omega}}u(t,x) > 0$ and by Lemma [Lem:Poincare], we obtain\n \n$$D[u](t) = \\int\\_{\\Omega} |u|^{\\alpha+2}|u\\_{xxx}|^{\\alpha+1} \\, dx \\geq \n m^{\\alpha+2}\\|u\\_{xxx}(t)\\|\\_{L\\_{\\alpha+1}(\\Omega)}^{\\alpha+1} \\geq Cm^{\\alpha+2}\\bigl(E[u](t)\\bigr)^{\\frac{\\alpha+1}{2}}$$ \nfor almost every *t* ∈ [0, *T*]. This concludes the proof.\nNext, we turn to *L*1-in-time bounds for the dissipation functional in terms of the energy. The proof is a simplified version of the one in for general degenerate parabolic problems of fourth order. In our case, it relies on testing the partial differential equation with a time cut-off of the second spatial derivative.\n[thm:L1-in-time] Fix *α* > 0 and a positive initial value *u*0 ∈ *H*1(Ω) with *u*0(*x*) > 0 for $x\\in \\bar{\\Omega}$. Let $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n \\cap\n L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ with $u\\_t \\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$ be a positive weak solution to on (0, *T*), as obtained in Theorem [thm:LocalExPL]. Then there exists a constant *C* > 0, independent of *t*, such that the dissipation functional *D*[*u*] enjoys the *L*1-in-time bound\n \n$$\\int\\_{t/2}^{t} D[u](s)\\, ds \\leq \\frac{C}{t} \\int\\_{t/4}^{t/2} E[u](s) \\, ds \\leq \\frac{C}{4} E[u]\\bigl(\\tfrac{t}{4}\\bigr).$$ \nWe choose a cut-off function *χ* ∈ *C*∞(R) in time such that 0 ≤ *χ* ≤ 1, *χ*(*s*) = 1 for *s* ≥ *t*/2, *χ*(*s*) = 0 for *s* ≤ *t*/4 and *χ*ʹ(*s*) ≤ *C*/*t* for some constant *C*, independent of *t*. Now we define the test function $\\varphi(s,x) = \\chi(s) u\\_{xx}(s,x)\\in L\\_{\\alpha+1}\\bigl((0,T);W^1\\_{\\alpha+1,B}(\\Omega)\\bigr)$. Since *u* is a weak solution to on the time interval [0, *t*], we obtain\n \n∫0*t*⟨*u**t*, *χ*(*s*)*u**x**x*⟩*W**α* + 11 *d**s* = ∫0*t*∫Ω*u**α* + 2*ψ*(*u**x**x**x*)*u**x**x**x**χ*(*s*) *d**x* *d**s* = ∫0*t**χ*(*s*)*D*[*u*](*s*) *d**s*. \nMoreover, since *χ*(0) = 0 and *χ*(*t*) = 1, we have the inequality\n \n$$\\label{eq:L\\_1\\_aux\\_2}\n 0 \\leq E[u](t) \n = \n \\int\\_{0}^{t} \\frac{d}{ds} \\bigl( \\chi(s) E[u](s) \\bigr) \\, ds \n = \n \\int\\_{0}^{t} \\chi'(s) E[u](s) \\, ds - \\int\\_{0}^{t} \\chi(s) \\langle u\\_{s}, u\\_{xx} \\rangle\\_{W^1\\_{\\alpha+1}} \\, ds.$$ \nCombining and and using that *χ* ≡ 1 on [*t*/2, *t*] and *D*[*u*](*s*) ≥ 0 for all 0 ≤ *s* ≤ *t*, we conclude that\n \n$$\\begin{aligned}\n \\int\\_{t/2}^{t} D[u](s) \\, ds & \\leq \\int\\_{0}^{t} \\chi(s)D[u](s) \\, ds \n \\leq \\int\\_{0}^{t} \\chi'(s) E[u](s) \\, ds \n \\leq \\frac{C}{t}\\int\\_{t/4}^{t/2} E[u](s) \\, ds.\n \\end{aligned}$$ \nFinally, since *E*[*u*] decreases along solutions, we may estimate\n \n$$\\frac{C}{t}\\int\\_{t/4}^{t/2} E[u](s) \\, ds \\leq \\frac{C}{4}E[u]\\bigl(\\tfrac{t}{4}\\bigr).$$ \nThis completes the proof.\nShear-Thickening Power-Law Fluids (*α* < 1) – Global Existence and Convergence to Steady States in Finite Time\n==============================================================================================================\nThis section deals with the long-time asymptotics of shear-thickening power-law fluids, i.e. we consider flow-behaviour exponents *α* < 1 in. We prove that for positive initial values *u*0 ∈ *H*1(Ω) that are close to a steady state in the sense that\n \n$$\\frac{1}{2}\\bar{u}\\_0 < u\\_0(x) < 2 \\bar{u}\\_0, \n \\quad x\\in \\bar{\\Omega}, \\qquad \\text{where} \\quad \\bar{u}\\_0 = \\fint\\_\\Omega u\\_0\\, dx,$$ \nproblem with *α* < 1 possesses a globally-in-time defined positive weak solution that converges to a steady state in finite time. As in the circular Taylor–Couette setting, the corresponding proof relies mainly on the differential inequality derived in Proposition [prop:EDestimate]. This differential inequality guarantees that the energy becomes zero in finite time 0 < *t*\\* < ∞. We construct a globally-in-time defined positive weak solution by constant extension at time *t*\\*.\nBy Theorem [thm:LocalExPL] and Remark [Rem:H1-initial] there exists a weak solution $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n \\cap\n L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ with $u\\_t \\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$ to. We define the time\n \n$$\\label{eq:time\\_tau}\n \\tau\n =\n \\sup\\{\\tilde{T} > 0;\\ \\text{$\\exists$ a weak solution $u$ to \\eqref{eq:PDE} on } [0,\\tilde{T}]\n \\text{ with } \\tfrac{1}{2} \\bar{u}\\_0 \\leq u(t,x) \\leq 2 \\bar{u}\\_0\\ \\forall\\ 0 \\leq t \\leq \\tilde{T}\\},$$ \nup to which solutions are bounded away from zero and bounded above. Note that by continuity of weak solutions, we have 0 < *τ*. By Remark [Rem:extension] we may also assume that *τ* ≤ *T*. In particular, we can apply the results of Section [sec:regularity-estimates] up to time *τ*.\n[Global existence and convergence in finite time][Thm:Shearthick] Fix 0 < *α* < 1. There exists *ɛ* > 0 such that, for all positive initial values *u*0 ∈ *H*1(Ω) with ∥*u*0 − *ū*0∥*H*1(Ω) < *ɛ*, there is a positive global weak solution\n \n$$u\\in C\\bigl([0,\\infty);H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,\\infty);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr) \\quad \\text{with} \\quad\n u\\_t \\in L\\_{\\frac{\\alpha+1}{\\alpha},\\text{loc}}\\bigl((0,\\infty);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr).$$ \nMoreover, there exists a time 0 < *t*\\* < ∞ such that \n$$u(t,\\cdot) \\longrightarrow \\bar{u}\\_0 \\text{ in } H^1(\\Omega), \\text{ as } t\\to t^\\*, \\quad \\text{and} \\quad u(t,x) = \\bar{u}\\_0, \\quad t \\geq t^\\*,\\ x\\in \\bar{\\Omega}.$$ \nLet $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr) $ the solution to provided by Theorem [thm:LocalExPL] and Remark [Rem:H1-initial] with initial datum *u*0 > *ū*0/2 in $ \\bar{\\Omega}$. Write *u*(*t*, *x*) = *ū*0 + *v*(*t*, *x*) for (*t*, *x*) ∈ [0, *T*] × Ω, where due to conservation of mass ∫Ω*v* *d**x* = 0 for all $t\\in [0,T]\\times \\bar{\\Omega}$. Then, by continuity and the definition of *τ*, we have ∣*v*(*t*, *x*)∣ ≤ *ū*0/2 for $(t,x)\\in [0,\\tau]\\times\\bar{\\Omega}$. Thus, there exists a constant *C* > 0 such that for almost every *t* ∈ [0, *τ*] it holds \n∫Ω∣*v**x**x**x*∣*α* + 1 *d**x* ≤ *C*∫Ω∣*u*∣*α* + 2∣*v**x**x**x*∣*α* + 1 *d**x*. \n Hence, using the energy-dissipation identity and Lemma [Lem:Poincare], we obtain \n$$\\frac{d}{dt} E[v](t) = \\frac{d}{dt} E[u](t) \n = \n - \\int\\_{\\Omega}|u|^{\\alpha+2} |v\\_{xxx}|^{\\alpha+1}\\, dx \n \\leq \n -C \\|v\\_{xxx}(t)\\|\\_{L\\_{\\alpha+1}(\\Omega)}^{\\alpha+1} \\leq -C \\bigl(E[v](t)\\bigr)^{\\frac{\\alpha+1}{2}}$$ \n for almost every *t* ∈ [0, *τ*]. This inequality implies that the energy *E*[*v*]( ⋅ ) = *E*[*u*]( ⋅ ) is decreasing and hence *τ* = *T*. Furthermore, it follows that \n$$\\frac{d}{dt} \\left( \\bigl(E[v](t)\\bigr)^{\\frac{1-\\alpha}{2}} \\right) \\leq -C\\_{\\alpha},$$ \n as long as *E*[*v*](*t*) > 0, and integration from 0 to *t* yields \n$$\\bigl(E[v](t)\\bigr)^{\\frac{1-\\alpha}{2}} \n \\leq \n \\bigl(E[v\\_0]\\bigr)^{\\frac{1-\\alpha}{2}} - C\\_{\\alpha}t, \\quad t\\in [0,T], \\text{ if } E[v](t) >0.$$ \n Thus, we conclude that \n$$E[v](t) \n \\leq \n \\left(\\bigl(E[v\\_0]\\bigr)^{\\frac{1-\\alpha}{2}} - C\\_{\\alpha}t\\right)^{\\frac{2}{1-\\alpha}}, \\quad t\\in [0,T], \\text{ if } E[v](t) >0,$$ \n which implies the existence of a finite time *t*\\* ≥ 0 with $t^\\* \\leq (E[v\\_0])^{\\frac{1-\\alpha}{2}}/C\\_{\\alpha}$ such that \n*E*[*v*](*t*) = 0, *t* ≥ *t*\\*. \n We may choose *ɛ* > 0 small enough so that we obtain *t*\\* < *T*. Finally, note that *E*[*v*](*t*) = 0 for *t* ≥ *t*\\* and *v̄*(*t*) = 0 implies that *v*(*t*, *x*) = 0 for all *t* ≥ *t*\\* and *x* ∈ Ω. Hence, the solution *u* may be extended by the constant solution *ū*0 for times *t* ≥ *t*\\* to a global-in-time weak solution $u\\in C\\bigl([0,\\infty);H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,\\infty);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ and we have \n*u*(*t*, *x*) → *ū*0 in *H*1(Ω) \n and uniformly as *t* → *t*\\* in finite time.\nShear-Thinning Power-Law Fluids (*α* > 1) – Global Existence and Polynomial Stability of Steady States\n======================================================================================================\nIn this section we study the long-time behaviour of solutions to the shear-thinning power-law equation. More precisely, we fix a flow-behaviour exponent *α* > 1 in and consider positive initial values *u*0 ∈ *H*1(Ω) that are close to a steady state in the sense that\n \n$$\\frac{1}{2}\\bar{u}\\_0 < u\\_0(x) < 2 \\bar{u}\\_0, \n \\quad x\\in \\bar{\\Omega},$$ \nwhere $\\bar{u}\\_0 = \\fint\\_\\Omega u\\_0\\, dx$. We show that there exist global positive weak solutions *u* to with *α* > 1 that remain *ɛ*-close to the steady state for all times and converge at rate $1/t^\\frac{1}{\\alpha-1}$ to equilibrium, as *t* → ∞. Note that convergence to equilibrium has already been proved for the global non-negative weak solutions constructed in, but with no rate of convergence. The result on the rate of convergence is the same as in for the cylindrical Taylor–Couette setting. The proof relies again on the differential inequality for the energy, derived in Proposition [prop:EDestimate]. However, in the shear-thinning case also the *L*1-in-time bound of Theorem [thm:L1-in-time] is crucial.\n[Global existence and polynomial stability][Thm:Shearthin] Fix 1 < *α* < ∞. There exists *ɛ* > 0 such that for all positive initial values *u*0 ∈ *H*1(Ω) with ∥*u*0 − *ū*0∥*H*1(Ω) < *ɛ*, there is a global positive weak solution\n \n$$u\\in C\\bigl([0,\\infty);H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,\\infty);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr) \\quad \\text{with} \\quad\n u\\_t \\in L\\_{\\frac{\\alpha+1}{\\alpha},\\text{loc}}\\bigl((0,\\infty);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr).$$ \nMoreover, there is a constant *C* > 0 such that\n \n$$\\|u(t)-\\bar{u}\\_0\\|\\_{H^1(\\Omega)} \\leq \\frac{C{\\varepsilon}}{\\bigl(1+C{\\varepsilon}^{\\alpha-1}t\\bigr)^{\\frac{1}{\\alpha-1}}},\n \\quad\n 0 \\leq t < \\infty.$$ \nFurthermore, the dissipation decreases polynomially along the solution in the following *L*1-in-time sense\n \n$$\\label{eq:L1-in-time-dissipation-thin}\n \\int\\_{t/2}^{t} D[u](s) \\, ds \\leq \\frac{C{\\varepsilon}^2}{\\bigl(1+C{\\varepsilon}^{\\alpha-1}t\\bigr)^{\\frac{2}{\\alpha-1}}}$$ \nfor all 0 ≤ *t* < ∞.\nNote that the weak solution $u\\in C\\bigl([0,\\infty);H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,\\infty);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ obtained in Theorem [Thm:Shearthin] satisfies *u*(*t*, *x*) ≥ *ū*0/2 for all $(t,x)\\in [0,\\infty)\\times\\bar{\\Omega}$. Hence, implies that the *W**α* + 13(Ω)-norm is also controlled in the *L*1-in-time sense by\n \n$$\\int\\_{t/2}^{t} \\int\\_{\\Omega} |u\\_{xxx}(s)|^{\\alpha+1} \\, dx\\,ds \\leq \\frac{C{\\varepsilon}}{\\bigl(1+C{\\varepsilon}^{\\alpha-1}t\\bigr)^{\\frac{1}{\\alpha-1}}}$$ \nfor all 0 ≤ *t* < ∞.\n[**Proof of Theorem [Thm:Shearthin]**] First, we show that there exists an *ɛ* > 0 such that for all initial values *u*0 ∈ *H*1(Ω) with *ū*0 = 0 and ∥*u*0 − *ū*0∥ < *ɛ*, there is a constant *C* > 0 independent of *ɛ* such that\n \n$$E[u](t) \\leq \\frac{C{\\varepsilon}^2}{\\bigl(1+{\\varepsilon}^{\\alpha-1}t\\bigr)^{\\frac{2}{\\alpha-1}}},\n \\quad 0 \\leq t < \\infty.$$ \nLet $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr) $ the solution to provided by Theorem [thm:LocalExPL] and Remark [Rem:H1-initial] with initial datum *u*0 ∈ *H*1(Ω) satisfying *u*0 > *ū*0/2 in $ \\bar{\\Omega}$. As in the proof of Theorem [Thm:Shearthick], we write *u*(*t*, *x*) = *ū*0 + *v*(*t*, *x*) for $ (t,x) \\in [0,T] \\times \\bar{\\Omega}$, where due to conservation of mass ∫Ω*v* *d**x* = 0 for all *t* ∈ [0, *T*]. Then, by continuity and the definition of *τ* (see ), we have ∣*v*(*t*, *x*)∣ ≤ *ū*0/2 for $(t,x) \\in [0,\\tau]\\times \\bar{\\Omega}$. By Lemma [Lem:Poincare] we then conclude that \n$$E[u](t)=E[v](t) \n \\leq \n C \\left(\\int\\_{\\Omega} |v\\_{xxx}|^{\\alpha+1}\\,dx\\right)^{\\frac{2}{\\alpha+1}} \n \\leq \n C \\left(\\int\\_{\\Omega} |u|^{\\alpha+2}|v\\_{xxx}|^{\\alpha+1}\\, dx\\right)^{\\frac{2}{\\alpha+1}} \n = C \\bigl(D[u](t)\\bigr)^{\\frac{2}{\\alpha+1}}$$ \n for almost every *t* ∈ [0, *τ*]. Inserting this into the energy-dissipation identity, we find that\n \n$$\\label{eq:Energy\\_dec\\_Shear\\_thin}\n \\frac{d}{dt} E[u](t) = - D[u](t) \\leq -C \\bigl(E[u](t)\\bigr)^{\\frac{\\alpha+1}{2}}$$ \nfor almost every *t* ∈ [0, *τ*]. This implies that the energy *E*[*u*]( ⋅ ) is decreasing and hence *τ* = *T*. Furthermore, we can rewrite estimate as \n$$\\frac{2}{1-\\alpha} \\frac{d}{dt} \\bigl( E[u](t) \\bigr)^{\\frac{1-\\alpha}{2}} \\leq - C,\n \\quad 0 \\leq t \\leq \\tau,$$ \n so that, after integration, we obtain \n$$\\frac{2}{1-\\alpha} \\bigl(E[u](t)\\bigr)^{\\frac{1-\\alpha}{2}} \\leq -Ct + \\frac{2}{1-\\alpha} \\bigl(E[u\\_0]\\bigr)^{\\frac{1-\\alpha}{2}}, \\quad 0\\leq t \\leq \\tau.$$ \n Since *α* > 1, we can rearrange this inequality to \n$$E[u](t) \\leq \\left(\\bigl(E[u\\_0]\\bigr)^{\\frac{1-\\alpha}{2}} + \\frac{C(\\alpha-1)}{2}t \\right)^{\\frac{2}{1-\\alpha}} = \\frac{E[u\\_0]}{\\left(1+C\\bigl(E[u\\_0]\\bigr)^{\\frac{\\alpha-1}{2}}t\\right)^{\\frac{2}{\\alpha-1}}}, \\quad 0 \\leq t \\leq \\tau.$$ \n Since the function $s\\mapsto \\frac{s}{\\bigl(1+Cs^{\\frac{\\alpha-1}{2}}t\\bigr)^{\\frac{2}{\\alpha-1}}}$ is increasing on [0, ∞) and *E*[*u*0] ≤ *ɛ*2 by assumption, we infer that \n$$E[u](t) \\leq \\frac{C{\\varepsilon}^2}{\\bigl(1+C{\\varepsilon}^{\\alpha-1}t\\bigr)^{\\frac{2}{\\alpha-1}}}, \\quad 0 \\leq t \\leq \\tau.$$ \n Now, we choose *ɛ* > 0 such that\n \n$$\\|u\\_0-\\bar{u}\\_0\\|\\_{L\\_{\\infty}(\\Omega)} \\leq C \\bigl(E[u\\_0]\\bigr)^{\\frac{1}{2}} \\leq \\frac{\\bar{u}\\_0}{2},$$ \nwhere the first estimate is due to the embedding $H^1(\\Omega) \\hookrightarrow L\\_{\\infty}(\\Omega)$ and the Poincaré inequality. This, together with the fact that *E*[*u*]( ⋅ ) is decreasing, guarantees that\n \n$$\\|u(t)- \\bar{u}\\_0\\|\\_{L\\_{\\infty}(\\Omega)} \n \\leq \n C \\bigl(E[u](t)\\bigr)^{\\frac{1}{2}} \n \\leq \n C \\bigl(E[u\\_0]\\bigr)^{\\frac{1}{2}} \n \\leq \\frac{\\bar{u}\\_0}{2},\n \\quad 0\\leq t \\leq \\tau.$$ \nHence, solutions *u* to on [0, *τ*] remain strictly bounded away from zero and by bootstrapping as in Remark [Rem:extension], we may extend it beyond time *τ* to a global-in-time weak solution $u\\in C\\bigl([0,\\infty);H^1(\\Omega)\\bigr)$ \n$\\cap L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,\\infty);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ that satisfies\n \n$$E[u](t) \\leq \\frac{C{\\varepsilon}^2}{\\bigl(1+C{\\varepsilon}^{\\alpha-1}t\\bigr)^{\\frac{2}{\\alpha-1}}}, \\quad 0 \\leq t <\\infty.$$ \nSince by Poincaré’s inequality we have\n \n$$\\|u(t)-\\bar{u}\\_0\\|\\_{H^1(\\Omega)} \\leq C \\sqrt{E[u](t)} \\leq \\frac{C{\\varepsilon}}{\\bigl(1+C{\\varepsilon}^{\\alpha-1}t\\bigr)^{\\frac{1}{\\alpha-1}}},\n \\quad 0 \\leq t <\\infty,$$ \nwe conclude the polynomial stability in *H*1(Ω). For the *L*1-in-time estimate, we apply Theorem [thm:L1-in-time] and obtain\n \n$$\\int\\_{t/2}^{t} D[u](s) \\, ds \\leq C E[u]\\bigl(\\tfrac{t}{4}\\bigr) \\leq \\frac{C{\\varepsilon}^2}{\\bigl(1+C{\\varepsilon}^{\\alpha-1}t\\bigr)^{\\frac{2}{\\alpha-1}}},\n \\quad 0 \\leq t <\\infty.$$ \nThis completes the proof.\nGlobal Existence and Exponential Stability for the Ellis-Law Thin-Film Equations\n================================================================================\nNow we turn to fluids with Ellis-law rheology. These are fluids whose viscosity approaches a Newtonian plateau for low shear rates, while for big shear rates the viscosity is shear-thinning. The corresponding thin-film equation is given by\n \n$$\\label{eq:PDE-Ellis}\n \\begin{cases}\n u\\_t + \\bigl(u^{3}(1+ |uu\\_{xxx}|^{\\alpha-1}) u\\_{xxx}\\bigr)\\_x\n =\n 0,\n & t>0,\\ x \\in \\Omega,\n \\\\\n u\\_x(t,x) = u\\_{xxx}(t,x) = 0, \n &\n t > 0,\\ x \\in \\partial\\Omega,\n \\\\\n u(0,x) = u\\_0(x),\n &\n x \\in \\Omega,\n \\end{cases}$$ \nfor flow-behaviour exponents *α* ≥ 1. Here Ω ⊂ R denotes, as before, a bounded interval.\n[def:weaksol-Ellis] Let *α* > 1. For a given *T* > 0 a weak solution to is defined as a function\n \n$$u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n \\cap\n L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\quad\n \\text{with}\n \\quad\n u\\_t \\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$$ \nthat has the following properties:\n* (Weak formulation) *u* satisfies the differential equation $\\eqref{eq:PDE-Ellis}\\_1$ in the weak sense, i.e.,\n \n$$\\int\\_0^T \\langle u\\_t,{\\varphi}\\rangle\\_{W^1\\_{\\alpha+1}(\\Omega)}\\, dt\n =\n \\int\\_0^T \\int\\_\\Omega u^3\\bigl(1+|uu\\_{xxx}|^{\\alpha-1}\\bigr)u\\_{xxx}\\, {\\varphi}\\_x\\, dx\\, dt$$ \nfor all test functions ${\\varphi}\\in L\\_{\\alpha+1}\\bigl((0,T);W^1\\_{\\alpha+1,B}(\\Omega)\\bigr)$.\n* (Initial and boundary values) *u* satisfies the contact angle condition *u**x* = 0 on ∂Ω and the initial condition $\\eqref{eq:PDE-Ellis}\\_3$ pointwise.\nIn the case of Ellis fluids we naturally obtain the dissipation functional\n \n$$D[u] = \\int\\_{\\Omega} u^3\\bigl(1+|uu\\_{xxx}|^{\\alpha-1}\\bigr)|u\\_{xxx}|^2 \\, dx.$$ \nFor general positive initial data in *H*1(Ω) we can show existence of local-in-time positive weak solutions.\n[Local existence of positive weak solutions] [thm:LocalExEllis] Let *α* > 1. Given a positive initial value *u*0 ∈ *H*1(Ω) with $u\\_0(x) > 0,\\ x \\in \\bar{\\Omega}$, there exists a time *T* > 0 such that problem admits at least one positive weak solution\n \n$$u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n \\cap\n L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\quad \\text{with} \\quad\n u\\_t \\in \n L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))^\\prime\\bigr)$$ \non (0, *T*) in the sense of Definition [def:weaksol-Ellis]. Moreover, such a solution has the following properties:\n* (Positivity) *u* is bounded away from zero\n \n$$0\n < C\\_T \\leq\n u(t,x),\n \\quad\n 0 \\leq t \\leq T,\\ x \\in \\bar{\\Omega}.$$\n* (Conservation of mass) *u* conserves its mass in the sense that\n \n∥*u*(*t*)∥*L*1(Ω) = ∥*u*0∥*L*1(Ω), 0 ≤ *t* ≤ *T*.\n* (Energy-dissipation identity) Energy is dissipated along solutions\n \n*E*[*u*](*t*) + ∫0*t**D*[*u*](*s*) *d**s* = *E*[*u*0] \nfor almost every *t* ∈ [0, *T*].\nFor positive initial datum *u*0 ∈ *W**α* + 1, *B*4*ρ*(Ω), 4*ρ* > 3 + 1/(*α* + 1) with *u*0(*x*) > 0, $x\\in \\bar{\\Omega}$, the problem actually possesses a unique maximal strong solution \n \n$$u \\in C\\bigl([0,T\\_{\\text{max}});W^{4\\rho}\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\cap\n C^{\\rho}\\bigl([0,T\\_{\\text{max}});L\\_{\\alpha+1}(\\Omega)\\bigr)\n \\cap\n C\\bigl((0,T\\_{\\text{max}});W^4\\_{\\alpha+1,B}(\\Omega)\\bigr)\n \\cap\n C^1\\bigl((0,T\\_{\\text{max}});L\\_{\\alpha+1}(\\Omega)\\bigr).$$ \nMoreover, the solution enjoys the following properties:\n* (Positivity) *u* is positive\n \n$$u(t,x) > 0,\n \\quad\n 0 \\leq t < T\\_{\\text{max}},\\, x\\in \\bar{\\Omega}.$$\n* (Conservation of mass) *u* conserves its mass in the sense that\n \n∥*u*(*t*)∥*L*1(Ω) = ∥*u*0∥*L*1(Ω), 0 ≤ *t* < *T*max.\n* (Energy-dissipation identity) *u* satisfies the energy-dissipation identity\n \n*E*[*u*](*t*) + ∫0*t**D*[*u*](*s*) = *E*[*u*0], 0 ≤ *t* < *T*max.\n* (Maximal time of existence) Suppose that *T*max < ∞. Then\n \n$$\\liminf\\_{t \\nearrow T\\_{\\text{max}}} \\frac{1}{\\min\\_{x \\in \\bar{\\Omega}} u(t)} \n + \n \\|u(t)\\|\\_{W^{4\\gamma}\\_{\\alpha+1,B}(\\Omega)} \n = \n \\infty$$ \nfor all *γ* ∈ (*θ*, 1].\n[**Proof of Theorem [thm:LocalExEllis]**] For initial data *u*0 ∈ *W**α* + 1, *B*4*ρ*(Ω), 4*ρ* > 3 + 1/(*α* + 1) with *u*0(*x*) > 0 we obtain local-in-time strong solutions. Choosing a sequence $\\big(u\\_0^{(k)}\\big)\\_{k \\in \\mathbb{N}}$ with $u\\_0^{(k)}(x) > 0,\\ x \\in \\bar{\\Omega}$, and *ū*0(*k*) = *ū*0 such that *u*0(*k*) → *u*0 strongly in *H*1(Ω) guarantees, together with the energy-dissipation identity and similar a-priori bounds as in Lemma [lem:uniformbounds] that the corresponding strong solutions *u*(*k*) converge weakly in $L\\_{\\infty}\\bigl((0,T);H^1(\\Omega)\\bigr)\\cap L\\_{\\alpha+1}\\bigr((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ to a weak solution $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \n \\cap\n L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$. Positivity, conservation of mass and the energy-dissipation identity for almost every *t* ∈ [0, *T*] are preserved under taking the weak limit.\n**Steady states of.** We now turn to stability. First, we find that the same characterisation of positive steady states as before holds true. This is the content of the following theorem which has already been proved in.\n[Characterisation of positive steady states] A function *u* ∈ *W**α* + 1, *B*3(Ω) is a positive steady-state solution of if and only if *u* ≡ *u*\\* ∈ R> 0 is given by positive constant.\n**Global existence and exponential stability for.** It is well-known that for the Newtonian thin-film equation\n \n$$\\label{eq:PDE-Newtonian}\n \\begin{cases}\n u\\_t + \\bigl(u^{3} u\\_{xxx}\\bigr)\\_x\n =\n 0,\n & t>0,\\ x \\in \\Omega,\n \\\\\n u\\_x(t,x) = u\\_{xxx}(t,x) = 0, \n &\n t > 0,\\ x \\in \\partial\\Omega,\n \\\\\n u(0,x) = u\\_0(x),\n &\n x \\in \\Omega,\n \\end{cases}$$ \nsolutions close to positive steady states converge exponentially fast to equilibrium. We now prove that the same behaviour can be found for Ellis-law thin films.\n[Global existence and exponential stability][Thm:ShearthinEllis] Fix 1 < *α* < ∞. There exists *ɛ* > 0 such that, for all positive initial values *u*0 ∈ *H*1(Ω) with ∥*u*0 − *ū*0∥*H*1(Ω) < *ɛ*, there is a global positive weak solution\n \n$$u\\in C\\bigl([0,\\infty);H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,\\infty);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr) \\quad \\text{with} \\quad\n u\\_t \\in L\\_{\\frac{\\alpha+1}{\\alpha},\\text{loc}}\\bigl((0,\\infty);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr).$$ \nMoreover, there is *λ* > 0 and a constant *C* > 0 such that \n∥*u*(*t*) − *ū*0∥*H*1(Ω) ≤ *C**e*− *λ**t*∥*u*0∥*H*1(Ω). \n Furthermore, we find that the dissipation decreases exponentially along the solution in the following *L*1-in-time sense:\n \n∫*t*/2*t**D*[*u*](*s*) *d**s* ≤ *C**e*− 2*λ**t*∥*u*0∥*H*1(Ω)2. \nLet *u*0 ∈ *H*1(Ω) with *ū*0/2 < *u*0(*x*) < 2*ū*0, $x\\in \\bar{\\Omega}$ and $u \\in C\\bigl([0,T];H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1}\\bigl((0,T);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr) $ the solution to provided by Theorem [thm:LocalExEllis]. We also define\n \n$$\\tau\n =\n \\sup\\{\\tilde{T} > 0;\\ \\text{$\\exists$ a weak solution $u$ to \\eqref{eq:PDE-Ellis} on } [0,\\tilde{T}]\n \\text{ with } \\tfrac{1}{2} \\bar{u}\\_0 \\leq u(t,x) \\leq 2 \\bar{u}\\_0\\ \\forall\\ 0 \\leq t \\leq \\tilde{T}\\}.$$ \nThen *τ* ≤ *T* because otherwise we can extend weak solutions beyond time *τ*.\nNext, write *u*(*t*, *x*) = *ū*0 + *v*(*t*, *x*) for (*t*, *x*) ∈ [0, *T*] × Ω, where due to conservation of mass ∫Ω*v*(*t*, *x*) *d**x* = 0 for all *t* ∈ [0, *T*]. Then, by continuity and the definition of *τ*, we have ∣*v*(*t*, *x*)∣ ≤ *ū*0/2 for 0 ≤ *t* ≤ *τ*.\nWe then find, by the energy-dissipation identity and the definition of *τ*, that\n \n$$\\begin{aligned}\n \\frac{d}{dt} E[u](t) &= -D[u](t) = -\\int\\_{\\Omega}u^3(t,x)\\bigl(1+|u(t,x)u\\_{xxx}(t,x)|^{\\alpha-1}\\bigr)|u\\_{xxx}(t,x)|^2 \\, dx \\\\\n &\\leq - \\int\\_{\\Omega} u^3(t,x) |u\\_{xxx}(t,x)|^2 \\, dx \\leq -C\\int\\_{\\Omega} |u\\_{xxx}(t,x)|^2 \\, dx \\leq - C E[u](t)\n \\end{aligned}$$ \nfor almost every *t* ∈ [0, *τ*], where in the last step we have applied Lemma [Lem:Poincare]. This yields that *E*[*u*](*t*) is decreasing and so *τ* = *T*. Applying Gronwall’s inequality, we deduce that\n \n*E*[*u*](*t*) ≤ *E*[*u*0]*e*− *C**t* \nfor all *t* ∈ [0, *τ*]. Now choose *ɛ* > 0 small enough so that\n \n$$\\|u\\_0-\\bar{u}\\_0\\|\\_{L^{\\infty}(\\Omega)} \\leq CE[u\\_0]^{\\frac{1}{2}} \\leq \\frac{\\bar{u}\\_0}{2},$$ \nwhere in the first estimate we have used the embedding $H^1(\\Omega) \\hookrightarrow L\\_{\\infty}(\\Omega)$ and Poincaré’s inequality. Using this and the fact that *E*[*u*](*t*) is decreasing, we get\n \n$$\\|u(t)-\\bar{u}\\_0\\|\\_{L^{\\infty}(\\Omega)} \\leq CE[u](t)^{\\frac{1}{2}} \\leq CE[u\\_0]^{\\frac{1}{2}} \\leq \\frac{\\bar{u}\\_0}{2}$$ \nfor all *t* ∈ [0, *T*]. We can then extend the solution beyond time *T* to a global-in-time weak solution $u\\in C\\bigl([0,\\infty);H^1(\\Omega)\\bigr) \\cap L\\_{\\alpha+1,\\mathrm{loc}}\\bigl((0,\\infty);W^3\\_{\\alpha+1,B}(\\Omega)\\bigr)$ to that satisfies\n \n*E*[*u*](*t*) ≤ *E*[*u*0]*e*− *C**t*, 0 ≤ *t* < ∞. \nBy Poincaré’s inequality, we then conclude that\n \n∥*u*(*t*) − *u*0∥*H*1(Ω) ≤ *C**E*[*u*(*t*)]1/2 ≤ *C*∥∇*u*0∥*L*2(Ω)*e*− *λ**t*, \nfor some *λ* > 0 and all *t* ∈ (0, ∞).\nThe *L*1-in-time estimate follows from adapting Theorem [thm:L1-in-time] to the new dissipation functional.\nProof of Lemma [lem:convergence] and Lemma [lem:limitflux]\n==========================================================\nHere we give precise proofs of the auxiliary results needed to establish local existence of positive weak solutions to the original problem in Section [sec:LocalExistence].\n[**Proof of Lemma [lem:convergence]**] **(i)** In Lemma [lem:uniformbounds] (i) and (iii) we have shown that\n \n$$\\begin{cases}\n (u^\\sigma)\\_\\sigma \\text{ is uniformly bounded in } L\\_\\infty\\bigl((0,T);H^1(\\Omega)\\bigr) & \\\\\n (u^\\sigma\\_t)\\_\\sigma \\text{ is uniformly bounded in } L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr).&\n \\end{cases}$$ \nMoreover, in view of the Rellich-Kondrachov theorem, see e.g., we have\n \n$$H^1(\\Omega) \\xhookrightarrow[]{c} C^{\\rho}(\\bar{\\Omega}) \\hookrightarrow (W^1\\_{\\alpha+1}(\\Omega))', \\quad \\rho \\in [0,1/2),$$ \nwhere $\\xhookrightarrow[]{c}$ indicates compactness of the embedding. This enables us to use, which gives that the sequence\n \n$$(u^\\sigma)\\_\\sigma \\text{ is relatively compact in } C\\bigl([0,T];C^\\rho(\\bar{\\Omega})\\bigr)$$ \nwith *ρ* ∈ [0, 1/2) as above.\n**(ii)** This is an immediate consequence of Lemma [lem:uniformbounds] (ii).\n**(iii)** By Lemma [lem:uniformbounds] (iii), we can extract a subsequence (*u**t**σ*)*σ* such that\n \n$$u^\\sigma\\_t \\rightharpoonup v \n \\quad \\text{weakly in } L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr) \\hookrightarrow {\\mathcal{D}}'\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$$ \nfor some limit function $v \\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$. Since, in addition,\n \n$$u^\\sigma \\longrightarrow u \\quad \\text{in } C\\bigl([0,T];C^\\rho(\\bar{\\Omega})\\bigr) \\hookrightarrow {\\mathcal{D}}'\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr) \\quad \n\\rho \\in [0,1/2),$$ \nwe conclude that\n \n$$u^\\sigma\\_t \\longrightarrow \n u\\_t\n \\quad \\text{in }\n {\\mathcal{D}}'\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)\\bigr),$$ \nand thus, $v = u\\_t \\in L\\_\\frac{\\alpha+1}{\\alpha}\\bigl((0,T);(W^1\\_{\\alpha+1,B}(\\Omega))'\\bigr)$.\n**(iv)** Note that the strong convergence *u**σ* → *u* in $C\\bigl([0,T];C^\\rho(\\bar{\\Omega})\\bigr),\\, \\rho \\in [0,1/2),$ in (i) implies uniform convergence\n \n$$\\label{eq:conv\\_1}\n u^\\sigma \n \\longrightarrow \n u \n \\quad\n \\text{in } \n C\\bigl([0,T]\\times \\bar{\\Omega}\\bigr"},"data_id":{"kind":"string","value":"arxiv_0002444"}}},{"rowIdx":594,"cells":{"content":{"kind":"string","value":" fragmentation brings in the membership disclosure protection. In general, the fragmentation process has the potential to be extended for other privacy models, because of its meta-approach, which is more easily generalizable. This might eventually provide unprecedented flexibility in using the fragmentation method as a general-purpose meta-algorithm in the context of a wide variety of scenarios. This paper is organized as follows. The remainder of this section discusses related work. Section 2 discusses the overview of the approach for the *k*-anonymity. Section 3 discusses details of the fragment-based *k*-anonymization method. Extending the fragmentation approach for ℓ-diversity (or *t*-closeness) is shown in Section 4. In addition, this section shows how the fragmentation-based anonymization can prevent the membership attack. The experimental results are presented in Section 5. Section 6 contains the conclusions and summary.\nRelated Work.\n=============\nThe problem of privacy preservation was first studied in. This approach was based on noise-based perturbation of the underlying data. Subsequently, the problem of *k*-anonymization of data was proposed in. Other models for data privacy have been proposed in. Numerous methods have been proposed for effective and efficient *k*-anonymization, and this continues to remain one of the most widely used models in the literature because of its simplicity. The theoretical results illustrating the degradation of privacy-preservation methods with increasing dimensionality have been discussed in. The work in proposed an anonymization method for high-dimensional data. However, it is only applicable to sparse transactional data, and is heavily dependent of the sparse structure of transaction data in order to achieve this goal. Kifer suggested the idea of releasing anonymized marginals beside the anonymized original table. However, the original table must still be anonymized as a whole which results in high information loss. Furthermore, publishing the marginals makes the published data highly workload dependant. The privacy models in can cope with the problem of curse of dimensionality by reducing the number of quasi-identifiers[4](#fn4). That is, they make an assumption about the number of quasi-identifiers known by an attacker, and apply anonymization only on limited number of quasi-identifiers. However, this assumption may not hold true in some cases. Other relevant works are able to provide some protection in the high dimensional case, though they can be challenged in some circumstances. Furthermore, any of these methods can be used in combination with our approach, which is designed as a more general purpose meta-algorithm. Finally, the concept of vertical partitioning and fragmentation has been explored in the context of distributed data privacy with cryptographic protocols, or for capturing confidential privacy constraints in the context of such methods. The goals and motivations of these methods are quite different, and are not at all focused on the problem of high dimensional anonymization.\nIn the context of increasing dimensionality, it is natural to explore feature selection as an alternative in order to reduce the data dimensionality. However, this is a rather drastic solution, which can impact the quality of the underlying results significantly. Therefore, this paper proposes the approach of fragmentation as a general-purpose meta-algorithm in order to improve the robustness of high-dimensional anonymization algorithms.\nOverview of The Approach for The Fragmentation *K*-Anonymity\n============================================================\nIn this section, we first introduce the most important notations used in the fragmentation-based *k*-anonymity in Table [table:notations]. An overview is then provided about vertical fragmentation, and its incorporation as a general-purpose meta-algorithm for privacy preservation.\n[H]\n[table:notations]\n|c|c|\n&\n${\\cal F}$ & a vertical fragmentation & *i**t**h* fragment in fragmentation ${\\cal F}$ & *j**t**h* equivalence class in fragments *F**i* & set of all class values in equivalence class *E**Q**i**j* & power set of set *S* & size of a set \nVertical Fragmentation\n----------------------\nLet *T* be a relation defined over a schema *T*(*A*1*f*, *A*2*f*, ..., *A**n**f*, *A**c*) where *A**i**f* represents the feature attributes, and *A**c* is the class attribute. A vertical fragmentation of relation *T* splits the feature variables into multiple non-overlapping fragments. Formally, a vertical fragmentation is defined as follows:\n(VERTICAL FRAGMENTATION). Given a relation schema *T*, a vertical fragmentation ${\\cal F}$ of *T* is a partitioning of the attributes into fragments ${\\cal F}=\\{\nF\\_1,F\\_2,...,F\\_m \\}$ such that each *F**i* contains a disjoint subset of the feature attributes. Therefore, it is the case that $\\forall\nF\\_i \\in {\\cal F}$, *F**i* ⊆ *T* and *F**i* ∩ *F**j* = ∅ (*i* ≠ *j*) and ⋃*F**i* = *T* (*i*=1, ..., *m*).\nIn addition, it is *implicitly assumed* that the class attribute *A**c* is associated with each fragment. As we will see later, the presence of this common attribute needs to be accounted for in a special way since it allows the re-construction of *some* correspondence between the attribute values of different fragments. Therefore, methods need to be designed to ensure that this correspondence cannot be used in order to attack the anonymity of the fragmented data. In some cases, this process requires the perturbation of a few class values, in order to ensure non-identifiability.\nNote that the fragmentation process is used as a meta-approach in conjunction with an off-the-shelf anonymization algorithm. A fragmentation ${\\cal F}$ is referred to as a **k*-anonymous fragmentation* after applying an anonymization algorithm to it, if and only if the following two conditions hold:\n**Fragment *k*-anonymity condition:** Each fragment in ${\\cal F}$ satisfies the *k*-anonymity condition. This condition can be easily satisfied by applying any off-the-shelf *k*-anonymity algorithm to each fragment.\n***K*-Anonymity non-reconstructability condition:** The relation resulting from joining any arbitrary fragments on the class variable satisfies the *k*-anonymity condition. The number of possible fragmentations of a set of features is rather large in the high-dimensional case. For example, for a set of *n* features, there may be *O*(*n**n*) possible fragmentations. Clearly, exhaustive search through all the possible fragmentations for the high-dimensional case may become prohibitive. Therefore, a systematic approach is required to search the space of possible fragmentations. Since this work is focussed on the classification problem, the fragmentation approach should attempt to maximize the amount of *non-redundant* information contained in each feature of a particular fragment, which is relevant for the classification process. Correspondingly, our systematic search approach utilizes a metric referred to as *Fragmentation Minimum Redundancy Maximum Relevance (FMRMR)* in order to create fragments.\nFragmentation Minimum Redundancy Maximum Relevance\n--------------------------------------------------\nThe ideal fragmentation is one in which the set of attributes in each fragment is a comprehensive representation of the information required for the mining process. Since this paper addresses the classification problem, the metric will explicitly use the class variable for quantification, though it is conceivable that the metric for other applications would be different. In the supervised context, a comprehensive representation refers to high predictability of the class variable from the features in each fragment, while minimizing redundancy. It is evident that the *simultaneous* incorporation of features with high mutual information within a given fragment does not provide any additional advantages, even when they are all highly relevant to the class attribute. This implies that a combination of the relevance to the class attribute and the mutual information with respect to one another can be useful for the process of constructing a fragment.\nTo this effect, we draw on the feature selection literature, which defines the concept of the *Minimum Redundancy Maximum Relevance (MRMR)* metric. This metric uses a dependency quantification (denoted as *W*) among the feature variables and a dependency quantification (denoted as *V*) between the feature variables and the class attribute in each fragment. Our proposed heuristic aims at maximizing the summation of MRMR for all the fragments in a given fragmentation. The Fragmentation MRMR (FMRMR) is the summation of the values of MRMR within a fragment. This value is defined for a particular fragmentation ${\\cal F}$ as follows:\n$ FMRMR({\\cal F})= \\sum\\_{t=1}^{|{\\cal F}|} (V\\_{t} - W\\_{t}) $\n$V\\_{t} = \\frac{1}{|A\\_t^f|} \\sum\\_{j \\in A\\_{t}^f} I(cls,j)$\n$W\\_{t} = \\frac{1}{|A\\_{t}^f|^2} \\sum\\_{k,j \\in A\\_{t}^f} I(k,j)$\nwhere:\n*A**t**f*: set of features in fragment *t* of fragmentation ${\\cal F}$\n*I*(*x*, *y*): mutual information between attributes *x* and *y*\n*V**t*: total mutual information between the features and the class attribute in fragment *t* of fragmentation ${\\cal F}$\n*W**t*: total pairwise mutual information between the features in fragment *t* of fragmentation ${\\cal F}$\n*c**l**s*: the class attribute\nThe overall approach for the *k*-anonymity uses a three-step technique for the fragmentation process. For a high-dimensional relation *T* with *n* features and one class attribute, these three steps are as follows:\n1. Use a carefully-designed search algorithm to decompose the relation into fragments. The constructed fragments have non-overlapping sets of features together with the class attribute. The fragmentation process uses the afore-mentioned measure in order to determine the optimal fragments.\n2. Anonymize each fragment separately using an existing anonymization algorithm, such as the Mondrian multi-dimensional *k*-anonymity algorithm.\n3. At this point, it should be noted that the anonymized fragments can be (partially) joined back using the common attribute, which in the supervised scenario is the class attribute. Depending on the distribution of values in the common attribute, the result might violate the *k*-anonymity constraint. This is essentially a **k*-anonymity non-reconstructability condition violation*. Therefore, additional steps are required in order to ensure non-reconstructability. The techniques for achieving this are slightly involved and distort the class variable in such a way that non-reconstructability is guaranteed. These methods will be described in the next section. It should be noted that the distortion of the class variable may result in some further reduction in accuracy. However, in *practice*, for most reasonable distributions, the required distortions are very limited, if any.\nThe second step in the afore-mentioned list does not require further explanation. Therefore, the exposition in this paper will describe the detailed methods for performing the first and the third steps. For the third step, three different alternatives will be proposed. It should also be noted that although the class and sensitive attributes have been considered the same in many works, they might be different, and data contains many sensitive attributes in practice. In such cases, the other sensitive attributes also need to be fragmented in order to ensure that the two fragments cannot be joined. However, they should be fragmented only *after* the quasi-identifiers have already been fragmented (using the same approach as discussed in the next section). This is because it is more critical to ensure that quasi-identifiers are evenly distributed among fragments. Therefore, what follows will only focus on quasi-identifiers for simplicity.\nFragmentation-based *K*-Anonymization\n=====================================\nIn this section, the first and third steps in the afore-mentioned fragmentation meta-algorithm will be discussed. First, the fragment construction heuristic will be introduced.\nFragment Construction Heuristic\n-------------------------------\nAs the number of features increases, the number of possible fragments grows exponentially. This explosion in the number of fragments makes exhaustive search in this space impractical. Therefore, we propose an algorithm which tries to form a fragmentation with maximum *F**M**R**M**R*. For simplicity, a binary fragmentation into two parts will be described, though it is possible in principle to fragment into multiple parts by repeating the process.\nWe define the *FMRMR contribution* of a feature attribute *A**i**f* with respect to fragment *F**j* of fragmentation ${\\cal F}$ as the difference between *F**M**R**M**R* of ${\\cal F}$ after and before adding *A**i**f* to *F**j*. The quantification of the mutual information between the *n* features and the class attribute is stored in the form of an (*n* + 1) × (*n* + 1) matrix denoted by [*M**I*](*n* + 1) × (*n* + 1).\nThe *F**M**R**M**R* metric attempts not to place features having high mutual information in one fragment. Therefore, as a starting point, two features having the highest mutual information are picked as seeds and placed in different fragments. Afterwards, in a greedy manner, and while there exists un-assigned features, the *FMRMR contributions* of all unassigned features with regards to both fragments are calculated. The unassigned feature with the highest *FMRMR contribution* is added to the relevant partition. Finally, the common attribute (class attribute in the supervised case) is added to each fragment separately. The overall approach is illustrated in Algorithm [alg:heuristic] in the Appendix A in the supplementary materials.\nThe Final Step: *K*-Anonymity non-Recons- tructability\n------------------------------------------------------\nAs indicated earlier, applying a *k*-anonymity algorithm on each fragment in order to satisfy the *fragment k-anonymity condition* is not sufficient for ensuring non-identifiability. This is because the common attribute (class attribute) can be used for (very approximate) joins, and such joins provide some additional information about fragment correspondence. Therefore, in theory, it may be possible that the overall anonymity level of the relation resulted from joining *k*-anonymized fragments is less than *k*, though in practice it is rather unlikely because of the approximate nature of the join.\nWe call a fragmentation in which all fragments satisfy the *k*-anonymity condition **k*-anonymity non-reconstructible* if the relations resulting from joining any arbitrary fragments on the class attribute satisfy *k*-anonymity. Similarly, a fragmentation is called *reconstructible* if *k*-anonymity is violated after joining some of its fragments. Definition [def:nonreconstructable] formally defines a **k*-anonymity non-reconstructible* fragmentation.\n(*K*-ANONYMITY NON-RECONSTRUCTIBLE FRAGMENTATION). [def:nonreconstructable] Fragmentation ${\\cal\nF}=\\{F\\_1,F\\_2,...,F\\_m\\}$ which satisfies the fragment *k*-anonymity condition is called *k*-anonymity non-reconstructible if and only if $\\forall s\n\\in \\mathcal P \\left({{\\cal F}}\\right)$, the relation resulting from joining members (fragments) of *s* satisfies the *k*-anonymity condition.\nThe power set $\\mathcal P \\left({{\\cal F}}\\right)$ has 2*m* members. However, the *k*-anonymity condition must be checked for members of size at least 2.\nIt should be noted that the joining process is only approximate and noisy, which is good for anonymization. Therefore, successful violation attacks of the type discussed above are often difficult to perform in practice. For example, joining will result in some tuples that do not have any corresponding tuple in the original table. These tuples are called *fake tuples*, and may sometimes be helpful for obfuscation of identification of relevant tuple identities.\nAs articulated in Definition [def:nonreconstructable], given a fragmentation ${\\cal F}$ with *m* fragments, the *k*-anonymity condition must be satisfied for (2*m*-*m*-1) possible relations resulted from joining arbitrary fragments. However, the relations resulted from joining more than two fragments can be simply obtained by consecutive binary joins. This paves the way to define the *k*-anonymity non-reconstructability condition by joining only two fragments. For instance, checking the *k*-anonymity non-reconstructability condition on a relation resulting from joining members of $\\{F\\_1,F\\_2,...,F\\_n\\} \\in \\mathcal P \\left({{\\cal\nF}}\\right)$ can be accomplished by checking the *k*-anonymity non-reconstructability condition in each of the following binary joins: *I*1 = *F*1 ⋈ *F*2, *I*2 = *I*1 ⋈ *F*3,..., *I**n* − 1 = *I**n* − 2 ⋈ *F**n*.\nTherefore, for simplicity, we can continue our discussion with only two fragments, without any loss of generality. This condition is formally stated in Theorem [thm:nonreconstructability]. An important concept to continue with the remaining of the paper is to understand the notion of equivalence classes resulting from the anonymization. In the anonymized table, records with the same value for their quasi-identifiers constitute an *equivalence class*.\n(*K*-ANONYMITY NON-RECONSTRUCTABILITY CONDITION). [thm:nonreconstructability] The condition for a given fragmentation ${\\cal F}=\\{F\\_1,F\\_2\\}$ which satisfies the fragment *k*-anonymity condition, and fragments *F*1 = {*E**Q*11, *E**Q*12, ..., *E**Q*1*n*} and *F*2 = {*E**Q*21, *E**Q*22, ..., *E**Q*2*m*}, to be non-reconstructible is that one of the following must be true for each joined pair *E**Q*1*i*, *E**Q*2*j*:\n∑*c* ∈ *C*1*i* ∩ *C*2*j**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) = 0\n∑*c* ∈ *C*1*i* ∩ *C*2*j**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) ≥ *k*\nThe proof of this theorem is presented in Appendix B in the supplementary materials.\nEnforcing and satisfying the *k*-anonymity non-reconstructability condition in a fragmentation may require some of the class values to be distorted. As explained shortly, the change in the class values can be performed using various strategies. However, minimizing the number of changes is always desirable in order to retain accuracy. The design of an algorithm which provably minimizes the changes is computationally intractable because of the exponential number of possibilities. Therefore, we propose three heuristic strategies to enforce the *k*-anonymity non-reconstructability condition. It is worth mentioning that the utility of each strategy is different from others. Before providing a more detailed exposition, we introduce the concept of a *dependency graph*, which provides the logical construct necessary for a good algorithmic design.\n### Dependency Graph\nA *dependency graph* is an undirected graph structure which captures the dependency among different equivalence classes in a given fragmentation. Nodes in the *dependency graph* are equivalence classes, and there exists an edge between node *E**Q**i**x* and *E**Q**j**y* provided that:\n1. *i* ≠ *j* that means *E**Q**i**x* and *E**Q**j**y* belong to two different fragments.\n2. *C**i**x* ∩ *C**j**y* ≠ ∅ that means *E**Q**i**x* and *E**Q**j**y* have at least one class value in common.\nThe set of all equivalence classes in a fragmentation may be divided into subsets having no dependency on each other. In other words, no equivalence class from one subset can be joined with equivalence classes in the other subset. Thus, the *dependency graph* is not connected, and the *dependency graph* components reflect the full dependencies among all equivalence classes, rather than a single connected *dependency graph*. The process of constructing the *dependency graph* is shown in Algorithm [alg:dependencygraph] in Appendix A in the supplementary materials.\nThe *k*-anonymity non-reconstructability condition is enforced on each connected component of the *dependency graph* separately, since there is no inter-component dependency. The *k*-anonymity non-reconstructability condition on each connected component can also be achieved by enforcing it on each edge. We introduce three different strategies in order to achieve this goal. \n### Naive Enforcement\nSatisfying the *k*-anonymity non-reconstructability condition for an edge between equivalence classes *E**Q*1*i* and *E**Q*2*j* can simply be done by enforcing each equivalence class to have only class value. In the naive *k*-anonymity non-reconstructability enforcement approach, class values in each equivalence class are changed to the majority class in that equivalence class. In this case, two given equivalence classes either cannot be joined, or their join generates at least *k*2 tuples. Such an approach is clearly suboptimal, and fails to take full advantage of the flexibility associated with distorting the class variable in a way which is sensitive to the behavior of the remaining data.\n### Dependency Graph-based Enforcement\nUnlike the naive approach, the class values in *only* equivalence classes violating the *k*-anonymity after being joined are changed to the majority class value in this approach. Another difference between this approach and the naive approach is that the dependency graph-based approach aims at minimizing the number of changes in each equivalence class. In order to achieve this goal, this approach changes only one attribute in each step.\nStarting from a random node (*current-node*) in the *dependency graph*, the *dependency graph* is explored in a breadth-first manner. The *k*-anonymity non-reconstructability condition is checked between *current-node* and every single unvisited neighbor nodes. If the condition does not hold between *current-node* and one of its unvisited neighbors, the class values with lowest frequency in the neighbor node is changed to the majority class value until the condition is satisfied. After satisfying *k*-anonymity non-reconstructability between *c**u**r**r**e**n**t*-*n**o**d**e* and all its neighbor nodes, *c**u**r**r**e**n**t*-*n**o**d**e* is marked *v**i**s**i**t**e**d*. This process must be repeated for all *components* in the *dependency graph* until all nodes are marked *v**i**s**i**t**e**d*. The pseudocode of this algorithm is demonstrated in Algorithm [alg:dgbe] in Appendix A in the supplementary materials.\n### Enforcement via *δ*-selectivity\nIn spite of the approximate nature of the join between different fragments, they are a potential threat to *k*-anonymity. Thus, the prevention of violating joins is important. Publishing the class values for *each and every* single tuple (row) in the anonymized fragment is a major cause of this violation.\nThe *δ*-selectivity approach changes the way in which class values are published. This enables a more relaxed *k*-anonymity non-reconstructability condition enforcement on the equivalence classes. Instead of publishing the class values on a *per tuple* basis, they are published on a *per equivalence class* basis with the use of *ambiguous values* (*s**l**o**t**s*). In an equivalence class, each class value has equal probability of being assigned to a tuple. This results in the possibility of assuming different instantiations (or versions) for a given equivalence class. Then, given two equivalence classes, there exist multiple ways to join them, corresponding to different assignments of class values to tuples. The modified *k*-anonymity non-reconstructability condition leverages this ambiguity effectively. Appendix C.I in the supplementary materials exemplifies tuple-level and equivalence class-level class value publishing.\nAs mentioned above, the ambiguous slots in an equivalence class *E**Q**i**j*, published at the equivalence class-level, can take any of the class values in *C**i**j*. In other words, different versions for *E**Q**i**j* can be assumed.\n(EQUIVALENCE CLASS VERSION). An arbitrary assignment of class values available in an equivalence class *E**Q**i**j* to ambiguous slots in *E**Q**i**j* generates a version of *E**Q**i**j* shown by *V*(*E**Q**i**j*).\nAlthough publishing the class values at the equivalence class-level reduces the risk of *k*-anonymity violation, the resulting equivalence classes are still vulnerable to be joined back and violate the *k*-anonymity. As an example, consider the extreme case where the class values are unique in an equivalence class. Each tuple in the equivalence class is assigned to a different class value which is similar to the case in which class values are released at the tuple level. Given two equivalence classes whose class values are published in equivalence class-level and have at least one class value in common, there exist different ways to join them. The number of tuples resulted from joining two equivalence classes are referred to as *equijoin selectivity*. Among all possible joins, those generating *k* tuples (or more) are referred to as **k*-anonymity-preserving equijoins*.\n(*K*-ANONYMITY-PRESERVING EQUIJOIN). Given two equivalence classes *E**Q*1*i* and *E**Q*2*j* whose class values are published at the equivalence class level, and which share at least one class value, the join between V(*E**Q*1*i*) and V(*E**Q*2*j*) is a *k*-anonymity-preserving equijoin if and only if it produces at least *k* tuples.\nWe can now define the equijoin selectivity privacy level in terms of the possible equijoins between two equivalence classes.\n(EQUIJOIN SELECTIVITY PRIVACY LEVEL). The ratio of number of *k*-anonymity-preserving equijoins in joining two equivalence classes *E**Q*1*i* and *E**Q*2*j* to the total number of possible equijoins in joining the same equivalence classes is referred to as equijoin selectivity privacy level of *E**Q*1*i* and *E**Q*2*j*. This value is denoted by *η*(*E**Q*1*i*, *E**Q*2*j*). In other words, *η*(*E**Q*1*i*, *E**Q*2*j*) = \n$ \\frac{|ds\\_p=\\{\\{ V(EQ\\_{1i})\\} \\Join\n\\{V(EQ\\_{2j})\\} |\\hspace{1mm}|\\{ V(EQ\\_{1i})\\} \\Join\n\\{V(EQ\\_{2j})\\}| \\geq k \\}|} {|ds\\_w=\\{\\{V(EQ\\_{1i})\\} \\Join\n\\{V(EQ\\_{2j}) \\} |}.$\nIntuitively, *η*(*E**Q*1*i*, *E**Q*2*j*) indicates the probability that the result of joining *E**Q*1*i* and *E**Q*2*j* is a *k*-anonymous equivalence class. As an example, when the value of *η* is 1, it indicates that ∀*v*1 ∈ *V*(*E**Q*1*i*), *v*2 ∈ *V*(*E**Q*2*j*), we have ∣*v*1 ⋈ *v*2∣ ≥ *k*. In other words, all possible instantiations will result in a *k*-anonymity-preserving equijoin.\n(*δ*-SELECTIVE *K*-ANONYMOUS FRAG.). A fragmentation ${\\cal F}=\\{F\\_1,F\\_2\\}$ that satisfies the *fragment *k*-anonymity condition* is called *δ*-selective if and only if ∀*E**Q*1*i*, *E**Q*2*j* we have *η*(*E**Q*1*i*, *E**Q*2*j*) ≥ *δ*.\nAlgorithm [alg:deltaselectivity] in Appendix A in the supplementary materials shows how *δ*-selectivity can be enforced on a fragmentation ${\\cal F}$.\nExtension to ℓ-Diversity\n========================\nAnalogous to the *k*-anonymity case, a fragmentation is called an ℓ-diverse fragmentation, if and only if 1) each fragment satisfies the ℓ-diversity requirement (fragment ℓ-diversity condition) and 2) joining the fragments does not violate the ℓ-diversity requirement (ℓ-diversity non-reconstructability condition). Satisfying the fragment ℓ-diversity condition is similar to that of the *k*-anonymity case. However, for ℓ-diversity, the non-reconstructability condition is different. Table [table:notationsdiversity] describes the notations used in this section. A brief overview of the steps for the fragmentation-based ℓ-diversity is provided below:\n[H]\n[table:notationsdiversity]\n|c|c|\n&\n${\\cal F}$ & a vertical fragmentation & *i**t**h* fragment in fragmentation ${\\cal F}$ & *i**t**h* segment & data chunk belongs to segment *S**i* and fragment *F**j* & *i*th equivalence class belongs to chunk *C**K**s**j* & set of class values for *C**K**i**j* & set of class values for *E**Q**i**j**s* & diversity level of segment *S**i* & size of a set \n1. Use the fragment construction algorithm proposed in Section 4.1 solely to *compute* the best fragmentation ${\\cal F}=\\{F\\_1,F\\_2,...,F\\_n\\}$. However, the fragmentation is not actually *executed* in this step.\n2. Cluster data records into *m* segments ${\\cal S}=\\{S\\_1,S\\_2,...,S\\_m\\}$ using a top-down clustering algorithm. Stop dividing each segment *S**i* into further sub-segments if the resulting sub-segments either violate the *k*-anonymity or ℓ-diversity requirement. Final segment *S**i* has the diversity level *l**i*.\n3. Vertically partition each segment *S**i*, using the fragments found in Step 1, into *n* data chunks. The diversity level of each data chunk *C**K**i**j* is equal to the diversity level of segment *S**i*, which is *l**i*.\n4. Use any off-the-shell ℓ-diversity algorithm to anonymize each chunk *C**K**i**j*. However, the diversity requirement of each *C**K**i**j* must be set to *l**i*.\n5. Merge equivalence classes belonging to the same vertical fragments and publish them as one fragment.\nIn the afore-mentioned algorithm, it is worth noting that the clustering algorithm in the second step should be adjusted to the relevant workload. Since the workload in our approach is classification, a classification-oriented clustering algorithm results in higher utility.\nThe ℓ-diversity non-reconstructability condition may be violated if at least two chunks *C**K**s**i* and *C**K**s**j* are joined and the resulting data set violates either the *k*-anonymity or the ℓ-diversity requirement. We prove that this will not be the case according to the way equivalence classes in chunks have been formed. For simplicity, we prove this for the case of two vertical fragments. However, the result is true in general.\nThe data set resulting from joining chunks *C**K**s*1 and *C**K**s*2 neither violates *k*-anonymity nor ℓ-diversity.\nAgain, the proof of this theorem is presented in Appendix B in the supplementary materials for the sake of brevity. It is easy to show that this extension can be utilized for *t*-closeness by simply enforcing *t*-closeness instead of ℓ-diversity in the aforementioned steps.\nMembership Disclosure Protection\n--------------------------------\nFragmenting the data can help protect against membership attack by disassociating different attributes. As discussed in, the ability to determine presence or absence of a subject’s record in the published data is a privacy threat. This can be done by comparing the subject’s quasi-identifiers with the published quasi-identifiers.\nConsider an attacker trying to find out the membership of subject *v* in the published fragmented data. As attributes are fragmented, the attacker must find the matching equivalence class in each fragment to which the subject’s attributes belong. This may not be possible considering the generalization applied on the attributes. However, provided that the attacker succeeds in finding the matching equivalence classes *E**Q*1*i*, *E**Q*2*j*,..., *E**Q**n**p*, the likelihood that the record pertaining to *v* exists in the published fragmented data is $\\frac{|EQ\\_{1i}\\Join EQ\\_{2j} \\Join...\\Join\nEQ\\_{np}|}{|F\\_1 \\Join F\\_2 \\Join... \\Join F\\_n|} = \\frac{|EQ\\_{1i}\\Join EQ\\_{2j} \\Join...\\Join\nEQ\\_{np}|}{\\sum\\_{i} \\sum\\_{j}...\\sum\\_{p} |EQ\\_{1i}\\Join EQ\\_{2j} \\Join...\\Join\nEQ\\_{np}|} $.\nThis likelihood is mostly impacted by *p* (the number of vertical fragments), anonymity level (either *k* or ℓ), and ∣*D*∣ (size of the data set). In most cases, the value of *p* is small, and ∣*D*∣ ≫ *k*. Therefore, the numerator of the likelihood formula becomes much smaller than the denominator. When the value of *p* increases, the number of common class values among specific equivalence classes *E**Q*1*i*, *E**Q*2*i*,..., and *E**Q**n**p* drops and even tends to zero in many cases. Therefore, the chance of a successful membership attack becomes negligible. In general, the fragmentation-based anonymization provides strong protection against membership attack.\nExperimental Results\n====================\nIn this section, we will present the experimental results showing the effectiveness of our method. The goal is to show that the fragmentation process is able to retain greater utility of the data both in terms of classification measures and information loss measures, at the same level of privacy.\nWe utilized two metrics to evaluate the effectiveness of our proposed method, *information loss* and *weighted F-measure* to capture the total amount of lost information and evaluate the utility of the data anonymized by our meta-algorithm, respectively. Details on how these metrics are used on fragmented data are available in Appendix D.I in the supplementary materials.\nBaselines\n---------\nSince the goal was to show the effectiveness of the fragmentation approach as a meta-algorithm, the baseline for the approach were the results for the anonymization process with and without fragmentation. The Mondrian multidimensional anonymity method was used for *both* the fragmented, *and* the unfragmented scenario. More accurately, in Step 2 of fragmented *k*-anonymity, we used *m**e**d**i**a**n* Mondrian and in Step 4 of fragmented ℓ-diversity, we utilized ℓ-diversity Mondrian. Therefore, the qualitative improvements show the effects of fragmentation, as a methodology to improve the effectiveness of an off-the-shelf approach.\nData Sets\n---------\nReal data set *Musk* from the *UCI Machine Learning Repository*[5](#fn5) was used. The detailed description of the data set is provided in the Appendix D.II of the supplementary materials.\nResults\n-------\nIn each case, the results were measured with varying dimensionality and anonymity level. In each case, the anonymity level was varied after fixing the dimensionality, and the dimensionality was varied on fixing the anonymity level. The anonymity level was fixed to 40, when the dimensionality was varied on the *X*-axis. While varying the anonymity level on the *X*-axis, the dimensionality was fixed to 40. In case of ℓ-diversity, the value of *l* is set to 2. In addition, we fixed *δ* to 0.5 in *δ*-selective enforcement approach. It is important to note that the information loss results do *not* vary with the different strategies for ensuring *k*-anonymity non-reconstructability, which affect only the class variable. Since the information loss metrics are based on the feature variables only, a single chart will be shown for the case of information loss, whereas the performance results for different approaches of *k*-anonymity (based on different strategies for ensuring *k*-anonymity non-reconstructability) and ℓ-diversity will be shown in the case of F-measure separately by means of solid and dashed lines, respectively.\nThe information loss with varying dimensionality is illustrated in Figure [fig:infolossdim]a. The dimensionality is illustrated on the *X*-axis, and the information loss is illustrated on the *Y*-axis in each case. Besides, diversity level is set to 2 for ℓ-diversity in all experiments. It is evident that the information loss of the *unfragmented* approach (for both *k*-anonymity and ℓ-diversity) increases with increasing dimensionality, which is in agreement of the results found earlier in. In fact, the error touches almost its upper bound, which implies that each generalized value starts losing more and more of its specificity in the unfragmented case. On the other hand, the fragmentation method shows drastic improvements in the amount of information loss, for both *k*-anonymity and ℓ-diversity. This implies that a significant amount of attribute specificity is retained in each fragment.\nThe information loss with increasing anonymity level is illustrated in Figure [fig:infolossdim]b. The anonymity level is illustrated on the *X*-axis, whereas the information level is illustrated on the *Y*-axis. It is not surprising that the information loss increases with the anonymity level, and enforcing diversity. However, as in the case of the results with increasing dimensionality, the improvements achieved by fragmentation were significant.\n[fig:infolossdim]\nThe comparisons for the F-measure with increasing dimensionality are much more tricky. This is because the addition of more dimensions to a data set affects the classification precision and recall (and hence F-measure) of the data in two mutually contradictory ways:\nA larger number of dimensions provides greater knowledge (in terms of more attributes) to the classifier in order to improve its precision and recall.\nData sets of larger dimensionality will have greater information loss on a *per attribute basis*, and this reduces the effectiveness of the classifiers.\nSo how does this tug-of-war between two mutually contradictory effects impact the final classification results, and how does the fragmentation process affect this tradeoff? Figure [fig:muskaccuracydim] compares the prediction F-measure of the unfragmented and fragmented anonymization methods with increasing dimensionality for two classifiers. In all cases, the different variants of the fragmentation scheme have higher classification accuracy than the unfragmented scheme. Even the naive scheme (in fragmented *k*-anonymity) was often able to perform better than the unfragmented approach in spite of its relative lack of sophistication in performing the class distortions. The difference in F-measure becomes even more drastic in case of ℓ-diversity and the fragmented anonymization achieves up to 28% improvement over unfragmented scenario.\n[fig:muskaccuracydim]\nIt is also immediately evident that the trend with increasing dimensionality is specific to the choice of data set, classifier, and specific approach. In particular, an interesting trend is that the F-measure changes only a little bit with increasing dimensionality in many cases, especially for unfragmented data. This is because the anonymization process in the unfragmented case changes values for many feature attributes to the very general value as the dimensionality grows. This change turns a high-dimensional data set into a data set with very few useful features for the classification. This phenomenon is reflected in the F-measure of unfragmented anonymization shown in Figure [fig:muskaccuracydim], which often does not vary much. In fact, only 6 feature attributes in *M**u**s**k* played a significant role in the classification. As a result, the F-measure does not vary too much with increasing dimensionality. Besides, as the equivalence classes have different class labels in the case of ℓ-diversity, the precision and recall degrade dramatically and cause the F-measure to be very low for unfragmented ℓ-diversity.\nThe effect of *k* on classification F-measure is illustrated in Figure [fig:muskaccuracyk]. The fragmented anonymization reveals a prominent improvement of up to 54% compared to the unfragmented anonymization. Normally, we expect the prediction F-measure to decline with increasing values of *k*. While this was often the case, there were also a few cases, where it has an unexpected rise. This trend has also sometimes been observed in earlier work, and is a result of the aggregation effects of the anonymization procedures (sometimes) removing the noisy artifacts in the data.\n[fig:muskaccuracyk]\nConclusions and Summary\n=======================\nThis paper presents a method for fragmentation-based anonymization for high-dimensional data. While the curse of dimensionality is a fundamental theoretical barrier, it is often possible to obtain effective results in practice. This paper uses fragmentation as a general purpose methodology to improve the effectiveness of any off-the-shelf algorithm for the anonymization process. Experimental results show significant improvements of the utility of the data after the fragmentation process. This meta-algorithm approach is fairly general and has the potential to be extended to a wider variety of scenarios and privacy models and workloads. This will be the focus of our future work.\n10\nSupplementary Materials\n=======================\nAppendix A. Pseudocodes\n=======================\nAlgorithm [alg:heuristic] depicts how a high-dimensional data set can be broken down into fragments.\n[alg:heuristic]\n[1] // *n* denotes the number of features\n: a (*n* + 1) by (*n* + 1) matrix storing the mutual information (between the features and the class attribute)\n*F**r**a**g**m**e**n**t*1, *F**r**a**g**m**e**n**t*2, *n**o**n**A**s**s**i**g**n**e**d**F**s*= **empty**\nSelect two features having the maximum mutual information as two seeds, *s**e**e**d*1 and *s**e**e**d*2. *F**r**a**g**m**e**n**t*1.add(*s**e**e**d*1) *F**r**a**g**m**e**n**t*2.add(*s**e**e**d*2) Add the rest of features to *n**o**n**A**s**s**i**g**n**e**d**F**s*\n(*n**o**n**A**s**s**i**g**n**e**d**F**s* not **empty**)\n*b**e**s**t**F*,*b**e**s**t**F**r**a**g**m**e**n**t*=**empty**\n*m**a**x**C**o**n**t**r**i**b**u**t**i**o**n*=0\n*a**i* in *n**o**n**A**s**s**i**g**n**e**d**F**s*\n*f* in *F**r**a**g**m**e**n**t*1,*F**r**a**g**m**e**n**t*2\n(*c**o**n**t**r**i**b**u**t**i**o**n*(*a**i*,*f*) > *m**a**x**C**o**n**t**r**i**b**u**t**i**o**n*)\n*b**e**s**t**F*=*a**i*\n*b**e**s**t**F**r**a**g**m**e**n**t*=*f*\n*m**a**x**C**o**n**t**r**i**b**u**t**i**o**n*=*c**o**n**t**r**i**b**u**t**i**o**n*(*a**i*,*f*)\nAdd *b**e**s**t**F* to *f*\nRemove *b**e**s**t**F* from *n**o**n**A**s**s**i**g**n**e**d**F**s* Add the class attribute to *F**r**a**g**m**e**n**t*1 and *F**r**a**g**m**e**n**t*2\n[H] [alg:dependencygraph]\n[1] mark all equivalence classes in *s**e**t*-*o**f*-*a**l**l*-*E**Q* as *u**n*-*v**i**s**i**t**e**d* *c*-*i**d*=1 *t**o**B**e**P**r**o**c**e**s**s**e**d*=**empty** *r**o**o**t*=random(*s**e**t*-*o**f*-*a**l**l*-*E**Q*) // choose an unvisited equivalence class randomly *t**o**B**e**P**r**o**c**e**s**s**e**d*.enqueue(*r**o**o**t*) (*t**o**B**e**P**r**o**c**e**s**s**e**d* not **empty**)\n*c**u**r**r**e**n**t*-*n**o**d**e*=*t**o**B**e**P**r**o**c**e**s**s**e**d*.dequeue()\nvisit(*c**u**r**r**e**n**t*-*n**o**d**e*)\n*E**Q* in dependant(*c**u**r**r**e**n**t*-*n**o**d**e*)\n(*E**Q* not visited)\ndraw an edge from *c**u**r**r**e**n**t*-*n**o**d**e* to *E**Q* in\ncomponent with id=*c*-*i**d*\n*t**o**B**e**P**r**o**c**e**s**s**e**d*.enqueue(*E**Q*) (any un-visited equivalence class left)\n*c*-*i**d*++\ngo to line 4\nAlgorithm [alg:dependencygraph] takes the set of all equivalence classes in a given fragmentation, and constructs the *dependency graph*. At the end of this process, the variable *c*-*i**d* indicates the number of components with no dependency (connection). The subroutine *d**e**p**e**n**d**a**n**t* takes an equivalence class *e**q* in either *F*1 or *F*2, and returns the set of equivalence classes in the other fragment that can be joined by *e**q*.\nAlgorithm [alg:dgbe] shows the procedure for enforcing the condition in Theorem [thm:nonreconstructability] in a *dependency graph*. This algorithm is invoked for each component the *dependency graph* in order to ensure non-identifiability. In Algorithm [alg:dgbe], the subroutine *c**h**a**n**g**e* (line 9) changes a class value having the lowest frequency in its equivalence class to the majority class value in that equivalence class. However, this change might affect the previously satisfied *k*-anonymity non-reconstructability condition between node *u**n* and its *v**i**s**i**t**e**d* neighbors. Thus, lines 10-14 re-check the previously satisfied reconstructability condition, and if needed, the subroutine *c**h**a**n**g**e* is called as necessary in order to re-satisfy it.\n[alg:dgbe]\n[1] *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e* = **empty** mark all nodes in *d*-*g**r**a**p**h* as *u**n**v**i**s**i**t**e**d* *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.enqueue(random(*d*-*g**r**a**p**h*)) (*p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e* **not empty**)\n*c**u**r**r**e**n**t*-*n**o**d**e* = *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.dequeue()\n*u**n* in unvisited-neighbors(*c**u**r**r**e**n**t*-*n**o**d**e*)\n((*c**u**r**r**e**n**t*-*n**o**d**e* and *u**n*) not satisfy the\n*k*-anonymity non-reconstructability condition)\nchange(*u**n*)\n(any previously-satisfied *k*-anonymity\nnon-reconstructability condition violates\nbetween *u**n* and visited- neighbors(*u**n*))\ncall change(*u**n*) until there is no violation\nbetween *u**n* and visisted-neighbors(*u**n*)\nvisit(*c**u**r**r**e**n**t*-*n**o**d**e*)\n*p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.enqueue(*u**n*)\nAlgorithm [alg:deltaselectivity] shows how *δ*-selectivity can be enforced on a fragmentation ${\\cal F}$. This algorithm is very similar to Algorithm [alg:dgbe]. It is worth noting that the class values of equivalence classes in the dependency graph (**d*-*g**r**a**p**h**) are at the equivalence class-level. Subroutine *c**h**a**n**g**e* simply removes a class value in an equivalence class. In order to reduce the class value distortion, the class value having the lowest frequency before converting the class values into equivalence class-level must be removed.\n[alg:deltaselectivity]\n[1] *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e* = **empty** mark all nodes in *d*-*g**r**a**p**h* as *u**n**v**i**s**i**t**e**d* *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.enqueue(random(*d*-*g**r**a**p**h*)) (*p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e* **not empty**)\n*c**u**r**r**e**n**t*-*n**o**d**e* = *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.dequeue()\n*u**n* in unvisited-neighbors(*c**u**r**r**e**n**t*-*n**o**d**e*)\n(*η*(*u**n*,*c**u**r**r**e**n**t*-*n**o**d**e*) < *δ*)\nchange(*u**n*)\n(*η* between *u**n* and any *v**n* ∈ visited-\nneighbors(*u**n*) becomes less than *δ*)\ncall change(*u**n*) until *η*(*u**n*,*v**n*) becomes greater\nthan *δ*\nvisit(*c**u**r**r**e**n**t*-*n**o**d**e*)\n*p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.enqueue(*u**n*)\nAppendix B. Proofs\n==================\nTHEOREM 4.1.\n*(*K*-ANONYMITY NON-RECONSTRUCTABILITY CONDITION). The condition for a given fragmentation ${\\cal F}=\\{F\\_1,F\\_2\\}$ which satisfies the fragment *k*-anonymity condition, and fragments *F*1 = {*E**Q*11, *E**Q*12, ..., *E**Q*1*n*} and *F*2 = {*E**Q*21, *E**Q*22, ..., *E**Q*2*m*}, to be non-reconstructible is that one of the following must be true for each joined pair *E**Q*1*i*, *E**Q*2*j*:* \n∑*c* ∈ *C*1*i* ∩ *C*2*j**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) = 0\n∑*c* ∈ *C*1*i* ∩ *C*2*j**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) ≥ *k*\nAccording to Definition [def:nonreconstructable], ${\\cal F}$ is *n**o**n*-*r**e**c**o**n**s**t**r**u**c**t**i**b**l**e* if and only if the relation *F*1 ⋈ *F*2 satisfies *k*-anonymity.\nJoining two equivalence classes *E**Q*1*i* and *E**Q*2*j* generates tuples with the same value for their quasi-identifiers. If the number of generated tuples is greater than *k*, joining *E**Q*1*i* and *E**Q*2*j* does not violate the *k*-anonymity property. For any given equivalence classes *E**Q*1*i* and *E**Q*2*j*, one of the following conditions is true:\n*C**L**S* = *C*1*i* ∩ *C*2*j* = ∅.\n*C**L**S* = *C*1*i* ∩ *C*2*j* = {*c*1, *c*2, ..., *c**w*}.\nIf the former condition is true, *E**Q*1*i* and *E**Q*2*j* do not have any common values to be joined on. Therefore, we have: \n∑*c* ∈ ∅*f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) = 0 \n If the latter condition is true, the joining of *E**Q*1*i* and *E**Q*2*j* will result in ∑*c* ∈ *C**L**S**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) tuples. Thus, the result of joining *E**Q*1*i* and *E**Q*2*j* is *k*-anonymous if the number of resulting tuples is at least *k*. Therefore, in either case the resulting fragmentation ${\\cal F}$ is *k*-anonymity non-reconstructible.\nTHEOREM 5.1. *The data set resulting from joining chunks *C**K**s*1 and *C**K**s*2 neither violates *k*-anonymity nor ℓ-diversity.*\nTo prove this theorem, we discriminate between different types of ℓ-diversity, and investigate each model individually.\n**Distinct ℓ-diversity**: The joining of *C**K**s*1 and *C**K**s*2 is performed based on joining equivalence classes *E**Q**i*1*s* and *E**Q**j*2*s*. According to the Step 4 of fragmentation-based ℓ-diversity, *E**Q**i*1*s* and *E**Q**j*2*s* have been constructed in such a way that have the same set (of size *l**s*) of class values with frequencies {*f*1*x*, *f*2*x*, ..., *f**l**s**x*} and {*f*1*y*, *f*2*y*, ..., *f**l**s**y*}, respectively. Since both *E**Q**i*1*s* and *E**Q**j*2*s* satisfy the *k*-anonymity, we have:\n1)∣*E**Q**i*1*s*∣ = *f*11 + *f*21 + ... + *f**l**s*1 ≥ *k*\n2)∣*E**Q**j*2*s*∣ = *f*12 + *f*22 + ... + *f**l**s*2 ≥ *k*\nEvidently, joining *E**Q**i*1*s* and *E**Q**j*2*s* are based on the common sensitive values, thus it results in a data set of size *f*11 × *f*12 + *f*21 × *f*22 + ... + *f**l**s*1 × *f**l**s*2 having exactly *l**s* sensitive values. Therefore, the data set resulting from this join satisfies the distinct ℓ-diversity. Since joining each pair of arbitrary equivalence classes *E**Q**i*1*s* and *E**Q**j*2*s* satisfies the distinct ℓ-diversity, joining *C**K**s*1 and *C**K**s*2 also satisfies this privacy requirement.\n**Entropy (or recursive) ℓ-diversity**: This part is a bit trickier. Joining each pair of equivalence classes *E**Q**i*1*s* and *E**Q**j*2*s* will result in a data set with at least *k* tuples (with the same reasoning as distinct ℓ-diversity), but most likely different level of diversity than *l**s* (it can be lower or higher than *l**s*). This might be considered as a privacy violation. However, joining all possible pairs of equivalence classes *E**Q**i*1*s* and *E**Q**j*2*s* generates a data set consisting of both real (all tuples in *S**s*) and fake tuples. There is an important observation here. If the attacker has enough background knowledge to rule out the faked tuples, the diversity level of real tuples are exactly equal to *l**s*. Hence, this is not a privacy violation.\nAppendix C. Enforcement via *δ*-selectivity\n===========================================\nI. Tuple-level vs. equivalence class-level class value publishing\n=================================================================\nTable [tuplelevelvsequilevel] exemplifies the difference between tuple-level and equivalence class-level publishing for a 5-anonymized fragment. As Table [tuplelevelvsequilevel]b illustrates, publishing the class values in the equivalence class-level usually results in an equivalence class with some *ambiguous slots* for its class values. For example, two ambiguous slots are illustrated in Table [tuplelevelvsequilevel]b. The only case in which publishing at the equivalence class-level does not result in an ambiguous slot is when the class values are unique in the equivalence class. As no further information is published regarding the frequency of the class values in each equivalence class, any class value available in the equivalence class can be placed in the ambiguous slots.\n[h] [tuplelevelvsequilevel]\nThe ambiguous slots in an equivalence class *E**Q**i**j*, published at the equivalence class-level, can take any of the class values in *C**i**j*. In other words, different versions for *E**Q**i**j* can be assumed. Table [table:versions] illustrates two possible versions for the equivalence class in Table [tuplelevelvsequilevel]b.\n[h] [table:versions]\nII. A subtle way to find *η*\n============================\nGiven two equivalence classes *E**Q*1*i* and *E**Q*2*j* whose class values are published in equivalence class-level and have at least one class value in common, we find the number of tuples resulted from their join which is referred to as *equijoin selectivity* in the literature. As ∣*E**Q*1*i*∣ and ∣*C*1*i*∣ refer to the number of tuples and the class values (in equivalence class-level) in *E**Q*1*i*, the number of ambiguous slots is equal to ∣*E**Q*1*i*∣ − ∣*C*1*i*∣. The set {*V*(*E**Q*1*i*)} represents different versions (or instantiations) of *E**Q*1*i*. Each member of this set is a combination of ∣*E**Q*1*i*∣ tuples in which there exist at least one tuple per each class value in *C*1*i*. The size of this set is equal to ${(|EQ\\_{1i}|-|C\\_{1i}|)+|C\\_{1i}|-1} \\choose {(|EQ\\_{1i}|-|C\\_{1i}|)}$.\nThere exist ∣{*V*(*E**Q*1*i*)}∣ × ∣{*V*(*E**Q*2*j*)}∣ different ways to join *E**Q*1*i* and *E**Q*2*j*. Among all possible joins, those generating *k* tuples (or more) are referred to as **k*-anonymity-preserving equijoins*.\nIn *η* formula, calculating the denominator is straightforward by multiplying ∣*V*(*E**Q*1*i*)∣ and ∣*V*(*E**Q*2*j*)∣. However, computing the numerator is more challenging. The simplest way is perform all joins, and count those which satisfy the minimum cardinality of *k*. A more subtle way is to consider the conditions (constraints) that exist on the number of class values in each equivalence class and map the problem as follows:\nThe number of pairs (*A*,*B*) of integer vectors *A* = (*a*1, *a*2, ..., *a**c*) and *B* = (*b*1, *b*2, ..., *b**c*) in which *a**d* and *b**d* depict the frequency of *d**t**h* common class value in *V*(*E**Q*1*i*) and *V*(*E**Q*2*j*) respectively, such that the following conditions are satisfied:\ncondition 1: ∑*a**d* × *b**d* ≥ *k*\ncondition 2: 1 ≤ *a**d* ≤ (1 + ∣*E**Q*1*i*∣ − ∣*C*1*i*∣)\ncondition 3: 1 ≤ *b**d* ≤ (1 + ∣*E**Q*2*j*∣ − ∣*C*2*j*∣)\ncondition 4: ∑*a**d* ≤ ∣*C**L**S*∣ + ∣*E**Q*1*i*∣ − ∣*C*1*i*∣\ncondition 5: ∑*b**d* ≤ ∣*C**L**S*∣ + ∣*E**Q*2*j*∣ − ∣*C*2*j*∣\nwhere *C**L**S* = *C*1*i* ∩ *C*2*j*.\nAppendix D. Experiments\n=======================\nI. Performance measures\n=======================\nWe utilized two metrics to evaluate the effectiveness of our proposed method. The first one is the widely-used metric known as *information loss*, which captures the total amount of lost information due to generalization. In fact, this metric shows the usefulness of data *on a per-attribute basis* for general workloads. For an anonymized data set with *n* tuples and *m* attributes, the information loss *I* is computed as follows:\n$I= \\sum\\_{i=1}^{n} \\sum\\_{j=1}^{m}\\frac{|upper\\_{ij}-lower\\_{ij}|}{n\n\\cdot m \\cdot |max\\_j-min\\_j|}$\nHere, *l**o**w**e**r**i**j* and *u**p**p**e**r**i**j* represent the lower and upper bounds of attribute *j* in tuple *i* after generalization, and *m**i**n**j* and *m**a**x**j* represent the maximum and minimum values taken by attribute *j* over all records. Note that the computation in the fragmented and unfragmented case is not different, as long as all the different attributes in the different fragments are used. This is the most direct measure of the data quality, since it normalizes the final result by the number of attributes.\nTo evaluate the utility of the data anonymized by our meta-algorithm, we calculated the weighted F-measure of a classifier trained on the anonymous fragmentation. This metric reflects the goodness of classifier more accurately in case on unevenly distributed test data. Each fragment was used to train the classifier separately. For a given test instance, the different fragments of the training data were trained separately, and the weighted majority label from the different classifiers was reported, where the weight used was score returned by the classifier from each fragment. Each class label was once considered as positive (and once as negative) and the weighted F-measure for each case was calculated by taking into account the fraction of the positive instances. We used J48[6](#fn6) and a *k*-NN[7](#fn7) classifier in Weka with the default setting. The value of *k* in *k*-NN was set to 5. The learning from generalized values was also done by the technique used in. In each case, the decomposition was performed into two fragments.\nIn addition, the amount of distortion required to the class values (for fragmentation *k*-anonymity) was measured. Specifically, the number of distorted class values were computed for each of the different techniques. The aim is to show that the amount of distortion required was relatively small.\nII. Data set description\n========================\nReal data set *Musk* from the *UCI Machine Learning Repository*[8](#fn8) was used. It contains 7074 instances with 168 numerical feature attributes describing a set of molecules. The goal is to predict whether a molecule is musk or non-musk. In order to test the effects of varying data dimensionality, we chose 10, 20, 30, and 40 features randomly and constructed four versions of *M**u**s**k* with different dimensionality. Around 70% of the *M**u**s**k* data set was used for training. The distribution of the two classes in the training and test data set is shown in Table [table:datasetdis]. Although the data set used in this work is numerical, the proposed meta-algorithm can be used to anonymize categorical values by simply using an off-the-shell anonymization technique capable to anonymize categorical values in Step 2 and 4 of fragmentation *k*-anonymity and ℓ-diversity, respectively.\n[table:datasetdis]\n|c|c|c| & & \nmusk & 1017 & 207 \nnon-musk & 3983 & 1867 \n---\n1. University of Calgary, hzakerza@ucalgary.ca[↩](#fnref1)\n2. IBM T.J. Watson Research Center, charu@us.ibm.com[↩](#fnref2)\n3. University of Calgary, kbarker@ucalgary.ca[↩](#fnref3)\n4. Although the works in are originally proposed for the set-valued (transaction) data, the relational data can be transformed to set-valuded data for anonymization.[↩](#fnref4)\n5. <http://archive.ics.uci.edu/ml.>[↩](#fnref5)\n6. J48 is an open source implementation of C4.5 in Java, <http://weka.sourceforge.net/doc/weka/classifiers/trees/J48.html.>[↩](#fnref6)\n7. <http://weka.sourceforge.net/doc/weka/classifiers/lazy/IBk.html.>[↩](#fnref7)\n8. <http://archive.ics.uci.edu/ml.>[↩](#fnref8)\n\nTowards Breaking the Curse of Dimensionality for High-Dimensional Privacy: An Extended Version\n==============================================================================================\nThe curse of dimensionality has remained a challenge for a wide variety of algorithms in data mining, clustering, classification and privacy. Recently, it was shown that an increasing dimensionality makes the data resistant to effective privacy. The theoretical results seem to suggest that the dimensionality curse is a fundamental barrier to privacy preservation. However, in practice, we show that some of the common properties of real data can be leveraged in order to greatly ameliorate the negative effects of the curse of dimensionality. In real data sets, many dimensions contain high levels of inter-attribute correlations. Such correlations enable the use of a process known as *vertical fragmentation* in order to decompose the data into vertical subsets of smaller dimensionality. An information-theoretic criterion of mutual information is used in the vertical decomposition process. This allows the use of an anonymization process, which is based on combining results from multiple independent fragments. We present a general approach which can be applied to the *k*-anonymity, ℓ-diversity, and *t*-closeness models. In the presence of inter-attribute correlations, such an approach continues to be much more robust in higher dimensionality, without losing accuracy. We present experimental results illustrating the effectiveness of the approach. This approach is resilient enough to prevent identity, attribute, and membership disclosure attack.\nIntroduction.\n=============\nThe problem of privacy-preservation has been studied extensively in recent years, because of the increasing amount of personal information which has become available in the context of a wide variety of applications. Starting with the seminal work in, a significant amount of work has been done on the problem of privacy preservation of different kinds of data. Numerous models have been proposed for the problem of privacy preservation. However, it has been shown that data anonymization is increasingly difficult with dimensionality, and the challenges extend to most privacy models.\nThe reason for the ineffectiveness of high-dimensional algorithms is simple. With increasing dimensionality, a larger number of attributes are available for background attacks, even when the perturbation on a single attribute is significant. As a result, it has been shown theoretically in that significantly larger perturbations are required with increasing dimensionality, and this reduces the effectiveness of the approach for privacy preservation. These results extend to a variety of models such as *k*-anonymity and ℓ-diversity.\nAn important observation about a *blind* anonymization process is that it often does not (fully) recognize that dependencies among the attributes may make a particular combination of dimensions more or less susceptible to anonymization. While such dependencies are *implicitly* utilized by many anonymization methods, their impact is often diluted by the overall anonymization procedure. Furthermore, the same dependencies impact the amount of information, which may be available in a particular subset of attributes for data mining applications. For example, an attribute such as *Age* and *Salary* may be highly correlated, and the *differential* impact of adding the attribute *Salary* may be less than adding another attribute such as *Sex* to the data.\nOne solution to the curse of dimensionality is to simply use feature selection in order to reduce the dimensionality of the data set, and retain a small subset of attributes which retains non-redundant information for a particular application. However, it is inevitable, that a pure feature selection approach will lose a significant amount of information for many application-specific scenarios. Therefore, a relevant question is as follows: “*Is it still somehow possible to retain all the attributes in the data, while using the non-redundancy of some subsets of attributes in the anonymization process to prevent identity and attribute disclosure attack, and also retain most of the utility in the data for mining scenarios?*”.\nA less drastic approach than feature selection is the concept of *vertical fragmentation*. The idea is to break up the data set into different subsets of attributes using vertical fragmentation, and anonymize each subset independently. The results from the different subsets of attributes then need to be combined for a particular application. Since all attributes are still retained, the amount of information loss of fragmentation is less than that of feature selection. The exact nature of the fragmentation may depend upon the specific application at hand. For example:\nIn a supervised application, the fragments may be completely disjoint and share no attributes other than the class attribute. Thus, while the correspondence information among different fragments is lost, this may not be as critical, since the class variable can be independently learned from each fragment. The amount of information lost is limited in such cases, especially if the individual fragments are carefully chosen based on information-theoretic principles. The results from the different fragments can then be combined carefully on *an aggregate basis* in order to obtain high quality classification results. Care needs to be taken in the fragmentation process that the common class attribute may not be used in order to partially join the fragments together, and reduce the anonymity.\nIn an unsupervised application, the fragments may need to have one or more common attributes in order to ensure a limited level of correspondence between different fragments. This case is actually not too different from the supervised case. The main difference is that instead of the class attribute, it is the common attribute which needs to be carefully accounted for during the fragmentation process. In this paper, we primarily focus on the supervised scenario of classification as a first application. The generalizations to other unsupervised scenarios will be handled in future work.\nIt should be emphasized that while the *theoretical* results of the dimensionality curse still hold true, their *practical* impact can be greatly alleviated by carefully accounting for the nature of the data set in a particular application. Pathological cases may exist in which every feature is independent of one another, and in such cases, the earlier theoretical results on the curse of dimensionality continue to hold true. However, such pathological cases rarely arise in practice. Therefore, the goal of fragmentation is to leverage the mutual information within different features in order to alleviate the dimensionality curse in the vast majority of cases. The experimental results of this paper show that the fragmentation method can achieve significant improvements over the currently available methods. It should also be emphasized that the fragmentation method is a *meta-algorithm* which can be *combined with any existing anonymization algorithm* in order to improve its effectiveness. We start with the *k*-anonymity model in this paper because we believe such a hard problem like curse of dimensionality must be first addressed in the simplest and most relaxed privacy model. Then, we explain how the fragmentation process can be generalized to satisfy the ℓ-diversity (or *t*-closeness) requirement. In addition, we discuss how the fragmentation brings in the membership disclosure protection. In general, the fragmentation process has the potential to be extended for other privacy models, because of its meta-approach, which is more easily generalizable. This might eventually provide unprecedented flexibility in using the fragmentation method as a general-purpose meta-algorithm in the context of a wide variety of scenarios. This paper is organized as follows. The remainder of this section discusses related work. Section 2 discusses the overview of the approach for the *k*-anonymity. Section 3 discusses details of the fragment-based *k*-anonymization method. Extending the fragmentation approach for ℓ-diversity (or *t*-closeness) is shown in Section 4. In addition, this section shows how the fragmentation-based anonymization can prevent the membership attack. The experimental results are presented in Section 5. Section 6 contains the conclusions and summary.\nRelated Work.\n=============\nThe problem of privacy preservation was first studied in. This approach was based on noise-based perturbation of the underlying data. Subsequently, the problem of *k*-anonymization of data was proposed in. Other models for data privacy have been proposed in. Numerous methods have been proposed for effective and efficient *k*-anonymization, and this continues to remain one of the most widely used models in the literature because of its simplicity. The theoretical results illustrating the degradation of privacy-preservation methods with increasing dimensionality have been discussed in. The work in proposed an anonymization method for high-dimensional data. However, it is only applicable to sparse transactional data, and is heavily dependent of the sparse structure of transaction data in order to achieve this goal. Kifer suggested the idea of releasing anonymized marginals beside the anonymized original table. However, the original table must still be anonymized as a whole which results in high information loss. Furthermore, publishing the marginals makes the published data highly workload dependant. The privacy models in can cope with the problem of curse of dimensionality by reducing the number of quasi-identifiers[4](#fn4). That is, they make an assumption about the number of quasi-identifiers known by an attacker, and apply anonymization only on limited number of quasi-identifiers. However, this assumption may not hold true in some cases. Other relevant works are able to provide some protection in the high dimensional case, though they can be challenged in some circumstances. Furthermore, any of these methods can be used in combination with our approach, which is designed as a more general purpose meta-algorithm. Finally, the concept of vertical partitioning and fragmentation has been explored in the context of distributed data privacy with cryptographic protocols, or for capturing confidential privacy constraints in the context of such methods. The goals and motivations of these methods are quite different, and are not at all focused on the problem of high dimensional anonymization.\nIn the context of increasing dimensionality, it is natural to explore feature selection as an alternative in order to reduce the data dimensionality. However, this is a rather drastic solution, which can impact the quality of the underlying results significantly. Therefore, this paper proposes the approach of fragmentation as a general-purpose meta-algorithm in order to improve the robustness of high-dimensional anonymization algorithms.\nOverview of The Approach for The Fragmentation *K*-Anonymity\n============================================================\nIn this section, we first introduce the most important notations used in the fragmentation-based *k*-anonymity in Table [table:notations]. An overview is then provided about vertical fragmentation, and its incorporation as a general-purpose meta-algorithm for privacy preservation.\n[H]\n[table:notations]\n|c|c|\n&\n${\\cal F}$ & a vertical fragmentation & *i**t**h* fragment in fragmentation ${\\cal F}$ & *j**t**h* equivalence class in fragments *F**i* & set of all class values in equivalence class *E**Q**i**j* & power set of set *S* & size of a set \nVertical Fragmentation\n----------------------\nLet *T* be a relation defined over a schema *T*(*A*1*f*, *A*2*f*, ..., *A**n**f*, *A**c*) where *A**i**f* represents the feature attributes, and *A**c* is the class attribute. A vertical fragmentation of relation *T* splits the feature variables into multiple non-overlapping fragments. Formally, a vertical fragmentation is defined as follows:\n(VERTICAL FRAGMENTATION). Given a relation schema *T*, a vertical fragmentation ${\\cal F}$ of *T* is a partitioning of the attributes into fragments ${\\cal F}=\\{\nF\\_1,F\\_2,...,F\\_m \\}$ such that each *F**i* contains a disjoint subset of the feature attributes. Therefore, it is the case that $\\forall\nF\\_i \\in {\\cal F}$, *F**i* ⊆ *T* and *F**i* ∩ *F**j* = ∅ (*i* ≠ *j*) and ⋃*F**i* = *T* (*i*=1, ..., *m*).\nIn addition, it is *implicitly assumed* that the class attribute *A**c* is associated with each fragment. As we will see later, the presence of this common attribute needs to be accounted for in a special way since it allows the re-construction of *some* correspondence between the attribute values of different fragments. Therefore, methods need to be designed to ensure that this correspondence cannot be used in order to attack the anonymity of the fragmented data. In some cases, this process requires the perturbation of a few class values, in order to ensure non-identifiability.\nNote that the fragmentation process is used as a meta-approach in conjunction with an off-the-shelf anonymization algorithm. A fragmentation ${\\cal F}$ is referred to as a **k*-anonymous fragmentation* after applying an anonymization algorithm to it, if and only if the following two conditions hold:\n**Fragment *k*-anonymity condition:** Each fragment in ${\\cal F}$ satisfies the *k*-anonymity condition. This condition can be easily satisfied by applying any off-the-shelf *k*-anonymity algorithm to each fragment.\n***K*-Anonymity non-reconstructability condition:** The relation resulting from joining any arbitrary fragments on the class variable satisfies the *k*-anonymity condition. The number of possible fragmentations of a set of features is rather large in the high-dimensional case. For example, for a set of *n* features, there may be *O*(*n**n*) possible fragmentations. Clearly, exhaustive search through all the possible fragmentations for the high-dimensional case may become prohibitive. Therefore, a systematic approach is required to search the space of possible fragmentations. Since this work is focussed on the classification problem, the fragmentation approach should attempt to maximize the amount of *non-redundant* information contained in each feature of a particular fragment, which is relevant for the classification process. Correspondingly, our systematic search approach utilizes a metric referred to as *Fragmentation Minimum Redundancy Maximum Relevance (FMRMR)* in order to create fragments.\nFragmentation Minimum Redundancy Maximum Relevance\n--------------------------------------------------\nThe ideal fragmentation is one in which the set of attributes in each fragment is a comprehensive representation of the information required for the mining process. Since this paper addresses the classification problem, the metric will explicitly use the class variable for quantification, though it is conceivable that the metric for other applications would be different. In the supervised context, a comprehensive representation refers to high predictability of the class variable from the features in each fragment, while minimizing redundancy. It is evident that the *simultaneous* incorporation of features with high mutual information within a given fragment does not provide any additional advantages, even when they are all highly relevant to the class attribute. This implies that a combination of the relevance to the class attribute and the mutual information with respect to one another can be useful for the process of constructing a fragment.\nTo this effect, we draw on the feature selection literature, which defines the concept of the *Minimum Redundancy Maximum Relevance (MRMR)* metric. This metric uses a dependency quantification (denoted as *W*) among the feature variables and a dependency quantification (denoted as *V*) between the feature variables and the class attribute in each fragment. Our proposed heuristic aims at maximizing the summation of MRMR for all the fragments in a given fragmentation. The Fragmentation MRMR (FMRMR) is the summation of the values of MRMR within a fragment. This value is defined for a particular fragmentation ${\\cal F}$ as follows:\n$ FMRMR({\\cal F})= \\sum\\_{t=1}^{|{\\cal F}|} (V\\_{t} - W\\_{t}) $\n$V\\_{t} = \\frac{1}{|A\\_t^f|} \\sum\\_{j \\in A\\_{t}^f} I(cls,j)$\n$W\\_{t} = \\frac{1}{|A\\_{t}^f|^2} \\sum\\_{k,j \\in A\\_{t}^f} I(k,j)$\nwhere:\n*A**t**f*: set of features in fragment *t* of fragmentation ${\\cal F}$\n*I*(*x*, *y*): mutual information between attributes *x* and *y*\n*V**t*: total mutual information between the features and the class attribute in fragment *t* of fragmentation ${\\cal F}$\n*W**t*: total pairwise mutual information between the features in fragment *t* of fragmentation ${\\cal F}$\n*c**l**s*: the class attribute\nThe overall approach for the *k*-anonymity uses a three-step technique for the fragmentation process. For a high-dimensional relation *T* with *n* features and one class attribute, these three steps are as follows:\n1. Use a carefully-designed search algorithm to decompose the relation into fragments. The constructed fragments have non-overlapping sets of features together with the class attribute. The fragmentation process uses the afore-mentioned measure in order to determine the optimal fragments.\n2. Anonymize each fragment separately using an existing anonymization algorithm, such as the Mondrian multi-dimensional *k*-anonymity algorithm.\n3. At this point, it should be noted that the anonymized fragments can be (partially) joined back using the common attribute, which in the supervised scenario is the class attribute. Depending on the distribution of values in the common attribute, the result might violate the *k*-anonymity constraint. This is essentially a **k*-anonymity non-reconstructability condition violation*. Therefore, additional steps are required in order to ensure non-reconstructability. The techniques for achieving this are slightly involved and distort the class variable in such a way that non-reconstructability is guaranteed. These methods will be described in the next section. It should be noted that the distortion of the class variable may result in some further reduction in accuracy. However, in *practice*, for most reasonable distributions, the required distortions are very limited, if any.\nThe second step in the afore-mentioned list does not require further explanation. Therefore, the exposition in this paper will describe the detailed methods for performing the first and the third steps. For the third step, three different alternatives will be proposed. It should also be noted that although the class and sensitive attributes have been considered the same in many works, they might be different, and data contains many sensitive attributes in practice. In such cases, the other sensitive attributes also need to be fragmented in order to ensure that the two fragments cannot be joined. However, they should be fragmented only *after* the quasi-identifiers have already been fragmented (using the same approach as discussed in the next section). This is because it is more critical to ensure that quasi-identifiers are evenly distributed among fragments. Therefore, what follows will only focus on quasi-identifiers for simplicity.\nFragmentation-based *K*-Anonymization\n=====================================\nIn this section, the first and third steps in the afore-mentioned fragmentation meta-algorithm will be discussed. First, the fragment construction heuristic will be introduced.\nFragment Construction Heuristic\n-------------------------------\nAs the number of features increases, the number of possible fragments grows exponentially. This explosion in the number of fragments makes exhaustive search in this space impractical. Therefore, we propose an algorithm which tries to form a fragmentation with maximum *F**M**R**M**R*. For simplicity, a binary fragmentation into two parts will be described, though it is possible in principle to fragment into multiple parts by repeating the process.\nWe define the *FMRMR contribution* of a feature attribute *A**i**f* with respect to fragment *F**j* of fragmentation ${\\cal F}$ as the difference between *F**M**R**M**R* of ${\\cal F}$ after and before adding *A**i**f* to *F**j*. The quantification of the mutual information between the *n* features and the class attribute is stored in the form of an (*n* + 1) × (*n* + 1) matrix denoted by [*M**I*](*n* + 1) × (*n* + 1).\nThe *F**M**R**M**R* metric attempts not to place features having high mutual information in one fragment. Therefore, as a starting point, two features having the highest mutual information are picked as seeds and placed in different fragments. Afterwards, in a greedy manner, and while there exists un-assigned features, the *FMRMR contributions* of all unassigned features with regards to both fragments are calculated. The unassigned feature with the highest *FMRMR contribution* is added to the relevant partition. Finally, the common attribute (class attribute in the supervised case) is added to each fragment separately. The overall approach is illustrated in Algorithm [alg:heuristic] in the Appendix A in the supplementary materials.\nThe Final Step: *K*-Anonymity non-Recons- tructability\n------------------------------------------------------\nAs indicated earlier, applying a *k*-anonymity algorithm on each fragment in order to satisfy the *fragment k-anonymity condition* is not sufficient for ensuring non-identifiability. This is because the common attribute (class attribute) can be used for (very approximate) joins, and such joins provide some additional information about fragment correspondence. Therefore, in theory, it may be possible that the overall anonymity level of the relation resulted from joining *k*-anonymized fragments is less than *k*, though in practice it is rather unlikely because of the approximate nature of the join.\nWe call a fragmentation in which all fragments satisfy the *k*-anonymity condition **k*-anonymity non-reconstructible* if the relations resulting from joining any arbitrary fragments on the class attribute satisfy *k*-anonymity. Similarly, a fragmentation is called *reconstructible* if *k*-anonymity is violated after joining some of its fragments. Definition [def:nonreconstructable] formally defines a **k*-anonymity non-reconstructible* fragmentation.\n(*K*-ANONYMITY NON-RECONSTRUCTIBLE FRAGMENTATION). [def:nonreconstructable] Fragmentation ${\\cal\nF}=\\{F\\_1,F\\_2,...,F\\_m\\}$ which satisfies the fragment *k*-anonymity condition is called *k*-anonymity non-reconstructible if and only if $\\forall s\n\\in \\mathcal P \\left({{\\cal F}}\\right)$, the relation resulting from joining members (fragments) of *s* satisfies the *k*-anonymity condition.\nThe power set $\\mathcal P \\left({{\\cal F}}\\right)$ has 2*m* members. However, the *k*-anonymity condition must be checked for members of size at least 2.\nIt should be noted that the joining process is only approximate and noisy, which is good for anonymization. Therefore, successful violation attacks of the type discussed above are often difficult to perform in practice. For example, joining will result in some tuples that do not have any corresponding tuple in the original table. These tuples are called *fake tuples*, and may sometimes be helpful for obfuscation of identification of relevant tuple identities.\nAs articulated in Definition [def:nonreconstructable], given a fragmentation ${\\cal F}$ with *m* fragments, the *k*-anonymity condition must be satisfied for (2*m*-*m*-1) possible relations resulted from joining arbitrary fragments. However, the relations resulted from joining more than two fragments can be simply obtained by consecutive binary joins. This paves the way to define the *k*-anonymity non-reconstructability condition by joining only two fragments. For instance, checking the *k*-anonymity non-reconstructability condition on a relation resulting from joining members of $\\{F\\_1,F\\_2,...,F\\_n\\} \\in \\mathcal P \\left({{\\cal\nF}}\\right)$ can be accomplished by checking the *k*-anonymity non-reconstructability condition in each of the following binary joins: *I*1 = *F*1 ⋈ *F*2, *I*2 = *I*1 ⋈ *F*3,..., *I**n* − 1 = *I**n* − 2 ⋈ *F**n*.\nTherefore, for simplicity, we can continue our discussion with only two fragments, without any loss of generality. This condition is formally stated in Theorem [thm:nonreconstructability]. An important concept to continue with the remaining of the paper is to understand the notion of equivalence classes resulting from the anonymization. In the anonymized table, records with the same value for their quasi-identifiers constitute an *equivalence class*.\n(*K*-ANONYMITY NON-RECONSTRUCTABILITY CONDITION). [thm:nonreconstructability] The condition for a given fragmentation ${\\cal F}=\\{F\\_1,F\\_2\\}$ which satisfies the fragment *k*-anonymity condition, and fragments *F*1 = {*E**Q*11, *E**Q*12, ..., *E**Q*1*n*} and *F*2 = {*E**Q*21, *E**Q*22, ..., *E**Q*2*m*}, to be non-reconstructible is that one of the following must be true for each joined pair *E**Q*1*i*, *E**Q*2*j*:\n∑*c* ∈ *C*1*i* ∩ *C*2*j**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) = 0\n∑*c* ∈ *C*1*i* ∩ *C*2*j**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) ≥ *k*\nThe proof of this theorem is presented in Appendix B in the supplementary materials.\nEnforcing and satisfying the *k*-anonymity non-reconstructability condition in a fragmentation may require some of the class values to be distorted. As explained shortly, the change in the class values can be performed using various strategies. However, minimizing the number of changes is always desirable in order to retain accuracy. The design of an algorithm which provably minimizes the changes is computationally intractable because of the exponential number of possibilities. Therefore, we propose three heuristic strategies to enforce the *k*-anonymity non-reconstructability condition. It is worth mentioning that the utility of each strategy is different from others. Before providing a more detailed exposition, we introduce the concept of a *dependency graph*, which provides the logical construct necessary for a good algorithmic design.\n### Dependency Graph\nA *dependency graph* is an undirected graph structure which captures the dependency among different equivalence classes in a given fragmentation. Nodes in the *dependency graph* are equivalence classes, and there exists an edge between node *E**Q**i**x* and *E**Q**j**y* provided that:\n1. *i* ≠ *j* that means *E**Q**i**x* and *E**Q**j**y* belong to two different fragments.\n2. *C**i**x* ∩ *C**j**y* ≠ ∅ that means *E**Q**i**x* and *E**Q**j**y* have at least one class value in common.\nThe set of all equivalence classes in a fragmentation may be divided into subsets having no dependency on each other. In other words, no equivalence class from one subset can be joined with equivalence classes in the other subset. Thus, the *dependency graph* is not connected, and the *dependency graph* components reflect the full dependencies among all equivalence classes, rather than a single connected *dependency graph*. The process of constructing the *dependency graph* is shown in Algorithm [alg:dependencygraph] in Appendix A in the supplementary materials.\nThe *k*-anonymity non-reconstructability condition is enforced on each connected component of the *dependency graph* separately, since there is no inter-component dependency. The *k*-anonymity non-reconstructability condition on each connected component can also be achieved by enforcing it on each edge. We introduce three different strategies in order to achieve this goal. \n### Naive Enforcement\nSatisfying the *k*-anonymity non-reconstructability condition for an edge between equivalence classes *E**Q*1*i* and *E**Q*2*j* can simply be done by enforcing each equivalence class to have only class value. In the naive *k*-anonymity non-reconstructability enforcement approach, class values in each equivalence class are changed to the majority class in that equivalence class. In this case, two given equivalence classes either cannot be joined, or their join generates at least *k*2 tuples. Such an approach is clearly suboptimal, and fails to take full advantage of the flexibility associated with distorting the class variable in a way which is sensitive to the behavior of the remaining data.\n### Dependency Graph-based Enforcement\nUnlike the naive approach, the class values in *only* equivalence classes violating the *k*-anonymity after being joined are changed to the majority class value in this approach. Another difference between this approach and the naive approach is that the dependency graph-based approach aims at minimizing the number of changes in each equivalence class. In order to achieve this goal, this approach changes only one attribute in each step.\nStarting from a random node (*current-node*) in the *dependency graph*, the *dependency graph* is explored in a breadth-first manner. The *k*-anonymity non-reconstructability condition is checked between *current-node* and every single unvisited neighbor nodes. If the condition does not hold between *current-node* and one of its unvisited neighbors, the class values with lowest frequency in the neighbor node is changed to the majority class value until the condition is satisfied. After satisfying *k*-anonymity non-reconstructability between *c**u**r**r**e**n**t*-*n**o**d**e* and all its neighbor nodes, *c**u**r**r**e**n**t*-*n**o**d**e* is marked *v**i**s**i**t**e**d*. This process must be repeated for all *components* in the *dependency graph* until all nodes are marked *v**i**s**i**t**e**d*. The pseudocode of this algorithm is demonstrated in Algorithm [alg:dgbe] in Appendix A in the supplementary materials.\n### Enforcement via *δ*-selectivity\nIn spite of the approximate nature of the join between different fragments, they are a potential threat to *k*-anonymity. Thus, the prevention of violating joins is important. Publishing the class values for *each and every* single tuple (row) in the anonymized fragment is a major cause of this violation.\nThe *δ*-selectivity approach changes the way in which class values are published. This enables a more relaxed *k*-anonymity non-reconstructability condition enforcement on the equivalence classes. Instead of publishing the class values on a *per tuple* basis, they are published on a *per equivalence class* basis with the use of *ambiguous values* (*s**l**o**t**s*). In an equivalence class, each class value has equal probability of being assigned to a tuple. This results in the possibility of assuming different instantiations (or versions) for a given equivalence class. Then, given two equivalence classes, there exist multiple ways to join them, corresponding to different assignments of class values to tuples. The modified *k*-anonymity non-reconstructability condition leverages this ambiguity effectively. Appendix C.I in the supplementary materials exemplifies tuple-level and equivalence class-level class value publishing.\nAs mentioned above, the ambiguous slots in an equivalence class *E**Q**i**j*, published at the equivalence class-level, can take any of the class values in *C**i**j*. In other words, different versions for *E**Q**i**j* can be assumed.\n(EQUIVALENCE CLASS VERSION). An arbitrary assignment of class values available in an equivalence class *E**Q**i**j* to ambiguous slots in *E**Q**i**j* generates a version of *E**Q**i**j* shown by *V*(*E**Q**i**j*).\nAlthough publishing the class values at the equivalence class-level reduces the risk of *k*-anonymity violation, the resulting equivalence classes are still vulnerable to be joined back and violate the *k*-anonymity. As an example, consider the extreme case where the class values are unique in an equivalence class. Each tuple in the equivalence class is assigned to a different class value which is similar to the case in which class values are released at the tuple level. Given two equivalence classes whose class values are published in equivalence class-level and have at least one class value in common, there exist different ways to join them. The number of tuples resulted from joining two equivalence classes are referred to as *equijoin selectivity*. Among all possible joins, those generating *k* tuples (or more) are referred to as **k*-anonymity-preserving equijoins*.\n(*K*-ANONYMITY-PRESERVING EQUIJOIN). Given two equivalence classes *E**Q*1*i* and *E**Q*2*j* whose class values are published at the equivalence class level, and which share at least one class value, the join between V(*E**Q*1*i*) and V(*E**Q*2*j*) is a *k*-anonymity-preserving equijoin if and only if it produces at least *k* tuples.\nWe can now define the equijoin selectivity privacy level in terms of the possible equijoins between two equivalence classes.\n(EQUIJOIN SELECTIVITY PRIVACY LEVEL). The ratio of number of *k*-anonymity-preserving equijoins in joining two equivalence classes *E**Q*1*i* and *E**Q*2*j* to the total number of possible equijoins in joining the same equivalence classes is referred to as equijoin selectivity privacy level of *E**Q*1*i* and *E**Q*2*j*. This value is denoted by *η*(*E**Q*1*i*, *E**Q*2*j*). In other words, *η*(*E**Q*1*i*, *E**Q*2*j*) = \n$ \\frac{|ds\\_p=\\{\\{ V(EQ\\_{1i})\\} \\Join\n\\{V(EQ\\_{2j})\\} |\\hspace{1mm}|\\{ V(EQ\\_{1i})\\} \\Join\n\\{V(EQ\\_{2j})\\}| \\geq k \\}|} {|ds\\_w=\\{\\{V(EQ\\_{1i})\\} \\Join\n\\{V(EQ\\_{2j}) \\} |}.$\nIntuitively, *η*(*E**Q*1*i*, *E**Q*2*j*) indicates the probability that the result of joining *E**Q*1*i* and *E**Q*2*j* is a *k*-anonymous equivalence class. As an example, when the value of *η* is 1, it indicates that ∀*v*1 ∈ *V*(*E**Q*1*i*), *v*2 ∈ *V*(*E**Q*2*j*), we have ∣*v*1 ⋈ *v*2∣ ≥ *k*. In other words, all possible instantiations will result in a *k*-anonymity-preserving equijoin.\n(*δ*-SELECTIVE *K*-ANONYMOUS FRAG.). A fragmentation ${\\cal F}=\\{F\\_1,F\\_2\\}$ that satisfies the *fragment *k*-anonymity condition* is called *δ*-selective if and only if ∀*E**Q*1*i*, *E**Q*2*j* we have *η*(*E**Q*1*i*, *E**Q*2*j*) ≥ *δ*.\nAlgorithm [alg:deltaselectivity] in Appendix A in the supplementary materials shows how *δ*-selectivity can be enforced on a fragmentation ${\\cal F}$.\nExtension to ℓ-Diversity\n========================\nAnalogous to the *k*-anonymity case, a fragmentation is called an ℓ-diverse fragmentation, if and only if 1) each fragment satisfies the ℓ-diversity requirement (fragment ℓ-diversity condition) and 2) joining the fragments does not violate the ℓ-diversity requirement (ℓ-diversity non-reconstructability condition). Satisfying the fragment ℓ-diversity condition is similar to that of the *k*-anonymity case. However, for ℓ-diversity, the non-reconstructability condition is different. Table [table:notationsdiversity] describes the notations used in this section. A brief overview of the steps for the fragmentation-based ℓ-diversity is provided below:\n[H]\n[table:notationsdiversity]\n|c|c|\n&\n${\\cal F}$ & a vertical fragmentation & *i**t**h* fragment in fragmentation ${\\cal F}$ & *i**t**h* segment & data chunk belongs to segment *S**i* and fragment *F**j* & *i*th equivalence class belongs to chunk *C**K**s**j* & set of class values for *C**K**i**j* & set of class values for *E**Q**i**j**s* & diversity level of segment *S**i* & size of a set \n1. Use the fragment construction algorithm proposed in Section 4.1 solely to *compute* the best fragmentation ${\\cal F}=\\{F\\_1,F\\_2,...,F\\_n\\}$. However, the fragmentation is not actually *executed* in this step.\n2. Cluster data records into *m* segments ${\\cal S}=\\{S\\_1,S\\_2,...,S\\_m\\}$ using a top-down clustering algorithm. Stop dividing each segment *S**i* into further sub-segments if the resulting sub-segments either violate the *k*-anonymity or ℓ-diversity requirement. Final segment *S**i* has the diversity level *l**i*.\n3. Vertically partition each segment *S**i*, using the fragments found in Step 1, into *n* data chunks. The diversity level of each data chunk *C**K**i**j* is equal to the diversity level of segment *S**i*, which is *l**i*.\n4. Use any off-the-shell ℓ-diversity algorithm to anonymize each chunk *C**K**i**j*. However, the diversity requirement of each *C**K**i**j* must be set to *l**i*.\n5. Merge equivalence classes belonging to the same vertical fragments and publish them as one fragment.\nIn the afore-mentioned algorithm, it is worth noting that the clustering algorithm in the second step should be adjusted to the relevant workload. Since the workload in our approach is classification, a classification-oriented clustering algorithm results in higher utility.\nThe ℓ-diversity non-reconstructability condition may be violated if at least two chunks *C**K**s**i* and *C**K**s**j* are joined and the resulting data set violates either the *k*-anonymity or the ℓ-diversity requirement. We prove that this will not be the case according to the way equivalence classes in chunks have been formed. For simplicity, we prove this for the case of two vertical fragments. However, the result is true in general.\nThe data set resulting from joining chunks *C**K**s*1 and *C**K**s*2 neither violates *k*-anonymity nor ℓ-diversity.\nAgain, the proof of this theorem is presented in Appendix B in the supplementary materials for the sake of brevity. It is easy to show that this extension can be utilized for *t*-closeness by simply enforcing *t*-closeness instead of ℓ-diversity in the aforementioned steps.\nMembership Disclosure Protection\n--------------------------------\nFragmenting the data can help protect against membership attack by disassociating different attributes. As discussed in, the ability to determine presence or absence of a subject’s record in the published data is a privacy threat. This can be done by comparing the subject’s quasi-identifiers with the published quasi-identifiers.\nConsider an attacker trying to find out the membership of subject *v* in the published fragmented data. As attributes are fragmented, the attacker must find the matching equivalence class in each fragment to which the subject’s attributes belong. This may not be possible considering the generalization applied on the attributes. However, provided that the attacker succeeds in finding the matching equivalence classes *E**Q*1*i*, *E**Q*2*j*,..., *E**Q**n**p*, the likelihood that the record pertaining to *v* exists in the published fragmented data is $\\frac{|EQ\\_{1i}\\Join EQ\\_{2j} \\Join...\\Join\nEQ\\_{np}|}{|F\\_1 \\Join F\\_2 \\Join... \\Join F\\_n|} = \\frac{|EQ\\_{1i}\\Join EQ\\_{2j} \\Join...\\Join\nEQ\\_{np}|}{\\sum\\_{i} \\sum\\_{j}...\\sum\\_{p} |EQ\\_{1i}\\Join EQ\\_{2j} \\Join...\\Join\nEQ\\_{np}|} $.\nThis likelihood is mostly impacted by *p* (the number of vertical fragments), anonymity level (either *k* or ℓ), and ∣*D*∣ (size of the data set). In most cases, the value of *p* is small, and ∣*D*∣ ≫ *k*. Therefore, the numerator of the likelihood formula becomes much smaller than the denominator. When the value of *p* increases, the number of common class values among specific equivalence classes *E**Q*1*i*, *E**Q*2*i*,..., and *E**Q**n**p* drops and even tends to zero in many cases. Therefore, the chance of a successful membership attack becomes negligible. In general, the fragmentation-based anonymization provides strong protection against membership attack.\nExperimental Results\n====================\nIn this section, we will present the experimental results showing the effectiveness of our method. The goal is to show that the fragmentation process is able to retain greater utility of the data both in terms of classification measures and information loss measures, at the same level of privacy.\nWe utilized two metrics to evaluate the effectiveness of our proposed method, *information loss* and *weighted F-measure* to capture the total amount of lost information and evaluate the utility of the data anonymized by our meta-algorithm, respectively. Details on how these metrics are used on fragmented data are available in Appendix D.I in the supplementary materials.\nBaselines\n---------\nSince the goal was to show the effectiveness of the fragmentation approach as a meta-algorithm, the baseline for the approach were the results for the anonymization process with and without fragmentation. The Mondrian multidimensional anonymity method was used for *both* the fragmented, *and* the unfragmented scenario. More accurately, in Step 2 of fragmented *k*-anonymity, we used *m**e**d**i**a**n* Mondrian and in Step 4 of fragmented ℓ-diversity, we utilized ℓ-diversity Mondrian. Therefore, the qualitative improvements show the effects of fragmentation, as a methodology to improve the effectiveness of an off-the-shelf approach.\nData Sets\n---------\nReal data set *Musk* from the *UCI Machine Learning Repository*[5](#fn5) was used. The detailed description of the data set is provided in the Appendix D.II of the supplementary materials.\nResults\n-------\nIn each case, the results were measured with varying dimensionality and anonymity level. In each case, the anonymity level was varied after fixing the dimensionality, and the dimensionality was varied on fixing the anonymity level. The anonymity level was fixed to 40, when the dimensionality was varied on the *X*-axis. While varying the anonymity level on the *X*-axis, the dimensionality was fixed to 40. In case of ℓ-diversity, the value of *l* is set to 2. In addition, we fixed *δ* to 0.5 in *δ*-selective enforcement approach. It is important to note that the information loss results do *not* vary with the different strategies for ensuring *k*-anonymity non-reconstructability, which affect only the class variable. Since the information loss metrics are based on the feature variables only, a single chart will be shown for the case of information loss, whereas the performance results for different approaches of *k*-anonymity (based on different strategies for ensuring *k*-anonymity non-reconstructability) and ℓ-diversity will be shown in the case of F-measure separately by means of solid and dashed lines, respectively.\nThe information loss with varying dimensionality is illustrated in Figure [fig:infolossdim]a. The dimensionality is illustrated on the *X*-axis, and the information loss is illustrated on the *Y*-axis in each case. Besides, diversity level is set to 2 for ℓ-diversity in all experiments. It is evident that the information loss of the *unfragmented* approach (for both *k*-anonymity and ℓ-diversity) increases with increasing dimensionality, which is in agreement of the results found earlier in. In fact, the error touches almost its upper bound, which implies that each generalized value starts losing more and more of its specificity in the unfragmented case. On the other hand, the fragmentation method shows drastic improvements in the amount of information loss, for both *k*-anonymity and ℓ-diversity. This implies that a significant amount of attribute specificity is retained in each fragment.\nThe information loss with increasing anonymity level is illustrated in Figure [fig:infolossdim]b. The anonymity level is illustrated on the *X*-axis, whereas the information level is illustrated on the *Y*-axis. It is not surprising that the information loss increases with the anonymity level, and enforcing diversity. However, as in the case of the results with increasing dimensionality, the improvements achieved by fragmentation were significant.\n[fig:infolossdim]\nThe comparisons for the F-measure with increasing dimensionality are much more tricky. This is because the addition of more dimensions to a data set affects the classification precision and recall (and hence F-measure) of the data in two mutually contradictory ways:\nA larger number of dimensions provides greater knowledge (in terms of more attributes) to the classifier in order to improve its precision and recall.\nData sets of larger dimensionality will have greater information loss on a *per attribute basis*, and this reduces the effectiveness of the classifiers.\nSo how does this tug-of-war between two mutually contradictory effects impact the final classification results, and how does the fragmentation process affect this tradeoff? Figure [fig:muskaccuracydim] compares the prediction F-measure of the unfragmented and fragmented anonymization methods with increasing dimensionality for two classifiers. In all cases, the different variants of the fragmentation scheme have higher classification accuracy than the unfragmented scheme. Even the naive scheme (in fragmented *k*-anonymity) was often able to perform better than the unfragmented approach in spite of its relative lack of sophistication in performing the class distortions. The difference in F-measure becomes even more drastic in case of ℓ-diversity and the fragmented anonymization achieves up to 28% improvement over unfragmented scenario.\n[fig:muskaccuracydim]\nIt is also immediately evident that the trend with increasing dimensionality is specific to the choice of data set, classifier, and specific approach. In particular, an interesting trend is that the F-measure changes only a little bit with increasing dimensionality in many cases, especially for unfragmented data. This is because the anonymization process in the unfragmented case changes values for many feature attributes to the very general value as the dimensionality grows. This change turns a high-dimensional data set into a data set with very few useful features for the classification. This phenomenon is reflected in the F-measure of unfragmented anonymization shown in Figure [fig:muskaccuracydim], which often does not vary much. In fact, only 6 feature attributes in *M**u**s**k* played a significant role in the classification. As a result, the F-measure does not vary too much with increasing dimensionality. Besides, as the equivalence classes have different class labels in the case of ℓ-diversity, the precision and recall degrade dramatically and cause the F-measure to be very low for unfragmented ℓ-diversity.\nThe effect of *k* on classification F-measure is illustrated in Figure [fig:muskaccuracyk]. The fragmented anonymization reveals a prominent improvement of up to 54% compared to the unfragmented anonymization. Normally, we expect the prediction F-measure to decline with increasing values of *k*. While this was often the case, there were also a few cases, where it has an unexpected rise. This trend has also sometimes been observed in earlier work, and is a result of the aggregation effects of the anonymization procedures (sometimes) removing the noisy artifacts in the data.\n[fig:muskaccuracyk]\nConclusions and Summary\n=======================\nThis paper presents a method for fragmentation-based anonymization for high-dimensional data. While the curse of dimensionality is a fundamental theoretical barrier, it is often possible to obtain effective results in practice. This paper uses fragmentation as a general purpose methodology to improve the effectiveness of any off-the-shelf algorithm for the anonymization process. Experimental results show significant improvements of the utility of the data after the fragmentation process. This meta-algorithm approach is fairly general and has the potential to be extended to a wider variety of scenarios and privacy models and workloads. This will be the focus of our future work.\n10\nSupplementary Materials\n=======================\nAppendix A. Pseudocodes\n=======================\nAlgorithm [alg:heuristic] depicts how a high-dimensional data set can be broken down into fragments.\n[alg:heuristic]\n[1] // *n* denotes the number of features\n: a (*n* + 1) by (*n* + 1) matrix storing the mutual information (between the features and the class attribute)\n*F**r**a**g**m**e**n**t*1, *F**r**a**g**m**e**n**t*2, *n**o**n**A**s**s**i**g**n**e**d**F**s*= **empty**\nSelect two features having the maximum mutual information as two seeds, *s**e**e**d*1 and *s**e**e**d*2. *F**r**a**g**m**e**n**t*1.add(*s**e**e**d*1) *F**r**a**g**m**e**n**t*2.add(*s**e**e**d*2) Add the rest of features to *n**o**n**A**s**s**i**g**n**e**d**F**s*\n(*n**o**n**A**s**s**i**g**n**e**d**F**s* not **empty**)\n*b**e**s**t**F*,*b**e**s**t**F**r**a**g**m**e**n**t*=**empty**\n*m**a**x**C**o**n**t**r**i**b**u**t**i**o**n*=0\n*a**i* in *n**o**n**A**s**s**i**g**n**e**d**F**s*\n*f* in *F**r**a**g**m**e**n**t*1,*F**r**a**g**m**e**n**t*2\n(*c**o**n**t**r**i**b**u**t**i**o**n*(*a**i*,*f*) > *m**a**x**C**o**n**t**r**i**b**u**t**i**o**n*)\n*b**e**s**t**F*=*a**i*\n*b**e**s**t**F**r**a**g**m**e**n**t*=*f*\n*m**a**x**C**o**n**t**r**i**b**u**t**i**o**n*=*c**o**n**t**r**i**b**u**t**i**o**n*(*a**i*,*f*)\nAdd *b**e**s**t**F* to *f*\nRemove *b**e**s**t**F* from *n**o**n**A**s**s**i**g**n**e**d**F**s* Add the class attribute to *F**r**a**g**m**e**n**t*1 and *F**r**a**g**m**e**n**t*2\n[H] [alg:dependencygraph]\n[1] mark all equivalence classes in *s**e**t*-*o**f*-*a**l**l*-*E**Q* as *u**n*-*v**i**s**i**t**e**d* *c*-*i**d*=1 *t**o**B**e**P**r**o**c**e**s**s**e**d*=**empty** *r**o**o**t*=random(*s**e**t*-*o**f*-*a**l**l*-*E**Q*) // choose an unvisited equivalence class randomly *t**o**B**e**P**r**o**c**e**s**s**e**d*.enqueue(*r**o**o**t*) (*t**o**B**e**P**r**o**c**e**s**s**e**d* not **empty**)\n*c**u**r**r**e**n**t*-*n**o**d**e*=*t**o**B**e**P**r**o**c**e**s**s**e**d*.dequeue()\nvisit(*c**u**r**r**e**n**t*-*n**o**d**e*)\n*E**Q* in dependant(*c**u**r**r**e**n**t*-*n**o**d**e*)\n(*E**Q* not visited)\ndraw an edge from *c**u**r**r**e**n**t*-*n**o**d**e* to *E**Q* in\ncomponent with id=*c*-*i**d*\n*t**o**B**e**P**r**o**c**e**s**s**e**d*.enqueue(*E**Q*) (any un-visited equivalence class left)\n*c*-*i**d*++\ngo to line 4\nAlgorithm [alg:dependencygraph] takes the set of all equivalence classes in a given fragmentation, and constructs the *dependency graph*. At the end of this process, the variable *c*-*i**d* indicates the number of components with no dependency (connection). The subroutine *d**e**p**e**n**d**a**n**t* takes an equivalence class *e**q* in either *F*1 or *F*2, and returns the set of equivalence classes in the other fragment that can be joined by *e**q*.\nAlgorithm [alg:dgbe] shows the procedure for enforcing the condition in Theorem [thm:nonreconstructability] in a *dependency graph*. This algorithm is invoked for each component the *dependency graph* in order to ensure non-identifiability. In Algorithm [alg:dgbe], the subroutine *c**h**a**n**g**e* (line 9) changes a class value having the lowest frequency in its equivalence class to the majority class value in that equivalence class. However, this change might affect the previously satisfied *k*-anonymity non-reconstructability condition between node *u**n* and its *v**i**s**i**t**e**d* neighbors. Thus, lines 10-14 re-check the previously satisfied reconstructability condition, and if needed, the subroutine *c**h**a**n**g**e* is called as necessary in order to re-satisfy it.\n[alg:dgbe]\n[1] *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e* = **empty** mark all nodes in *d*-*g**r**a**p**h* as *u**n**v**i**s**i**t**e**d* *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.enqueue(random(*d*-*g**r**a**p**h*)) (*p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e* **not empty**)\n*c**u**r**r**e**n**t*-*n**o**d**e* = *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.dequeue()\n*u**n* in unvisited-neighbors(*c**u**r**r**e**n**t*-*n**o**d**e*)\n((*c**u**r**r**e**n**t*-*n**o**d**e* and *u**n*) not satisfy the\n*k*-anonymity non-reconstructability condition)\nchange(*u**n*)\n(any previously-satisfied *k*-anonymity\nnon-reconstructability condition violates\nbetween *u**n* and visited- neighbors(*u**n*))\ncall change(*u**n*) until there is no violation\nbetween *u**n* and visisted-neighbors(*u**n*)\nvisit(*c**u**r**r**e**n**t*-*n**o**d**e*)\n*p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.enqueue(*u**n*)\nAlgorithm [alg:deltaselectivity] shows how *δ*-selectivity can be enforced on a fragmentation ${\\cal F}$. This algorithm is very similar to Algorithm [alg:dgbe]. It is worth noting that the class values of equivalence classes in the dependency graph (**d*-*g**r**a**p**h**) are at the equivalence class-level. Subroutine *c**h**a**n**g**e* simply removes a class value in an equivalence class. In order to reduce the class value distortion, the class value having the lowest frequency before converting the class values into equivalence class-level must be removed.\n[alg:deltaselectivity]\n[1] *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e* = **empty** mark all nodes in *d*-*g**r**a**p**h* as *u**n**v**i**s**i**t**e**d* *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.enqueue(random(*d*-*g**r**a**p**h*)) (*p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e* **not empty**)\n*c**u**r**r**e**n**t*-*n**o**d**e* = *p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.dequeue()\n*u**n* in unvisited-neighbors(*c**u**r**r**e**n**t*-*n**o**d**e*)\n(*η*(*u**n*,*c**u**r**r**e**n**t*-*n**o**d**e*) < *δ*)\nchange(*u**n*)\n(*η* between *u**n* and any *v**n* ∈ visited-\nneighbors(*u**n*) becomes less than *δ*)\ncall change(*u**n*) until *η*(*u**n*,*v**n*) becomes greater\nthan *δ*\nvisit(*c**u**r**r**e**n**t*-*n**o**d**e*)\n*p**r**o**c**e**s**s**i**n**g*-*q**u**e**u**e*.enqueue(*u**n*)\nAppendix B. Proofs\n==================\nTHEOREM 4.1.\n*(*K*-ANONYMITY NON-RECONSTRUCTABILITY CONDITION). The condition for a given fragmentation ${\\cal F}=\\{F\\_1,F\\_2\\}$ which satisfies the fragment *k*-anonymity condition, and fragments *F*1 = {*E**Q*11, *E**Q*12, ..., *E**Q*1*n*} and *F*2 = {*E**Q*21, *E**Q*22, ..., *E**Q*2*m*}, to be non-reconstructible is that one of the following must be true for each joined pair *E**Q*1*i*, *E**Q*2*j*:* \n∑*c* ∈ *C*1*i* ∩ *C*2*j**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) = 0\n∑*c* ∈ *C*1*i* ∩ *C*2*j**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) ≥ *k*\nAccording to Definition [def:nonreconstructable], ${\\cal F}$ is *n**o**n*-*r**e**c**o**n**s**t**r**u**c**t**i**b**l**e* if and only if the relation *F*1 ⋈ *F*2 satisfies *k*-anonymity.\nJoining two equivalence classes *E**Q*1*i* and *E**Q*2*j* generates tuples with the same value for their quasi-identifiers. If the number of generated tuples is greater than *k*, joining *E**Q*1*i* and *E**Q*2*j* does not violate the *k*-anonymity property. For any given equivalence classes *E**Q*1*i* and *E**Q*2*j*, one of the following conditions is true:\n*C**L**S* = *C*1*i* ∩ *C*2*j* = ∅.\n*C**L**S* = *C*1*i* ∩ *C*2*j* = {*c*1, *c*2, ..., *c**w*}.\nIf the former condition is true, *E**Q*1*i* and *E**Q*2*j* do not have any common values to be joined on. Therefore, we have: \n∑*c* ∈ ∅*f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) = 0 \n If the latter condition is true, the joining of *E**Q*1*i* and *E**Q*2*j* will result in ∑*c* ∈ *C**L**S**f**r**e**q*(*c*, *E**Q*1*i*) × *f**r**e**q*(*c*, *E**Q*2*j*) tuples. Thus, the result of joining *E**Q*1*i* and *E**Q*2*j* is *k*-anonymous if the number of resulting tuples is at least *k*. Therefore, in either case the resulting fragmentation ${\\cal F}$ is *k*-anonymity non-reconstructible.\nTHEOREM 5.1. *The data set resulting from joining chunks *C**K**s*1 and *C**K**s*2 neither violates *k*-anonymity nor ℓ-diversity.*\nTo prove this theorem, we discriminate between different types of ℓ-diversity, and investigate each model individually.\n**Distinct ℓ-diversity**: The joining of *C**K**s*1 and *C**K**s*2 is performed based on joining equivalence classes *E**Q**i*1*s* and *E**Q**j*2*s*. According to the Step 4 of fragmentation-based ℓ-diversity, *E**Q**i*1*s* and *E**Q**j*2*s* have been constructed in such a way that have the same set (of size *l**s*) of class values with frequencies {*f*1*x*, *f*2*x*, ..., *f**l**s**x*} and {*f*1*y*, *f*2*y*, ..., *f**l**s**y*}, respectively. Since both *E**Q**i*1*s* and *E**Q**j*2*s* satisfy the *k*-anonymity, we have:\n1)∣*E**Q**i*1*s*∣ = *f*11 + *f*21 + ... + *f**l**s*1 ≥ *k*\n2)∣*E**Q**j*2*s*∣ = *f*12 + *f*22 + ... + *f**l**s*2 ≥ *k*\nEvidently, joining *E**Q**i*1*s* and *E**Q**j*2*s* are based on the common sensitive values, thus it results in a data set of size *f*11 × *f*12 + *f*21 × *f*22 + ... + *f**l**s*1 × *f**l**s*2 having exactly *l**s* sensitive values. Therefore, the data set resulting from this join satisfies the distinct ℓ-diversity. Since joining each pair of arbitrary equivalence classes *E**Q**i*1*s* and *E**Q**j*2*s* satisfies the distinct ℓ-diversity, joining *C**K**s*1 and *C**K**s*2 also satisfies this privacy requirement.\n**Entropy (or recursive) ℓ-diversity**: This part is a bit trickier. Joining each pair of equivalence classes *E**Q**i*1*s* and *E**Q**j*2*s* will result in a data set with at least *k* tuples (with the same reasoning as distinct ℓ-diversity), but most likely different level of diversity than *l**s* (it can be lower or higher than *l**s*). This might be considered as a privacy violation. However, joining all possible pairs of equivalence classes *E**Q**i*1*s* and *E**Q**j*2*s* generates a data set consisting of both real (all tuples in *S**s*) and fake tuples. There is an important observation here. If the attacker has enough background knowledge to rule out the faked tuples, the diversity level of real tuples are exactly equal to *l**s*. Hence, this is not a privacy violation.\nAppendix C. Enforcement via *δ*-selectivity\n===========================================\nI. Tuple-level vs. equivalence class-level class value publishing\n=================================================================\nTable [tuplelevelvsequilevel] exemplifies the difference between tuple-level and equivalence class-level publishing for a 5-anonymized fragment. As Table [tuplelevelvsequilevel]b illustrates, publishing the class values in the equivalence class-level usually results in an equivalence class with some *ambiguous slots* for its class values. For example, two ambiguous slots are illustrated in Table [tuplelevelvsequilevel]b. The only case in which publishing at the equivalence class-level does not result in an ambiguous slot is when the class values are unique in the equivalence class. As no further information is published regarding the frequency of the class values in each equivalence class, any class value available in the equivalence class can be placed in the ambiguous slots.\n[h] [tuplelevelvsequilevel]\nThe ambiguous slots in an equivalence class *E**Q**i**j*, published at the equivalence class-level, can take any of the class values in *C**i**j*. In other words, different versions for *E**Q**i**j* can be assumed. Table [table:versions] illustrates two possible versions for the equivalence class in Table [tuplelevelvsequilevel]b.\n[h] [table:versions]\nII. A subtle way to find *η*\n============================\nGiven two equivalence classes *E**Q*1*i* and *E**Q*2*j* whose class values are published in equivalence class-level and have at least one class value in common, we find the number of tuples resulted from their join which is referred to as *equijoin selectivity* in the literature. As ∣*E**Q*1*i*∣ and ∣*C*1*i*∣ refer to the number of tuples and the class values (in equivalence class-level) in *E**Q*1*i*, the number of ambiguous slots is equal to ∣*E**Q*1*i*∣ − ∣*C*1*i*∣. The set {*V*(*E**Q*1*i*)} represents different versions (or instantiations) of *E**Q*1*i*. Each member of this set is a combination of ∣*E**Q*1*i*∣ tuples in which there exist at least one tuple per each class value in *C*1*i*. The size of this set is equal to ${(|EQ\\_{1i}|-|C\\_{1i}|)+|C\\_{1i}|-1} \\choose {(|EQ\\_{1i}|-|C\\_{1i}|)}$.\nThere exist ∣{*V*(*E**Q*1*i*)}∣ × ∣{*V*(*E**Q*2*j*)}∣ different ways to join *E**Q*1*i* and *E**Q*2*j*. Among all possible joins, those generating *k* tuples (or more) are referred to as **k*-anonymity-preserving equijoins*.\nIn *η* formula, calculating the denominator is straightforward by multiplying ∣*V*(*E**Q*1*i*)∣ and ∣*V*(*E**Q*2*j*)∣. However, computing the numerator is more challenging. The simplest way is perform all joins, and count those which satisfy the minimum cardinality of *k*. A more subtle way is to consider the conditions (constraints) that exist on the number of class values in each equivalence class and map the problem as follows:\nThe number of pairs (*A*,*B*) of integer vectors *A* = (*a*1, *a*2, ..., *a**c*) and *B* = (*b*1, *b*2, ..., *b**c*) in which *a**d* and *b**d* depict the frequency of *d**t**h* common class value in *V*(*E**Q*1*i*) and *V*(*E**Q*2*j*) respectively, such that the following conditions are satisfied:\ncondition 1: ∑*a**d* × *b**d* ≥ *k*\ncondition 2: 1 ≤ *a**d* ≤ (1 + ∣*E**Q*1*i*∣ − ∣*C*1*i*∣)\ncondition 3: 1 ≤ *b**d* ≤ (1 + ∣*E**Q*2*j*∣ − ∣*C*2*j*∣)\ncondition 4: ∑*a**d* ≤ ∣*C**L**S*∣ + ∣*E**Q*1*i*∣ − ∣*C*1*i*∣\ncondition 5: ∑*b**d* ≤ ∣*C**L**S*∣ + ∣*E**Q*2*j*∣ − ∣*C*2*j*∣\nwhere *C**L**S* = *C*1*i* ∩ *C*2*j*.\nAppendix D. Experiments\n=======================\nI. Performance measures\n=======================\nWe utilized two metrics to evaluate the effectiveness of our proposed method. The first one is the widely-used metric known as *information loss*, which captures the total amount of lost information due to generalization. In fact, this metric shows the usefulness of data *on a per-attribute basis* for general workloads. For an anonymized data set with *n* tuples and *m* attributes, the information loss *I* is computed as follows:\n$I= \\sum\\_{i=1}^{n} \\sum\\_{j=1}^{m}\\frac{|upper\\_{ij}-lower\\_{ij}|}{n\n\\cdot m \\cdot |max\\_j-min\\_j|}$\nHere, *l**o**w**e**r**i**j* and *u**p**p**e**r**i**j* represent the lower and upper bounds of attribute *j* in tuple *i* after generalization, and *m**i**n**j* and *m**a**x**j* represent the maximum and minimum values taken by attribute *j* over all records. Note that the computation in the fragmented and unfragmented case is not different, as long as all the different attributes in the different fragments are used. This is the most direct measure of the data quality, since it normalizes the final result by the number of attributes.\nTo evaluate the utility of the data anonymized by our meta-algorithm, we calculated the weighted F-measure of a classifier trained on the anonymous fragmentation. This metric reflects the goodness of classifier more accurately in case on unevenly distributed test data. Each fragment was used to train the classifier separately. For a given test instance, the different fragments of the training data were trained separately, and the weighted majority label from the different classifiers was reported, where the weight used was score returned by the classifier from each fragment. Each class label was once considered as positive (and once as negative) and the weighted F-measure for each case was calculated by taking into account the fraction of the positive instances. We used J48[6](#fn6) and a *k*-NN[7](#fn7) classifier in Weka with the default setting. The value of *k* in *k*-NN was set to 5. The learning from generalized values was also done by the technique used in. In each case, the decomposition was performed into two fragments.\nIn addition, the amount of distortion required to the class values (for fragmentation *k*-anonymity) was measured. Specifically, the number of distorted class values were computed for each of the different techniques. The aim is to show that the amount of distortion required was relatively small.\nII. Data set description\n========================\nReal data set *Musk* from the *UCI Machine Learning Repository*[8](#fn8) was used. It contains 7074 instances with 168 numerical feature attributes describing a set of molecules. The goal is to predict whether a molecule is musk or non-musk. In order to test the effects of varying data dimensionality, we chose 10, 20, 30, and 40 features randomly and constructed four versions of *M**u**s**k* with different dimensionality. Around 70% of the *M**u**s**k* data set was used for training. The distribution of the two classes in the training and test data set is shown in Table [table:datasetdis]. Although the data set used in this work is numerical, the proposed meta-algorithm can be used to anonymize categorical values by simply using an off-the-shell anonymization technique capable to anonymize categorical values in Step 2 and 4 of fragmentation *k*-anonymity and ℓ-diversity, respectively.\n[table:datasetdis]\n|c|c|c| & & \nmusk & 1017 & 207 \nnon-musk & 3983 & 1867 \n---\n1. University of Calgary, hzakerza@ucalgary.ca[↩](#fnref1)\n2. IBM T.J. Watson Research Center, charu@us.ibm.com[↩](#fnref2)\n3. University of Calgary, kbarker@ucalgary.ca[↩](#fnref3)\n4. Although the works in are originally proposed for the set-valued (transaction) data, the relational data can be transformed to set-valuded data for anonymization.[↩](#fnref4)\n5. <http://archive.ics.uci.edu/ml.>[↩](#fnref5)\n6. J48 is an open source implementation of C4.5 in Java, <http://weka.sourceforge.net/doc/weka/classifiers/trees/J48.html.>[↩](#fnref6)\n7. <http://weka.sourceforge.net/doc/weka/classifiers/lazy/IBk.html.>[↩](#fnref7)\n8. <http://archive.ics.uci.edu/ml.>[↩](#fnref8)\n"},"data_id":{"kind":"string","value":"arxiv_0002453"}}},{"rowIdx":595,"cells":{"content":{"kind":"string","value":" provides insight and instruction into more recent changes to the article template.\nThe “`acmart`” document class can be used to prepare articles for any ACM publication — conference or journal, and for any stage of publication, from review to final “camera-ready” copy, to the author’s own version, with *very* few changes to the source.\nTemplate Overview\n=================\nAs noted in the introduction, the “`acmart`” document class can be used to prepare many different kinds of documentation — a double-blind initial submission of a full-length technical paper, a two-page SIGGRAPH Emerging Technologies abstract, a “camera-ready” journal article, a SIGCHI Extended Abstract, and more — all by selecting the appropriate *template style* and *template parameters*.\nThis document will explain the major features of the document class. For further information, the *LaTeX User’s Guide* is available from <https://www.acm.org/publications/proceedings-template>.\nTemplate Styles\n---------------\nThe primary parameter given to the “`acmart`” document class is the *template style* which corresponds to the kind of publication or SIG publishing the work. This parameter is enclosed in square brackets and is a part of the `documentclass` command:\n```\n \\documentclass[STYLE]{acmart}\n```\nJournals use one of three template styles. All but three ACM journals use the `acmsmall` template style:\n* `acmsmall`: The default journal template style.\n* `acmlarge`: Used by JOCCH and TAP.\n* `acmtog`: Used by TOG.\nThe majority of conference proceedings documentation will use the `acmconf` template style.\n* `acmconf`: The default proceedings template style.\n* `sigchi`: Used for SIGCHI conference articles.\n* `sigchi-a`: Used for SIGCHI “Extended Abstract” articles.\n* `sigplan`: Used for SIGPLAN conference articles.\nTemplate Parameters\n-------------------\nIn addition to specifying the *template style* to be used in formatting your work, there are a number of *template parameters* which modify some part of the applied template style. A complete list of these parameters can be found in the *LaTeX User’s Guide.*\nFrequently-used parameters, or combinations of parameters, include:\n* `anonymous,review`: Suitable for a “double-blind” conference submission. Anonymizes the work and includes line numbers. Use with the command to print the submission’s unique ID on each page of the work.\n* `authorversion`: Produces a version of the work suitable for posting by the author.\n* `screen`: Produces colored hyperlinks.\nThis document uses the following string as the first command in the source file:\n```\n\\documentclass[sigconf, language=french,\nlanguage=german, language=spanish, language=english]{acmart}\n```\nModifications\n=============\nModifying the template — including but not limited to: adjusting margins, typeface sizes, line spacing, paragraph and list definitions, and the use of the `\\vspace` command to manually adjust the vertical spacing between elements of your work — is not allowed.\n**Your document will be returned to you for revision if modifications are discovered.**\nTypefaces\n=========\nThe “`acmart`” document class requires the use of the “Libertine” typeface family. Your TeX installation should include this set of packages. Please do not substitute other typefaces. The “`lmodern`” and “`ltimes`” packages should not be used, as they will override the built-in typeface families.\nTitle Information\n=================\nThe title of your work should use capital letters appropriately - <https://capitalizemytitle.com/> has useful rules for capitalization. Use the `title` command to define the title of your work. If your work has a subtitle, define it with the `subtitle` command. Do not insert line breaks in your title.\nIf your title is lengthy, you must define a short version to be used in the page headers, to prevent overlapping text. The `title` command has a “short title” parameter:\n```\n \\title[short title]{full title}\n```\nAuthors and Affiliations\n========================\nEach author must be defined separately for accurate metadata identification. As an exception, multiple authors may share one affiliation. Authors’ names should not be abbreviated; use full first names wherever possible. Include authors’ e-mail addresses whenever possible.\nGrouping authors’ names or e-mail addresses, or providing an “e-mail alias,” as shown below, is not acceptable:\n```\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n```\nThe `authornote` and `authornotemark` commands allow a note to apply to multiple authors — for example, if the first two authors of an article contributed equally to the work.\nIf your author list is lengthy, you must define a shortened version of the list of authors to be used in the page headers, to prevent overlapping text. The following command should be placed just after the last `\\author{}` definition:\n```\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n```\nOmitting this command will force the use of a concatenated list of all of the authors’ names, which may result in overlapping text in the page headers.\nThe article template’s documentation, available at <https://www.acm.org/publications/proceedings-template>, has a complete explanation of these commands and tips for their effective use.\nNote that authors’ addresses are mandatory for journal articles.\nRights Information\n==================\nAuthors of any work published by ACM will need to complete a rights form. Depending on the kind of work, and the rights management choice made by the author, this may be copyright transfer, permission, license, or an OA (open access) agreement.\nRegardless of the rights management choice, the author will receive a copy of the completed rights form once it has been submitted. This form contains LaTeX commands that must be copied into the source document. When the document source is compiled, these commands and their parameters add formatted text to several areas of the final document:\n* the “ACM Reference Format” text on the first page.\n* the “rights management” text on the first page.\n* the conference information in the page header(s).\nRights information is unique to the work; if you are preparing several works for an event, make sure to use the correct set of commands with each of the works.\nThe ACM Reference Format text is required for all articles over one page in length, and is optional for one-page articles (abstracts).\nCCS Concepts and User-Defined Keywords\n======================================\nTwo elements of the “acmart” document class provide powerful taxonomic tools for you to help readers find your work in an online search.\nThe ACM Computing Classification System — <https://www.acm.org/publications/class-2012> — is a set of classifiers and concepts that describe the computing discipline. Authors can select entries from this classification system, via <https://dl.acm.org/ccs/ccs.cfm>, and generate the commands to be included in the LaTeX source.\nUser-defined keywords are a comma-separated list of words and phrases of the authors’ choosing, providing a more flexible way of describing the research being presented.\nCCS concepts and user-defined keywords are required for for all articles over two pages in length, and are optional for one- and two-page articles (or abstracts).\nSectioning Commands\n===================\nYour work should use standard LaTeX sectioning commands: `section`, `subsection`, `subsubsection`, and `paragraph`. They should be numbered; do not remove the numbering from the commands.\nSimulating a sectioning command by setting the first word or words of a paragraph in boldface or italicized text is **not allowed.**\nTables\n======\nThe “`acmart`” document class includes the “`booktabs`” package — <https://ctan.org/pkg/booktabs> — for preparing high-quality tables.\nTable captions are placed *above* the table.\nBecause tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite. To ensure this proper “floating” placement of tables, use the environment **table** to enclose the table’s contents and the table caption. The contents of the table itself must go in the **tabular** environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules. Again, detailed instructions on **tabular** material are found in the *LaTeX User’s Guide*.\nImmediately following this sentence is the point at which Table [tab:freq] is included in the input file; compare the placement of the table here with the table in the printed output of this document.\n[tab:freq]\nccl Non-English or Math&Frequency&Comments \nØ& 1 in 1,000& For Swedish names \n*π* & 1 in 5& Common in math \n$ & 4 in 5 & Used in business \nΨ12 & 1 in 40,000& Unexplained usage \nTo set a wider table, which takes up the whole width of the page’s live area, use the environment **table\\*** to enclose the table’s contents and the table caption. As with a single-column table, this wide table will “float” to a location deemed more desirable. Immediately following this sentence is the point at which Table [tab:commands] is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document.\n[tab:commands]\nccl Command &A Number & Comments \n`’134author` & 100& Author \n`’134table`& 300 & For tables \n`’134table*`& 400& For wider tables \nAlways use midrule to separate table header rows from data rows, and use it only for this purpose. This enables assistive technologies to recognise table headers and support their users in navigating tables more easily.\nMath Equations\n==============\nYou may want to display math equations in three distinct styles: inline, numbered or non-numbered display. Each of the three are discussed in the next sections.\nInline (In-text) Equations\n--------------------------\nA formula that appears in the running text is called an inline or in-text formula. It is produced by the **math** environment, which can be invoked with the usual `’134begin…’134end` construction or with the short form `$…$`. You can use any of the symbols and structures, from *α* to *ω*, available in LaTeX ; this section will simply show a few examples of in-text equations in context. Notice how this equation:\nnx=0\n, set here in in-line math style, looks slightly different when set in display style. (See next section).\nDisplay Equations\n-----------------\nA numbered display equation—one set off by vertical space from the text and centered horizontally—is produced by the **equation** environment. An unnumbered display equation is produced by the **displaymath** environment.\nAgain, in either environment, you can use any of the symbols and structures available in LaTeX; this section will just give a couple of examples of display equations in context. First, consider the equation, shown as an inline equation above:\n \nlim*n* → ∞*x* = 0 \nNotice how it is formatted somewhat differently in the **displaymath** environment. Now, we’ll enter an unnumbered equation:\n \n∑*i* = 0∞*x* + 1 \nand follow it with another numbered equation:\n \n∑*i* = 0∞*x**i* = ∫0*π* + 2*f* \njust to demonstrate LaTeX’s able handling of numbering.\nFigures\n=======\nThe “`figure`” environment should be used for figures. One or more images can be placed within a figure. If your figure contains third-party material, you must clearly identify it as such, as shown in the example below.\n![1907 Franklin Model D roadster. Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons. (https://goo.gl/VLCRBB).](sample-franklin)1907 Franklin Model D roadster. Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons. (<https://goo.gl/VLCRBB>).\nYour figures should contain a caption which describes the figure to the reader.\nFigure captions are placed *below* the figure.\nEvery figure should also have a figure description unless it is purely decorative. These descriptions convey what’s in the image to someone who cannot see it. They are also used by search engine crawlers for indexing images, and when images cannot be loaded.\nA figure description must be unformatted plain text less than 2000 characters long (including spaces). **Figure descriptions should not repeat the figure caption – their purpose is to capture important information that is not already provided in the caption or the main text of the paper.** For figures that convey important and complex new information, a short text description may not be adequate. More complex alternative descriptions can be placed in an appendix and referenced in a short figure description. For example, provide a data table capturing the information in a bar chart, or a structured list representing a graph. For additional information regarding how best to write figure descriptions and why doing this is so important, please see <https://www.acm.org/publications/taps/describing-figures/>.\nThe “Teaser Figure”\n-------------------\nA “teaser figure” is an image, or set of images in one figure, that are placed after all author and affiliation information, and before the body of the article, spanning the page. If you wish to have such a figure in your article, place the command immediately before the `\\maketitle` command:\n```\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n```\nCitations and Bibliographies\n============================\nThe use of for the preparation and formatting of one’s references is strongly recommended. Authors’ names should be complete — use full first names (“Donald E. Knuth”) not initials (“D. E. Knuth”) — and the salient identifying features of a reference should be included: title, year, volume, number, pages, article DOI, etc.\nThe bibliography is included in your source document with these two commands, placed just before the `\\end{document}` command:\n```\n \\bibliographystyle{ACM-Reference-Format}\n \\bibliography{bibfile}\n```\nwhere “`bibfile`” is the name, without the “`.bib`” suffix, of the file.\nCitations and references are numbered by default. A small number of ACM publications have citations and references formatted in the “author year” style; for these exceptions, please include this command in the **preamble** (before the command “`\\begin{document}`”) of your LaTeX source:\n```\n \\citestyle{acmauthoryear}\n```\nSome examples. A paginated journal article, an enumerated journal article, a reference to an entire issue, a monograph (whole book), a monograph/whole book in a series (see 2a in spec. document), a divisible-book such as an anthology or compilation followed by the same example, however we only output the series if the volume number is given (so Editor00a’s series should NOT be present since it has no vol. no.), a chapter in a divisible book, a chapter in a divisible book in a series, a multi-volume work as book, a couple of articles in a proceedings (of a conference, symposium, workshop for example) (paginated proceedings article), a proceedings article with all possible elements, an example of an enumerated proceedings article, an informally published work, a couple of preprints, a doctoral dissertation, a master’s thesis:, an online document / world wide web resource, a video game (Case 1) and (Case 2) and and (Case 3) a patent, work accepted for publication, ’YYYYb’-test for prolific author and. Other cites might contain ’duplicate’ DOI and URLs (some SIAM articles). Boris / Barbara Beeton: multi-volume works as books and. A couple of citations with DOIs:. Online citations:. Artifacts: and.\nAcknowledgments\n===============\nIdentification of funding sources and other support, and thanks to individuals and groups that assisted in the research and the preparation of the work should be included in an acknowledgment section, which is placed just before the reference section in your document.\nThis section has a special environment:\n```\n \\begin{acks}\n ...\n \\end{acks}\n```\nso that the information contained therein can be more easily collected during the article metadata extraction phase, and to ensure consistency in the spelling of the section heading.\nAuthors should not prepare this section as a numbered or unnumbered `\\section`; please use the “`acks`” environment.\nAppendices\n==========\nIf your work needs an appendix, add it before the “`\\end{document}`” command at the conclusion of your source document.\nStart the appendix with the “`appendix`” command:\n```\n \\appendix\n```\nand note that in the appendix, sections are lettered, not numbered. This document has two appendices, demonstrating the section and subsection identification method.\nMulti-language papers\n=====================\nPapers may be written in languages other than English or include titles, subtitles, keywords and abstracts in different languages (as a rule, a paper in a language other than English should include an English title and an English abstract). Use `language=...` for every language used in the paper. The last language indicated is the main language of the paper. For example, a French paper with additional titles and abstracts in English and German may start with the following command\n```\n\\documentclass[sigconf, language=english, language=german,\n language=french]{acmart}\n```\nThe title, subtitle, keywords and abstract will be typeset in the main language of the paper. The commands `\\translatedXXX`, `XXX` begin title, subtitle and keywords, can be used to set these elements in the other languages. The environment `translatedabstract` is used to set the translation of the abstract. These commands and environment have a mandatory first argument: the language of the second argument. See `sample-sigconf-i13n.tex` file for examples of their usage.\nSIGCHI Extended Abstracts\n=========================\nThe “`sigchi-a`” template style (available only in LaTeX and not in Word) produces a landscape-orientation formatted article, with a wide left margin. Three environments are available for use with the “`sigchi-a`” template style, and produce formatted output in the margin:\n`sidebar`:\nPlace formatted text in the margin.\n`marginfigure`:\nPlace a figure in the margin.\n`margintable`:\nPlace a table in the margin.\nTo Robert, for the bagels and explaining CMYK and color spaces.\nResearch Methods\n================\nPart One\n--------\nLorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi malesuada, quam in pulvinar varius, metus nunc fermentum urna, id sollicitudin purus odio sit amet enim. Aliquam ullamcorper eu ipsum vel mollis. Curabitur quis dictum nisl. Phasellus vel semper risus, et lacinia dolor. Integer ultricies commodo sem nec semper.\nPart Two\n--------\nEtiam commodo feugiat nisl pulvinar pellentesque. Etiam auctor sodales ligula, non varius nibh pulvinar semper. Suspendisse nec lectus non ipsum convallis congue hendrerit vitae sapien. Donec at laoreet eros. Vivamus non purus placerat, scelerisque diam eu, cursus ante. Etiam aliquam tortor auctor efficitur mattis.\nOnline Resources\n================\nNam id fermentum dui. Suspendisse sagittis tortor a nulla mollis, in pulvinar ex pretium. Sed interdum orci quis metus euismod, et sagittis enim maximus. Vestibulum gravida massa ut felis suscipit congue. Quisque mattis elit a risus ultrices commodo venenatis eget dui. Etiam sagittis eleifend elementum.\nNam interdum magna at lectus dignissim, ac dignissim lorem rhoncus. Maecenas eu arcu ac neque placerat aliquam. Nunc pulvinar massa et mattis lacinia.\n\nThe Name of the Title is Hope\n=============================\nA clear and well-documented LaTeX document is presented as an article formatted for publication by ACM in a conference proceedings or journal publication. Based on the “acmart” document class, this article presents and explains many of the common variations, as well as many of the formatting elements an author may use in the preparation of the documentation of their work.\n<ccs2012> <concept> <conceptid>10010520.10010553.10010562</conceptid> <conceptdesc>Computer systems organization Embedded systems</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10010520.10010575.10010755</conceptid> <conceptdesc>Computer systems organization Redundancy</conceptdesc> <conceptsignificance>300</conceptsignificance> </concept> <concept> <conceptid>10010520.10010553.10010554</conceptid> <conceptdesc>Computer systems organization Robotics</conceptdesc> <conceptsignificance>100</conceptsignificance> </concept> <concept> <conceptid>10003033.10003083.10003095</conceptid> <conceptdesc>Networks Network reliability</conceptdesc> <conceptsignificance>100</conceptsignificance> </concept> </ccs2012>\n![Seattle Mariners at Spring Training, 2010.](sampleteaser \"fig:\") [fig:teaser]\nIntroduction\n============\nACM’s consolidated article template, introduced in 2017, provides a consistent LaTeX style for use across ACM publications, and incorporates accessibility and metadata-extraction functionality necessary for future Digital Library endeavors. Numerous ACM and SIG-specific LaTeX templates have been examined, and their unique features incorporated into this single new template.\nIf you are new to publishing with ACM, this document is a valuable guide to the process of preparing your work for publication. If you have published with ACM before, this document provides insight and instruction into more recent changes to the article template.\nThe “`acmart`” document class can be used to prepare articles for any ACM publication — conference or journal, and for any stage of publication, from review to final “camera-ready” copy, to the author’s own version, with *very* few changes to the source.\nTemplate Overview\n=================\nAs noted in the introduction, the “`acmart`” document class can be used to prepare many different kinds of documentation — a double-blind initial submission of a full-length technical paper, a two-page SIGGRAPH Emerging Technologies abstract, a “camera-ready” journal article, a SIGCHI Extended Abstract, and more — all by selecting the appropriate *template style* and *template parameters*.\nThis document will explain the major features of the document class. For further information, the *LaTeX User’s Guide* is available from <https://www.acm.org/publications/proceedings-template>.\nTemplate Styles\n---------------\nThe primary parameter given to the “`acmart`” document class is the *template style* which corresponds to the kind of publication or SIG publishing the work. This parameter is enclosed in square brackets and is a part of the `documentclass` command:\n```\n \\documentclass[STYLE]{acmart}\n```\nJournals use one of three template styles. All but three ACM journals use the `acmsmall` template style:\n* `acmsmall`: The default journal template style.\n* `acmlarge`: Used by JOCCH and TAP.\n* `acmtog`: Used by TOG.\nThe majority of conference proceedings documentation will use the `acmconf` template style.\n* `acmconf`: The default proceedings template style.\n* `sigchi`: Used for SIGCHI conference articles.\n* `sigchi-a`: Used for SIGCHI “Extended Abstract” articles.\n* `sigplan`: Used for SIGPLAN conference articles.\nTemplate Parameters\n-------------------\nIn addition to specifying the *template style* to be used in formatting your work, there are a number of *template parameters* which modify some part of the applied template style. A complete list of these parameters can be found in the *LaTeX User’s Guide.*\nFrequently-used parameters, or combinations of parameters, include:\n* `anonymous,review`: Suitable for a “double-blind” conference submission. Anonymizes the work and includes line numbers. Use with the `\\acmSubmissionID` command to print the submission’s unique ID on each page of the work.\n* `authorversion`: Produces a version of the work suitable for posting by the author.\n* `screen`: Produces colored hyperlinks.\nThis document uses the following string as the first command in the source file:\n```\n\\documentclass[sigconf]{acmart}\n```\nModifications\n=============\nModifying the template — including but not limited to: adjusting margins, typeface sizes, line spacing, paragraph and list definitions, and the use of the `\\vspace` command to manually adjust the vertical spacing between elements of your work — is not allowed.\n**Your document will be returned to you for revision if modifications are discovered.**\nTypefaces\n=========\nThe “`acmart`” document class requires the use of the “Libertine” typeface family. Your TeX installation should include this set of packages. Please do not substitute other typefaces. The “`lmodern`” and “`ltimes`” packages should not be used, as they will override the built-in typeface families.\nTitle Information\n=================\nThe title of your work should use capital letters appropriately - <https://capitalizemytitle.com/> has useful rules for capitalization. Use the `title` command to define the title of your work. If your work has a subtitle, define it with the `subtitle` command. Do not insert line breaks in your title.\nIf your title is lengthy, you must define a short version to be used in the page headers, to prevent overlapping text. The `title` command has a “short title” parameter:\n```\n \\title[short title]{full title}\n```\nAuthors and Affiliations\n========================\nEach author must be defined separately for accurate metadata identification. Multiple authors may share one affiliation. Authors’ names should not be abbreviated; use full first names wherever possible. Include authors’ e-mail addresses whenever possible.\nGrouping authors’ names or e-mail addresses, or providing an “e-mail alias,” as shown below, is not acceptable:\n```\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n```\nThe `authornote` and `authornotemark` commands allow a note to apply to multiple authors — for example, if the first two authors of an article contributed equally to the work.\nIf your author list is lengthy, you must define a shortened version of the list of authors to be used in the page headers, to prevent overlapping text. The following command should be placed just after the last `\\author{}` definition:\n```\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n```\nOmitting this command will force the use of a concatenated list of all of the authors’ names, which may result in overlapping text in the page headers.\nThe article template’s documentation, available at <https://www.acm.org/publications/proceedings-template>, has a complete explanation of these commands and tips for their effective use.\nNote that authors’ addresses are mandatory for journal articles.\nRights Information\n==================\nAuthors of any work published by ACM will need to complete a rights form. Depending on the kind of work, and the rights management choice made by the author, this may be copyright transfer, permission, license, or an OA (open access) agreement.\nRegardless of the rights management choice, the author will receive a copy of the completed rights form once it has been submitted. This form contains LaTeX commands that must be copied into the source document. When the document source is compiled, these commands and their parameters add formatted text to several areas of the final document:\n* the “ACM Reference Format” text on the first page.\n* the “rights management” text on the first page.\n* the conference information in the page header(s).\nRights information is unique to the work; if you are preparing several works for an event, make sure to use the correct set of commands with each of the works.\nThe ACM Reference Format text is required for all articles over one page in length, and is optional for one-page articles (abstracts).\nCCS Concepts and User-Defined Keywords\n======================================\nTwo elements of the “acmart” document class provide powerful taxonomic tools for you to help readers find your work in an online search.\nThe ACM Computing Classification System — <https://www.acm.org/publications/class-2012> — is a set of classifiers and concepts that describe the computing discipline. Authors can select entries from this classification system, via <https://dl.acm.org/ccs/ccs.cfm>, and generate the commands to be included in the LaTeX source.\nUser-defined keywords are a comma-separated list of words and phrases of the authors’ choosing, providing a more flexible way of describing the research being presented.\nCCS concepts and user-defined keywords are required for for all articles over two pages in length, and are optional for one- and two-page articles (or abstracts).\nSectioning Commands\n===================\nYour work should use standard LaTeX sectioning commands: `section`, `subsection`, `subsubsection`, and `paragraph`. They should be numbered; do not remove the numbering from the commands.\nSimulating a sectioning command by setting the first word or words of a paragraph in boldface or italicized text is **not allowed.**\nTables\n======\nThe “`acmart`” document class includes the “`booktabs`” package — <https://ctan.org/pkg/booktabs> — for preparing high-quality tables.\nTable captions are placed *above* the table.\nBecause tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite. To ensure this proper “floating” placement of tables, use the environment **table** to enclose the table’s contents and the table caption. The contents of the table itself must go in the **tabular** environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules. Again, detailed instructions on **tabular** material are found in the *LaTeX User’s Guide*.\nImmediately following this sentence is the point at which Table [tab:freq] is included in the input file; compare the placement of the table here with the table in the printed output of this document.\n[tab:freq]\nccl Non-English or Math&Frequency&Comments \nØ& 1 in 1,000& For Swedish names \n*π* & 1 in 5& Common in math \n$ & 4 in 5 & Used in business \nΨ12 & 1 in 40,000& Unexplained usage \nTo set a wider table, which takes up the whole width of the page’s live area, use the environment **table\\*** to enclose the table’s contents and the table caption. As with a single-column table, this wide table will “float” to a location deemed more desirable. Immediately following this sentence is the point at which Table [tab:commands] is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document.\n[tab:commands]\nccl Command &A Number & Comments \n`’134author` & 100& Author \n`’134table`& 300 & For tables \n`’134table*`& 400& For wider tables \nAlways use midrule to separate table header rows from data rows, and use it only for this purpose. This enables assistive technologies to recognise table headers and support their users in navigating tables more easily.\nMath Equations\n==============\nYou may want to display math equations in three distinct styles: inline, numbered or non-numbered display. Each of the three are discussed in the next sections.\nInline (In-text) Equations\n--------------------------\nA formula that appears in the running text is called an inline or in-text formula. It is produced by the **math** environment, which can be invoked with the usual `’134begin…’134end` construction or with the short form `$…$`. You can use any of the symbols and structures, from *α* to *ω*, available in LaTeX ; this section will simply show a few examples of in-text equations in context. Notice how this equation:\nnx=0\n, set here in in-line math style, looks slightly different when set in display style. (See next section).\nDisplay Equations\n-----------------\nA numbered display equation—one set off by vertical space from the text and centered horizontally—is produced by the **equation** environment. An unnumbered display equation is produced by the **displaymath** environment.\nAgain, in either environment, you can use any of the symbols and structures available in LaTeX; this section will just give a couple of examples of display equations in context. First, consider the equation, shown as an inline equation above:\n \nlim*n* → ∞*x* = 0 \nNotice how it is formatted somewhat differently in the **displaymath** environment. Now, we’ll enter an unnumbered equation:\n \n∑*i* = 0∞*x* + 1 \nand follow it with another numbered equation:\n \n∑*i* = 0∞*x**i* = ∫0*π* + 2*f* \njust to demonstrate LaTeX’s able handling of numbering.\nFigures\n=======\nThe “`figure`” environment should be used for figures. One or more images can be placed within a figure. If your figure contains third-party material, you must clearly identify it as such, as shown in the example below.\n![1907 Franklin Model D roadster. Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons. (https://goo.gl/VLCRBB).](sample-franklin)1907 Franklin Model D roadster. Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons. (<https://goo.gl/VLCRBB>).\nYour figures should contain a caption which describes the figure to the reader.\nFigure captions are placed *below* the figure.\nEvery figure should also have a figure description unless it is purely decorative. These descriptions convey what’s in the image to someone who cannot see it. They are also used by search engine crawlers for indexing images, and when images cannot be loaded.\nA figure description must be unformatted plain text less than 2000 characters long (including spaces). **Figure descriptions should not repeat the figure caption – their purpose is to capture important information that is not already provided in the caption or the main text of the paper.** For figures that convey important and complex new information, a short text description may not be adequate. More complex alternative descriptions can be placed in an appendix and referenced in a short figure description. For example, provide a data table capturing the information in a bar chart, or a structured list representing a graph. For additional information regarding how best to write figure descriptions and why doing this is so important, please see <https://www.acm.org/publications/taps/describing-figures/>.\nThe “Teaser Figure”\n-------------------\nA “teaser figure” is an image, or set of images in one figure, that are placed after all author and affiliation information, and before the body of the article, spanning the page. If you wish to have such a figure in your article, place the command immediately before the `\\maketitle` command:\n```\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n```\nCitations and Bibliographies\n============================\nThe use of for the preparation and formatting of one’s references is strongly recommended. Authors’ names should be complete — use full first names (“Donald E. Knuth”) not initials (“D. E. Knuth”) — and the salient identifying features of a reference should be included: title, year, volume, number, pages, article DOI, etc.\nThe bibliography is included in your source document with these two commands, placed just before the `\\end{document}` command:\n```\n \\bibliographystyle{ACM-Reference-Format}\n \\bibliography{bibfile}\n```\nwhere “`bibfile`” is the name, without the “`.bib`” suffix, of the file.\nCitations and references are numbered by default. A small number of ACM publications have citations and references formatted in the “author year” style; for these exceptions, please include this command in the **preamble** (before the command “`\\begin{document}`”) of your LaTeX source:\n```\n \\citestyle{acmauthoryear}\n```\nSome examples. A paginated journal article, an enumerated journal article, a reference to an entire issue, a monograph (whole book), a monograph/whole book in a series (see 2a in spec. document), a divisible-book such as an anthology or compilation followed by the same example, however we only output the series if the volume number is given (so Editor00a’s series should NOT be present since it has no vol. no.), a chapter in a divisible book, a chapter in a divisible book in a series, a multi-volume work as book, a couple of articles in a proceedings (of a conference, symposium, workshop for example) (paginated proceedings article), a proceedings article with all possible elements, an example of an enumerated proceedings article, an informally published work, a couple of preprints, a doctoral dissertation, a master’s thesis:, an online document / world wide web resource, a video game (Case 1) and (Case 2) and and (Case 3) a patent, work accepted for publication, ’YYYYb’-test for prolific author and. Other cites might contain ’duplicate’ DOI and URLs (some SIAM articles). Boris / Barbara Beeton: multi-volume works as books and. A couple of citations with DOIs:. Online citations:. Artifacts: and.\nAcknowledgments\n===============\nIdentification of funding sources and other support, and thanks to individuals and groups that assisted in the research and the preparation of the work should be included in an acknowledgment section, which is placed just before the reference section in your document.\nThis section has a special environment:\n```\n \\begin{acks}\n ...\n \\end{acks}\n```\nso that the information contained therein can be more easily collected during the article metadata extraction phase, and to ensure consistency in the spelling of the section heading.\nAuthors should not prepare this section as a numbered or unnumbered `\\section`; please use the “`acks`” environment.\nAppendices\n==========\nIf your work needs an appendix, add it before the “`\\end{document}`” command at the conclusion of your source document.\nStart the appendix with the “`appendix`” command:\n```\n \\appendix\n```\nand note that in the appendix, sections are lettered, not numbered. This document has two appendices, demonstrating the section and subsection identification method.\nMulti-language papers\n=====================\nPapers may be written in languages other than English or include titles, subtitles, keywords and abstracts in different languages (as a rule, a paper in a language other than English should include an English title and an English abstract). Use `language=...` for every language used in the paper. The last language indicated is the main language of the paper. For example, a French paper with additional titles and abstracts in English and German may start with the following command\n```\n\\documentclass[sigconf, language=english, language=german,\n language=french]{acmart}\n```\nThe title, subtitle, keywords and abstract will be typeset in the main language of the paper. The commands `\\translatedXXX`, `XXX` begin title, subtitle and keywords, can be used to set these elements in the other languages. The environment `translatedabstract` is used to set the translation of the abstract. These commands and environment have a mandatory first argument: the language of the second argument. See `sample-sigconf-i13n.tex` file for examples of their usage.\nSIGCHI Extended Abstracts\n=========================\nThe “`sigchi-a`” template style (available only in LaTeX and not in Word) produces a landscape-orientation formatted article, with a wide left margin. Three environments are available for use with the “`sigchi-a`” template style, and produce formatted output in the margin:\n* `sidebar`: Place formatted text in the margin.\n* `marginfigure`: Place a figure in the margin.\n* `margintable`: Place a table in the margin.\nTo Robert, for the bagels and explaining CMYK and color spaces.\nResearch Methods\n================\nPart One\n--------\nLorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi malesuada, quam in pulvinar varius, metus nunc fermentum urna, id sollicitudin purus odio sit amet enim. Aliquam ullamcorper eu ipsum vel mollis. Curabitur quis dictum nisl. Phasellus vel semper risus, et lacinia dolor. Integer ultricies commodo sem nec semper.\nPart Two\n--------\nEtiam commodo feugiat nisl pulvinar pellentesque. Etiam auctor sodales ligula, non varius nibh pulvinar semper. Suspendisse nec lectus non ipsum convallis congue hendrerit vitae sapien. Donec at laoreet eros. Vivamus non purus placerat, scelerisque diam eu, cursus ante. Etiam aliquam tortor auctor efficitur mattis.\nOnline Resources\n================\nNam id fermentum dui. Suspendisse sagittis tortor a nulla mollis, in pulvinar ex pretium. Sed interdum orci quis metus euismod, et sagittis enim maximus. Vestibulum gravida massa ut felis suscipit congue. Quisque mattis elit a risus ultrices commodo venenatis eget dui. Etiam sagittis eleifend elementum.\nNam interdum magna at lectus dignissim, ac dignissim lorem rhoncus. Maecenas eu arcu ac neque placerat aliquam. Nunc pulvinar massa et mattis lacinia.\n\nThe Name of the Title is Hope\n=============================\nA clear and well-documented LaTeX document is presented as an article formatted for publication by ACM in a conference proceedings or journal publication. Based on the “acmart” document class, this article presents and explains many of the common variations, as well as many of the formatting elements an author may use in the preparation of the documentation of their work.\n<ccs2012> <concept> <conceptid>10010520.10010553.10010562</conceptid> <conceptdesc>Computer systems organization Embedded systems</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10010520.10010575.10010755</conceptid> <conceptdesc>Computer systems organization Redundancy</conceptdesc> <conceptsignificance>300</conceptsignificance> </concept> <concept> <conceptid>10010520.10010553.10010554</conceptid> <conceptdesc>Computer systems organization Robotics</conceptdesc> <conceptsignificance>100</conceptsignificance> </concept> <concept> <conceptid>10003033.10003083.10003095</conceptid> <conceptdesc>Networks Network reliability</conceptdesc> <conceptsignificance>100</conceptsignificance> </concept> </ccs2012>\n![Seattle Mariners at Spring Training, 2010.](sampleteaser \"fig:\") [fig:teaser]\nIntroduction\n============\nACM’s consolidated article template, introduced in 2017, provides a consistent LaTeX style for use across ACM publications, and incorporates accessibility and metadata-extraction functionality necessary for future Digital Library endeavors. Numerous ACM and SIG-specific LaTeX templates have been examined, and their unique features incorporated into this single new template.\nIf you are new to publishing with ACM, this document is a valuable guide to the process of preparing your work for publication. If you have published with ACM before, this document provides insight and instruction into more recent changes to the article template.\nThe “`acmart`” document class can be used to prepare articles for any ACM publication — conference or journal, and for any stage of publication, from review to final “camera-ready” copy, to the author’s own version, with *very* few changes to the source.\nTemplate Overview\n=================\nAs noted in the introduction, the “`acmart`” document class can be used to prepare many different kinds of documentation — a double-blind initial submission of a full-length technical paper, a two-page SIGGRAPH Emerging Technologies abstract, a “camera-ready” journal article, a SIGCHI Extended Abstract, and more — all by selecting the appropriate *template style* and *template parameters*.\nThis document will explain the major features of the document class. For further information, the *LaTeX User’s Guide* is available from <https://www.acm.org/publications/proceedings-template>.\nTemplate Styles\n---------------\nThe primary parameter given to the “`acmart`” document class is the *template style* which corresponds to the kind of publication or SIG publishing the work. This parameter is enclosed in square brackets and is a part of the `documentclass` command:\n```\n \\documentclass[STYLE]{acmart}\n```\nJournals use one of three template styles. All but three ACM journals use the `acmsmall` template style:\n* `acmsmall`: The default journal template style.\n* `acmlarge`: Used by JOCCH and TAP.\n* `acmtog`: Used by TOG.\nThe majority of conference proceedings documentation will use the `acmconf` template style.\n* `acmconf`: The default proceedings template style.\n* `sigchi`: Used for SIGCHI conference articles.\n* `sigchi-a`: Used for SIGCHI “Extended Abstract” articles.\n* `sigplan`: Used for SIGPLAN conference articles.\nTemplate Parameters\n-------------------\nIn addition to specifying the *template style* to be used in formatting your work, there are a number of *template parameters* which modify some part of the applied template style. A complete list of these parameters can be found in the *LaTeX User’s Guide.*\nFrequently-used parameters, or combinations of parameters, include:\n* `anonymous,review`: Suitable for a “double-blind” conference submission. Anonymizes the work and includes line numbers. Use with the `\\acmSubmissionID` command to print the submission’s unique ID on each page of the work.\n* `authorversion`: Produces a version of the work suitable for posting by the author.\n* `screen`: Produces colored hyperlinks.\nThis document uses the following string as the first command in the source file:\n```\n\\documentclass[sigconf]{acmart}\n```\nModifications\n=============\nModifying the template — including but not limited to: adjusting margins, typeface sizes, line spacing, paragraph and list definitions, and the use of the `\\vspace` command to manually adjust the vertical spacing between elements of your work — is not allowed.\n**Your document will be returned to you for revision if modifications are discovered.**\nTypefaces\n=========\nThe “`acmart`” document class requires the use of the “Libertine” typeface family. Your TeX installation should include this set of packages. Please do not substitute other typefaces. The “`lmodern`” and “`ltimes`” packages should not be used, as they will override the built-in typeface families.\nTitle Information\n=================\nThe title of your work should use capital letters appropriately - <https://capitalizemytitle.com/> has useful rules for capitalization. Use the `title` command to define the title of your work. If your work has a subtitle, define it with the `subtitle` command. Do not insert line breaks in your title.\nIf your title is lengthy, you must define a short version to be used in the page headers, to prevent overlapping text. The `title` command has a “short title” parameter:\n```\n \\title[short title]{full title}\n```\nAuthors and Affiliations\n========================\nEach author must be defined separately for accurate metadata identification. Multiple authors may share one affiliation. Authors’ names should not be abbreviated; use full first names wherever possible. Include authors’ e-mail addresses whenever possible.\nGrouping authors’ names or e-mail addresses, or providing an “e-mail alias,” as shown below, is not acceptable:\n```\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n```\nThe `authornote` and `authornotemark` commands allow a note to apply to multiple authors — for example, if the first two authors of an article contributed equally to the work.\nIf your author list is lengthy, you must define a shortened version of the list of authors to be used in the page headers, to prevent overlapping text. The following command should be placed just after the last `\\author{}` definition:\n```\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n```\nOmitting this command will force the use of a concatenated list of all of the authors’ names, which may result in overlapping text in the page headers.\nThe article template’s documentation, available at <https://www.acm.org/publications/proceedings-template>, has a complete explanation of these commands and tips for their effective use.\nNote that authors’ addresses are mandatory for journal articles.\nRights Information\n==================\nAuthors of any work published by ACM will need to complete a rights form. Depending on the kind of work, and the rights management choice made by the author, this may be copyright transfer, permission, license, or an OA (open access) agreement.\nRegardless of the rights management choice, the author will receive a copy of the completed rights form once it has been submitted. This form contains LaTeX commands that must be copied into the source document. When the document source is compiled, these commands and their parameters add formatted text to several areas of the final document:\n* the “ACM Reference Format” text on the first page.\n* the “rights management” text on the first page.\n* the conference information in the page header(s).\nRights information is unique to the work; if you are preparing several works for an event, make sure to use the correct set of commands with each of the works.\nThe ACM Reference Format text is required for all articles over one page in length, and is optional for one-page articles (abstracts).\nCCS Concepts and User-Defined Keywords\n======================================\nTwo elements of the “acmart” document class provide powerful taxonomic tools for you to help readers find your work in an online search.\nThe ACM Computing Classification System — <https://www.acm.org/publications/class-2012> — is a set of classifiers and concepts that describe the computing discipline. Authors can select entries from this classification system, via <https://dl.acm.org/ccs/ccs.cfm>, and generate the commands to be included in the LaTeX source.\nUser-defined keywords are a comma-separated list of words and phrases of the authors’ choosing, providing a more flexible way of describing the research being presented.\nCCS concepts and user-defined keywords are required for for all articles over two pages in length, and are optional for one- and two-page articles (or abstracts).\nSectioning Commands\n===================\nYour work should use standard LaTeX sectioning commands: `section`, `subsection`, `subsubsection`, and `paragraph`. They should be numbered; do not remove the numbering from the commands.\nSimulating a sectioning command by setting the first word or words of a paragraph in boldface or italicized text is **not allowed.**\nTables\n======\nThe “`acmart`” document class includes the “`booktabs`” package — <https://ctan.org/pkg/booktabs> — for preparing high-quality tables.\nTable captions are placed *above* the table.\nBecause tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite. To ensure this proper “floating” placement of tables, use the environment **table** to enclose the table’s contents and the table caption. The contents of the table itself must go in the **tabular** environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules. Again, detailed instructions on **tabular** material are found in the *LaTeX User’s Guide*.\nImmediately following this sentence is the point at which Table [tab:freq] is included in the input file; compare the placement of the table here with the table in the printed output of this document.\n[tab:freq]\nccl Non-English or Math&Frequency&Comments \nØ& 1 in 1,000& For Swedish names \n*π* & 1 in 5& Common in math \n$ & 4 in 5 & Used in business \nΨ12 & 1 in 40,000& Unexplained usage \nTo set a wider table, which takes up the whole width of the page’s live area, use the environment **table\\*** to enclose the table’s contents and the table caption. As with a single-column table, this wide table will “float” to a location deemed more desirable. Immediately following this sentence is the point at which Table [tab:commands] is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document.\n[tab:commands]\nccl Command &A Number & Comments \n`’134author` & 100& Author \n`’134table`& 300 & For tables \n`’134table*`& 400& For wider tables \nAlways use midrule to separate table header rows from data rows, and use it only for this purpose. This enables assistive technologies to recognise table headers and support their users in navigating tables more easily.\nMath Equations\n==============\nYou may want to display math equations in three distinct styles: inline, numbered or non-numbered display. Each of the three are discussed in the next sections.\nInline (In-text) Equations\n--------------------------\nA formula that appears in the running text is called an inline or in-text formula. It is produced by the **math** environment, which can be invoked with the usual `’134begin…’134end` construction or with the short form `$…$`. You can use any of the symbols and structures, from *α* to *ω*, available in LaTeX ; this section will simply show a few examples of in-text equations in context. Notice how this equation:\nnx=0\n, set here in in-line math style, looks slightly different when set in display style. (See next section).\nDisplay Equations\n-----------------\nA numbered display equation—one set off by vertical space from the text and centered horizontally—is produced by the **equation** environment. An unnumbered display equation is produced by the **displaymath** environment.\nAgain, in either environment, you can use any of the symbols and structures available in LaTeX; this section will just give a couple of examples of display equations in context. First, consider the equation, shown as an inline equation above:\n \nlim*n* → ∞*x* = 0 \nNotice how it is formatted somewhat differently in the **displaymath** environment. Now, we’ll enter an unnumbered equation:\n \n∑*i* = 0∞*x* + 1 \nand follow it with another numbered equation:\n \n∑*i* = 0∞*x**i* = ∫0*π* + 2*f* \njust to demonstrate LaTeX’s able handling of numbering.\nFigures\n=======\nThe “`figure`” environment should be used for figures. One or more images can be placed within a figure. If your figure contains third-party material, you must clearly identify it as such, as shown in the example below.\n![1907 Franklin Model D roadster. Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons. (https://goo.gl/VLCRBB).](sample-franklin)1907 Franklin Model D roadster. Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons. (<https://goo.gl/VLCRBB>).\nYour figures should contain a caption which describes the figure to the reader.\nFigure captions are placed *below* the figure.\nEvery figure should also have a figure description unless it is purely decorative. These descriptions convey what’s in the image to someone who cannot see it. They are also used by search engine crawlers for indexing images, and when images cannot be loaded.\nA figure description must be unformatted plain text less than 2000 characters long (including spaces). **Figure descriptions should not repeat the figure caption – their purpose is to capture important information that is not already provided in the caption or the main text of the paper.** For figures that convey important and complex new information, a short text description may not be adequate. More complex alternative descriptions can be placed in an appendix and referenced in a short figure description. For example, provide a data table capturing the information in a bar chart, or a structured list representing a graph. For additional information regarding how best to write figure descriptions and why doing this is so important, please see <https://www.acm.org/publications/taps/describing-figures/>.\nThe “Teaser Figure”\n-------------------\nA “teaser figure” is an image, or set of images in one figure, that are placed after all author and affiliation information, and before the body of the article, spanning the page. If you wish to have such a figure in your article, place the command immediately before the `\\maketitle` command:\n```\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n```\nCitations and Bibliographies\n============================\nThe use of for the preparation and formatting of one’s references is strongly recommended. Authors’ names should be complete — use full first names (“Donald E. Knuth”) not initials (“D. E. Knuth”) — and the salient identifying features of a reference should be included: title, year, volume, number, pages, article DOI, etc.\nThe bibliography is included in your source document with these two commands, placed just before the `\\end{document}` command:\n```\n \\bibliographystyle{ACM-Reference-Format}\n \\bibliography{bibfile}\n```\nwhere “`bibfile`” is the name, without the “`.bib`” suffix, of the file.\nCitations and references are numbered by default. A small number of ACM publications have citations and references formatted in the “author year” style; for these exceptions, please include this command in the **preamble** (before the command “`\\begin{document}`”) of your LaTeX source:\n```\n \\citestyle{acmauthoryear}\n```\nSome examples. A paginated journal article, an enumerated journal article, a reference to an entire issue, a monograph (whole book), a monograph/whole book in a series (see 2a in spec. document), a divisible-book such as an anthology or compilation followed by the same example, however we only output the series if the volume number is given (so Editor00a’s series should NOT be present since it has no vol. no.), a chapter in a divisible book, a chapter in a divisible book in a series, a multi-volume work as book, a couple of articles in a proceedings (of a conference, symposium, workshop for example) (paginated proceedings article), a proceedings article with all possible elements, an example of an enumerated proceedings article, an informally published work, a couple of preprints, a doctoral dissertation, a master’s thesis:, an online document / world wide web resource, a video game (Case 1) and (Case 2) and and (Case 3) a patent, work accepted for publication, ’YYYYb’-test for prolific author and. Other cites might contain ’duplicate’ DOI and URLs (some SIAM articles). Boris / Barbara Beeton: multi-volume works as books and. A couple of citations with DOIs:. Online citations:. Artifacts: and.\nAcknowledgments\n===============\nIdentification of funding sources and other support, and thanks to individuals and groups that assisted in the research and the preparation of the work should be included in an acknowledgment section, which is placed just before the reference section in your document.\nThis section has a special environment:\n```\n \\begin{acks}\n ...\n \\end{acks}\n```\nso that the information contained therein can be more easily collected during the article metadata extraction phase, and to ensure consistency in the spelling of the section heading.\nAuthors should not prepare this section as a numbered or unnumbered `\\section`; please use the “`acks`” environment.\nAppendices\n==========\nIf your work needs an appendix, add it before the “`\\end{document}`” command at the conclusion of your source document.\nStart the appendix with the “`appendix`” command:\n```\n \\appendix\n```\nand note that in the appendix, sections are lettered, not numbered. This document has two appendices, demonstrating the section and subsection identification method.\nMulti-language papers\n=====================\nPapers may be written in languages other than English or include titles, subtitles, keywords and abstracts in different languages (as a rule, a paper in a language other than English should include an English title and an English abstract). Use `language=...` for every language used in the paper. The last language indicated is the main language of the paper. For example, a French paper with additional titles and abstracts in English and German may start with the following command\n```\n\\documentclass[sigconf, language=english, language=german,\n language=french]{acmart}\n```\nThe title, subtitle, keywords and abstract will be typeset in the main language of the paper. The commands `\\translatedXXX`, `XXX` begin title, subtitle and keywords, can be used to set these elements in the other languages. The environment `translatedabstract` is used to set the translation of the abstract. These commands and environment have a mandatory first argument: the language of the second argument. See `sample-sigconf-i13n.tex` file for examples of their usage.\nSIGCHI Extended Abstracts\n=========================\nThe “`sigchi-a`” template style (available only in LaTeX and not in Word) produces a landscape-orientation formatted article, with a wide left margin. Three environments are available for use with the “`sigchi-a`” template style, and produce formatted output in the margin:\n* `sidebar`: Place formatted text in the margin.\n* `marginfigure`: Place a figure in the margin.\n* `margintable`: Place a table in the margin.\nTo Robert, for the bagels and explaining CMYK and color spaces.\nResearch Methods\n================\nPart One\n--------\nLorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi malesuada, quam in pulvinar varius, metus nunc fermentum urna, id sollicitudin purus odio sit amet enim. Aliquam ullamcorper eu ipsum vel mollis. Curabitur quis dictum nisl. Phasellus vel semper risus, et lacinia dolor. Integer ultricies commodo sem nec semper.\nPart Two\n--------\nEtiam commodo feugiat nisl pulvinar pellentesque. Etiam auctor sodales ligula, non varius nibh pulvinar semper. Suspendisse nec lectus non ipsum convallis congue hendrerit vitae sapien. Donec at laoreet eros. Vivamus non purus placerat, scelerisque diam eu, cursus ante. Etiam aliquam tortor auctor efficitur mattis.\nOnline Resources\n================\nNam id fermentum dui. Suspendisse sagittis tortor a nulla mollis, in pulvinar ex pretium. Sed interdum orci quis metus euismod, et sagittis enim maximus. Vestibulum gravida massa ut felis suscipit congue. Quisque mattis elit a risus ultrices commodo venenatis eget dui. Etiam sagittis eleifend elementum.\nNam interdum magna at lectus dignissim, ac dignissim lorem rhoncus. Maecenas eu arcu ac neque placerat aliquam. Nunc pulvinar massa et mattis lacinia.\n\nThe Name of the Title is Hope\n=============================\nA clear and well-documented LaTeX document is presented as an article formatted for publication by ACM in a conference proceedings or journal publication. Based on the “acmart” document class, this article presents and explains many of the common variations, as well as many of the formatting elements an author may use in the preparation of the documentation of their work.\n<ccs2012> <concept> <conceptid>10010520.10010553.10010562</conceptid> <conceptdesc>Computer systems organization Embedded systems</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10010520.10010575.10010755</conceptid> <conceptdesc>Computer systems organization Redundancy</conceptdesc> <conceptsignificance>300</conceptsignificance> </concept> <concept> <conceptid>10010520.10010553.10010554</conceptid> <conceptdesc>Computer systems organization Robotics</conceptdesc> <conceptsignificance>100</conceptsignificance> </concept> <concept> <conceptid>10003033.10003083.10003095</conceptid> <conceptdesc>Networks Network reliability</conceptdesc> <conceptsignificance>100</conceptsignificance> </concept> </ccs2012>\n![Seattle Mariners at Spring Training, 2010.](sampleteaser \"fig:\") [fig:teaser]\nIntroduction\n============\nACM’s consolidated article template, introduced in 2017, provides a consistent LaTeX style for use across ACM publications, and incorporates accessibility and metadata-extraction functionality necessary for future Digital Library endeavors. Numerous ACM and SIG-specific LaTeX templates have been examined, and their unique features incorporated into this single new template.\nIf you are new to publishing with ACM, this document is a valuable guide to the process of preparing your work for publication. If you have published with ACM before, this document provides insight and instruction into more recent changes to the article template.\nThe “`acmart`” document class can be used to prepare articles for any ACM publication — conference or journal, and for any stage of publication, from review to final “camera-ready” copy, to the author’s own version, with *very* few changes to the source.\nTemplate Overview\n=================\nAs noted in the introduction, the “`acmart`” document class can be used to prepare many different kinds of documentation — a double-blind initial submission of a full-length technical paper, a two-page SIGGRAPH Emerging Technologies abstract, a “camera-ready” journal article, a SIGCHI Extended Abstract, and more — all by selecting the appropriate *template style* and *template parameters*.\nThis document will explain the major features of the document class. For further information, the *LaTeX User’s Guide* is available from <https://www.acm.org/publications/proceedings-template>.\nTemplate Styles\n---------------\nThe primary parameter given to the “`acmart`” document class is the *template style* which corresponds to the kind of publication or SIG publishing the work. This parameter is enclosed in square brackets and is a part of the `documentclass` command:\n```\n \\documentclass[STYLE]{acmart}\n```\nJournals use one of three template styles. All but three ACM journals use the `acmsmall` template style:\n* `acmsmall`: The default journal template style.\n* `acmlarge`: Used by JOCCH and TAP.\n* `acmtog`: Used by TOG.\nThe majority of conference proceedings documentation will use the `acmconf` template style.\n* `acmconf`: The default proceedings template style.\n* `sigchi`: Used for SIGCHI conference articles.\n* `sigchi-a`: Used for SIGCHI “Extended Abstract” articles.\n* `sigplan`: Used for SIGPLAN conference articles.\nTemplate Parameters\n-------------------\nIn addition to specifying the *template style* to be used in formatting your work, there are a number of *template parameters* which modify some part of the applied template style. A complete list of these parameters can be found in the *LaTeX User’s Guide.*\nFrequently-used parameters, or combinations of parameters, include:\n* `anonymous,review`: Suitable for a “double-blind” conference submission. Anonymizes the work and includes line numbers. Use with the `\\acmSubmissionID` command to print the submission’s unique ID on each page of the work.\n* `authorversion`: Produces a version of the work suitable for posting by the author.\n* `screen`: Produces colored hyperlinks.\nThis document uses the following string as the first command in the source file:\n```\n\\documentclass[sigconf]{acmart}\n```\nModifications\n=============\nModifying the template — including but not limited to: adjusting margins, typeface sizes, line spacing, paragraph and list definitions, and the use of the `\\vspace` command to manually adjust the vertical spacing between elements of your work — is not allowed.\n**Your document will be returned to you for revision if modifications are discovered.**\nTypefaces\n=========\nThe “`acmart`” document class requires the use of the “Libertine” typeface family. Your TeX installation should include this set of packages. Please do not substitute other typefaces. The “`lmodern`” and “`ltimes`” packages should not be used, as they will override the built-in typeface families.\nTitle Information\n=================\nThe title of your work should use capital letters appropriately - <https://capitalizemytitle.com/> has useful rules for capitalization. Use the `title` command to define the title of your work. If your work has a subtitle, define it with the `subtitle` command. Do not insert line breaks in your title.\nIf your title is lengthy, you must define a short version to be used in the page headers, to prevent overlapping text. The `title` command has a “short title” parameter:\n```\n \\title[short title]{full title}\n```\nAuthors and Affiliations\n========================\nEach author must be defined separately for accurate metadata identification. Multiple authors may share one affiliation. Authors’ names should not be abbreviated; use full first names wherever possible. Include authors’ e-mail addresses whenever possible.\nGrouping authors’ names or e-mail addresses, or providing an “e-mail alias,” as shown below, is not acceptable:\n```\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n```\nThe `authornote` and `authornotemark` commands allow a note to apply to multiple authors — for example, if the first two authors of an article contributed equally to the work.\nIf your author list is lengthy, you must define a shortened version of the list of authors to be used in the page headers, to prevent overlapping text. The following command should be placed just after the last `\\author{}` definition:\n```\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n```\nOmitting this command will force the use of a concatenated list of all of the authors’ names, which may result in overlapping text in the page headers.\nThe article template’s documentation, available at <https://www.acm.org/publications/proceedings-template>, has a complete explanation of these commands and tips for their effective use.\nNote that authors’ addresses are mandatory for journal articles.\nRights Information\n==================\nAuthors of any work published by ACM will need to complete a rights form. Depending on the kind of work, and the rights management choice made by the author, this may be copyright transfer, permission, license, or an OA (open access) agreement.\nRegardless of the rights management choice, the author will receive a copy of the completed rights form once it has been submitted. This form contains LaTeX commands that must be copied into the source document. When the document source is compiled, these commands and their parameters add formatted text to several areas of the final document:\n* the “ACM Reference Format” text on the first page.\n* the “rights management” text on the first page.\n* the conference information in the page header(s).\nRights information is unique to the work; if you are preparing several works for an event, make sure to use the correct set of commands with each of the works.\nThe ACM Reference Format text is required for all articles over one page in length, and is optional for one-page articles (abstracts).\nCCS Concepts and User-Defined Keywords\n======================================\nTwo elements of the “acmart” document class provide powerful taxonomic tools for you to help readers find your work in an online search.\nThe ACM Computing Classification System — <https://www.acm.org/publications/class-2012> — is a set of classifiers and concepts that describe the computing discipline. Authors can select entries from this classification system, via <https://dl.acm.org/ccs/ccs.cfm>, and generate the commands to be included in the LaTeX source.\nUser-defined keywords are a comma-separated list of words and phrases of the authors’ choosing, providing a more flexible way of describing the research being presented.\nCCS concepts and user-defined keywords are required for for all articles over two pages in length, and are optional for one- and two-page articles (or abstracts).\nSectioning Commands\n===================\nYour work should use standard LaTeX sectioning commands: `section`, `subsection`, `subsubsection`, and `paragraph`. They should be numbered; do not remove the numbering from the commands.\nSimulating a sectioning command by setting the first word or words of a paragraph in boldface or italicized text is **not allowed.**\nTables\n======\nThe “`acmart`” document class includes the “`booktabs`” package — <https://ctan.org/pkg/booktabs> — for preparing high-quality tables.\nTable captions are placed *above* the table.\nBecause tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite. To ensure this proper “floating” placement of tables, use the environment **table** to enclose the table’s contents and the table caption. The contents of the table itself must go in the **tabular** environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules. Again, detailed instructions on **tabular** material are found in the *LaTeX User’s Guide*.\nImmediately following this sentence is the point at which Table [tab:freq] is included in the input file; compare the placement of the table here with the table in the printed output of this document.\n[tab:freq]\nccl Non-English or Math&Frequency&Comments \nØ& 1 in 1,000& For Swedish names \n*π* & 1 in 5& Common in math \n$ & 4 in 5 & Used in business \nΨ12 & 1 in 40,000& Unexplained usage \nTo set a wider table, which takes up the whole width of the page’s live area, use the environment **table\\*** to enclose the table’s contents and the table caption. As with a single-column table, this wide table will “float” to a location deemed more desirable. Immediately following this sentence is the point at which Table [tab:commands] is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document.\n[tab:commands]\nccl Command &A Number & Comments \n`’134author` & 100& Author \n`’134table`& 300 & For tables \n`’134table*`& 400& For wider tables \nAlways use midrule to separate table header rows from data rows, and use it only for this purpose. This enables assistive technologies to recognise table headers and support their users in navigating tables more easily.\nMath Equations\n==============\nYou may want to display math equations in three distinct styles: inline, numbered or non-numbered display. Each of the three are discussed in the next sections.\nInline (In-text) Equations\n--------------------------\nA formula that appears in the running text is called an inline or in-text formula. It is produced by the **math** environment, which can be invoked with the usual `’134begin…’134end` construction or with the short form `$…$`. You can use any of the symbols and structures, from *α* to *ω*, available in LaTeX ; this section will simply show a few examples of in-text equations in context. Notice how this equation:\nnx=0\n, set here in in-line math style, looks slightly different when set in display style. (See next section).\nDisplay Equations\n-----------------\nA numbered display equation—one set off by vertical space from the text and centered horizontally—is produced by the **equation** environment. An unnumbered display equation is produced by the **displaymath** environment.\nAgain, in either environment, you can use any of the symbols and structures available in LaTeX; this section will just give a couple of examples of display equations in context. First, consider the equation, shown as an inline equation above:\n \nlim*n* → ∞*x* = 0 \nNotice how it is formatted somewhat differently in the **displaymath** environment. Now, we’ll enter an unnumbered equation:\n \n∑*i* = 0∞*x* + 1 \nand follow it with another numbered equation:\n \n∑*i* = 0∞*x**i* = ∫0*π* + 2*f* \njust to demonstrate LaTeX’s able handling of numbering.\nFigures\n=======\nThe “`figure`” environment should be used for figures. One or more images can be placed within a figure. If your figure contains third-party material, you must clearly identify it as such, as shown in the example below.\n![1907 Franklin Model D roadster. Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons. (https://goo.gl/VLCRBB).](sample-franklin)1907 Franklin Model D roadster. Photograph by Harris & Ewing, Inc. [Public domain], via Wikimedia Commons. (<https://goo.gl/VLCRBB>).\nYour figures should contain a caption which describes the figure to the reader.\nFigure captions are placed *below* the figure.\nEvery figure should also have a figure description unless it is purely decorative. These descriptions convey what’s in the image to someone who cannot see it. They are also used by search engine crawlers for indexing images, and when images cannot be loaded.\nA figure description must be unformatted plain text less than 2000 characters long (including spaces). **Figure descriptions should not repeat the figure caption – their purpose is to capture important information that is not already provided in the caption or the main text of the paper.** For figures that convey important and complex new information, a short text description may not be adequate. More complex alternative descriptions can be placed in an appendix and referenced in a short figure description. For example, provide a data table capturing the information in a bar chart, or a structured list representing a graph. For additional information regarding how best to write figure descriptions and why doing this is so important, please see <https://www.acm.org/publications/taps/describing-figures/>.\nThe “Teaser Figure”\n-------------------\nA “teaser figure” is an image, or set of images in one figure, that are placed after all author and affiliation information, and before the body of the article, spanning the page. If you wish to have such a figure in your article, place the command immediately before the `\\maketitle` command:\n```\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n```\nCitations and Bibliographies\n============================\nThe use of for the preparation and formatting of one’s references is strongly recommended. Authors’ names should be complete — use full first names (“Donald E. Knuth”) not initials (“D. E. Knuth”) — and the salient identifying features of a reference should be included: title, year, volume, number, pages, article DOI, etc.\nThe bibliography is included in your source document with these two commands, placed just before the `\\end{document}` command:\n```\n \\bibliographystyle{ACM-Reference-Format}\n \\bibliography{bibfile}\n```\nwhere “`bibfile`” is the name, without the “`.bib`” suffix, of the file.\nCitations and references are numbered by default. A small number of ACM publications have citations and references formatted in the “author year” style; for these exceptions, please include this command in the **preamble** (before the command “`\\begin{document}`”) of your LaTeX source:\n```\n \\citestyle{acmauthoryear}\n```\nSome examples. A paginated journal article, an enumerated journal article, a reference to an entire issue, a monograph (whole book), a monograph/whole book in a series (see 2a in spec. document), a divisible-book such as an anthology or compilation followed by the same example, however we only output the series if the volume number is given (so Editor00a’s series should NOT be present since it has no vol. no.), a chapter in a divisible book, a chapter in a divisible book in a series, a multi-volume work as book, a couple of articles in a proceedings (of a conference, symposium, workshop for example) (paginated proceedings article), a proceedings article with all possible elements, an example of an enumerated proceedings article, an informally published work, a couple of preprints, a doctoral dissertation, a master’s thesis:, an online document / world wide web resource, a video game (Case 1) and (Case 2) and and (Case 3) a patent, work accepted for publication, ’YYYYb’-test for prolific author and. Other cites might contain ’duplicate’ DOI and URLs (some SIAM articles). Boris / Barbara Beeton: multi-volume works as books and. A couple of citations with DOIs:. Online citations:. Artifacts: and.\nAcknowledgments\n===============\nIdentification of funding sources and other support, and thanks to individuals and groups that assisted in the research and the preparation of the work should be included in an acknowledgment section, which is placed just before the reference section in your document.\nThis section has a special environment:\n```\n \\begin{acks}\n ...\n \\end{acks}\n```\nso that the information contained therein can be more easily collected during the article metadata extraction phase, and to ensure consistency in the spelling of the section heading.\nAuthors should not prepare this section as a numbered or unnumbered `\\section`; please use the “`acks`” environment.\nAppendices\n==========\nIf your work needs an appendix, add it before the “`\\end{document}`” command at the conclusion of your source document.\nStart the appendix with the “`appendix`” command:\n```\n \\appendix\n```\nand note that in the appendix, sections are lettered, not numbered. This document has two appendices, demonstrating the section and subsection identification method.\nMulti-language papers\n=====================\nPapers may be written in languages other than English or include titles, subtitles, keywords and abstracts in different languages (as a rule, a paper in a language other than English should include an English title and an English abstract). Use `language=...` for every language used in the paper. The last language indicated is the main language of the paper. For example, a French paper with additional titles and abstracts in English and German may start with the following command\n```\n\\documentclass[sigconf, language=english, language=german,\n language=french]{acmart}\n```\nThe title, subtitle, keywords and abstract will be typeset in the main language of the paper. The commands `\\translatedXXX`, `XXX` begin title, subtitle and keywords, can be used to set these elements in the other languages. The environment `translatedabstract` is used to set the translation of the abstract. These commands and environment have a mandatory first argument: the language of the second argument. See `sample-sigconf-i13n.tex` file for examples of their usage.\nSIGCHI Extended Abstracts\n=========================\nThe “`sigchi-a`” template style (available only in LaTeX and not in Word) produces a landscape-orientation formatted article, with a wide left margin. Three environments are available for use with the “`sigchi-a`” template style, and produce formatted output in the margin:\n* `sidebar`: Place formatted text in the margin.\n* `marginfigure`: Place a figure in the margin.\n* `margintable`: Place a table in the margin.\nTo Robert, for the bagels and explaining CMYK and color spaces.\nResearch Methods\n================\nPart One\n--------\nLorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi malesuada, quam in pulvinar varius, metus nunc fermentum urna, id sollicitudin purus odio sit amet enim. Aliquam ullamcorper eu ipsum vel mollis. Curabitur quis dictum nisl. Phasellus vel semper risus, et lacinia dolor. Integer ultricies commodo sem nec semper.\nPart Two\n--------\nEtiam commodo feugiat nisl pulvinar pellentesque. Etiam auctor sodales ligula, non varius nibh pulvinar semper. Suspendisse nec lectus non ipsum convallis congue hendrerit vitae sapien. Donec at laoreet eros. Vivamus non purus placerat, scelerisque diam eu, cursus ante. Etiam aliquam tortor auctor efficitur mattis.\nOnline Resources\n================\nNam id fermentum dui. Suspendisse sagittis tortor a nulla mollis, in pulvinar ex pretium. Sed interdum orci quis metus euismod, et sagittis enim maximus. Vestibulum gravida massa ut felis suscipit congue. Quisque mattis elit a risus ultrices commodo venenatis eget dui. Etiam sagittis eleifend elementum.\nNam interdum magna at lectus dignissim, ac dignissim lorem rhoncus. Maecenas eu arcu ac neque placerat aliquam. Nunc pulvinar massa et mattis lacinia.\n\nEverything Perturbed All at Once:Enabling Differentiable Graph Attacks\n======================================================================\nAs powerful tools for representation learning on graphs, graph neural networks (GNNs) have played an important role in applications including social networks, recommendation systems, and online web services. However, GNNs have been shown to be vulnerable to adversarial attacks, which can significantly degrade their effectiveness. Recent state-of-the-art approaches in adversarial attacks rely on gradient-based meta-learning to selectively perturb a single edge with the highest attack score until they reach the budget constraint. While effective in identifying vulnerable links, these methods are plagued by high computational costs. By leveraging continuous relaxation and parameterization of the graph structure, we propose a novel attack method called Differentiable Graph Attack (DGA) to efficiently generate effective attacks and meanwhile eliminate the need for costly retraining. Compared to the state-of-the-art, DGA achieves nearly equivalent attack performance with 6 times less training time and 11 times smaller GPU memory footprint on different benchmark datasets. Additionally, we provide extensive experimental analyses of the transferability of the DGA among different graph models, as well as its robustness against widely-used defense mechanisms.\nIntroduction\n============\nGraph Neural Networks (GNNs) are powerful in modeling graph-structured data and show remarkable performance on many real-world applications such as social networks , recommendation systems , and drug discovery . Given the successful applications of GNNs, there are also growing concerns about their robustness under adversarial attacks . For example, toxic behavior detectors with GNN backbones could be vulnerable to adversarial attacks, leading to undetected instances of harassment, extremism, or radicalization targeting innocent individuals on social media . To ensure the reliability and safety of GNN-based systems in practice, it becomes paramount to understand their vulnerabilities to adversarial attacks as a foundation for their robust deployment. In a nutshell, developing more effective adversarial attack methods on GNNs not only aids in assessing the robustness and defense strategies of GNNs but also enhances the understanding of the underlying properties of current GNN models.\nPresently, adversarial attacks on GNNs can be categorized based on the attacker’s capacity, goal, level of knowledge, and perturbation type . One of the most practical setups among these is the **gray-box attack**, in which attackers have complete knowledge of the training data but no information on the victim model. Following this setup as in previous works , within the budget limit, we seek for attacking strategies which perform topology attacks (adding or removing edges) on poisoning training data to compromise the overall node classification performance of the victim model.\nDespite the effectiveness in identifying vulnerable edges via the gradient-based *“learn to attack”* approaches, the state-of-the-art methods in graph adversarial attacks are faced with two major bottlenecks: (i) **Non-convexity and discrete structure**: Meta-learning approaches usually formulate the attack as a bi-level optimization problem, which is challenging to solve due to the non-convexity of both levels, making it difficult to derive a closed-form solution. Additionally, the discrete nature of the graph structure poses another challenge for directly applying widely-used gradient-based techniques such as FGSM and PGD , which are typically employed on image data; and (ii) **Computational and resource costs**: Existing methods selectively perturb a single edge with the highest attack score and require retraining the surrogate model from scratch, incurring high computational costs. Additionally, it is unrealistic to scale these techniques to handle large datasets.\n [fig:comparecomplex]\n[tp] \n[fig:dgaframework]\nTo tackle the above-mentioned bottlenecks, we propose the ifferentiable raph ttack (DGA) with continuous relaxation on the graph structure. It offers several advantages with its novel *train-then-sample* attack scheme. During the training process, instead of searching over a discrete set of candidate perturbations, we *relax* the search space to a continuous space, so that the graph structure can be optimized with respect to the attack objective through gradient descent, allowing for fine-grained adjustments and improved effectiveness of the attack. Then, we simulate the lower-level optimization process with one-step fine-tuning on the surrogate model to avoid calculating the accumulation of meta-gradients and further reduce computation complexity. This offers more control over the attack process compared to methods with fixed attack epochs. Essentially, DGA utilizes a single-step adaption approach to simulate the poisoning process, offering a trade-off between the poisoning simulation procedure and training time. For the sampling stage, our sampling scheme enables DGA to generate poisoned graphs with varying budgets while being trained only once, further streamlining the attack process. Theoretically, we show that the estimation error in forward/backward passes of our algorithm, as well as the convergence of the method, depends on the error of edge sampling. These benefits collectively make DGA a powerful and efficient method for generating adversarial attacks on graph structures, as shown in Figure [fig:comparecomplex]. “FOA” and “FDA” refer to first-order approximation and finite-difference approximation, respectively (See Section [sec:cntsrelax] for details).\nWe conduct empirical analysis of DGA with common benchmark datasets. Our main contributions are summarized as follows:\n* **Effectiveness.** DGA outperforms or performs comparably to the SOTA methods on widely-used datasets. Notably, our method is effective even when the perturbation budget is low, which is more aligned with real-world scenarios and maintains the unnoticeability of the attack.\n* **Time and Resource Efficiency.** DGA is more than 6 times faster and requires approximately 11 times less GPU memory than SOTA methods. Importantly, the attack time of DGA remains constant even as the perturbation budget increases. Additionally, DGA can be easily adapted to large-scale graphs, making it a practical choice for real-world scenarios.\n* **Transferability and Imperceptibility.** DGA exhibits excellent transferability to a variety of GNN models and demonstrates imperceptibility toward commonly-used defense methods. These findings highlight the valuable role of DGA in assessing the robustness of existing GNN methods and providing insights for the development of novel defense strategies against adversarial attacks.\nBackground\n==========\nWe consider the task of semi-supervised node classification, in which a model is trained to predict the labels of nodes in a graph, using both labeled and unlabeled data. We denote an undirected graph as *G* = (V, E), where V = {*i* = 1, …, *N*} is the set of *N* nodes, and E ⊆ V × V is the edge set with cardinality ∣E∣ = *M*. In the semi-supervised setting, we are given a subset V*L* ⊆ V of labeled nodes, in which the nodes are associated with class labels from *C* = {*c*1, *c*2, ⋯, *c**k*}. The goal is to learn a function *f* to infer labels of nodes in the unlabeled node set V*U* = V \\ V*L*. Without loss of generality, we collectively use *X* ∈ R*N* × *D* to denote the node features of all nodes in the graph, where *D* is the dimension of feature vectors. Additionally, we denote the adjacency matrix associated with this graph as *A* ∈ {0, 1}*N* × *N*. For each entry (*i*, *j*) in *A*, *A**i*, *j* = 1 if (*i*, *j*) ∈ E and 0 otherwise. In this context, the neighborhood of node *v* is denoted as N(*v*) = {*i* ∈ V : (*i*, *v*) ∈ E}. Given a graph *G*, the node embeddings are obtained by using a GNN model *f**θ*(*X*, *A*), parameterized by *θ*. Particularly, the GNN model takes the node features *X* and adjacency information *A* as input and outputs the logits of each node.\nProblem Setting\n---------------\nWe follow and focus on the *combination* of the following specific attack setting: (1) **gray-box**, in which the attackers have access to complete information about the training data but zero knowledge about the specifics of the underlying model. To overcome this challenge, surrogate models are utilized to approximate and simulate the behavior of the target model; (2) **poisoning**, in which the attacker’s goal is to increase the classification error (one minus accuracy) by training on modified (poisoned) data; (3) **graph structure attack**, in which the attacker’s perturbation type is adding/removing edges. To ensure the attack remains undetected, a maximum perturbation budget denoted as Δ, which restricts the difference between the perturbed graph structure *A* and the original structure *A**o**r**i**g* such that ∥*A* − *A**o**r**i**g*∥0 ≤ 2Δ; (4) **untargeted/global**, in which the attacker’s goal is to compromise the *overall* node classification performance of the model instead of targeting individual nodes.\nFormally, this specific graph attack setting can be formulated as a *bi-level optimization* problem\n \n$$\\begin{aligned}\n& \\min\\_{A\\in \\mathcal{A}\\_{\\Delta}} \\ell\\_{atk}\\left(f\\_{\\theta^\\*}(X, A), Y\\right), \\\\\n& \\textrm{s.t.} \\quad \\theta^\\* = \\arg \\min\\_\\theta \\ell\\_{train}(f\\_\\theta(X, A), Y),\n \\end{aligned} \n\\label{eq:bi-level}$$ \nwhere $\\mathcal{A}\\_{\\Delta}\\coloneqq \\{A' \\in \\{0,1\\}^{N\\times N}\\mid \\|A'-A\\_{orig}\\|\\_0 \\leq 2\\Delta\\}$ denotes the search space containing all possible adjacency matrices with the given modification budget Δ. Here, the upper-level optimization aims to find the optimal edge perturbations which result in a new adjacency matrix *A* that maximizes the attack success. The attack objective ℓ*a**t**k* could be the inverse of either training loss ℓ*t**r**a**i**n* or self-training loss ℓ*s**e**l**f*. Specifically, ℓ*t**r**a**i**n* is the cross-entropy loss on the training set and ℓ*s**e**l**f* is computed with pseudo labels for the unlabeled nodes. The pseudo labels are predicted by a well-trained surrogate model given the clean graph. In a sense, ℓ*s**e**l**f* can be used to estimate the generalization loss after the attack. As for the nested inner problem, the lower-level optimization targets finding the optimal GNN parameters *θ*\\* through an optimization process of training the GNN on the perturbed graph from scratch. Essentially, finding a good attack method for GNNs involves solving this bi-level optimization problem in a effective and efficient manner.\nDifferentiable Graph Attack\n===========================\nIn light of the non-convex and combinatorial characteristics of the optimization problem presented in Equation, it is infeasible to acquire a closed-form solution or employ gradient-based iterative algorithms commonly used in deep learning. To address this challenge, we propose a novel approach that is both effective and efficient by continuous relaxing and parameterizing the graph structure as a continuous edge probability map. An illustration of the DGA framework is shown in Figure [fig:dgaframework].\nContinuous Relaxation on Graph Structure\n----------------------------------------\nWe model each edge with a Bernoulli random variable and transform the discrete graph adjacency matrix *A* to an unnormalized probability matrix *P*, where the (*i*, *j*)-th element *p**i**j* indicates the (unnormalized) edge probability between node *i* and node *j* under the attack setting. Then, the search space for modified adjacency matrix {0, 1}*N* × *N* is relaxed to the positive orthant R+ +*N* × *N*. As shown in Figure [fig:dgaframework], there is no need to enforce the perturbation budget constraint during the DGA training process. The technique outlined in Section [sec:sample] ensures that the generated poisoned graph consistently adheres to the budget. We consider the following continuous relaxation of.\n \n$$\\begin{aligned}\n \\label{eq:bi-level-cnts}\\min\\_{P\\in\\mathbb{R}\\_{++}^{N\\times N}} \\ell\\_{atk}(f\\_{\\theta^\\*}(X,P),Y),\\quad \\quad \\\\\n \\mathrm{s.t.} \\quad \\theta^\\* = \\operatorname\\*{argmin}\\_{\\theta} \\ell\\_{train}(f\\_\\theta(X,P),Y). \n\\end{aligned}$$ \nInstead of exactly solving the lower-level problem, we approximate the optimal surrogate model *θ*\\* by a single-step adaptation with step size *α* following the techniques in . By changing variables *Q* = log*P*, the optimization problem in becomes unconstrained w.r.t. unnormalized log-probabilities *Q*. We define that $\\mathbf{q}\\coloneqq \\mathrm{vec} (Q)$, $\\mathcal{L}\\_{atk} (\\theta,\\mathbf{q}) \\coloneqq \\ell\\_{atk}(f\\_{\\theta}(X,P),Y)$, $\\mathcal{L}\\_{train} (\\theta,\\mathbf{q}) \\coloneqq \\ell\\_{train}(f\\_{\\theta}(X,P),Y)$, and ∇1L( ⋅ , ⋅ ), ∇2L( ⋅ , ⋅ ) are gradients of L with respect to its first and second arguments, respectively. Then, the bi-level optimization in becomes a simpler compositional optimization problem.\n \nmin**q** ∈ R*N* × *N*L*a**t**k*(*θ̂*, **q**), *θ̂* = *θ* − *α*∇*θ*L*t**r**a**i**n*(*θ*, **q**). \nThe hyper-gradient with respect to the log probabilities **q** can be computed as\n \n$$\\begin{aligned}\n\\label{eq:hyper-grad}\n\\nabla\\_{\\mathbf{q}} \\mathcal{L}\\_{atk}(\\hat{\\theta},\\mathbf{q}) = \\nabla\\_2\\mathcal{L}\\_{atk}(\\hat{\\theta},\\mathbf{q}) - \\alpha \\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta,\\mathbf{q}) \\nabla\\_1 \\mathcal{L}\\_{atk}(\\hat{\\theta},\\mathbf{q}).\\end{aligned}$$ \nBased on the hyper-gradient, we can update the log-probability matrix *Q* by a gradient descent step. It is worth noting that the Hessian matrix computation in can be costly when the number of nodes is large. To this end, we can either set *α* = 0 in the hyper-gradient (“first-order approximation”) or use the finite difference approximation similar to. To be specific, with *θ*± = *θ* ± *δ*∇1L*a**t**k*(*θ̂*, **q**), we can approximate the last term in as\n \n$$\\begin{aligned}\n\\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta,\\mathbf{q}) \\nabla\\_1 \\mathcal{L}\\_{atk}(\\hat{\\theta},\\mathbf{q}) \\approx \\frac{\\nabla\\_2 \\mathcal{L}\\_{train}(\\theta\\_+,\\mathbf{q})-\\nabla\\_2 \\mathcal{L}\\_{train}(\\theta\\_-,\\mathbf{q})}{2\\delta}.\\end{aligned}$$ \nIn the rest of the paper, we refer to the variant of our algorithm using first-order approximation (*α* = 0 in Eq. [eq:hyper-grad]) as **DGA-FOA**, and the variant using the finite difference approximation as **DGA-FDA**.\n[1]\n**Initialize:**\n[tbp] [alg:DGA-train]\n[1] **Input:** Graph structure *A**o**r**i**g*, step size *η*, total num. of iterations *T*, node features *X*, labels *Y* Sample a sparse graph *Ã**t* from unnormalized log-probabilities *Q**t* by Gumbel top-*k* sampling Compute the single-step adaptation by *θ̂**t* = *θ* − *α*∇*θ*ℓ*t**r**a**i**n*(*f**θ*(*X*, *Ã**t*), *Y*) Compute the approximated hyper-gradient $\\tilde{\\nabla}\\_t$ based on *θ̂**t* and *Ã**t*: Option I: first-order approximation (FOA) or Option II: finite-difference approximation (FDA) Update the probability matrix by $Q\\_{t+1} = Q\\_t - \\eta \\tilde{\\nabla}\\_t$ and symmetrize *Q**t* + 1 Obtain the probability matrix *P* from the unnormalized log-probability matrix *Q**T* **Return:** Probability matrix *P* for graph poisoning\nEdge Sampling for Expressiveness and Efficiency\n-----------------------------------------------\nNote that the gradient-based continuous optimization of results in a dense probability matrix *P*. Utilizing this matrix as input may lead to over-smoothing or over-squashing of the surrogate graph neural network *θ*. Moreover, message passing on a dense graph exacerbates the issue of “neighborhood explosion”, which significantly increases computational costs. To resolve this problem, we use the Gumbel-Top-*k* trick to sample a sparser graph *Ã* from the unnormalized log-probabilities *Q*. Formally, for the *i*-th node *v**i*, we construct *k* edges between *v**i* and each of the first *k* elements of\n \n$$\\begin{aligned}\n \\operatorname\\*{argsort}\\_j \\left(- \\log(- \\log(u\\_j))+Q\\_{ij}\\right),\n \\end{aligned}$$ \nwhere *u**j* is a vector randomly sampled from the uniform distribution. The top-*k* operation can be replaced by the differentiable relaxed top-*k* operation in. Intuitively, such sampling can be regarded as a stochastic relaxation of the *k*-nearest neighbors rule.\nObtaining the Poisoned Graph\n----------------------------\nUtilizing the learned edge probability map *P* (output of Alg. [alg:DGA-train]), we generate perturbations by sampling from the discrepancy between the final and initial edge probability maps. This process serves to test the performance of the DGA attack, as illustrated in the blue box in Figure [fig:dgaframework]. To prepare for the attack, DGA first symmetrizes the edge probability matrix *P* for undirected graphs as $\\bar{P} = \\frac{1}{2}(P + P^\\top)$. Next, we compute the difference score matrix *S* = (*P̄* − *A**o**r**i**g*) ⊙ (1 − 2*A**o**r**i**g*), where ⊙ indicates element-wise multiplication. Here we flip the sign of difference score for every connected edge in the original graph *A**o**r**i**g*, which allows us to identify existing edges with the most significant negative impact, as well as the non-existent edges with the most potent positive influence on the attack objective. The resulting difference score matrix *S* is employed to construct a categorical distribution, from which we can sample perturbations in order to obtain the poisoned graph.\nTaking into account the allocated attack budget, we sample Δ edges from the categorical distribution without replacement and flip them, thereby generating a poisoned graph. Subsequently, we proceed to train models from scratch using this poisoned graph, allowing us to assess the attack’s performance on a specific model. In practice, we repeat this sampling process multiple times and select the instance that yields the best ℓ*a**t**k* (attack loss) value.\nConvergence Analysis\n====================\nBefore the *t*-th iteration, our algorithm samples edges for each node *i* by the Gumbel top-*k* trick, which is equivalent to sampling *k* edges from $\\mathbf{p}\\_t^i = \\left(p\\_t^{i,1}, \\dotsc, p\\_t^{i,N}\\right)$ without replacement , where **p***t**i* is the normalized probabilities corresponding to the *i*-th line of *Q**t*. This result in the sparsified log-probability matrix $\\tilde{Q}\\_t = \\textrm{Diag}(\\tilde{\\mathbf{q}}\\_t)$ and $\\tilde{\\mathbf{q}} \\in \\{0,1\\}^{N^2}$ is a vector that stores sampled edges for each node, i.e. $\\tilde{\\mathbf{q}}\\_t^{i\\cdot j} =\\mathbf{q}\\_t^{i\\cdot j}$ if the edge (*i*, *j*) is sampled and $\\tilde{\\mathbf{q}}\\_t^{i\\cdot j} = 0$ otherwise. Note that the sampled adjacency matrix *Ã**t* is just the binarized copy of *Q̃**t*. For brevity, we define that $\\Phi(\\mathbf{q})\\coloneqq \\mathcal{L}\\_{atk}(\\hat{\\theta},\\mathbf{q})$, *θ̂* = *θ* − *α*∇*θ*L*t**r**a**i**n*(*θ*, **q**). We make some regularity assumptions of the loss functions.\n[asm:lipshort] Suppose that the loss function L*a**t**k* is Lipschitz continuous and has Lipschitz-continuous gradient while L*a**t**k* has Lipschitz continuous gradient and Hessian.\n[lem:smooth] Under Assumption [asm:lipshort], there exists *G* > 0 and *L* > 0 such that Φ is *G* − Lipschitz continuous and ∇Φ is *L*-Lipschitz continuous.\nThe proof of Lemma [lem:smooth] can be found in Appendix [appendix:proofs]. With the sparsified $\\tilde{\\mathbf{q}}$, we approximate the objective function Φ(**q**) by $\\Phi(\\mathbf{\\tilde{q}})$ and the error is\n \n$$\\begin{aligned}\n\\label{eq:est\\_err}\n\\|\\Phi(\\mathbf{\\tilde{q}\\_t}) - \\Phi(\\mathbf{q}\\_t)\\|\\_2 \\leq G\\|\\mathbf{\\tilde{q}}\\_t - \\mathbf{q}\\_t\\|\\_2,\\end{aligned}$$ \nwhere $\\|\\mathbf{\\tilde{q}}\\_t - \\mathbf{q}\\_t\\|\\_2$ is the estimation error by cause of the edge sampling per iteration. Besides, the hyper-gradient ∇Φ(**q**) in can be estimated by\n \n$$\\begin{aligned}\n\\nabla \\Phi(\\mathbf{\\tilde{q}}) &= \\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\tilde{\\theta}},\\mathbf{\\tilde{q}}) - \\alpha \\nabla^2\\_{1,2}\\mathcal{L}\\_{train} (\\theta,\\mathbf{\\tilde{q}})\\nabla\\_1 \\mathcal{L}\\_{atk}(\\hat{\\theta},\\mathbf{\\tilde{q}}),\\quad \\\\\n\\hat{\\tilde{\\theta}} &= \\theta - \\alpha \\nabla\\_\\theta \\mathcal{L}\\_{train}(\\theta,\\mathbf{\\tilde{q}}),\\end{aligned}$$ \nwhere $\\nabla \\Phi(\\mathbf{\\tilde{q}})$ is the vectorized $\\tilde{\\nabla}\\_t$ in Alg. [alg:DGA-train]. Next, we present the main theorem on convergence.\n[thm:convergence] Under assumptions above, our algorithm with proper step size *η* leads to\n \n$$\\begin{aligned}\n\\label{eq:bound}\n\\mathbb{E}\\|\\nabla \\Phi(\\mathbf{q}\\_\\tau)\\|\\_2^2 \\leq \\mathcal{O}\\left(\\frac{\\Phi(\\mathbf{q}\\_0) - \\inf\\_{\\mathbf{q}}\\Phi}{T} + \\mathrm{Avg. Err.}\\right),\\end{aligned}$$ \nwhere *τ* is sampled from $\\{0,1,\\dotsc, T-1\\}$ uniformly at random and Avg. Err. is the average error due to edge sampling across the iterations. To be specific, $\\mathrm{Avg. Err.}\\coloneqq \\frac{1}{T}\\sum\\_{t=0}^{T-1}\\mathbb{E} \\|\\tilde{\\mathbf{q}}\\_t - \\mathbf{q}\\_t\\|\\_2^2$.\nThe proof of Theorem [thm:convergence] can be found in Appendix [appendix:proofs]. Theorem [thm:convergence] demonstrates that our algorithm exhibits an O(1/*T*) non-asymptotic convergence rate when *k* = *N*, the same as that of gradient descent. While opting for a larger value of *k* can be advantageous from the estimation/optimization perspective, it results in worse expressiveness and efficiency as explained in Section [sectin:edgesampling]. Therefore, we can manage the trade-off between estimation/optimization error and expressiveness/efficiency by adjusting the hyper-parameter *k*. Fortunately, the estimation error $\\|\\mathbf{\\tilde{q}}\\_t - \\mathbf{q}\\_t\\|\\_2$ could be negligible even with a small *k* when the distribution constructed by **p***i* is light-tailed (e.g., power law), thanks to the Gumbel top-*k* sampling.\n[tbp]\n[tab:datastat]\n[tbp]\n[tab:main-res]\n[bp][tab: computationcomplexity]\nExperiments\n===========\nOur proposed attack method, DGA (Differentiable Graph Attack), is evaluated through a series of experiments aimed at demonstrating its effectiveness and efficiency. The experimental settings are introduced in Section [sec:expsetting]. The attack performance and the generalizability and transferability of DGA is analyzed in Section [sec:attack]. We also show the computation complexity of DGA in Section [sec:complex]. Furthermore, we conduct experiments with defense methods in Section [sec:defense] to demonstrate the robustness of our approach. Additionally, we present a study on Gumbel-*k* in Section [sec:gumbel-k] and provide visualization of the distribution of the generated poisoned graph in Section [sec:visual].\nExperimental Settings\n---------------------\n**Dataset.** The experiments are conducted on three widely-used datasets: two citation network datasets, namely CiteSeer and Cora , and one social network dataset PolBlogs . All experiments are performed on the largest connected component of the graphs. Following previous works , we randomly split the datasets into the train, validation, and test sets using a 10%, 10%, and 80% ratio, respectively. For each experiment, we report the average performance of 10 runs. The experiments are conducted on the largest connected component (LCC) of the graphs, ensuring that only the main connected portion of the graphs is considered for analysis and evaluation. The statistics for the LCC of these datasets are summarized in Table [tab:datastat].\n**Victim Models.** Following previous methods, we first take the widely-used GCN as our victim model. Moreover, as we adopt a gray-box setting, in which the model architecture is considered unknown, we further conduct experiments with Graph Attention Network (GAT) and DeepWalk to measure the transferability of DGA. Additionally, to assess the effectiveness of our proposed DGA attack method, we conduct experiments to evaluate its robustness against existing defense methods, namely GCN-Jaccard and GCN-SVD.\n**Perturbation Budgets.** To be closer to real-world scenarios and demonstrate the imperceptibility of the attack method, we set 1%, 3%, and 5% as the perturbation budget in our experiments. Each method is allowed to modify 1%, 3%, and 5% of the number of edges in the original graph, respectively.\n**Baselines.** To assess the effectiveness and applicability of DGA, we conduct a comprehensive comparison with six representative baselines: including heuristic-based method DICE and learning-based methods Meta-Self, Meta-Train, A-Meta-Self, A-Meta-Train , and GraD .\nAdditional information about the experimental setup, implementation, and baselines can be found in Appendix [appendix:experimentsetup]. [1](#fn1)\n[tbp] [tab:moreres]\nAttack Performance\n------------------\nTable [tab:main-res] presents the test accuracy results of the proposed DGA method compared to baselines. The accuracy values for the original clean graphs are reported under the 0% perturbation rate column. Remarkably, DGA outperforms or closely approximates the best performance in 8 out of 9 metrics, while exhibiting a significantly reduced training time (over 6 times faster) and GPU memory usage (over 11 times less) compared to the previous SOTA Meta-Self and Meta-Train methods. Additionally, DGA demonstrates strong attack performance even at low perturbation rates, causing a mere 3% to 4% drop in test accuracy with only a 1% budget. Notably, the DGA-FOA method consistently achieves better results than its counterpart, DGA-FDA, on 7 out of 9 metrics. This observation suggests that the first-order approximation exhibits greater robustness and efficacy in practical applications. Detailed descriptions and reproduction information for these baselines can be found in Appendix [appendix:baseline]. Note that the GraD paper evaluates their performance on the overall graph, distinct from the commonly used LCC setting. Standard deviations of these experiments are provided in Appendix [appendix: resultsstd].\n**Generalizability and Transferability.** In order to be consistent with the gray-box attack setting, in which attackers have no information on the victim model, we evaluate the generalizability and transferability of DGA with two different GNN models. We conduct experiments with both supervised attention-based method GAT and unsupervised random walk-based method DeepWalk trained with the DGA poisoned graph. These graphs are derived using GCN as the surrogate model. Detailed descriptions for these base models are provided in Appendix [appendix:basemodel]. The results, shown in the GAT and DeepWalk section in Table [tab:moreres], demonstrate that our method achieves good transferability performance on both GAT and DeepWalk models. Our method achieved the best or second-best performance in 11 out of 18 metrics. Note that since the graph poisoned by DICE contains isolated nodes, we do not evaluate DeepWalk with this method.\nTraining Time and Memory Usage.\n-------------------------------\nTable [tab: computationcomplexity] presents a comparison of training time and maximum GPU memory usage between the proposed DGA method and baselines, considering a 5% perturbation rate. Our approach demonstrates significantly shorter training time and lower memory usage compared to the baselines. Regarding training time, our DGA methods demonstrate significant efficiency. Specifically, when compared to the previous SOTA methods, DGA achieves training time reductions of over 5, 6, and 16 times on the Citeseer, Cora, and PolBlogs datasets, respectively. Furthermore, our DGA methods exhibit superior efficiency in terms of GPU memory usage. Compared to the SOTA methods, DGA showcases reductions of over 11, 12, and 15 times on the Citeseer, Cora, and PolBlogs datasets, respectively. Additional comparisons with other budgets can be found in Appendix [appendix: computationcomplexity]. It is worth noting that DICE is a random perturbation method that runs quickly and does not utilize GPU resources. These findings highlight the computational advantages of DGA, making it a highly efficient and scalable solution for adversarial attacks on graph structures. The substantial reductions in training time and GPU memory usage signify the potential of DGA for practical applications in real-world scenarios, as it can be easily applied to large-scale graphs.\nRobustness against Existing Defense Methods\n-------------------------------------------\nTo evaluate the effectiveness of our proposed DGA attack method, we conduct experiments to test its robustness against existing defense methods. By doing so, we aim to assess whether our attack method could overcome these defense methods and reveal vulnerabilities in the system that were previously undiscovered. We test DGA against two popular defense methods, namely GCN-Jaccard and GCN-SVD . Detailed descriptions for these defense models are provided in Appendix [appendix:basemodel]. In these experiments, the poisoned graphs are first vaccinated with defense models. Then a GCN model is trained from scratch on the vaccinated graph. We report the classification accuracy of the well-trained GCN model. Results are shown in the Low-rank SVD and Jaccard section in Table [tab:moreres]. Note that we cannot perform Jaccard on the PolBlogs dataset, as this dataset does not contain node features. Our method achieves the best or second-best performance compared with the baseline on 11 out of 15 metrics. These results demonstrate the effectiveness of our DGA attack method in overcoming existing defense methods and exposing potential vulnerabilities of GNN models.\n[bp] [tab:gumbelk]\n[btp] [fig:attackvis]\nImpact of Gumbel-\n-----------------\nIn this section, we conduct ablation studies on the hyperparameter *k* for the Gumbel-*k* trick in DGA-FOA using the Cora dataset. As shown in Table [tab:gumbelk], the performance of DGA remains relatively stable across different perturbation rates for varying values of “k.” However, we observe a trend where the performance first improves and then declines as *k* increases. Increasing *k* initially enhances DGA’s performance, indicating its potential for better results. Nevertheless, we find that beyond a certain threshold, the performance starts to decline. This drop in performance can be attributed to overfitting and over-squashing of the GNNs. When *k* becomes excessively large, the graph becomes dense and closely resembles a fully-connected graph, leading to over-squashing and reduced generalization. Notably, the average degree of the dataset is around 2, as indicated in Table [tab:datastat]. The ablation study highlights that the best-performing DGA configurations are achieved when setting *k* around the average degree of the dataset. While higher *k* values may improve DGA’s ability to exploit complex structures, they come at the cost of increased computational resources. Hence, we suggest choosing *k* near the average degree of the graph to achieve a good balance between performance and computational efficiency.\nVisualization and Analysis of DGA Attacked Graphs\n-------------------------------------------------\n[sec:visual] To elucidate the impact of the attack on the graph structure and its implications for the performance and behavior of the targeted model, we visualize three properties for the clean graph and poisoned graph on the Citeseer dataset, as shown in Figure [fig:attackvis]. Additional visualizations on other datasets are provided in Appendix [appendix: visualization]. Firstly, the node degree distributions for the clean graph and perturbed graph are shown in Figure [fig:degreedist]. Notably, we observe that the degree distributions of the two graphs exhibit high similarities. This observation provides evidence for the imperceptibility of our proposed DGA attack. Furthermore, we present the feature similarity and label equality analysis for different types of edges in the graphs, as shown in Figure [fig:featurediff] and Figure [fig:labelequal], respectively. We can observe that DGA exhibits a tendency to establish connections between nodes that possess dissimilar features and have different labels while removing edges that link nodes sharing similar features and belonging to the same label. These observations align with the homophily property in social networks and citation networks and the pattern demonstrated in .\nRelated Work\n============\n**Adversarial Attacks on Graphs** aim to interfere with the performance of graph neural networks by introducing subtle perturbations to graph data. Following the taxonomy in recent surveys , adversarial attacks on GNNs can be classified based on several factors, including the level of the task being performed (node-level tasks or graph-level tasks ), the goal of the attack (targeted or untargeted/global attack ), the phase of the attack (poisoning attack or evasion attack ), the level of knowledge that the attacker has about the model (white-box , black-box , or gray-box ), and the perturbation type of the attack (perturbing node features , graph structure , or injecting new nodes ). Different combinations of these scenarios present unique challenges and require different approaches to design effective attacks and defenses.\nIn this paper, we focus on a practical problem setting of *node-level, untargeted, gray-box poisoning* attack on *graph structure* . Previous works formulate this setting as a bi-level optimization problem and try to solve it on the discrete graph structure. MetAttack first employs a meta-learning approach and treat the input data as a hyper-parameter to learn. They adopt a greedy algorithm to obtain the discrete graph structure by selecting one bit to flip at a time, iteratively for each bit in the perturbation budget. Building on the meta-learning framework, AtkSE approximates the continuous distribution of hyper-gradients using discrete edge flipping intervals. GraD introduced a novel attack objective to address the gradient bias. However, as illustrated in Figure [fig:comparecomplex], the meta-gradient calculation process of these approaches requires computing the Hessian matrix and accumulating trained-from-scratch inner-loop iteration, resulting in time and resource-intensive operations that are repeated until the perturbation budget is filled. To achieve effective attacks while minimizing time and resource costs, we propose a novel approach that relaxes the graph structure and learns a continuous edge probability map with attack gradients. By simultaneously updating the graph structure and surrogate model parameters, we can efficiently sample optimal perturbation sets from the learned probability map, enabling continuous budget-constrained search.\n**Continuous Relaxation on Graph Structure** was first explored in graph structure learning . For instance, LDS incorporates a probabilistic map as a graph generator to model the adjacency matrix of the graph DGM augments the graph structure using continuous updates to the graph structure with a differentiable graph module and a diffusion module. NeuralSparse learns *k*-neighbor subgraphs for robust graph representation learning by selecting at most *k* edges for each nodes. PTDNet introduces a denoising map drawn from a Bernoulli distribution and learns to drop task-irrelevant edges. In contrast to these previous works, our proposed DGA, taking an adversarial learning perspective, is the first to employ continuous relaxation on graph structure and utilize learnable probabilities for attacking GNNs. Notably, proposed a method for white-box graph attack with continuous relaxation on the perturbation map and use projected gradient descent to update the perturbations. In contrast, DGA adopts stochastic gradient descent to directly update the graph structure, enabling better flexibility, transferability and interpretibility.\nConclusions, Limitations and Social Impact\n==========================================\nThis paper proposes a poisoning attack model on graph-structured data. We propose a novel approach DGA, which leverages continuous relaxation and parameterization of the graph structure to generate effective and efficient attacks. DGA outperforms or approximates the state-of-the-art performance on a variety of benchmark datasets, with significantly less training time and GPU memory occupation compared to existing methods. We also provide extensive analysis of the transferability of our approach to other graph models, as well as its robustness against widely-used defense mechanisms. These can help assess the robustness of existing GNN methods, as well as guide the development of new defense strategies for adversarial attacks. Additionally, DGA can be expand to large-scale graphs simply with neighbor-sampling training mechanisms, which remains a direction for future exploration.\nEthical Considerations\n======================\nIt is worth noting the potential negative social impact of this work. The proposed poisoning attack model, if misused, can significantly impact the robustness and integrity of GNNs. This is a reminder of the importance of protecting the privacy of data, including attributes of nodes, training labels, and graph structure, to prevent malicious exploitation and unauthorized attacks.\nProof of Theorem [thm:convergence]\n==================================\nFirst, we formally state the assumptions of Theorem [thm:convergence].\n[asm:smooth] For any *θ*, *θ*ʹ ∈ R*D*, **q**, **q**ʹ ∈ R*N*2, there exist *G*1, *G*2, *L*1, 2, *L*2, 1, *L*2, 2, *H*1, 2 > 0 such that\n \n$$\\begin{aligned}\n&\\|\\mathcal{L}\\_{atk}(\\theta,\\mathbf{q}) - \\mathcal{L}\\_{atk}(\\theta',\\mathbf{q})\\|\\_2\\leq G\\_1\\|\\theta-\\theta'\\|\\_2,\\\\\n& \\max\\left\\{ \\|\\mathcal{L}\\_{atk}(\\theta,\\mathbf{q}) - \\mathcal{L}\\_{atk}(\\theta,\\mathbf{q}')\\|\\_2, \\right. \\\\\n&\\qquad \\left. \\|\\mathcal{L}\\_{train}(\\theta,\\mathbf{q}) - \\mathcal{L}\\_{train}(\\theta,\\mathbf{q}')\\|\\_2\\right\\}\\leq G\\_2\\|\\mathbf{q}-\\mathbf{q}'\\|\\_2,\\\\\n& \\|\\nabla\\_2 \\mathcal{L}\\_{atk}(\\theta,\\mathbf{q})-\\nabla\\_2 \\mathcal{L}\\_{atk}(\\theta',\\mathbf{q})\\|\\_2 \\leq L\\_{2,1} \\|\\theta-\\theta'\\|\\_2, \\\\\n& \\|\\nabla\\_2 \\mathcal{L}\\_{atk}(\\theta,\\mathbf{q})-\\nabla\\_2 \\mathcal{L}\\_{atk}(\\theta,\\mathbf{q}')\\|\\_2 \\leq L\\_{2,2}\\|\\mathbf{q} - \\mathbf{q}'\\|\\_2,\\\\\n& \\|\\nabla\\_1 \\mathcal{L}\\_{train}(\\theta,\\mathbf{q})-\\nabla\\_1 \\mathcal{L}\\_{train}(\\theta,\\mathbf{q}')\\|\\_2 \\leq L\\_{1,2}\\|\\mathbf{q} - \\mathbf{q}'\\|\\_2,\\\\\n& \\|\\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta, \\mathbf{q}) - \\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta, \\mathbf{q}')\\|\\_F \\leq H\\_{1,2}\\|\\mathbf{q} - \\mathbf{q}'\\|\\_2,\\end{aligned}$$ \nwhere ∥ ⋅ ∥*F* refers to the Frobenius norm.\n[lem:smooth-2] Under Assumption [asm:smooth], there exists *G* > 0 and *L* > 0 such that Φ is *G* − Lipschitz continuous and ∇Φ is *L*-Lipschitz continuous.\nNote that ∇Φ(**q**) = ∇2L*a**t**k*(*θ̂*, **q**) − *α*∇1, 22L*t**r**a**i**n*(*θ*, **q**) ∇1L*a**t**k*(*θ̂*, **q**).\nFirst, we can obtain that\n \n$$\\begin{aligned}\n \\|\\nabla \\Phi(\\mathbf{q})\\|\\leq G\\_2 + \\sqrt{\\min(D,N^2)} \\alpha L\\_{1,2} G\\_1.\\end{aligned}$$ \nFor any **q**, **q'**, we define that *θ̂*ʹ = *θ* − *α*∇*θ*L*t**r**a**i**n*(*θ*, **q**ʹ) such that\n \n$$\\begin{aligned}\n& \\|\\nabla \\Phi(\\mathbf{q}) - \\nabla \\Phi(\\mathbf{q}')\\|\\_2 \\leq \\|\\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}, \\mathbf{q}) - \\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}', \\mathbf{q}')\\|\\_2 \\\\\n& \\quad\\quad\\quad\\quad\\quad\\quad\\quad + \\alpha \\|\\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta, \\mathbf{q}) \\nabla\\_1 \\mathcal{L}\\_{atk}(\\hat{\\theta},\\mathbf{q}) \\\\\n& \\quad\\quad\\quad\\quad\\quad\\quad\\quad - \\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta, \\mathbf{q}') \\nabla\\_1 \\mathcal{L}\\_{atk}(\\hat{\\theta}',\\mathbf{q}')\\|\\_2\\\\\n& \\leq \\|\\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}, \\mathbf{q}) - \\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}', \\mathbf{q}')\\|\\_2 \\\\\n& \\quad\\quad + \\alpha \\|\\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta, \\mathbf{q})\\|\\_F \\|\\nabla\\_1 \\mathcal{L}\\_{atk}(\\hat{\\theta},\\mathbf{q}) - \\nabla\\_1 \\mathcal{L}\\_{atk}(\\hat{\\theta}',\\mathbf{q}')\\|\\_2\\\\\n& \\quad\\quad + \\alpha \\|\\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta, \\mathbf{q}) - \\nabla^2\\_{1,2}\\mathcal{L}\\_{train}(\\theta, \\mathbf{q}')\\|\\_F \\| \\nabla\\_1 \\mathcal{L}\\_{atk}(\\hat{\\theta}',\\mathbf{q}')\\|\\_2\\\\\n& \\leq (1+\\alpha \\sqrt{\\min(D,N^2)}L\\_{1,2})\\|\\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}, \\mathbf{q}) \\\\\n& \\quad\\quad - \\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}', \\mathbf{q}')\\|\\_2 + \\alpha G\\_1 H\\_{1,2} \\|\\mathbf{q} - \\mathbf{q}'\\|\\_2.\\end{aligned}$$ \nWe further have\n \n$$\\begin{aligned}\n& \\|\\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}, \\mathbf{q}) - \\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}', \\mathbf{q}')\\|\\_2\\\\\n& \\leq \\|\\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}, \\mathbf{q}) - \\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}, \\mathbf{q}')\\|\\_2 \\\\\n&\\quad\\quad + \\|\\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}, \\mathbf{q}') - \\nabla\\_2 \\mathcal{L}\\_{atk}(\\hat{\\theta}', \\mathbf{q}')\\|\\_2\\\\\n& \\leq L\\_{2,2} \\|\\mathbf{q} - \\mathbf{q}'\\|\\_2 + L\\_{2,1} \\|\\hat{\\theta}-\\hat{\\theta}'\\|\\_2 \\\\\n& = L\\_{2,2} \\|\\mathbf{q} - \\mathbf{q}'\\|\\_2 + \\alpha L\\_{2,1}\\| \\nabla\\_{\\theta} \\mathcal{L}\\_{train}(\\theta,\\mathbf{q}) \\\\\n& \\quad\\quad - \\nabla\\_{\\theta} \\mathcal{L}\\_{train}(\\theta,\\mathbf{q}')\\|\\_2 \\leq (L\\_{2,2}+ \\alpha L\\_{1,2} L\\_{2,1}) \\|\\mathbf{q} - \\mathbf{q}'\\|\\_2\\end{aligned}$$ \nWe define $G\\coloneqq G\\_2 + \\alpha \\sqrt{\\min(D,N^2)} \\, L\\_{1,2} \\, G\\_1$ and $L \\coloneqq (1 + \\allowbreak\\alpha \\sqrt{\\min(D,N^2)} \\, L\\_{1,2}) \\allowbreak\\,(L\\_{2,2} + \\alpha L\\_{1,2} L\\_{2,1}) + \\alpha G\\_1 H\\_{1,2}$.\n[Proof of Theorem [thm:convergence]] By the *L*-Lipschitzness of ∇Φ as shown Lemma [lem:smooth-2], we have\n \n$$\\begin{aligned}\n\\nonumber\n \\mathbb{E}\\_t\\Phi(\\mathbf{q}\\_{t+1}) & \\leq \\Phi(\\mathbf{q}\\_t) - \\eta \\nabla \\Phi(\\mathbf{q}\\_t) \\cdot \\mathbb{E}\\_t \\nabla\\Phi(\\tilde{\\mathbf{q}}\\_t) \\\\ \n \\nonumber\n & \\quad\\quad + \\eta^2 L \\mathbb{E}\\_t \\|\\nabla\\Phi(\\tilde{\\mathbf{q}}\\_t) - \\nabla \\Phi(\\mathbf{q}\\_t)\\|\\_2^2 + \\eta^2 L\\| \\Phi(\\mathbf{q}\\_t)\\|\\_2^2\\\\\n \\nonumber\n & \\leq \\Phi(\\mathbf{q}\\_t) - \\eta \\nabla \\Phi(\\mathbf{q}\\_t) \\cdot \\mathbb{E}\\_t \\nabla\\Phi(\\tilde{\\mathbf{q}}\\_t)\\\\ & \\quad\\quad + \\eta^2 L^3 \\mathbb{E}\\_t \\|\\tilde{\\mathbf{q}}\\_t - \\mathbf{q}\\_t\\|\\_2^2 + \\eta^2 L\\| \\Phi(\\mathbf{q}\\_t)\\|\\_2^2,\n \\label{eq:smoothness\\_starter}\\end{aligned}$$ \nwhere E*t*[ ⋅ ] refers to the expectation conditioned on the randomness before iteration *t*. Young’s inequalities leads to\n \n$$\\begin{aligned}\n\\nonumber\n& - \\eta \\nabla \\Phi(\\mathbf{q}\\_t) \\cdot \\mathbb{E}\\_t \\nabla\\Phi(\\tilde{\\mathbf{q}}\\_t) \\\\\n\\nonumber\n& = - \\eta \\| \\nabla \\Phi(\\mathbf{q}\\_t)\\|\\_2^2 - \\eta \\mathbb{E}\\_t\\left[\\nabla \\Phi(\\mathbf{q}\\_t) \\cdot (\\nabla\\Phi(\\tilde{\\mathbf{q}}\\_t)-\\nabla \\Phi(\\mathbf{q}\\_t))\\right]\\\\\n\\nonumber\n& \\leq - \\frac{\\eta}{2} \\| \\nabla \\Phi(\\mathbf{q}\\_t)\\|\\_2^2 + \\frac{\\eta}{2} \\mathbb{E}\\_t[\\|\\nabla\\Phi(\\tilde{\\mathbf{q}}\\_t)-\\nabla \\Phi(\\mathbf{q}\\_t))\\|\\_2^2]\\\\\n\\label{eq:cross\\_term}\n& \\leq - \\frac{\\eta}{2} \\| \\nabla \\Phi(\\mathbf{q}\\_t)\\|\\_2^2 + \\frac{\\eta L^2}{2} \\mathbb{E}\\_t[\\|\\tilde{\\mathbf{q}}\\_t-\\mathbf{q}\\_t\\|\\_2^2].\\end{aligned}$$ \nPlug into and re-arrange the terms.\n \n$$\\begin{aligned}\n& \\mathbb{E}\\_t\\Phi(\\mathbf{q}\\_{t+1}) \\\\\n& \\leq \\Phi(\\mathbf{q}\\_t) -\\frac{\\eta}{2}\\left(1-2\\eta L\\right)\\| \\nabla \\Phi(\\mathbf{q}\\_t)\\|\\_2^2 + \\frac{\\eta L^2}{2}\\left(1+2\\eta L\\right) \\mathbb{E}\\_t \\|\\tilde{\\mathbf{q}}\\_t - \\mathbf{q}\\_t\\|\\_2^2.\\end{aligned}$$ \nSet $\\eta = \\frac{1}{4L}$ and use the tower property of expectation.\n \n$$\\begin{aligned}\n\\mathbb{E}\\|\\nabla \\Phi(\\mathbf{q}\\_t)\\|\\_2^2 \\leq 16 L\\mathbb{E}[\\Phi(\\mathbf{q}\\_t) - \\Phi(\\mathbf{q}\\_{t+1})] + 3L^2 \\mathbb{E} \\|\\tilde{\\mathbf{q}}\\_t - \\mathbf{q}\\_t\\|\\_2^2. \\end{aligned}$$ \nDo telescoping sum from *t* = 0 to *T* − 1 and divide *T* on both sides.\n \n$$\\begin{aligned}\n \\frac{1}{T}\\sum\\_{t=0}^{T-1} \\mathbb{E}\\|\\nabla \\Phi(\\mathbf{q}\\_t)\\|\\_2^2 & \\leq \\frac{16 L(\\Phi(\\mathbf{q}\\_0) - \\inf\\_{\\mathbf{q}}\\Phi )}{T} + \\frac{3L^2}{T}\\sum\\_{t=0}^{T-1}\\mathbb{E} \\|\\tilde{\\mathbf{q}}\\_t - \\mathbf{q}\\_t\\|\\_2^2.\\end{aligned}$$ \nExperimental Setup\n==================\nDataset Description\n-------------------\n**CiteSeer** is a citation network containing 3,312 scientific publications classified into 6 classes. The network consists of 4,732 links, and each publication is represented by a binary word vector indicating the presence or absence of words from a dictionary of 3,703 unique words.\n**Cora** is another citation network comprising 2,708 scientific publications classified into seven classes. The network includes 5,429 links, and each publication is represented by a binary word vector indicating the presence or absence of words from a dictionary of 1,433 unique words.\n**PolBlogs** is a graph with 1,490 vertices representing political blogs and 19,025 edges representing links between blogs. The links are automatically extracted from the front pages of blogs. Each vertex is labeled as either liberal or conservative, indicating the political leaning of the blog.\nImplementation Details.\n-----------------------\nOur method is implemented using PyTorch and PyTorch Geometric frameworks, with training conducted using the Adam optimizer . The experiments are carried out on a single NVIDIA RTX A5000 24GB GPU. The search space for model and training hyperparameters can be found in Table [tab: hyper-para]. Note that the number of finetuning iterations is applicable to DGA-FOA only, whereas for DGA-FDA, it is set to 1, indicating one-step optimization. For all experiments, optimal hyperparameters are selected based on the performance of the validation set.\n[!htbp][tab: hyper-para]\nBase Model Description\n----------------------\n**Graph Attention Network (GAT)** is a neural network architecture that performs graph convolutions using attention mechanisms to selectively aggregate information from neighboring nodes. It is frequently employed as a foundational layer of defense against adversarial attacks.\n**DeepWalk** is an unsupervised learning method that aims to learn low-dimensional representations of nodes in a graph. It generates random walks within the graph and applies the skip-gram model to learn node embeddings. As DeepWalk is trained in an unsupervised manner without node characteristics or graph convolutions, this transfer setting is more challenging.\n**GCN-Jaccard** is a defense method that focuses on identifying and removing adversarial nodes in a graph. The Jaccard similarity between the neighborhood sets of two nodes is used to determine if a node is likely to be adversarial. By identifying and removing such nodes, GCN-Jaccard aims to improve the robustness of graph models against adversarial attacks.\n**GCN-SVD** is a defense method that mitigates adversarial attacks by approximating the graph Laplacian with a low-rank matrix. By reducing the dimensionality of the graph Laplacian, GCN-SVD aims to preserve the essential structural information while suppressing the influence of potential adversarial perturbations.\nBaseline Reproduction Detail\n----------------------------\n**DICE** (Delete Internally, Connect Externally) is an attack method that focuses on modifying the graph structure to undermine the performance of targeted models. It achieves this by randomly connecting nodes with different labels or removing edges between nodes with the same label. This manipulation of the graph aims to disrupt the original connectivity patterns and induce misclassification errors in the targeted model. For our implementation of the DICE attack, we use the code provided in the DeepRobust package .\n**MetAttack** is an attack method that utilizes meta-learning to solve a bi-level optimization problem. It employs a greedy approach to selectively perturb one edge at a time in order to maximize the adversarial impact on the targeted model. The method includes four variants of MetAttack, each employing different loss functions and incorporating first-order approximation: Meta-Train, Meta-Self, A-Meta-Train, and A-Meta-Self. Note that the “Train” variants use cross-entropy loss on the training set, while the “Self” variants use self-training loss with pseudo labels. The “A-” variants indicate the use of first-order approximations during the optimization process. For reproduction, we rely on the code available in the DeepRobust package with the default hyperparameters included with the code.\n**GraD** is a recent attack method that leverages the meta-learning framework and introduces a novel attack objective to mitigate gradient bias. It claims that it outperforms MetAttack in terms of overall graph performance across the datasets, rather than only focusing on the largest connected component. For reproduction, we use the official code and the default hyperparameters provided at <https://github.com/Zihan-Liu-00/GraD--NeurIPS22>.\nAdditional Experimental Results\n===============================\nResults with standard deviation\n-------------------------------\nThe standard deviations of the experiments conducted on the GCN, GAT, DeepWalk, and GCN model vaccinated with low-rank SVD approximation (GCN-SVD), and Jaccard (GCN-Jaccard) methods, as described in Section [sec: exp], are presented in Tables [tab:main-res-std] to [tab: defense-svd-std], respectively. The standard deviations are calculated based on 10 runs, providing a reliable estimation of the variation in the experimental results. This indicates that our method consistently performs well and exhibits relatively low variability, thereby highlighting its effectiveness and reliability. These results demonstrate the robustness and stability of our proposed method.\n[!htbp] [tab:main-res-std]\n[!htbp] [tab:transfer-gat-std]\n[!htbp] [tab:transfer-deepwalk-std]\n[!htbp] [tab: defense-jaccard-std]\n[!htbp] [tab: defense-svd-std]\nComputation Complexity with 1% and 3% Perturbation Rates\n--------------------------------------------------------\nTable [tab: computationcomplexity-addiitional-1] and Table [tab: computationcomplexity-addiitional-3] provide a comparison of training time and peak GPU memory usage between the proposed DGA method and the baselines, considering perturbation rates of 1% and 3% respectively. Our approach demonstrates notable advantages over baselines, including significantly reduced training time and lower memory usage. Importantly, our method maintains consistent computational efficiency even as the perturbation rate increases. These results highlight the computational benefits of DGA, making it an efficient and scalable solution for adversarial attacks on graph structures. Furthermore, the substantial reductions in training time and GPU memory usage also imply the promising potential of DGA for real-world applications, particularly when dealing with large-scale graphs.\n[!htbp][tab: computationcomplexity-addiitional-1]\n[!htbp][tab: computationcomplexity-addiitional-3]\nVisualization for Cora and Polblogs Datasets\n--------------------------------------------\nWe present a visual comparison of statistics between the poisoned graph generated by DGA and the original clean graph. Specifically, we analyze the node degree distribution, node feature similarity, and label equality on the Cora and Polblogs datasets, as shown in Figure [fig:attackvisother]. To enhance clarity and improve visualization, we scale the x-axis of the node degree distribution plot. Notably, we consistently observe similar trends and patterns in these statistics across the Cora and Polblogs datasets, reaffirming our previous observations on the Citeseer dataset as outlined in Sec. [sec:attack]. By conducting a visual comparison of these statistics, we gain a better understanding of the attack’s influence on the graph structure and its implications for the performance and behavior of the targeted model. This analysis significantly contributes to our comprehensive evaluation and assessment of the effectiveness and impacts of the proposed DGA method.\n[!htbp]\n[fig:attackvisother]\n---\n1. The code will be be made publicly available once accepted.[↩](#fnref1)\n"},"data_id":{"kind":"string","value":"arxiv_0002458"}}},{"rowIdx":596,"cells":{"content":{"kind":"string","value":" already defined in 1951, it was not until 1980 that the origin of the observed overabundances was understood as the result of mass transfer in a binary system. The polluting heavy elements were formerly produced within an asymptotic giant branch (AGB) companion, which is now a very faint white dwarf (WD) that cannot be directly observed in most cases.\nAlthough they have been less intensively studied, barium stars are also found in the main sequence (i.e. barium dwarfs, dBa; ; ). Additionally, closely related to Ba stars, CH stars are their low metallicity counterparts. The latter have similar enhancement of s-process elements and strong CH molecular bands, but weaker lines of other metals.\nThe exact mode of mass transfer responsible for the formation of these families of polluted binaries remains uncertain. Many systems have orbital periods in a range (form 100 to 1000 days) that cannot be accounted for by simple models of orbital evolution (;, and references therein). Nevertheless, the white-dwarf nature of the companion is beyond doubt, thanks to the analysis of the orbital mass functions performed by,, and. The latter analysis reveals a mass distribution for the WD companion which peaks around 0.6 M⊙, in accordance with the expectation. However, the exact value of the derived WD mass depends upon the mass of the primary star (the barium star), which is difficult to derive with a good accuracy.\nThe mass of barium stars may be derived from their location in the Hertzsprung-Russell diagram (HRD) and from a comparison with evolutionary tracks. used a Bayesian method to infer barium star masses, based on a HRD constructed from Hipparcos parallaxes. These authors concluded that mild and strong barium stars have somewhat different mass distributions, which are characterized by masses in the range 2.5 – 4.5 M⊙ and 1 – 3 M⊙, respectively. The distinction between mild and strong barium stars is made on the *Ba index* introduced by. The index reflects the strength of the barium spectral lines, based on visual inspection, on a scale from Ba1 to Ba5, where Ba5 corresponds to the strongest lines. In this and our past studies, we associate Ba1 - Ba2 indices with mild barium stars and Ba3 - Ba5 indices with strong barium stars. The catalogue of barium stars by introduces many barium stars with a *Ba index* smaller than one, which we denote as Ba0. These targets deserve special attention as they may turn out not to be barium stars.\n![[Fig:varpi-errpi] Parallax and its error for the sample of Ba and related stars studied in this paper that were part of TGAS. The red line represents the threshold \\varpi /\\sigma(\\varpi) = 3. Only stars to the right of the line are considered for inclusion in the HRD. ](parplot.png)[Fig:varpi-errpi] Parallax and its error for the sample of Ba and related stars studied in this paper that were part of TGAS. The red line represents the threshold $\\varpi /\\sigma(\\varpi) = 3$. Only stars to the right of the line are considered for inclusion in the HRD. \nThanks to the recent data release (DR1; ) of the more accurate parallaxes provided by the Tycho-Gaia Astrometric Solution (TGAS; ; ), it is possible to locate barium stars in the HRD with a much better accuracy than with the Hipparcos data. Stellar models still suffer from large uncertainties, but the Gaia mission will mark a significant step forward for the derivation of barium star masses.\nThe relation between the location of barium stars in the HRD and the orbital parameters also provides valuable insights into the binary interaction processes at play in these systems. For instance, one may expect that for barium stars located in the red clump, the distribution of orbital periods exhibits a higher lower limit caused by the large radii reached by the star at the tip of the red giant branch (RGB). One may also expect that short-period systems for which the red giant has gone through the RGB tip, show smaller or null eccentricities due to tidal circularization along the RGB.\nThe paper is organized as follows: The sample is described in Sect. [Sect:TGAS] and the HRD of barium stars and related systems is then constructed in Sect. [Sect:HRD]. We discuss the resulting mass distribution of barium giants in Sect. [Sect:masses] and we investigate the possible relationship between location in the HRD and orbital period in Sect. [Sect:P-HRD].\nThe sample\n==========\nThe sample was constructed by selecting targets with Tycho-2 identifiers in the barium star lists of and, in the list of CH and related stars of, and in the list of dwarf barium stars of and. The sample of (dwarf and giant) barium, CH and related stars with a Tycho-2 identifier amounts to 546 entries. The TGAS catalogue provides a parallax value for 400 of them. We removed from the list 11 high proper motion dwarf stars, labelled PM\\* in the SIMBAD database, for which we could not find any confirmation that they are either carbon or barium stars (e.g. HD 208998, neither a carbon star nor a Ba star according to ). The list of stars removed for that reason includes HD 11397, HD 15206, HD 24508, HD 89668, HD 108564, HD 145417, HD 153075, HD 154276, HD 161612, HD 164922, and HD 208998.\n![[Fig:Vhisto] V magnitude distribution of our stellar sample according to the presence or absence of the star in the TGAS catalogue. ](TGAS_hist.png)[Fig:Vhisto] *V* magnitude distribution of our stellar sample according to the presence or absence of the star in the TGAS catalogue. \nFig. [Fig:varpi-errpi] shows in logarithm scale the relation between the parallax ($\\varpi$) and its error ($\\sigma(\\varpi)$) for the TGAS targets. The red line represents the limit where $\\varpi /\\sigma(\\varpi) \\ge 3$, which was used as precision condition. Among the initial 389 barium and related stars with a TGAS parallax, after removing the 11 high proper motion stars, only 52 (13%) had to be rejected because they do not fulfil the precision criterion. Since this rejection rate is small, and since the analysis of these data do not significantly affect any statistical property of the sample, such as its average luminosity, there is no need to investigate how this rejection rate could bias the results. Nine more targets were excluded because their position within the Galaxy made it difficult to constrain the interstellar extinction on the line of sight, which, as explained below, prevents a reliable derivation of the stellar effective temperature.\nWhen the TGAS parallax was not available, we used the Hipparcos value, always imposing the condition that $\\varpi/\\sigma(\\varpi)$ exceeds 3. Additionally, for 21 confirmed astrometric binaries, we used the parallax rederived by instead of the TGAS (available for 15 of them) or Hipparcos values. The reason to do this for the astrometric binaries is that the TGAS parallax, obtained by applying a single star solution, is possibly disturbed by the orbital motion. The possible presence of astrometric binaries in the sample is discussed at the end of the present section. This results in 103 entries from Hipparcos and 313 from TGAS, in addition to the 21 astrometric binaries. Our final sample thus contains 437 entries. Of these, 77% are barium giants while the other samples are much smaller; i.e. 10% of the stars are barium dwarfs, 5% are CH giants, 5% are carbon stars, and the remaining 3% are CH subgiants.\n![[Fig:EB-Vfit] Error ellipse resulting from fitting the SED of HD 183915, revealing the strong correlation between T_{\\rm eff} and E_{\\rm B-V}. The red cross identifies the best fit, predicting E_{\\rm B-V} = 0.203 whereas the selective extinction derived from for that star is E_{\\rm B-V} = 0.225 (leading to A_V = 0.70). ](HD_183915.png)[Fig:EB-Vfit] Error ellipse resulting from fitting the SED of HD 183915, revealing the strong correlation between $T\\_{\\rm eff}$ and $E\\_{\\rm B-V}$. The red cross identifies the best fit, predicting $E\\_{\\rm B-V} = 0.203$ whereas the selective extinction derived from for that star is $E\\_{\\rm B-V} = 0.225$ (leading to *A**V* = 0.70). \nAs shown in Fig. [Fig:Vhisto], the major cause for a star not to be included in the TGAS catalogue is because it is brighter than *V* ∼ 6.5. A small fraction of fainter stars are missing as well. In an attempt to identify whether their absence in TGAS could be related to their astrometric binary nature, stars with a missing TGAS parallax and a known orbital period (see Jorissen et al. 2017, in preparation) were collected in Table [Tab:missing]. In that Table, stars are ordered by increasing $a\\_1/\\varpi$ (seventh column), the ratio between the angular semi-major axis of the astrometric orbit of the primary component around the centre of mass of the system and the parallax. This ratio may be estimated from Eq. (13) of, which depends only on the masses and the orbital period\n \n$$\\label{Eq:a1varpi}\n\\frac{a\\_1}{\\varpi} = P^{2/3} \\frac{M\\_2}{(M\\_1+M\\_2)^{2/3}},$$ \nwhere $a\\_1/\\varpi$ is in AU, *P* is expressed in years and masses in M⊙. This quantity corresponds in fact to the semi-major axis of the absolute orbit of the primary component. In case the astrometric orbit was detected from the Hipparcos data, the values listed in the seventh column of Table [Tab:missing] are the observed values, which correspond to the orbit of the photocentre of the system around its centre of mass. However, since the cases considered here correspond to a WD companion, the photocentric orbit is identical to that of the primary component. If the astrometric orbit is not seen in the Hipparcos data, the ratio $a\\_1/\\varpi$ is estimated from Eq. [Eq:a1varpi] by adopting *M*1 = 2.5 M⊙ and *M*2 = 0.62 M⊙, along with the observed orbital period.\nA high $a\\_1/\\varpi$ value means that the binary motion with respect to the parallactic motion is large. If, in addition, the orbital period is close to 1 yr, the parallactic and binary motions are difficult to disentangle. This could be a possible cause for the absence of a star in the TGAS catalogue. However, this is not observed because the fainter stars with high $a\\_1/\\varpi$ values absent from TGAS often have orbital periods much longer than 1 yr. Moreover, several systems with similarly high $a\\_1/\\varpi$ ratios do have TGAS parallaxes (like HD 50264, HD 87080, HD 107574...). Another possible cause for the absence of a star in the TGAS catalogue could be its colour, as very red and very blue stars were excluded. The identification of the exact reason for the absence of several faint barium and related stars has to await the availability of the next data releases when quality flags of the astrometric solution become available.\n![[Fig:FeH] Metallicity distribution of the barium and related stars with a value of [Fe/H] available in the literature. ](FeH.png)[Fig:FeH] Metallicity distribution of the barium and related stars with a value of [Fe/H] available in the literature. \nHertzsprung-Russell diagram\n===========================\nAtmospheric parameters\n----------------------\n[Sect:params]\nlllllllllllll HD/BD & TYC & *V* & *G* & *G* − *K**s* & $\\varpi$(HIP) (mas) & $a\\_1/\\varpi$ (AU) & *P* (d) & Rem \n \n \n121447 & 6140-641-1 & 7.80 & 7.13 & 2.98 & 2.2 ± 1.0 & 0.2 & 186 \n46407 & 5369-220-1 & 6.24 & - & - & 6.6${+1.3}\\atop{-1.1}$ & 0.3 & 457 & PJ,\\* \n92626 & 8201-1209-1 & 7.09 & 6.67 & 2.25 & 6.6${+0.9}\\atop{-0.6}$ & 0.5 & 918 & PJ,\\* \nNGC 2420 250 & 1373-1426-1 & 11.14 & 11.01 & 2.37& - & 0.7 & 1404 \n101013 & 3454-2188-1 & 6.12 & - & - & 7.1${+0.7}\\atop{-0.6}$ & 0.8 & 1711 & PJ,\\* \n+38 118 & 2797-46-1 & 8.86 & 8.32 & 2.76 & - & 1.4 & 3877 \n \n \n77247 & 3805-1493-1 & 6.86 & 6.54 & 1.89 & 2.9 ± 1.0 & 0.1 & 80 & \\* \n218356 & 2239-1475-1 & 4.54 & - & - & 6.1 ± 0.7 & 0.1 & 111 & \\* \n288174 & 119-1058-1 & 9.02 & - & - & 2.9 ± 1.3 & 0.8 & 1818 \n204075 & 6372-1278-1 & 3.74 & - & - & 8.6${+1.1}\\atop{-1.0}$& 1.0 & 2378 & PJ,\\* \n131670 & 4999-334-1 & 8.01 & 7.62 & 2.31 & 2.3 ± 1.2 & 1.2 & 2930 \n139195 & 933-1240-1 & 5.26 & - & - & 13.9 ± 0.7 &1.7 & 5324 &\\* \n53199 & 761-980-1 & 9.07 & 8.79 & 1.79 & 3.7 ± 1.3 & 2.3 & 8300 &\\* \n51959 & 4813-1015-1 & 8.92 & 8.61 & 2.07 & 6.5 ± 1.3 & 2.5 & 9488 &\\* \n104979 & 866-1180-1 & 4.12 & - & - & 19.1 ± 0.8 & 3.3 & 13940 &\\* \n98839 & 3015-2321-1 & 5.03 & - & - & 6.6 ± 0.6 & 3.7 & 16419 & \\* \n119185 & 5552-1079-1 & 8.91 & 8.57 & 2.02 & 3.9 ± 1.1 &4.1 & 19467 & \\* \n \n \n89948 & 6631-715-1 & 7.55 & 7.31 & 1.12 & 23.4 ± 0.9 & 0.7 & 667.8 & PJ,\\* \n76225 & 6580-2586-1 & 9.20 & - & - & 3.4 ± 1.1 &1.0 & 2411 & \\* \n98991 & 6088-2156-1 & 5.09 & - & - & 22.0 ± 0.8 &1.1 & 2834 & \\* \n221531 & 5832-970-1 & 8.36 & 8.21 & 0.98 &9.6${+1.4}\\atop{-1.3}$ & 1.2 & 1416 & PJ,\\* \n95241 & 3012-2522-1 & 6.03 & - & - & 22.0 ± 0.8 &1.8 & 5448 & \\* \n \n![[Fig:LandT] Luminosity and T_{\\rm eff}(and their 1\\sigma error bar) derived from SED fitting with models of solar metallicity and metallicity [Fe/H] = -0.5. ](T_MARCS.png \"fig:\")\n![[Fig:Pastel] Effective temperatures obtained with the SED fitting method compared with those obtained from high-resolution spectra collected from the PASTEL catalogue. The dashed red lines indicate a \\pm200 K uncertainty. ](PastelPaper.png)[Fig:Pastel] Effective temperatures obtained with the SED fitting method compared with those obtained from high-resolution spectra collected from the PASTEL catalogue. The dashed red lines indicate a ± 200 K uncertainty. \nThe atmospheric parameters of barium and related stars were derived by modelling the spectral energy distribution (SED) obtained by collecting magnitudes listed in the Simbad database. The best-fitting MARCS model was determined from a parameter-grid search using a *χ*2 minimization method (see for details). The stellar temperature was then assigned from the best-fitting model and the luminosity was obtained by integrating the SED over all wavelengths spanned by the model, and applying the distance modulus derived from the parallax. The error bar on the luminosity is propagated from the parallax uncertainty and is thus asymmetric. The error on the temperature is the 1-sigma error enclosing 67% of the model fits (see Fig. [Fig:EB-Vfit]).\nThe reddening $E\\_{\\rm B-V}$ was initially left free during the fitting process. Its best value was estimated by looking for the amount of reddening to be applied to the MARCS models to match the observed magnitudes. A value of 3.1 was used for the ratio of the total to the selective extinction $R = A\\_V/E\\_{\\rm B-V}$, from which *A**V* is derived. It turned out that the temperature and extinction derived in this manner are strongly correlated, as shown in Fig. [Fig:EB-Vfit].\nIn a second run, the extinction was fixed at the value computed by, in his three-dimensional map of the extinction within the nearest kiloparsec. The location of the target in the Galaxy was computed from its galactic coordinates and its parallax. The resulting $T\\_{\\rm eff}$ and $E\\_{\\rm B-V}$ values were often found to fall at the lower edge of the 1*σ* ellipse error. We decided to let $E\\_{\\rm B-V}$ vary freely between 0 and Gontcharov’s value plus 0.07 mag, which is the 1*σ* error bar suggested in their paper.\nAnother parameter that has an impact on the determination of the stellar parameters is the metallicity. As they are difficult to constrain by fitting SEDs, metallicities of barium and related stars were collected from the literature and their distribution is presented in Fig. [Fig:FeH]. The barium-giant distribution peaks at [Fe/H] ∼ − 0.14 with a standard deviation of 0.2. determined the metallicity of 182 Ba giants and obtained a similar mean value of [Fe/H] ∼ − 0.12 ± 0.14. As expected, CH stars have lower metallicities, and it seems that barium dwarfs may be slightly more metal poor than barium giants as well, but the dwarf sample is much smaller. We decided to use MARCS models with fixed [Fe/H] = − 0.25 to determine the stellar parameters. To evaluate the impact of the metallicity on the derived parameters, two more SED fits have been carried out: one imposing a solar metallicity and another one imposing [Fe/H] = − 0.5. The resulting $T\\_{\\rm eff}$ and luminosities of the limiting cases are presented in Fig. [Fig:LandT], and reveal that the uncertainty introduced on the stellar parameters by the metallicity of the best-fitting MARCS model is negligible. The impact of the metallicity on the stellar tracks and hence on the derived mass is of course much larger and is discussed in Sect. [Sect:masses].\nFinally, in order to evaluate our method, we include (Fig. [Fig:Pastel]) a comparison of the effective temperatures, which we obtained for the Ba giants with the SED fitting method with temperatures for the same stars determined from high-resolution spectra. We collected the latter values from the PASTEL catalogue. Fig. [Fig:Pastel] shows that most of the 102 Ba giants that we compare lie within the ± 200K uncertainty, indicated by the dashed red lines. All the parameters described so far are listed in Table [Tab:master].\nInput physics for the stellar grid calculations\n-----------------------------------------------\nTo determine the fundamental parameters of our sample stars from their location in the HR diagram, we computed extended grids of stellar models covering the mass range 0.6 ≤ *M*zams/M⊙ ≤ 6. All the models were evolved from the pre-main sequence up to the end of the AGB phase for stars of initial mass *M*zams ≤ 4M⊙ and up to the occurrence of convergence problems in the higher mass tracks. The models were computed with the STAREVOL code with the following input physics. We used the radiative opacity tables of above 8000 K and of at lower temperatures. We took the modifications of the opacity due to the formation of H2, H2O, OH, C2, CN, and CO molecules in the atmosphere of C-rich stars (C/O>1) into account following the formulation of. The nuclear network includes 182 reactions coupling 55 species from H to Cl. The nuclear rates have been recently updated with NACRE II compilation. We used the mixing length theory to determine the temperature gradient in the convective regions with *α*MLT = 1.75 and adopt the Schwarzschild criterion. For the mass loss rate, we considered the prescription up to the beginning of the AGB phase and then switched to the formulation. We also included some overshooting at the base of the convective envelope, following the exponential decay expression of with *f*over = 0.1. Finally we used a grey atmosphere surface boundary condition.\nDiscussion\n----------\n![[Fig:HRDStarevol] Hertzsprung-Russell diagram for sample of barium and related stars labelled as in queried catalogues. Green crosses indicate barium giants; pink squares indicate barium dwarfs; black dots indicate CH giants; and blue triangles indicate subgiant CH stars. Typical error bars are shown over the legend for two situations: a favourable case (left: \\varpi /\\sigma(\\varpi) = 8), and limiting case for \\varpi /\\sigma(\\varpi) = 3 (right). Stellar tracks from the STAREVOL code were overplotted for models of masses 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0 M_{\\odot}, as labelled and metallicity [Fe/H] = -0.25. The solid black tracks correspond to the post-main-sequence evolution up to the tip of the RGB (low-mass stars) or to the onset of core He-burning (intermediate-mass stars), dotted blue tracks correspond to core He-burning, and dashed red tracks to the early and thermally pulsing AGB (TP-AGB) phase. ](HRDm25.png)[Fig:HRDStarevol] Hertzsprung-Russell diagram for sample of barium and related stars labelled as in queried catalogues. Green crosses indicate barium giants; pink squares indicate barium dwarfs; black dots indicate CH giants; and blue triangles indicate subgiant CH stars. Typical error bars are shown over the legend for two situations: a favourable case (left: $\\varpi /\\sigma(\\varpi) = 8$), and limiting case for $\\varpi /\\sigma(\\varpi) = 3$ (right). Stellar tracks from the STAREVOL code were overplotted for models of masses 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0 M⊙, as labelled and metallicity [Fe/H] = − 0.25. The solid black tracks correspond to the post-main-sequence evolution up to the tip of the RGB (low-mass stars) or to the onset of core He-burning (intermediate-mass stars), dotted blue tracks correspond to core He-burning, and dashed red tracks to the early and thermally pulsing AGB (TP-AGB) phase. \nFig. [Fig:HRDStarevol] shows the Ba and related star sample in the HRD with parameters obtained as described in Sect. [Sect:atmosphere], i.e. leaving the extinction free (between 0. and Gontcharov’s value), fixing the metallicity of the atmospheric models to [Fe/H] = − 0.25, and by restricting the sample to stars with $\\varpi /\\sigma(\\varpi) \\ge 3$. We kept the original category (Ba giant, Ba dwarf, CH giant, or CH subgiant) given in the queried catalogues for each star shown in Fig. [Fig:HRDStarevol]. Typical error bars are shown for two situations: a favourable case with $\\varpi /\\sigma(\\varpi) = 8$ (left) and limiting case with $\\varpi /\\sigma(\\varpi) = 3$ (right). In the latter case, the asymmetric nature of the errors on the luminosity starts to be noticeable, so that biases are manifested (e.g. ). However, as apparent on Fig. [Fig:varpi-errpi], the number of targets with $\\varpi /\\sigma(\\varpi) < 3$ is not large, and a discussion on the effect of the biases does not seem to be required here.\nEvolutionary tracks for [Fe/H] = − 0.25 have been superimposed for models of masses 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0 M⊙, as labelled. Fig. [Fig:HRDFeH] shows the impact of metallicity on the 2.0 M⊙ and 3.0 M⊙ tracks (blue and red tracks, respectively). It appears that, for the RGB and the region occupied by the red clump, the [Fe/H] = − 0.5 track of 2.0 M⊙ covers the same region as the [Fe/H] = 0 track of 3.0 M⊙, leading to a strong degeneracy in the mass determination of a given star, which can only be lifted by knowing its metallicity.\nAnother important source of uncertainties is in the physics of the evolutionary models. Figure [Fig:tracks] shows the difference among three sets of evolutionary tracks. The solid black tracks were computed with STAREVOL as described in Sect. [Sect:STAREVOL] using the solar composition corresponding to Z⊙=0.0134. The dotted green lines show the Geneva models from for solar composition as well, but with Z⊙=0.014. Finally, the dashed red tracks correspond to the tracks from for solar metallicity with Z⊙=0.019. Apart from the scatter at the zero-age main sequence caused by the diverse solar metallicity values, varying descriptions of convection, overshooting, rotation, mass loss, etc., lead to uncertainties in the models in addition to our observational uncertainties. However, comparing evolutionary models is beyond the scope of this publication. Finally, barium stars are post-mass-transfer objects which, while on the main sequence, accreted mass from an AGB companion. This effect, which is not taken into account in the single-star evolutionary models that we use for comparison, could also affect their evolution.\nA first interesting result apparent in the HRD of Fig. [Fig:HRDStarevol] is that several stars previously classified as Ba giants need to be reclassified as they are in fact dwarfs or subgiants. These are listed in Table [Tab:Bad] and were selected from the criterion *L* ≤ 10L⊙.\nWe also report that most of subgiant CH stars (blue triangles) populate the region of the HRD associated with Ba dwarfs. Some of these stars, located around the 1.0 M⊙ track, seem to lie on the main sequence, contrary to what is implied by their designation as subgiants. This could have been guessed from the high gravities (log*g* ≥ 4.0) inferred by in their spectroscopic analysis of subgiant CH stars. Other subgiant CH stars seem to be genuine subgiants, but they fall amidst stars classified as Ba dwarfs. The exact designation of these stars depends on the evolutionary models used for comparison, as the extension of the main sequence depends on the chosen overshoot parameter and the consideration or not of internal rotation. However, it seems clear that there is no clear separation between Ba dwarfs and CH subgiants in the HRD, so that these designations should not be taken literally.\n![[Fig:HRDFeH] Same as Fig. [Fig:HRDStarevol] with the 2.0 and 3.0 M_{\\odot} (blue and red, respectively) evolutionary tracks (not covering the AGB) computed with [Fe/H] = 0.0 (left) and [Fe/H] = -0.5 (right). In each panel the solid tracks correspond to the indicated metallicity, which is also used to derive the atmospheric parameters of the targets; the dashed tracks correspond to the other metallicity ([Fe/H] = -0.5 and 0.0 in the left and right panel, respectively). ](HRD_Zcomp.png)[Fig:HRDFeH] Same as Fig. [Fig:HRDStarevol] with the 2.0 and 3.0 M⊙ (blue and red, respectively) evolutionary tracks (not covering the AGB) computed with [Fe/H] = 0.0 (left) and [Fe/H] = − 0.5 (right). In each panel the solid tracks correspond to the indicated metallicity, which is also used to derive the atmospheric parameters of the targets; the dashed tracks correspond to the other metallicity ([Fe/H] = − 0.5 and 0.0 in the left and right panel, respectively). \n![[Fig:tracks] Comparison among STAREVOL (solid black lines), Geneva (dotted green lines), and Padova (dashed red lines) evolutionary tracks up to the beginning of the AGB phase. ](comparingtracks.png)[Fig:tracks] Comparison among STAREVOL (solid black lines), Geneva (dotted green lines), and Padova (dashed red lines) evolutionary tracks up to the beginning of the AGB phase. \nllp0.3p0.3 Name & TYC & Ref. & Rem. \nHD 8270 & 8036-564-1 &, dwarf nature confirmed by \nHD 13551 & 8851-37-1 &, dwarf nature confirmed by \nHD 22589 & 4722-19-1 &, dwarf nature confirmed by & orbit available \nCpD -44 5038 & 7735-447-1 & \nBD -10 4311 & 5630-641-1 & & orbit available \nHD 197481 & 7457-641-1 & \nCS 22180-0013 &5279-303-1 & Bartkevicius \nHIP 19050 & 1814-348-1 & Bartkevicius, classified as R \nHD 175179 & 5123-323-1 & Bartkevicius PM\\*, moderate enhancement (0.2 – 0.3 dex of Y and Ba) \nA strong concentration of barium-rich stars is also found in the red clump, as is further discussed in Sect. [Sect:masses].\nMass distribution\n=================\nWe determine the masses from a comparison with stellar evolutionary tracks from the STAREVOL code. Three different metallicities have been considered: [Fe/H] = − 0.5, − 0.25, and 0. To ensure self-consistency, when comparing the location of the target stars in the HRD with these tracks, we use the stellar parameters derived from the SED fit associated with the same metallicity (Sect. [Sect:atmosphere] and Table [Tab:master]).\nTo derive stellar masses, we use a statistical approach that takes into account the time that a star spends at a given location in the HRD. We first interpolated all the evolutionary tracks using a constant time step of 10,000 yrs. This interpolation thus produces more points in a long-lasting phase (such as the main sequence or the red clump). This allows us to fill the HRD with many points (around 17 × 106), each characterized by a value for $T\\_{\\rm e}$, *L*, and mass *m**T*, *L*; our method takes into account the mass lost by the star when ascending the RGB. For a given star, with a known temperature *T**o* and luminosity *L**o*, along with their associated errors (*σ**T**o*, *σ**L**o*), we then estimate its mass, *m**o*, by computing a Gaussian average of the masses for all the points located in a restricted neighbourhood around *T**o*, *L**o*. More specifically, the mass of the star is given by\n \n$$m\\_o = \\frac{1}{2 \\pi \\sigma\\_{To} \\sigma\\_{Lo}} \\sum\\_{T\\_e} \\sum\\_L \\frac{m\\_{T,L}^2}{\\rho\\_{T,L}} e^{-\\frac{(T\\_e-T\\_o)^2}{2 \\sigma^2\\_{To}}}e^{-\\frac{(L-L\\_o)^2}{2 \\sigma^2\\_{Lo}}},$$ \nwhere *ρ**T*, *L* is a normalization factor given by\n \n$$\\rho\\_{T,L} = \\frac{1}{2 \\pi \\sigma\\_{To} \\sigma\\_{Lo}} \\sum\\_{T\\_e} \\sum\\_L m\\_{T,L} e^{-\\frac{(T\\_e-T\\_o)^2}{2 \\sigma^2\\_{To}}}e^{-\\frac{(L-L\\_o)^2}{2 \\sigma^2\\_{Lo}}}.$$ \nWe thereby derive the mass for all our objects, for the three computed metallicities. The distribution for giant barium stars (thus excluding barium dwarfs, CH stars, and CH subgiants) is shown in Fig. [Fig:massesSTAREVOL] to highlight the sensitivity on the metallicity: if one adopts the solar metallicity for all the targets, the mass distribution exhibits a modest peak at 2.8 M⊙, and this peak is superimposed on a broader distribution extending from 2.5 to 5.5 M⊙. Adopting instead a lower metallicity [Fe/H] = − 0.5 for all targets, the peak of the mass distribution is better marked and shifted towards lower masses (2.3 M⊙). As shown in Fig. [Fig:FeH], the metallicity distribution of barium stars peaks at [Fe/H] ∼ − 0.14. Hence, we conclude that the mass distribution obtained by adopting the average metallicity of -0.25 should be close to reality. In that case, the distribution can be described by two Gaussians: a main peak at 2.5 M⊙ with a standard deviation of 0.18 M⊙ and a broader tail at higher masses (up to 4.5 M⊙), which peaks at 3 M⊙ with a standard deviation of 1 M⊙.\n![[Fig:massesSTAREVOL] Mass distribution for giant barium stars (thus excluding barium dwarfs, CH stars, and CH subgiants), from a comparison with the STAREVOL evolutionary tracks, for the three metallicities [Fe/H] = 0, [Fe/H] =-0.25, and [Fe/H] =-0.5. Adopting T_{\\rm eff} \\pm \\sigma_{\\rm Teff} and L \\pm \\sigma_L, we determine the red (-) and green (+) histograms. The vertical dashed line indicates the peak of the [Fe/H] =-0.25 distribution. ](Badistr_py.png)[Fig:massesSTAREVOL] Mass distribution for giant barium stars (thus excluding barium dwarfs, CH stars, and CH subgiants), from a comparison with the STAREVOL evolutionary tracks, for the three metallicities [Fe/H] = 0, [Fe/H] = − 0.25, and [Fe/H] = − 0.5. Adopting $T\\_{\\rm eff} \\pm \\sigma\\_{\\rm Teff}$ and *L* ± *σ**L*, we determine the red (-) and green (+) histograms. The vertical dashed line indicates the peak of the [Fe/H] = − 0.25 distribution. \n![[Fig:HRDFamaey] Distribution of a comparison sample of K and M giants from in the HRD. Data are presented as a contour density plot. STAREVOL evolutionary tracks for stars with initial masses 1, 2, 3, 4, 5, and 6 M_{\\odot} (from bottom to top) and [Fe/H] = -0.25 are superimposed; white colours correspond to Hertzsprung-gap and RGB evolution, green colours to core He-burning, and red to AGB evolution. Stars from the Y group correspond to the faint extension of the stellar density in the regions covered by the 4, 5, and 6 M_{\\odot} evolutionary tracks. ](PP44_Teff_MV_STAREVOL_FeHm25_contour.png)[Fig:HRDFamaey] Distribution of a comparison sample of K and M giants from in the HRD. Data are presented as a contour density plot. STAREVOL evolutionary tracks for stars with initial masses 1, 2, 3, 4, 5, and 6 M⊙ (from bottom to top) and [Fe/H] = -0.25 are superimposed; white colours correspond to Hertzsprung-gap and RGB evolution, green colours to core He-burning, and red to AGB evolution. Stars from the Y group correspond to the faint extension of the stellar density in the regions covered by the 4, 5, and 6 M⊙ evolutionary tracks. \n![[Fig:massPP44] Same as Fig. [Fig:massesSTAREVOL] for the comparison sample of K and M giants (solid red line) from. The hatched histogram corresponds to barium stars. ](FaBanew_py.png)[Fig:massPP44] Same as Fig. [Fig:massesSTAREVOL] for the comparison sample of K and M giants (solid red line) from. The hatched histogram corresponds to barium stars. \nComparison with M- and K-type giants\n------------------------------------\nWe compare the mass distribution of barium giants with the sample of 5952 K and 739 M giants from. These authors performed a Bayesian classification of this large sample into different kinematical groups, based not only on their kinematics (Tycho-2 proper motions and CORAVEL radial velocities; ;, respectively) but also on their luminosity, which is derived from a Bayesian estimate based on the Hipparcos parallax. The B (‘Background’) group constitutes the smooth velocity ellipsoid, whereas the Y (‘Young’) group exhibits all signatures of young stars, i.e. small velocity dispersions and scale height. The average metallicity of stars belonging to the B group was estimated to be about − 0.2 from Fig. 15 of and − 0.12 ± 0.18 from as quoted by. The close match between the average metallicities of barium giants (Fig. [Fig:FeH]) and the B group of comparison giants is encouraging, since this match indicates that B group giants are an appropriate sample to compare with the barium giants. To locate B-group giants in the HRD, it was necessary to use various calibrations from, notably (*V* − *I*, *V* − *K*) and (*V* − *K*, $T\\_{\\rm eff}$) to convert Hipparcos *V* − *I* indices into temperatures, and (*V* − *K*, *B**C**K*) to convert visual absolute magnitudes into luminosities. The resulting HRD is shown in Fig. [Fig:HRDFamaey] and indicates a strong concentration of solar-metallicity stars with masses in the range 2 – 3 M⊙in the red clump.\nDiscussion\n----------\nThe derived mass distribution (Fig. [Fig:massPP44]) reveals that the samples of field K and M and barium giants behave similarly in terms of mass, except for a deficit of low-mass stars ($M \\la 2.0$ M⊙, the exact threshold value depending on metallicity) among barium stars. In the mass distribution of the comparison sample, the Y group populates the high-mass tail, whereas the B group is associated with the lower mass population, including the strong peak around 2.4 M⊙ with a standard deviation of 0.26 M⊙ (for [Fe/H] = − 0.25), where stars residing in the red clump accumulate. have also found a peak around 2.3 M⊙ in the mass distribution of a sample of binary red giants in open clusters. The position of the peak should therefore not be considered as a distinctive feature of barium giants.\nThe deficit of low-mass giants ($M \\la 2.0$ M⊙) could, however, be specific to Ba stars. One could think that it is caused by the longer evolutionary timescales needed to form a low-mass barium star whose companion could then also be of low mass. However, this argument does not hold because a star with $M \\gtrsim 1.3$ M⊙ reaches the tip of the AGB within 5 Gyr and has time to pollute its companion and form a low-mass Ba star of solar metallicity. Additionally, low-mass Ba stars do not necessarily have low-mass companions. Neither can we claim that the deficit of low-mass barium stars is attributed to the absence of third dredge-up in low-mass ($M \\lesssim$ 1.3 M⊙) AGB stars because evidence for s-process enrichment in stars with such low masses has been suggested based on the analysis of post-AGB stars. The most probable explanation for the deficit of low-mass Ba giants is that they reached a large radius at the tip of the RGB, resulting in a premature shrinkage of the orbit caused by a unstable Roche-lobe overflow (unless they had a very long orbital period). We elaborate on the method to determine $R\\_{\\rm RGB tip}$ in Sect. [Sect:P-HRD].\n![[Fig:Bamm] Comparison of the mass distributions for 164 mild (Ba1 – Ba2) and 110 very mild (Ba0) barium stars (red and black curves, respectively) computed using evolutionary tracks with [Fe/H] = -0.25. Very mild barium stars are defined as having a Ba index smaller than 1 in the catalogue of. Upper panel: Normalized histograms are shown. Lower panel: Cumulative frequency distributions are shown with the maximum absolute difference between these distributions amounting to 0.168, corresponding to a first-kind error of 13% when rejecting the null hypothesis of identical distributions in a Kolmogorov-Smirnov test with an effective sample size of 48.2 [= 164 \\times 110 / (164 + 110)]. ](bamm_bam_py.png)[Fig:Bamm] Comparison of the mass distributions for 164 mild (Ba1 – Ba2) and 110 very mild (Ba0) barium stars (red and black curves, respectively) computed using evolutionary tracks with [Fe/H] = − 0.25. Very mild barium stars are defined as having a *Ba index* smaller than 1 in the catalogue of. Upper panel: Normalized histograms are shown. Lower panel: Cumulative frequency distributions are shown with the maximum absolute difference between these distributions amounting to 0.168, corresponding to a first-kind error of 13% when rejecting the null hypothesis of identical distributions in a Kolmogorov-Smirnov test with an effective sample size of 48.2 [ = 164 × 110/(164 + 110)]. \nAmong the barium-star sample, the high-mass tail is dominated by stars with a barium index, based on a visual inspection of the strength of the barium lines in the spectrum, lower than unity (named Ba0 in this paper), i.e. with a very moderate (if any) s-process enhancement (Fig. [Fig:Bamm]). Some of these stars, which appeared in the second edition (1991) of the barium-star catalogue of L\" u, could be bright giants, or even supergiants, where the barium lines are strengthened as a simple positive luminosity effect (i.e. an increase of the line strength with increasing luminosity) instead of being genuine barium stars.\nFinally, we re-investigate the statement by and that mild (Ba1-Ba2) and strong (Ba3-Ba5) barium stars have somewhat different mass distributions, as they are characterized by masses in the range 2.5 – 4.5 M⊙ and 1 – 3 M⊙, respectively. As shown by Fig. [Fig:basm], our analysis does not support this claim, since the mass distributions of 164 mild and 50 strong barium giants are undistinguishable under the assumption that their average metallicities are similar. The maximum absolute difference between the two distributions is 0.091, which for an effective sample size of 38.3 ( = 164 × 50/214) corresponds to a very large first-kind error of 91% for the Kolmogorov-Smirnov test.\n![[Fig:basm] Comparison of the mass distributions for mild and strong barium stars (red and black curves, respectively) computed from evolutionary tracks for [Fe/H] = -0.25. Upper panel: Normalized histograms are shown. Lower panel: Cumulative frequency distributions with the maximum absolute difference between them amounting to 0.091. ](bas_bam_py.png)[Fig:basm] Comparison of the mass distributions for mild and strong barium stars (red and black curves, respectively) computed from evolutionary tracks for [Fe/H] = − 0.25. Upper panel: Normalized histograms are shown. Lower panel: Cumulative frequency distributions with the maximum absolute difference between them amounting to 0.091. \nLocation in the HRD and orbital period\n======================================\nFig. [Fig:HRP] is a first attempt to correlate the location of a binary system in the HRD with its orbital period. We only include barium star systems for which the orbital period is known, and make the size of the symbols proportional to this value. We use the periods determined by and Jorissen et al. (in preparation) for the giants (see also Table [Tab:master]) and Escorza et al. (in preparation) for the main sequence and subgiant stars. We still keep the original labels they had in the queried catalogues and in Fig. [Fig:HRDStarevol].\nThere seems to be a tendency for larger orbital periods in the red clump, as compared to systems lying below (i.e. with luminosities from the main sequence up to the red clump). In an attempt to quantify this effect, the sample has been split into two subsamples: one with *L*/L⊙ ≤ 25 (log*L*/L⊙ ≤ 1.4) and the other with 25 < *L*/L⊙ ≤ 160 (1.5 < log*L*/L⊙ ≤ 2.2). The first sample comprises main-sequence and subgiant stars whose current primary components never evolved up the RGB. The second sample, however, comprises systems in the red clump after their passage through the RGB tip, at least for systems with low-mass primaries, and a few less evolved objects, which happen to cross the red clump on their way to the RGB tip. Given the large radius reached at the RGB tip (see below), one may suspect that in the shorter period systems the giant filled its Roche lobe, and possibly evolved through a common-envelope stage into a cataclysmic variable or even a coalesced pair. Fig. [Fig:histoP] confirms that very few systems (6/36 = 17%) located in the red clump (green) have periods shorter than 1000 d and none of the systems have a period below 300 d. Compared to the less evolved systems (pink), there is indeed a deficit of short-period systems among red clump stars.\n![[Fig:HRP] Same as Fig. [Fig:HRDStarevol], but with the symbol size proportional to the orbital period. ](HRD_Pscale.png)[Fig:HRP] Same as Fig. [Fig:HRDStarevol], but with the symbol size proportional to the orbital period. \n![[Fig:histoP] Normalized histogram (left scale, with bins of size 0.5 in log scale) and cumulative frequency distribution (right scale) for orbital periods of barium and related systems. Systems with L/L_{\\odot} \\le 25 are depicted in pink, and systems with 25 < L/L_{\\odot} \\le 160) are depicted in green.](Phisto.png)[Fig:histoP] Normalized histogram (left scale, with bins of size 0.5 in log scale) and cumulative frequency distribution (right scale) for orbital periods of barium and related systems. Systems with *L*/L⊙ ≤ 25 are depicted in pink, and systems with 25 < *L*/L⊙ ≤ 160) are depicted in green.\nTo apply the Kolmogorov-Smirnov (K-S) test, the maximum difference between the two cumulative frequency distributions was evaluated and amounts to *D* = 0.2, for an effective sample size of *N* = 18 = *m* × *n*/(*m* + *n*), where *m* = 35 and *n* = 36 are the sizes of the pre-RGB and red clump samples, respectively. The resulting effective difference *D**N*1/2 = 0.85 corresponds to a first-risk error of 47% to reject the null hypothesis incorrectly. Thus, the K-S test does not give us sufficient evidence to reject the null hypothesis and to conclude that the two distributions are different. However, it is known that the K-S test tends to be more sensitive near the centre of the distribution, while our samples seem to differ in the tails. Moreover, since the second-kind error has not been assessed because of the risk of incorrectly accepting the null hypothesis, the above result does not constitute a definite proof that the samples are extracted from the same parent population.\nFor this reason, we turned to another test (“four-box test”) relying on the hypergeometric probability distribution. We divided our two samples (pre-clump and clump) into two subsamples with periods shorter and longer than 1000 days. Among the 35 pre-clump systems, 20 have short periods and 15 have long periods. Among the 36 ( ≡ *N*1) systems in the clump, 30 have short periods and 6 have long periods. Hence, among the 71 ( ≡ *N*) binaries, 21 binaries have short periods, so the probability of finding a short-period binary is *p* = 21/71 = 0.3. From this, we can now calculate the expected number of short-period objects among the clump sample as *x̃* = *p* × *N*1 = 10.8 instead of 6 that are observed. In order to check whether or not this difference is significant, we compute the variance on *x̃* as follows from the hypergeometric probability distribution (“small-sample statistics”): \n$\\sigma^{2}\\_{\\tilde x} = N1 \\times p \\times (1-p) \\times \n\\frac{N-N1}{N-1} = 3.78,$ i.e. $\\sigma\\_{\\tilde x} = 1.9$. \nHence, the expectation is *x̃* = 10.8 ± 1.9 or 8.9 < *x* < 12.7 at 1 *σ*, meaning that the pre-clump and clump samples are different with a 2.5*σ* significance.\nWe show in Table [Tab:RGB] that the 1000 d period corresponds to the period threshold below which a $\\backsim$1M⊙ Ba dwarf fills its Roche lobe when it reaches the RGB tip, and that the less populated 400 – 1000 d period range can be associated with the period cut-offs for stars in the mass range [1.0 – 2.5] M⊙. The critical periods for Roche-lobe overflow (RLOF) listed in Table [Tab:RGB] are derived as follows, assuming that the giant star fills its Roche lobe ($R\\_{\\rm L\\_{1}}$) at the RGB tip (with radius $R\\_{\\rm tip}$):\n \n$$R\\_{L\\_{1}} = a\\;\\frac{0.49 q^{2/3}}{0.6 q^{2/3} + \\ln( 1 + q^{1/3})} \\equiv a\\;f = R\\_{\\rm tip},$$ \nwhere *q* = *M*1/*M*2, with *M*1 the mass of the giant star, and *M*2 that of its WD companion. The critical semi-major axis of a circular orbit is then expressed as $a = R\\_{\\rm L\\_{1}}/f = R\\_{\\rm tip}/f$ and the corresponding period cut-off is\n \n$$\\label{Eq:Pcut}\nP = \\frac{(R\\_{\\rm tip} / f)^{3/2}} {(M\\_1+M\\_2)^{1/2}},$$ \nwhere *P* is expressed in years, $R\\_{\\rm tip}$ in AU (and is identified with the Roche radius $R\\_{\\rm L\\_{1}}$ around star 1), and the component masses *M*1, 2 in M⊙. The radius $R\\_{\\rm tip}$ is given by the STAREVOL models. A direct comparison between this predictions and the observed values is not straightforward. The periods entering the above equation are pre-mass-transfer values, while the observed periods are the post-mass-transfer values. A detailed discussion of the relationship between initial and final periods is beyond the scope of this paper (see for example, or ). Nevertheless, this analysis offers an order of magnitude estimate, although it should be considered with caution.\nThe absence of short-period barium stars in the red clump strongly suggests that RLOF on the RGB leads to systems that disappear from the barium star family. They could be the progenitors of short-period cataclysmic variables, or mergers. It is also tempting to associate these RGB-coalesced systems with early R-type carbon stars, which are another family of peculiar stars with a strong concentration in the red clump, and none of these stars are members of a binary system. The major problem with the hypothesis of a link between barium and R stars, however, is the lack of s-process element overabundances in early R-type carbon stars and of 12C in Ba stars.\nThe segregation in terms of orbital periods between pre-red clump and red clump systems requires further discussion, because all these barium systems host a WD whose progenitor went through the thermally pulsing AGB (TP-AGB) phase. We should in principle expect a cut-off in the period distribution of the barium dwarf systems as well because AGB stars also reach very large radii.\nIn fact, three different critical periods must thus be considered: (i) $P\\_{\\rm RGB, 1}$, when the initially more massive component (now a WD) reaches the RGB tip, i.e. when in the above equations, $M\\_1 = M\\_{\\rm 1@RGBtip}$ and $M\\_2 = M\\_{\\rm Ba}$; (ii) $P\\_{\\rm AGB, 1}$, when the initially more massive component reaches the AGB tip, i.e. when $M\\_1 = M\\_{\\rm 1@AGBtip} = M\\_{\\rm WD}$ and $M\\_2 = M\\_{\\rm Ba}$; and (iii) $P\\_{\\rm RGB, 2}$, when the initially less massive component (the current barium star) reaches the RGB tip, its companion being a WD, i.e. when $M\\_1 = M\\_{\\rm Ba}$ and $M\\_2 = M\\_{\\rm WD}$. A supplementary condition is that the barium-star mass ($M\\_{\\rm Ba}$) is lower than the initial mass of its companion, the WD progenitor; we neglected the mass transferred onto the barium star to make the computations easier. Periods $P\\_{\\rm RGB, 1}$ and $P\\_{\\rm AGB, 1}$ are shown in Fig. [Fig:Pcrit], whereas $P\\_{\\rm RGB, 2}$ are listed in Table [Tab:RGB] (assuming $M\\_2 = M\\_{\\rm WD}$ = 0.6 M⊙). There is a weak dependence of $P\\_{\\rm RGB, 1}$ on the initial mass ratio.\n[Fig:Pcrit] Critical periods below which one of the components fills its Roche lobe at either the tip of the RGB ($P\\_{\\rm RGB, 1}$, see text; dashed line at the bottom of the figure) or at the tip of the AGB ($P\\_{\\rm AGB, 1}$, solid line at the top of the figure). The input data $R\\_{\\rm RGB,tip}$ and $R\\_{\\rm AGB,tip}$ are taken from the STAREVOL grid (red curves solar metallicity *Z* = 0.0134; black curves: [Fe/H] = − 0.5 or *Z* = 0.0043). Various superimposed lines in a series correspond to various barium-star masses. The corresponding $M\\_{\\rm Ba}$ value may be identified from the starting point of the curve, which then extends towards increasing masses; for a given barium-star mass, the AGB component of mass $M\\_{\\rm AGB}$ – the former primary of the binary system – is necessarily larger. The critical period $P\\_{\\rm RGB, 2}$ (see text) does not strongly depend upon the mass of the WD companion. Therefore, its value can simply be read off the $P\\_{\\rm RGB, 1}$ curve (bold dashed line) at the corresponding $M\\_{\\rm Ba}$ value, which determines the key value $R\\_{\\rm RGB,tip}$. Although $P\\_{\\rm RGB, 1}$ and $P\\_{\\rm RGB, 2}$ are not strictly equal, they are not different enough to warrant a specific $P\\_{\\rm RGB, 2}$ curve that would jeopardize the clarity of the figure (see also Table [Tab:RGB]). This is a corrected version of the original Fig. 17.\nWe clearly see that RLOF at the RGB tip occurs for all systems with a low-mass ($M \\lesssim 2.2$M⊙) primary and periods shorter than 110 d (and shorter than 1000 d for 1 M⊙ stars; see also Table [Tab:RGB]). This period threshold is consistent with the orbital-period segregation observed in the barium stars. On the other hand, intermediate-mass stars ($2.2 \\lesssim M$/M$\\_{\\odot} \\lesssim 8 $) do not expand as much as their lower mass counterparts and the critical period is too short to affect the barium star population.\nAnother interesting conclusion may be drawn from the other (apparent) puzzle emerging from the present analysis; that is how the long-period thresholds (600 to 7500 d; $P\\_{\\rm AGB, 1}$ in Fig. [Fig:Pcrit]) imposed by the passage of the WD progenitor through the AGB tip can be reconciled with the much shorter periods currently observed among barium systems. This is the long-standing puzzle identified by, for example,, and. Most barium systems should thus have gone through RLOF at the AGB tip, but this RLOF cannot have led to a catastrophic orbital shrinkage. In that case, we seek to understand why the barium systems seemingly disappeared after RLOF occurring on the RGB, but not after RLOF occurring along the AGB of the former primary and current WD. The key difference between RLOF close to AGB and RGB tip is the mass of the residual envelope, which is generally much larger at the RGB tip. Hence, RLOF close to AGB tip does not involve a massive envelope, and we may expect a modest orbital shrinkage at that stage. Alternatively, if RLOF starts when the mass ratio is close to, or below, unity (a likely possibility with a stripped AGB star), a common evolution can be avoided, since the orbital separation expands soon after the beginning of mass transfer. This moderate shrinkage could then account for the fact that the observed periods for RGB systems shown in Fig. [Fig:histoP] are shorter than the AGB threshold periods indicated in Fig. [Fig:Pcrit]. The possibility of intense mass loss and angular momentum loss on the AGB should also be considered (e.g. ; ).\nAn interesting conclusion ensues by applying the previous arguments to the early stage of the binary system, when the former primary ascended the RGB with the future barium star still on the main sequence. For such systems with an orbital period shorter than $P\\_{\\rm RGB, 1}$, RLOF with dramatic orbital shrinkage must have occurred, thus probably removing from the barium-star family those systems that had an initial period shorter than 1000 d with a primary mass of 1 M⊙ or shorter than 200 d with a primary mass of 2.2 M⊙. Stars in the mass range 1 – 2.2 M⊙ would have led to a WD with a mass below 0.58 M⊙, according to the initial - final mass relationship inferred from the STAREVOL tracks. Therefore, we may expect a deficit of low-mass WDs among barium systems. This issue will be investigated in a forthcoming paper.\nllllllll $M\\_{\\rm now}$ & $M\\_{\\rm initial}$ &log(*L*/L⊙) & $R\\_{\\rm RGB tip}$ & *a*& *P* \n(M⊙) & (M⊙) & & (R⊙) & (R⊙) & (d) \n1.0 & 1.2 & 3.42 & 184 & 438 & 835 \n1.5 & 1.6 & 3.43 & 162 & 352 & 524 \n2.0 & 2.1 & 3.39 & 136 & 283 & 340 \n2.5 & 2.6 & 3.64 & 190 & 380 & 484 \n3.0 & 3.0 & 3.21 & 92 & 177 & 141 \nConclusions\n===========\nWe have constructed an HRD of a sample of barium and related stars with unprecedented accuracy thanks to the TGAS parallaxes. From the comparison of the location of these stars on the HRD with STAREVOL evolutionary models, we find that the reclassification of some of these stars is needed. Several stars classified as Ba giants in the past are in fact dwarfs or subgiants. Additionally, there is no separation between subgiant CH stars and dwarf Ba in the HRD, contrary to what is implied by their designations.\nThe mass distribution derived for barium giants shows a deficit of low-mass stars ($M \\la 2.0$ M⊙; the exact threshold depends on metallicity), as compared to normal K and M field giants. This deficit is probably a consequence of a premature shrinkage of the orbits with shorter periods, as less massive stars reach large radii on the tip of the RGB. Additionally, stars with a barium index below Ba1, introduced by in the second edition of his catalogue, populate the high-mass tail of the distribution and probably are not genuine Ba stars but bright giants where the Ba lines are strengthened by a positive luminosity effect. Finally, and in contrast to previous claims, we do not see any difference between the mass distributions of mild (Ba1 – Ba2) and strong (Ba3 – Ba5) barium stars, at least when adopting the same average metallicity [Fe/H] = − 0.25 for both groups. It seems that there is no correlation between Ba abundance and Ba stellar mass.\nThere is a deficit of barium systems with short periods among those populating the red clump, as a result of the period threshold set by the large radii reached at the tip of the RGB. The systems missing from the red clump population could have coalesced as a result of unstable RLOF close to the tip of the RGB.\nThe accuracy of the TGAS parallaxes marks a significant step forward for the derivation of barium star masses. However, to obtain information on individual stars, the derivation of metallicities from spectroscopic data for each target is crucial. A consistent derivation of the metallicities would also allow us to conclude if the apparent difference between the metallicity distributions of dwarf and giant barium stars is real.\nThis research has been funded by the Belgian Science Policy Office under contract BR/143/A2/STARLAB and the Fonds voor Wetenschappelijk Onderzoek Vlaanderen (FWO). A.E. acknowledges support from the FWO. S.V.E acknowledges the Fondation ULB. L.S and D.P. are Senior FNRS Research Associates. D.K acknowledges support from BELSPO. The authors want to acknowledge the referee, Claudio B. Pereira, for his interest and very valuable input. This work has made use of data from the European Space Agency (ESA) mission *Gaia* (<https://www.cosmos.esa.int/gaia>), processed by the *Gaia* Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the *Gaia* Multilateral Agreement. This research has also made use of the SIMBAD database, operated at CDS, Strasbourg, France.\nSample of barium and related stars with accurate parallaxes\n===========================================================\nTable [Tab:master] lists the barium and related stars with good-precision Gaia and Hipparcos parallaxes (i.e. $\\varpi / \\sigma(\\varpi) > 3$). The full list is available at CDS, Strasbourg.\nrrrrrrrrrrrrrrrrrrrrrrr \nName & & & type & [Fe/H] & & & source & [Fe/H] & $T\\_{\\rm eff, min}$ & &$T\\_{\\rm eff, max}$ & & & \n& & & & & & & && & & & & & \n \nHD 26 & 19364 & 0.08 & CH & -0.45 & 1986ApJ...303..226S & 3.40 ± 0.30 & Gaia & 0 & 5136 & 5290 & 5342& 52 & 61 & 74 & \n& & & & & & & & -0.25& 4898 & 5003 & 5193& 44 & 52 & 63 & \n& & & & & & & & -0.5 & 4773 & 4901 & 5077 & 42 & 49 & 59 \nHD 218 & - & - & Ba3-5 & N/A & N/A & 1.71 ± 0.27 & Gaia & 0 & 3787 & 3979 & 4058 & 507 & 680 &960 \n& & & & & & & & -0.25& 3782 & 3958 & 4045 & 511 & 686 &969 \n& & & & & & & & -0.5 & 3768 & 3943 & 4035 & 508 & 681 &963 \nHD 749 & - & - & Ba1-2 & -0.06& 2006A&A...454..895 &7.11 ± 1.08 & Hipparcos & 0& 4687 &4909 & 4967 & 14.0 & 18.5 & 25.8 \n& & & & & & & & -0.25&4657& 4892 & 4979 & 13.8 & 18.3 & 25.4 \n& & & & & & & & -0.5 &4662&4795 & 4990 & 12.9 & 17.0 & 23.7 \nHD 4084 & - & - &Ba3-5& -0.42 & 2016MNRAS..459...4299 & 1.81 ± 0.25& Gaia & 0&4631 &4874& 4970& 126& 164& 221& \n& & & & & & & & -0.25&4609 &4894 & 4992& 127& 165& 223 \n& & & & & & & & -0.5 &4574 &4941 & 4991& 128& 166& 224 \nHD 5322 & - & - &Ba1-2& -0.18 & 2016MNRAS..459...4299 & 1.43 ± 0.26 & Gaia & 0 &4827 &5083& 5326 &497& 696 &1046 \n& & & & & & & & -0.25&4832 &5065& 5237 &487& 683 &1026 \n& & & & & & & & -0.5 &4858 &4957& 5224 &457& 641 &964 \nHD 5395 & - & - & Ba1-2& -0.24 & 2016MNRAS..440..1095 & 15.84 ± 0.58 &Hipparcos & 0&5139 &5201 &5263 & 76& 82 &88 \n& & & & & & & & -0.25&4903 &5081 &5178 & 77& 83 &89 \n& & & & & & & & -0.5 &4660 &4836 &5087& 70& 76 &82 \nHD 5424 &1881& 0.23 &dBa1-2& -0.51 & 2006A&A...454..895 & 2.50 ± 0.27 & Gaia & 0&4727&4888& 5015& 29& 36& 45 \n& & & & & & & & -0.25&4667&4870& 4987& 29& 36& 45 \n& & & & & & & & -0.5&4575& 4804& 4953& 29& 35& 44 \n... \n \n---\n1. Table A.1 is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via <http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/>[↩](#fnref1)\n\nHertzsprung-Russell diagram and mass distributionof barium stars [1](#fn1)\n==========================================================================\n### Received; Accepted\nIntroduction\n============\nBarium (Ba) giants are a class of G- and K-type giants with strong spectral lines of elements, such as barium or strontium, produced by the slow neutron capture (s-) process of nucleosynthesis. Although the class was already defined in 1951, it was not until 1980 that the origin of the observed overabundances was understood as the result of mass transfer in a binary system. The polluting heavy elements were formerly produced within an asymptotic giant branch (AGB) companion, which is now a very faint white dwarf (WD) that cannot be directly observed in most cases.\nAlthough they have been less intensively studied, barium stars are also found in the main sequence (i.e. barium dwarfs, dBa; ; ). Additionally, closely related to Ba stars, CH stars are their low metallicity counterparts. The latter have similar enhancement of s-process elements and strong CH molecular bands, but weaker lines of other metals.\nThe exact mode of mass transfer responsible for the formation of these families of polluted binaries remains uncertain. Many systems have orbital periods in a range (form 100 to 1000 days) that cannot be accounted for by simple models of orbital evolution (;, and references therein). Nevertheless, the white-dwarf nature of the companion is beyond doubt, thanks to the analysis of the orbital mass functions performed by,, and. The latter analysis reveals a mass distribution for the WD companion which peaks around 0.6 M⊙, in accordance with the expectation. However, the exact value of the derived WD mass depends upon the mass of the primary star (the barium star), which is difficult to derive with a good accuracy.\nThe mass of barium stars may be derived from their location in the Hertzsprung-Russell diagram (HRD) and from a comparison with evolutionary tracks. used a Bayesian method to infer barium star masses, based on a HRD constructed from Hipparcos parallaxes. These authors concluded that mild and strong barium stars have somewhat different mass distributions, which are characterized by masses in the range 2.5 – 4.5 M⊙ and 1 – 3 M⊙, respectively. The distinction between mild and strong barium stars is made on the *Ba index* introduced by. The index reflects the strength of the barium spectral lines, based on visual inspection, on a scale from Ba1 to Ba5, where Ba5 corresponds to the strongest lines. In this and our past studies, we associate Ba1 - Ba2 indices with mild barium stars and Ba3 - Ba5 indices with strong barium stars. The catalogue of barium stars by introduces many barium stars with a *Ba index* smaller than one, which we denote as Ba0. These targets deserve special attention as they may turn out not to be barium stars.\n![[Fig:varpi-errpi] Parallax and its error for the sample of Ba and related stars studied in this paper that were part of TGAS. The red line represents the threshold \\varpi /\\sigma(\\varpi) = 3. Only stars to the right of the line are considered for inclusion in the HRD. ](parplot.png)[Fig:varpi-errpi] Parallax and its error for the sample of Ba and related stars studied in this paper that were part of TGAS. The red line represents the threshold $\\varpi /\\sigma(\\varpi) = 3$. Only stars to the right of the line are considered for inclusion in the HRD. \nThanks to the recent data release (DR1; ) of the more accurate parallaxes provided by the Tycho-Gaia Astrometric Solution (TGAS; ; ), it is possible to locate barium stars in the HRD with a much better accuracy than with the Hipparcos data. Stellar models still suffer from large uncertainties, but the Gaia mission will mark a significant step forward for the derivation of barium star masses.\nThe relation between the location of barium stars in the HRD and the orbital parameters also provides valuable insights into the binary interaction processes at play in these systems. For instance, one may expect that for barium stars located in the red clump, the distribution of orbital periods exhibits a higher lower limit caused by the large radii reached by the star at the tip of the red giant branch (RGB). One may also expect that short-period systems for which the red giant has gone through the RGB tip, show smaller or null eccentricities due to tidal circularization along the RGB.\nThe paper is organized as follows: The sample is described in Sect. [Sect:TGAS] and the HRD of barium stars and related systems is then constructed in Sect. [Sect:HRD]. We discuss the resulting mass distribution of barium giants in Sect. [Sect:masses] and we investigate the possible relationship between location in the HRD and orbital period in Sect. [Sect:P-HRD].\nThe sample\n==========\nThe sample was constructed by selecting targets with Tycho-2 identifiers in the barium star lists of and, in the list of CH and related stars of, and in the list of dwarf barium stars of and. The sample of (dwarf and giant) barium, CH and related stars with a Tycho-2 identifier amounts to 546 entries. The TGAS catalogue provides a parallax value for 400 of them. We removed from the list 11 high proper motion dwarf stars, labelled PM\\* in the SIMBAD database, for which we could not find any confirmation that they are either carbon or barium stars (e.g. HD 208998, neither a carbon star nor a Ba star according to ). The list of stars removed for that reason includes HD 11397, HD 15206, HD 24508, HD 89668, HD 108564, HD 145417, HD 153075, HD 154276, HD 161612, HD 164922, and HD 208998.\n![[Fig:Vhisto] V magnitude distribution of our stellar sample according to the presence or absence of the star in the TGAS catalogue. ](TGAS_hist.png)[Fig:Vhisto] *V* magnitude distribution of our stellar sample according to the presence or absence of the star in the TGAS catalogue. \nFig. [Fig:varpi-errpi] shows in logarithm scale the relation between the parallax ($\\varpi$) and its error ($\\sigma(\\varpi)$) for the TGAS targets. The red line represents the limit where $\\varpi /\\sigma(\\varpi) \\ge 3$, which was used as precision condition. Among the initial 389 barium and related stars with a TGAS parallax, after removing the 11 high proper motion stars, only 52 (13%) had to be rejected because they do not fulfil the precision criterion. Since this rejection rate is small, and since the analysis of these data do not significantly affect any statistical property of the sample, such as its average luminosity, there is no need to investigate how this rejection rate could bias the results. Nine more targets were excluded because their position within the Galaxy made it difficult to constrain the interstellar extinction on the line of sight, which, as explained below, prevents a reliable derivation of the stellar effective temperature.\nWhen the TGAS parallax was not available, we used the Hipparcos value, always imposing the condition that $\\varpi/\\sigma(\\varpi)$ exceeds 3. Additionally, for 21 confirmed astrometric binaries, we used the parallax rederived by instead of the TGAS (available for 15 of them) or Hipparcos values. The reason to do this for the astrometric binaries is that the TGAS parallax, obtained by applying a single star solution, is possibly disturbed by the orbital motion. The possible presence of astrometric binaries in the sample is discussed at the end of the present section. This results in 103 entries from Hipparcos and 313 from TGAS, in addition to the 21 astrometric binaries. Our final sample thus contains 437 entries. Of these, 77% are barium giants while the other samples are much smaller; i.e. 10% of the stars are barium dwarfs, 5% are CH giants, 5% are carbon stars, and the remaining 3% are CH subgiants.\n![[Fig:EB-Vfit] Error ellipse resulting from fitting the SED of HD 183915, revealing the strong correlation between T_{\\rm eff} and E_{\\rm B-V}. The red cross identifies the best fit, predicting E_{\\rm B-V} = 0.203 whereas the selective extinction derived from for that star is E_{\\rm B-V} = 0.225 (leading to A_V = 0.70). ](HD_183915.png)[Fig:EB-Vfit] Error ellipse resulting from fitting the SED of HD 183915, revealing the strong correlation between $T\\_{\\rm eff}$ and $E\\_{\\rm B-V}$. The red cross identifies the best fit, predicting $E\\_{\\rm B-V} = 0.203$ whereas the selective extinction derived from for that star is $E\\_{\\rm B-V} = 0.225$ (leading to *A**V* = 0.70). \nAs shown in Fig. [Fig:Vhisto], the major cause for a star not to be included in the TGAS catalogue is because it is brighter than *V* ∼ 6.5. A small fraction of fainter stars are missing as well. In an attempt to identify whether their absence in TGAS could be related to their astrometric binary nature, stars with a missing TGAS parallax and a known orbital period (see Jorissen et al. 2017, in preparation) were collected in Table [Tab:missing]. In that Table, stars are ordered by increasing $a\\_1/\\varpi$ (seventh column), the ratio between the angular semi-major axis of the astrometric orbit of the primary component around the centre of mass of the system and the parallax. This ratio may be estimated from Eq. (13) of, which depends only on the masses and the orbital period\n \n$$\\label{Eq:a1varpi}\n\\frac{a\\_1}{\\varpi} = P^{2/3} \\frac{M\\_2}{(M\\_1+M\\_2)^{2/3}},$$ \nwhere $a\\_1/\\varpi$ is in AU, *P* is expressed in years and masses in M⊙. This quantity corresponds in fact to the semi-major axis of the absolute orbit of the primary component. In case the astrometric orbit was detected from the Hipparcos data, the values listed in the seventh column of Table [Tab:missing] are the observed values, which correspond to the orbit of the photocentre of the system around its centre of mass. However, since the cases considered here correspond to a WD companion, the photocentric orbit is identical to that of the primary component. If the astrometric orbit is not seen in the Hipparcos data, the ratio $a\\_1/\\varpi$ is estimated from Eq. [Eq:a1varpi] by adopting *M*1 = 2.5 M⊙ and *M*2 = 0.62 M⊙, along with the observed orbital period.\nA high $a\\_1/\\varpi$ value means that the binary motion with respect to the parallactic motion is large. If, in addition, the orbital period is close to 1 yr, the parallactic and binary motions are difficult to disentangle. This could be a possible cause for the absence of a star in the TGAS catalogue. However, this is not observed because the fainter stars with high $a\\_1/\\varpi$ values absent from TGAS often have orbital periods much longer than 1 yr. Moreover, several systems with similarly high $a\\_1/\\varpi$ ratios do have TGAS parallaxes (like HD 50264, HD 87080, HD 107574...). Another possible cause for the absence of a star in the TGAS catalogue could be its colour, as very red and very blue stars were excluded. The identification of the exact reason for the absence of several faint barium and related stars has to await the availability of the next data releases when quality flags of the astrometric solution become available.\n![[Fig:FeH] Metallicity distribution of the barium and related stars with a value of [Fe/H] available in the literature. ](FeH.png)[Fig:FeH] Metallicity distribution of the barium and related stars with a value of [Fe/H] available in the literature. \nHertzsprung-Russell diagram\n===========================\nAtmospheric parameters\n----------------------\n[Sect:params]\nlllllllllllll HD/BD & TYC & *V* & *G* & *G* − *K**s* & $\\varpi$(HIP) (mas) & $a\\_1/\\varpi$ (AU) & *P* (d) & Rem \n \n \n121447 & 6140-641-1 & 7.80 & 7.13 & 2.98 & 2.2 ± 1.0 & 0.2 & 186 \n46407 & 5369-220-1 & 6.24 & - & - & 6.6${+1.3}\\atop{-1.1}$ & 0.3 & 457 & PJ,\\* \n92626 & 8201-1209-1 & 7.09 & 6.67 & 2.25 & 6.6${+0.9}\\atop{-0.6}$ & 0.5 & 918 & PJ,\\* \nNGC 2420 250 & 1373-1426-1 & 11.14 & 11.01 & 2.37& - & 0.7 & 1404 \n101013 & 3454-2188-1 & 6.12 & - & - & 7.1${+0.7}\\atop{-0.6}$ & 0.8 & 1711 & PJ,\\* \n+38 118 & 2797-46-1 & 8.86 & 8.32 & 2.76 & - & 1.4 & 3877 \n \n \n77247 & 3805-1493-1 & 6.86 & 6.54 & 1.89 & 2.9 ± 1.0 & 0.1 & 80 & \\* \n218356 & 2239-1475-1 & 4.54 & - & - & 6.1 ± 0.7 & 0.1 & 111 & \\* \n288174 & 119-1058-1 & 9.02 & - & - & 2.9 ± 1.3 & 0.8 & 1818 \n204075 & 6372-1278-1 & 3.74 & - & - & 8.6${+1.1}\\atop{-1.0}$& 1.0 & 2378 & PJ,\\* \n131670 & 4999-334-1 & 8.01 & 7.62 & 2.31 & 2.3 ± 1.2 & 1.2 & 2930 \n139195 & 933-1240-1 & 5.26 & - & - & 13.9 ± 0.7 &1.7 & 5324 &\\* \n53199 & 761-980-1 & 9.07 & 8.79 & 1.79 & 3.7 ± 1.3 & 2.3 & 8300 &\\* \n51959 & 4813-1015-1 & 8.92 & 8.61 & 2.07 & 6.5 ± 1.3 & 2.5 & 9488 &\\* \n104979 & 866-1180-1 & 4.12 & - & - & 19.1 ± 0.8 & 3.3 & 13940 &\\* \n98839 & 3015-2321-1 & 5.03 & - & - & 6.6 ± 0.6 & 3.7 & 16419 & \\* \n119185 & 5552-1079-1 & 8.91 & 8.57 & 2.02 & 3.9 ± 1.1 &4.1 & 19467 & \\* \n \n \n89948 & 6631-715-1 & 7.55 & 7.31 & 1.12 & 23.4 ± 0.9 & 0.7 & 667.8 & PJ,\\* \n76225 & 6580-2586-1 & 9.20 & - & - & 3.4 ± 1.1 &1.0 & 2411 & \\* \n98991 & 6088-2156-1 & 5.09 & - & - & 22.0 ± 0.8 &1.1 & 2834 & \\* \n221531 & 5832-970-1 & 8.36 & 8.21 & 0.98 &9.6${+1.4}\\atop{-1.3}$ & 1.2 & 1416 & PJ,\\* \n95241 & 3012-2522-1 & 6.03 & - & - & 22.0 ± 0.8 &1.8 & 5448 & \\* \n \n![[Fig:LandT] Luminosity and T_{\\rm eff}(and their 1\\sigma error bar) derived from SED fitting with models of solar metallicity and metallicity [Fe/H] = -0.5. ](T_MARCS.png \"fig:\")\n![[Fig:Pastel] Effective temperatures obtained with the SED fitting method compared with those obtained from high-resolution spectra collected from the PASTEL catalogue. The dashed red lines indicate a \\pm200 K uncertainty. ](PastelPaper.png)[Fig:Pastel] Effective temperatures obtained with the SED fitting method compared with those obtained from high-resolution spectra collected from the PASTEL catalogue. The dashed red lines indicate a ± 200 K uncertainty. \nThe atmospheric parameters of barium and related stars were derived by modelling the spectral energy distribution (SED) obtained by collecting magnitudes listed in the Simbad database. The best-fitting MARCS model was determined from a parameter-grid search using a *χ*2 minimization method (see for details). The stellar temperature was then assigned from the best-fitting model and the luminosity was obtained by integrating the SED over all wavelengths spanned by the model, and applying the distance modulus derived from the parallax. The error bar on the luminosity is propagated from the parallax uncertainty and is thus asymmetric. The error on the temperature is the 1-sigma error enclosing 67% of the model fits (see Fig. [Fig:EB-Vfit]).\nThe reddening $E\\_{\\rm B-V}$ was initially left free during the fitting process. Its best value was estimated by looking for the amount of reddening to be applied to the MARCS models to match the observed magnitudes. A value of 3.1 was used for the ratio of the total to the selective extinction $R = A\\_V/E\\_{\\rm B-V}$, from which *A**V* is derived. It turned out that the temperature and extinction derived in this manner are strongly correlated, as shown in Fig. [Fig:EB-Vfit].\nIn a second run, the extinction was fixed at the value computed by, in his three-dimensional map of the extinction within the nearest kiloparsec. The location of the target in the Galaxy was computed from its galactic coordinates and its parallax. The resulting $T\\_{\\rm eff}$ and $E\\_{\\rm B-V}$ values were often found to fall at the lower edge of the 1*σ* ellipse error. We decided to let $E\\_{\\rm B-V}$ vary freely between 0 and Gontcharov’s value plus 0.07 mag, which is the 1*σ* error bar suggested in their paper.\nAnother parameter that has an impact on the determination of the stellar parameters is the metallicity. As they are difficult to constrain by fitting SEDs, metallicities of barium and related stars were collected from the literature and their distribution is presented in Fig. [Fig:FeH]. The barium-giant distribution peaks at [Fe/H] ∼ − 0.14 with a standard deviation of 0.2. determined the metallicity of 182 Ba giants and obtained a similar mean value of [Fe/H] ∼ − 0.12 ± 0.14. As expected, CH stars have lower metallicities, and it seems that barium dwarfs may be slightly more metal poor than barium giants as well, but the dwarf sample is much smaller. We decided to use MARCS models with fixed [Fe/H] = − 0.25 to determine the stellar parameters. To evaluate the impact of the metallicity on the derived parameters, two more SED fits have been carried out: one imposing a solar metallicity and another one imposing [Fe/H] = − 0.5. The resulting $T\\_{\\rm eff}$ and luminosities of the limiting cases are presented in Fig. [Fig:LandT], and reveal that the uncertainty introduced on the stellar parameters by the metallicity of the best-fitting MARCS model is negligible. The impact of the metallicity on the stellar tracks and hence on the derived mass is of course much larger and is discussed in Sect. [Sect:masses].\nFinally, in order to evaluate our method, we include (Fig. [Fig:Pastel]) a comparison of the effective temperatures, which we obtained for the Ba giants with the SED fitting method with temperatures for the same stars determined from high-resolution spectra. We collected the latter values from the PASTEL catalogue. Fig. [Fig:Pastel] shows that most of the 102 Ba giants that we compare lie within the ± 200K uncertainty, indicated by the dashed red lines. All the parameters described so far are listed in Table [Tab:master].\nInput physics for the stellar grid calculations\n-----------------------------------------------\nTo determine the fundamental parameters of our sample stars from their location in the HR diagram, we computed extended grids of stellar models covering the mass range 0.6 ≤ *M*zams/M⊙ ≤ 6. All the models were evolved from the pre-main sequence up to the end of the AGB phase for stars of initial mass *M*zams ≤ 4M⊙ and up to the occurrence of convergence problems in the higher mass tracks. The models were computed with the STAREVOL code with the following input physics. We used the radiative opacity tables of above 8000 K and of at lower temperatures. We took the modifications of the opacity due to the formation of H2, H2O, OH, C2, CN, and CO molecules in the atmosphere of C-rich stars (C/O>1) into account following the formulation of. The nuclear network includes 182 reactions coupling 55 species from H to Cl. The nuclear rates have been recently updated with NACRE II compilation. We used the mixing length theory to determine the temperature gradient in the convective regions with *α*MLT = 1.75 and adopt the Schwarzschild criterion. For the mass loss rate, we considered the prescription up to the beginning of the AGB phase and then switched to the formulation. We also included some overshooting at the base of the convective envelope, following the exponential decay expression of with *f*over = 0.1. Finally we used a grey atmosphere surface boundary condition.\nDiscussion\n----------\n![[Fig:HRDStarevol] Hertzsprung-Russell diagram for sample of barium and related stars labelled as in queried catalogues. Green crosses indicate barium giants; pink squares indicate barium dwarfs; black dots indicate CH giants; and blue triangles indicate subgiant CH stars. Typical error bars are shown over the legend for two situations: a favourable case (left: \\varpi /\\sigma(\\varpi) = 8), and limiting case for \\varpi /\\sigma(\\varpi) = 3 (right). Stellar tracks from the STAREVOL code were overplotted for models of masses 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0 M_{\\odot}, as labelled and metallicity [Fe/H] = -0.25. The solid black tracks correspond to the post-main-sequence evolution up to the tip of the RGB (low-mass stars) or to the onset of core He-burning (intermediate-mass stars), dotted blue tracks correspond to core He-burning, and dashed red tracks to the early and thermally pulsing AGB (TP-AGB) phase. ](HRDm25.png)[Fig:HRDStarevol] Hertzsprung-Russell diagram for sample of barium and related stars labelled as in queried catalogues. Green crosses indicate barium giants; pink squares indicate barium dwarfs; black dots indicate CH giants; and blue triangles indicate subgiant CH stars. Typical error bars are shown over the legend for two situations: a favourable case (left: $\\varpi /\\sigma(\\varpi) = 8$), and limiting case for $\\varpi /\\sigma(\\varpi) = 3$ (right). Stellar tracks from the STAREVOL code were overplotted for models of masses 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0 M⊙, as labelled and metallicity [Fe/H] = − 0.25. The solid black tracks correspond to the post-main-sequence evolution up to the tip of the RGB (low-mass stars) or to the onset of core He-burning (intermediate-mass stars), dotted blue tracks correspond to core He-burning, and dashed red tracks to the early and thermally pulsing AGB (TP-AGB) phase. \nFig. [Fig:HRDStarevol] shows the Ba and related star sample in the HRD with parameters obtained as described in Sect. [Sect:atmosphere], i.e. leaving the extinction free (between 0. and Gontcharov’s value), fixing the metallicity of the atmospheric models to [Fe/H] = − 0.25, and by restricting the sample to stars with $\\varpi /\\sigma(\\varpi) \\ge 3$. We kept the original category (Ba giant, Ba dwarf, CH giant, or CH subgiant) given in the queried catalogues for each star shown in Fig. [Fig:HRDStarevol]. Typical error bars are shown for two situations: a favourable case with $\\varpi /\\sigma(\\varpi) = 8$ (left) and limiting case with $\\varpi /\\sigma(\\varpi) = 3$ (right). In the latter case, the asymmetric nature of the errors on the luminosity starts to be noticeable, so that biases are manifested (e.g. ). However, as apparent on Fig. [Fig:varpi-errpi], the number of targets with $\\varpi /\\sigma(\\varpi) < 3$ is not large, and a discussion on the effect of the biases does not seem to be required here.\nEvolutionary tracks for [Fe/H] = − 0.25 have been superimposed for models of masses 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0 M⊙, as labelled. Fig. [Fig:HRDFeH] shows the impact of metallicity on the 2.0 M⊙ and 3.0 M⊙ tracks (blue and red tracks, respectively). It appears that, for the RGB and the region occupied by the red clump, the [Fe/H] = − 0.5 track of 2.0 M⊙ covers the same region as the [Fe/H] = 0 track of 3.0 M⊙, leading to a strong degeneracy in the mass determination of a given star, which can only be lifted by knowing its metallicity.\nAnother important source of uncertainties is in the physics of the evolutionary models. Figure [Fig:tracks] shows the difference among three sets of evolutionary tracks. The solid black tracks were computed with STAREVOL as described in Sect. [Sect:STAREVOL] using the solar composition corresponding to Z⊙=0.0134. The dotted green lines show the Geneva models from for solar composition as well, but with Z⊙=0.014. Finally, the dashed red tracks correspond to the tracks from for solar metallicity with Z⊙=0.019. Apart from the scatter at the zero-age main sequence caused by the diverse solar metallicity values, varying descriptions of convection, overshooting, rotation, mass loss, etc., lead to uncertainties in the models in addition to our observational uncertainties. However, comparing evolutionary models is beyond the scope of this publication. Finally, barium stars are post-mass-transfer objects which, while on the main sequence, accreted mass from an AGB companion. This effect, which is not taken into account in the single-star evolutionary models that we use for comparison, could also affect their evolution.\nA first interesting result apparent in the HRD of Fig. [Fig:HRDStarevol] is that several stars previously classified as Ba giants need to be reclassified as they are in fact dwarfs or subgiants. These are listed in Table [Tab:Bad] and were selected from the criterion *L* ≤ 10L⊙.\nWe also report that most of subgiant CH stars (blue triangles) populate the region of the HRD associated with Ba dwarfs. Some of these stars, located around the 1.0 M⊙ track, seem to lie on the main sequence, contrary to what is implied by their designation as subgiants. This could have been guessed from the high gravities (log*g* ≥ 4.0) inferred by in their spectroscopic analysis of subgiant CH stars. Other subgiant CH stars seem to be genuine subgiants, but they fall amidst stars classified as Ba dwarfs. The exact designation of these stars depends on the evolutionary models used for comparison, as the extension of the main sequence depends on the chosen overshoot parameter and the consideration or not of internal rotation. However, it seems clear that there is no clear separation between Ba dwarfs and CH subgiants in the HRD, so that these designations should not be taken literally.\n![[Fig:HRDFeH] Same as Fig. [Fig:HRDStarevol] with the 2.0 and 3.0 M_{\\odot} (blue and red, respectively) evolutionary tracks (not covering the AGB) computed with [Fe/H] = 0.0 (left) and [Fe/H] = -0.5 (right). In each panel the solid tracks correspond to the indicated metallicity, which is also used to derive the atmospheric parameters of the targets; the dashed tracks correspond to the other metallicity ([Fe/H] = -0.5 and 0.0 in the left and right panel, respectively). ](HRD_Zcomp.png)[Fig:HRDFeH] Same as Fig. [Fig:HRDStarevol] with the 2.0 and 3.0 M⊙ (blue and red, respectively) evolutionary tracks (not covering the AGB) computed with [Fe/H] = 0.0 (left) and [Fe/H] = − 0.5 (right). In each panel the solid tracks correspond to the indicated metallicity, which is also used to derive the atmospheric parameters of the targets; the dashed tracks correspond to the other metallicity ([Fe/H] = − 0.5 and 0.0 in the left and right panel, respectively). \n![[Fig:tracks] Comparison among STAREVOL (solid black lines), Geneva (dotted green lines), and Padova (dashed red lines) evolutionary tracks up to the beginning of the AGB phase. ](comparingtracks.png)[Fig:tracks] Comparison among STAREVOL (solid black lines), Geneva (dotted green lines), and Padova (dashed red lines) evolutionary tracks up to the beginning of the AGB phase. \nllp0.3p0.3 Name & TYC & Ref. & Rem. \nHD 8270 & 8036-564-1 &, dwarf nature confirmed by \nHD 13551 & 8851-37-1 &, dwarf nature confirmed by \nHD 22589 & 4722-19-1 &, dwarf nature confirmed by & orbit available \nCpD -44 5038 & 7735-447-1 & \nBD -10 4311 & 5630-641-1 & & orbit available \nHD 197481 & 7457-641-1 & \nCS 22180-0013 &5279-303-1 & Bartkevicius \nHIP 19050 & 1814-348-1 & Bartkevicius, classified as R \nHD 175179 & 5123-323-1 & Bartkevicius PM\\*, moderate enhancement (0.2 – 0.3 dex of Y and Ba) \nA strong concentration of barium-rich stars is also found in the red clump, as is further discussed in Sect. [Sect:masses].\nMass distribution\n=================\nWe determine the masses from a comparison with stellar evolutionary tracks from the STAREVOL code. Three different metallicities have been considered: [Fe/H] = − 0.5, − 0.25, and 0. To ensure self-consistency, when comparing the location of the target stars in the HRD with these tracks, we use the stellar parameters derived from the SED fit associated with the same metallicity (Sect. [Sect:atmosphere] and Table [Tab:master]).\nTo derive stellar masses, we use a statistical approach that takes into account the time that a star spends at a given location in the HRD. We first interpolated all the evolutionary tracks using a constant time step of 10,000 yrs. This interpolation thus produces more points in a long-lasting phase (such as the main sequence or the red clump). This allows us to fill the HRD with many points (around 17 × 106), each characterized by a value for $T\\_{\\rm e}$, *L*, and mass *m**T*, *L*; our method takes into account the mass lost by the star when ascending the RGB. For a given star, with a known temperature *T**o* and luminosity *L**o*, along with their associated errors (*σ**T**o*, *σ**L**o*), we then estimate its mass, *m**o*, by computing a Gaussian average of the masses for all the points located in a restricted neighbourhood around *T**o*, *L**o*. More specifically, the mass of the star is given by\n \n$$m\\_o = \\frac{1}{2 \\pi \\sigma\\_{To} \\sigma\\_{Lo}} \\sum\\_{T\\_e} \\sum\\_L \\frac{m\\_{T,L}^2}{\\rho\\_{T,L}} e^{-\\frac{(T\\_e-T\\_o)^2}{2 \\sigma^2\\_{To}}}e^{-\\frac{(L-L\\_o)^2}{2 \\sigma^2\\_{Lo}}},$$ \nwhere *ρ**T*, *L* is a normalization factor given by\n \n$$\\rho\\_{T,L} = \\frac{1}{2 \\pi \\sigma\\_{To} \\sigma\\_{Lo}} \\sum\\_{T\\_e} \\sum\\_L m\\_{T,L} e^{-\\frac{(T\\_e-T\\_o)^2}{2 \\sigma^2\\_{To}}}e^{-\\frac{(L-L\\_o)^2}{2 \\sigma^2\\_{Lo}}}.$$ \nWe thereby derive the mass for all our objects, for the three computed metallicities. The distribution for giant barium stars (thus excluding barium dwarfs, CH stars, and CH subgiants) is shown in Fig. [Fig:massesSTAREVOL] to highlight the sensitivity on the metallicity: if one adopts the solar metallicity for all the targets, the mass distribution exhibits a modest peak at 2.8 M⊙, and this peak is superimposed on a broader distribution extending from 2.5 to 5.5 M⊙. Adopting instead a lower metallicity [Fe/H] = − 0.5 for all targets, the peak of the mass distribution is better marked and shifted towards lower masses (2.3 M⊙). As shown in Fig. [Fig:FeH], the metallicity distribution of barium stars peaks at [Fe/H] ∼ − 0.14. Hence, we conclude that the mass distribution obtained by adopting the average metallicity of -0.25 should be close to reality. In that case, the distribution can be described by two Gaussians: a main peak at 2.5 M⊙ with a standard deviation of 0.18 M⊙ and a broader tail at higher masses (up to 4.5 M⊙), which peaks at 3 M⊙ with a standard deviation of 1 M⊙.\n![[Fig:massesSTAREVOL] Mass distribution for giant barium stars (thus excluding barium dwarfs, CH stars, and CH subgiants), from a comparison with the STAREVOL evolutionary tracks, for the three metallicities [Fe/H] = 0, [Fe/H] =-0.25, and [Fe/H] =-0.5. Adopting T_{\\rm eff} \\pm \\sigma_{\\rm Teff} and L \\pm \\sigma_L, we determine the red (-) and green (+) histograms. The vertical dashed line indicates the peak of the [Fe/H] =-0.25 distribution. ](Badistr_py.png)[Fig:massesSTAREVOL] Mass distribution for giant barium stars (thus excluding barium dwarfs, CH stars, and CH subgiants), from a comparison with the STAREVOL evolutionary tracks, for the three metallicities [Fe/H] = 0, [Fe/H] = − 0.25, and [Fe/H] = − 0.5. Adopting $T\\_{\\rm eff} \\pm \\sigma\\_{\\rm Teff}$ and *L* ± *σ**L*, we determine the red (-) and green (+) histograms. The vertical dashed line indicates the peak of the [Fe/H] = − 0.25 distribution. \n![[Fig:HRDFamaey] Distribution of a comparison sample of K and M giants from in the HRD. Data are presented as a contour density plot. STAREVOL evolutionary tracks for stars with initial masses 1, 2, 3, 4, 5, and 6 M_{\\odot} (from bottom to top) and [Fe/H] = -0.25 are superimposed; white colours correspond to Hertzsprung-gap and RGB evolution, green colours to core He-burning, and red to AGB evolution. Stars from the Y group correspond to the faint extension of the stellar density in the regions covered by the 4, 5, and 6 M_{\\odot} evolutionary tracks. ](PP44_Teff_MV_STAREVOL_FeHm25_contour.png)[Fig:HRDFamaey] Distribution of a comparison sample of K and M giants from in the HRD. Data are presented as a contour density plot. STAREVOL evolutionary tracks for stars with initial masses 1, 2, 3, 4, 5, and 6 M⊙ (from bottom to top) and [Fe/H] = -0.25 are superimposed; white colours correspond to Hertzsprung-gap and RGB evolution, green colours to core He-burning, and red to AGB evolution. Stars from the Y group correspond to the faint extension of the stellar density in the regions covered by the 4, 5, and 6 M⊙ evolutionary tracks. \n![[Fig:massPP44] Same as Fig. [Fig:massesSTAREVOL] for the comparison sample of K and M giants (solid red line) from. The hatched histogram corresponds to barium stars. ](FaBanew_py.png)[Fig:massPP44] Same as Fig. [Fig:massesSTAREVOL] for the comparison sample of K and M giants (solid red line) from. The hatched histogram corresponds to barium stars. \nComparison with M- and K-type giants\n------------------------------------\nWe compare the mass distribution of barium giants with the sample of 5952 K and 739 M giants from. These authors performed a Bayesian classification of this large sample into different kinematical groups, based not only on their kinematics (Tycho-2 proper motions and CORAVEL radial velocities; ;, respectively) but also on their luminosity, which is derived from a Bayesian estimate based on the Hipparcos parallax. The B (‘Background’) group constitutes the smooth velocity ellipsoid, whereas the Y (‘Young’) group exhibits all signatures of young stars, i.e. small velocity dispersions and scale height. The average metallicity of stars belonging to the B group was estimated to be about − 0.2 from Fig. 15 of and − 0.12 ± 0.18 from as quoted by. The close match between the average metallicities of barium giants (Fig. [Fig:FeH]) and the B group of comparison giants is encouraging, since this match indicates that B group giants are an appropriate sample to compare with the barium giants. To locate B-group giants in the HRD, it was necessary to use various calibrations from, notably (*V* − *I*, *V* − *K*) and (*V* − *K*, $T\\_{\\rm eff}$) to convert Hipparcos *V* − *I* indices into temperatures, and (*V* − *K*, *B**C**K*) to convert visual absolute magnitudes into luminosities. The resulting HRD is shown in Fig. [Fig:HRDFamaey] and indicates a strong concentration of solar-metallicity stars with masses in the range 2 – 3 M⊙in the red clump.\nDiscussion\n----------\nThe derived mass distribution (Fig. [Fig:massPP44]) reveals that the samples of field K and M and barium giants behave similarly in terms of mass, except for a deficit of low-mass stars ($M \\la 2.0$ M⊙, the exact threshold value depending on metallicity) among barium stars. In the mass distribution of the comparison sample, the Y group populates the high-mass tail, whereas the B group is associated with the lower mass population, including the strong peak around 2.4 M⊙ with a standard deviation of 0.26 M⊙ (for [Fe/H] = − 0.25), where stars residing in the red clump accumulate. have also found a peak around 2.3 M⊙ in the mass distribution of a sample of binary red giants in open clusters. The position of the peak should therefore not be considered as a distinctive feature of barium giants.\nThe deficit of low-mass giants ($M \\la 2.0$ M⊙) could, however, be specific to Ba stars. One could think that it is caused by the longer evolutionary timescales needed to form a low-mass barium star whose companion could then also be of low mass. However, this argument does not hold because a star with $M \\gtrsim 1.3$ M⊙ reaches the tip of the AGB within 5 Gyr and has time to pollute its companion and form a low-mass Ba star of solar metallicity. Additionally, low-mass Ba stars do not necessarily have low-mass companions. Neither can we claim that the deficit of low-mass barium stars is attributed to the absence of third dredge-up in low-mass ($M \\lesssim$ 1.3 M⊙) AGB stars because evidence for s-process enrichment in stars with such low masses has been suggested based on the analysis of post-AGB stars. The most probable explanation for the deficit of low-mass Ba giants is that they reached a large radius at the tip of the RGB, resulting in a premature shrinkage of the orbit caused by a unstable Roche-lobe overflow (unless they had a very long orbital period). We elaborate on the method to determine $R\\_{\\rm RGB tip}$ in Sect. [Sect:P-HRD].\n![[Fig:Bamm] Comparison of the mass distributions for 164 mild (Ba1 – Ba2) and 110 very mild (Ba0) barium stars (red and black curves, respectively) computed using evolutionary tracks with [Fe/H] = -0.25. Very mild barium stars are defined as having a Ba index smaller than 1 in the catalogue of. Upper panel: Normalized histograms are shown. Lower panel: Cumulative frequency distributions are shown with the maximum absolute difference between these distributions amounting to 0.168, corresponding to a first-kind error of 13% when rejecting the null hypothesis of identical distributions in a Kolmogorov-Smirnov test with an effective sample size of 48.2 [= 164 \\times 110 / (164 + 110)]. ](bamm_bam_py.png)[Fig:Bamm] Comparison of the mass distributions for 164 mild (Ba1 – Ba2) and 110 very mild (Ba0) barium stars (red and black curves, respectively) computed using evolutionary tracks with [Fe/H] = − 0.25. Very mild barium stars are defined as having a *Ba index* smaller than 1 in the catalogue of. Upper panel: Normalized histograms are shown. Lower panel: Cumulative frequency distributions are shown with the maximum absolute difference between these distributions amounting to 0.168, corresponding to a first-kind error of 13% when rejecting the null hypothesis of identical distributions in a Kolmogorov-Smirnov test with an effective sample size of 48.2 [ = 164 × 110/(164 + 110)]. \nAmong the barium-star sample, the high-mass tail is dominated by stars with a barium index, based on a visual inspection of the strength of the barium lines in the spectrum, lower than unity (named Ba0 in this paper), i.e. with a very moderate (if any) s-process enhancement (Fig. [Fig:Bamm]). Some of these stars, which appeared in the second edition (1991) of the barium-star catalogue of L\" u, could be bright giants, or even supergiants, where the barium lines are strengthened as a simple positive luminosity effect (i.e. an increase of the line strength with increasing luminosity) instead of being genuine barium stars.\nFinally, we re-investigate the statement by and that mild (Ba1-Ba2) and strong (Ba3-Ba5) barium stars have somewhat different mass distributions, as they are characterized by masses in the range 2.5 – 4.5 M⊙ and 1 – 3 M⊙, respectively. As shown by Fig. [Fig:basm], our analysis does not support this claim, since the mass distributions of 164 mild and 50 strong barium giants are undistinguishable under the assumption that their average metallicities are similar. The maximum absolute difference between the two distributions is 0.091, which for an effective sample size of 38.3 ( = 164 × 50/214) corresponds to a very large first-kind error of 91% for the Kolmogorov-Smirnov test.\n![[Fig:basm] Comparison of the mass distributions for mild and strong barium stars (red and black curves, respectively) computed from evolutionary tracks for [Fe/H] = -0.25. Upper panel: Normalized histograms are shown. Lower panel: Cumulative frequency distributions with the maximum absolute difference between them amounting to 0.091. ](bas_bam_py.png)[Fig:basm] Comparison of the mass distributions for mild and strong barium stars (red and black curves, respectively) computed from evolutionary tracks for [Fe/H] = − 0.25. Upper panel: Normalized histograms are shown. Lower panel: Cumulative frequency distributions with the maximum absolute difference between them amounting to 0.091. \nLocation in the HRD and orbital period\n======================================\nFig. [Fig:HRP] is a first attempt to correlate the location of a binary system in the HRD with its orbital period. We only include barium star systems for which the orbital period is known, and make the size of the symbols proportional to this value. We use the periods determined by and Jorissen et al. (in preparation) for the giants (see also Table [Tab:master]) and Escorza et al. (in preparation) for the main sequence and subgiant stars. We still keep the original labels they had in the queried catalogues and in Fig. [Fig:HRDStarevol].\nThere seems to be a tendency for larger orbital periods in the red clump, as compared to systems lying below (i.e. with luminosities from the main sequence up to the red clump). In an attempt to quantify this effect, the sample has been split into two subsamples: one with *L*/L⊙ ≤ 25 (log*L*/L⊙ ≤ 1.4) and the other with 25 < *L*/L⊙ ≤ 160 (1.5 < log*L*/L⊙ ≤ 2.2). The first sample comprises main-sequence and subgiant stars whose current primary components never evolved up the RGB. The second sample, however, comprises systems in the red clump after their passage through the RGB tip, at least for systems with low-mass primaries, and a few less evolved objects, which happen to cross the red clump on their way to the RGB tip. Given the large radius reached at the RGB tip (see below), one may suspect that in the shorter period systems the giant filled its Roche lobe, and possibly evolved through a common-envelope stage into a cataclysmic variable or even a coalesced pair. Fig. [Fig:histoP] confirms that very few systems (6/36 = 17%) located in the red clump (green) have periods shorter than 1000 d and none of the systems have a period below 300 d. Compared to the less evolved systems (pink), there is indeed a deficit of short-period systems among red clump stars.\n![[Fig:HRP] Same as Fig. [Fig:HRDStarevol], but with the symbol size proportional to the orbital period. ](HRD_Pscale.png)[Fig:HRP] Same as Fig. [Fig:HRDStarevol], but with the symbol size proportional to the orbital period. \n![[Fig:histoP] Normalized histogram (left scale, with bins of size 0.5 in log scale) and cumulative frequency distribution (right scale) for orbital periods of barium and related systems. Systems with L/L_{\\odot} \\le 25 are depicted in pink, and systems with 25 < L/L_{\\odot} \\le 160) are depicted in green.](Phisto.png)[Fig:histoP] Normalized histogram (left scale, with bins of size 0.5 in log scale) and cumulative frequency distribution (right scale) for orbital periods of barium and related systems. Systems with *L*/L⊙ ≤ 25 are depicted in pink, and systems with 25 < *L*/L⊙ ≤ 160) are depicted in green.\nTo apply the Kolmogorov-Smirnov (K-S) test, the maximum difference between the two cumulative frequency distributions was evaluated and amounts to *D* = 0.2, for an effective sample size of *N* = 18 = *m* × *n*/(*m* + *n*), where *m* = 35 and *n* = 36 are the sizes of the pre-RGB and red clump samples, respectively. The resulting effective difference *D**N*1/2 = 0.85 corresponds to a first-risk error of 47% to reject the null hypothesis incorrectly. Thus, the K-S test does not give us sufficient evidence to reject the null hypothesis and to conclude that the two distributions are different. However, it is known that the K-S test tends to be more sensitive near the centre of the distribution, while our samples seem to differ in the tails. Moreover, since the second-kind error has not been assessed because of the risk of incorrectly accepting the null hypothesis, the above result does not constitute a definite proof that the samples are extracted from the same parent population.\nFor this reason, we turned to another test (“four-box test”) relying on the hypergeometric probability distribution. We divided our two samples (pre-clump and clump) into two subsamples with periods shorter and longer than 1000 days. Among the 35 pre-clump systems, 20 have short periods and 15 have long periods. Among the 36 ( ≡ *N*1) systems in the clump, 30 have short periods and 6 have long periods. Hence, among the 71 ( ≡ *N*) binaries, 21 binaries have short periods, so the probability of finding a short-period binary is *p* = 21/71 = 0.3. From this, we can now calculate the expected number of short-period objects among the clump sample as *x̃* = *p* × *N*1 = 10.8 instead of 6 that are observed. In order to check whether or not this difference is significant, we compute the variance on *x̃* as follows from the hypergeometric probability distribution (“small-sample statistics”): \n$\\sigma^{2}\\_{\\tilde x} = N1 \\times p \\times (1-p) \\times \n\\frac{N-N1}{N-1} = 3.78,$ i.e. $\\sigma\\_{\\tilde x} = 1.9$. \nHence, the expectation is *x̃* = 10.8 ± 1.9 or 8.9 < *x* < 12.7 at 1 *σ*, meaning that the pre-clump and clump samples are different with a 2.5*σ* significance.\nWe show in Table [Tab:RGB] that the 1000 d period corresponds to the period threshold below which a $\\backsim$1M⊙ Ba dwarf fills its Roche lobe when it reaches the RGB tip, and that the less populated 400 – 1000 d period range can be associated with the period cut-offs for stars in the mass range [1.0 – 2.5] M⊙. The critical periods for Roche-lobe overflow (RLOF) listed in Table [Tab:RGB] are derived as follows, assuming that the giant star fills its Roche lobe ($R\\_{\\rm L\\_{1}}$) at the RGB tip (with radius $R\\_{\\rm tip}$):\n \n$$R\\_{L\\_{1}} = a\\;\\frac{0.49 q^{2/3}}{0.6 q^{2/3} + \\ln( 1 + q^{1/3})} \\equiv a\\;f = R\\_{\\rm tip},$$ \nwhere *q* = *M*1/*M*2, with *M*1 the mass of the giant star, and *M*2 that of its WD companion. The critical semi-major axis of a circular orbit is then expressed as $a = R\\_{\\rm L\\_{1}}/f = R\\_{\\rm tip}/f$ and the corresponding period cut-off is\n \n$$\\label{Eq:Pcut}\nP = \\frac{(R\\_{\\rm tip} / f)^{3/2}} {(M\\_1+M\\_2)^{1/2}},$$ \nwhere *P* is expressed in years, $R\\_{\\rm tip}$ in AU (and is identified with the Roche radius $R\\_{\\rm L\\_{1}}$ around star 1), and the component masses *M*1, 2 in M⊙. The radius $R\\_{\\rm tip}$ is given by the STAREVOL models. A direct comparison between this predictions and the observed values is not straightforward. The periods entering the above equation are pre-mass-transfer values, while the observed periods are the post-mass-transfer values. A detailed discussion of the relationship between initial and final periods is beyond the scope of this paper (see for example, or ). Nevertheless, this analysis offers an order of magnitude estimate, although it should be considered with caution.\nThe absence of short-period barium stars in the red clump strongly suggests that RLOF on the RGB leads to systems that disappear from the barium star family. They could be the progenitors of short-period cataclysmic variables, or mergers. It is also tempting to associate these RGB-coalesced systems with early R-type carbon stars, which are another family of peculiar stars with a strong concentration in the red clump, and none of these stars are members of a binary system. The major problem with the hypothesis of a link between barium and R stars, however, is the lack of s-process element overabundances in early R-type carbon stars and of 12C in Ba stars.\nThe segregation in terms of orbital periods between pre-red clump and red clump systems requires further discussion, because all these barium systems host a WD whose progenitor went through the thermally pulsing AGB (TP-AGB) phase. We should in principle expect a cut-off in the period distribution of the barium dwarf systems as well because AGB stars also reach very large radii.\nIn fact, three different critical periods must thus be considered: (i) $P\\_{\\rm RGB, 1}$, when the initially more massive component (now a WD) reaches the RGB tip, i.e. when in the above equations, $M\\_1 = M\\_{\\rm 1@RGBtip}$ and $M\\_2 = M\\_{\\rm Ba}$; (ii) $P\\_{\\rm AGB, 1}$, when the initially more massive component reaches the AGB tip, i.e. when $M\\_1 = M\\_{\\rm 1@AGBtip} = M\\_{\\rm WD}$ and $M\\_2 = M\\_{\\rm Ba}$; and (iii) $P\\_{\\rm RGB, 2}$, when the initially less massive component (the current barium star) reaches the RGB tip, its companion being a WD, i.e. when $M\\_1 = M\\_{\\rm Ba}$ and $M\\_2 = M\\_{\\rm WD}$. A supplementary condition is that the barium-star mass ($M\\_{\\rm Ba}$) is lower than the initial mass of its companion, the WD progenitor; we neglected the mass transferred onto the barium star to make the computations easier. Periods $P\\_{\\rm RGB, 1}$ and $P\\_{\\rm AGB, 1}$ are shown in Fig. [Fig:Pcrit], whereas $P\\_{\\rm RGB, 2}$ are listed in Table [Tab:RGB] (assuming $M\\_2 = M\\_{\\rm WD}$ = 0.6 M⊙). There is a weak dependence of $P\\_{\\rm RGB, 1}$ on the initial mass ratio.\n[Fig:Pcrit] Critical periods below which one of the components fills its Roche lobe at either the tip of the RGB ($P\\_{\\rm RGB, 1}$, see text; dashed line at the bottom of the figure) or at the tip of the AGB ($P\\_{\\rm AGB, 1}$, solid line at the top of the figure). The input data $R\\_{\\rm RGB,tip}$ and $R\\_{\\rm AGB,tip}$ are taken from the STAREVOL grid (red curves solar metallicity *Z* = 0.0134; black curves: [Fe/H] = − 0.5 or *Z* = 0.0043). Various superimposed lines in a series correspond to various barium-star masses. The corresponding $M\\_{\\rm Ba}$ value may be identified from the starting point of the curve, which then extends towards increasing masses; for a given barium-star mass, the AGB component of mass $M\\_{\\rm AGB}$ – the former primary of the binary system – is necessarily larger. The critical period $P\\_{\\rm RGB, 2}$ (see text) does not strongly depend upon the mass of the WD companion. Therefore, its value can simply be read off the $P\\_{\\rm RGB, 1}$ curve (bold dashed line) at the corresponding $M\\_{\\rm Ba}$ value, which determines the key value $R\\_{\\rm RGB,tip}$. Although $P\\_{\\rm RGB, 1}$ and $P\\_{\\rm RGB, 2}$ are not strictly equal, they are not different enough to warrant a specific $P\\_{\\rm RGB, 2}$ curve that would jeopardize the clarity of the figure (see also Table [Tab:RGB]). This is a corrected version of the original Fig. 17.\nWe clearly see that RLOF at the RGB tip occurs for all systems with a low-mass ($M \\lesssim 2.2$M⊙) primary and periods shorter than 110 d (and shorter than 1000 d for 1 M⊙ stars; see also Table [Tab:RGB]). This period threshold is consistent with the orbital-period segregation observed in the barium stars. On the other hand, intermediate-mass stars ($2.2 \\lesssim M$/M$\\_{\\odot} \\lesssim 8 $) do not expand as much as their lower mass counterparts and the critical period is too short to affect the barium star population.\nAnother interesting conclusion may be drawn from the other (apparent) puzzle emerging from the present analysis; that is how the long-period thresholds (600 to 7500 d; $P\\_{\\rm AGB, 1}$ in Fig. [Fig:Pcrit]) imposed by the passage of the WD progenitor through the AGB tip can be reconciled with the much shorter periods currently observed among barium systems. This is the long-standing puzzle identified by, for example,, and. Most barium systems should thus have gone through RLOF at the AGB tip, but this RLOF cannot have led to a catastrophic orbital shrinkage. In that case, we seek to understand why the barium systems seemingly disappeared after RLOF occurring on the RGB, but not after RLOF occurring along the AGB of the former primary and current WD. The key difference between RLOF close to AGB and RGB tip is the mass of the residual envelope, which is generally much larger at the RGB tip. Hence, RLOF close to AGB tip does not involve a massive envelope, and we may expect a modest orbital shrinkage at that stage. Alternatively, if RLOF starts when the mass ratio is close to, or below, unity (a likely possibility with a stripped AGB star), a common evolution can be avoided, since the orbital separation expands soon after the beginning of mass transfer. This moderate shrinkage could then account for the fact that the observed periods for RGB systems shown in Fig. [Fig:histoP] are shorter than the AGB threshold periods indicated in Fig. [Fig:Pcrit]. The possibility of intense mass loss and angular momentum loss on the AGB should also be considered (e.g. ; ).\nAn interesting conclusion ensues by applying the previous arguments to the early stage of the binary system, when the former primary ascended the RGB with the future barium star still on the main sequence. For such systems with an orbital period shorter than $P\\_{\\rm RGB, 1}$, RLOF with dramatic orbital shrinkage must have occurred, thus probably removing from the barium-star family those systems that had an initial period shorter than 1000 d with a primary mass of 1 M⊙ or shorter than 200 d with a primary mass of 2.2 M⊙. Stars in the mass range 1 – 2.2 M⊙ would have led to a WD with a mass below 0.58 M⊙, according to the initial - final mass relationship inferred from the STAREVOL tracks. Therefore, we may expect a deficit of low-mass WDs among barium systems. This issue will be investigated in a forthcoming paper.\nllllllll $M\\_{\\rm now}$ & $M\\_{\\rm initial}$ &log(*L*/L⊙) & $R\\_{\\rm RGB tip}$ & *a*& *P* \n(M⊙) & (M⊙) & & (R⊙) & (R⊙) & (d) \n1.0 & 1.2 & 3.42 & 184 & 438 & 835 \n1.5 & 1.6 & 3.43 & 162 & 352 & 524 \n2.0 & 2.1 & 3.39 & 136 & 283 & 340 \n2.5 & 2.6 & 3.64 & 190 & 380 & 484 \n3.0 & 3.0 & 3.21 & 92 & 177 & 141 \nConclusions\n===========\nWe have constructed an HRD of a sample of barium and related stars with unprecedented accuracy thanks to the TGAS parallaxes. From the comparison of the location of these stars on the HRD with STAREVOL evolutionary models, we find that the reclassification of some of these stars is needed. Several stars classified as Ba giants in the past are in fact dwarfs or subgiants. Additionally, there is no separation between subgiant CH stars and dwarf Ba in the HRD, contrary to what is implied by their designations.\nThe mass distribution derived for barium giants shows a deficit of low-mass stars ($M \\la 2.0$ M⊙; the exact threshold depends on metallicity), as compared to normal K and M field giants. This deficit is probably a consequence of a premature shrinkage of the orbits with shorter periods, as less massive stars reach large radii on the tip of the RGB. Additionally, stars with a barium index below Ba1, introduced by in the second edition of his catalogue, populate the high-mass tail of the distribution and probably are not genuine Ba stars but bright giants where the Ba lines are strengthened by a positive luminosity effect. Finally, and in contrast to previous claims, we do not see any difference between the mass distributions of mild (Ba1 – Ba2) and strong (Ba3 – Ba5) barium stars, at least when adopting the same average metallicity [Fe/H] = − 0.25 for both groups. It seems that there is no correlation between Ba abundance and Ba stellar mass.\nThere is a deficit of barium systems with short periods among those populating the red clump, as a result of the period threshold set by the large radii reached at the tip of the RGB. The systems missing from the red clump population could have coalesced as a result of unstable RLOF close to the tip of the RGB.\nThe accuracy of the TGAS parallaxes marks a significant step forward for the derivation of barium star masses. However, to obtain information on individual stars, the derivation of metallicities from spectroscopic data for each target is crucial. A consistent derivation of the metallicities would also allow us to conclude if the apparent difference between the metallicity distributions of dwarf and giant barium stars is real.\nThis research has been funded by the Belgian Science Policy Office under contract BR/143/A2/STARLAB and the Fonds voor Wetenschappelijk Onderzoek Vlaanderen (FWO). A.E. acknowledges support from the FWO. S.V.E acknowledges the Fondation ULB. L.S and D.P. are Senior FNRS Research Associates. D.K acknowledges support from BELSPO. The authors want to acknowledge the referee, Claudio B. Pereira, for his interest and very valuable input. This work has made use of data from the European Space Agency (ESA) mission *Gaia* (<https://www.cosmos.esa.int/gaia>), processed by the *Gaia* Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the *Gaia* Multilateral Agreement. This research has also made use of the SIMBAD database, operated at CDS, Strasbourg, France.\nSample of barium and related stars with accurate parallaxes\n===========================================================\nTable [Tab:master] lists the barium and related stars with good-precision Gaia and Hipparcos parallaxes (i.e. $\\varpi / \\sigma(\\varpi) > 3$). The full list is available at CDS, Strasbourg.\nrrrrrrrrrrrrrrrrrrrrrrr \nName & & & type & [Fe/H] & & & source & [Fe/H] & $T\\_{\\rm eff, min}$ & &$T\\_{\\rm eff, max}$ & & & \n& & & & & & & && & & & & & \n \nHD 26 & 19364 & 0.08 & CH & -0.45 & 1986ApJ...303..226S & 3.40 ± 0.30 & Gaia & 0 & 5136 & 5290 & 5342& 52 & 61 & 74 & \n& & & & & & & & -0.25& 4898 & 5003 & 5193& 44 & 52 & 63 & \n& & & & & & & & -0.5 & 4773 & 4901 & 5077 & 42 & 49 & 59 \nHD 218 & - & - & Ba3-5 & N/A & N/A & 1.71 ± 0.27 & Gaia & 0 & 3787 & 3979 & 4058 & 507 & 680 &960 \n& & & & & & & & -0.25& 3782 & 3958 & 4045 & 511 & 686 &969 \n& & & & & & & & -0.5 & 3768 & 3943 & 4035 & 508 & 681 &963 \nHD 749 & - & - & Ba1-2 & -0.06& 2006A&A...454..895 &7.11 ± 1.08 & Hipparcos & 0& 4687 &4909 & 4967 & 14.0 & 18.5 & 25.8 \n& & & & & & & & -0.25&4657& 4892 & 4979 & 13.8 & 18.3 & 25.4 \n& & & & & & & & -0.5 &4662&4795 & 4990 & 12.9 & 17.0 & 23.7 \nHD 4084 & - & - &Ba3-5& -0.42 & 2016MNRAS..459...4299 & 1.81 ± 0.25& Gaia & 0&4631 &4874& 4970& 126& 164& 221& \n& & & & & & & & -0.25&4609 &4894 & 4992& 127& 165& 223 \n& & & & & & & & -0.5 &4574 &4941 & 4991& 128& 166& 224 \nHD 5322 & - & - &Ba1-2& -0.18 & 2016MNRAS..459...4299 & 1.43 ± 0.26 & Gaia & 0 &4827 &5083& 5326 &497& 696 &1046 \n& & & & & & & & -0.25&4832 &5065& 5237 &487& 683 &1026 \n& & & & & & & & -0.5 &4858 &4957& 5224 &457& 641 &964 \nHD 5395 & - & - & Ba1-2& -0.24 & 2016MNRAS..440..1095 & 15.84 ± 0.58 &Hipparcos & 0&5139 &5201 &5263 & 76& 82 &88 \n& & & & & & & & -0.25&4903 &5081 &5178 & 77& 83 &89 \n& & & & & & & & -0.5 &4660 &4836 &5087& 70& 76 &82 \nHD 5424 &1881& 0.23 &dBa1-2& -0.51 & 2006A&A...454..895 & 2.50 ± 0.27 & Gaia & 0&4727&4888& 5015& 29& 36& 45 \n& & & & & & & & -0.25&4667&4870& 4987& 29& 36& 45 \n& & & & & & & & -0.5&4575& 4804& 4953& 29& 35& 44 \n... \n \n---\n1. Table A.1 is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via <http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/>[↩](#fnref1)\n"},"data_id":{"kind":"string","value":"arxiv_0002461"}}},{"rowIdx":597,"cells":{"content":{"kind":"string","value":"Study of LINER sources with broad H*α* emission.X-ray properties and comparison to luminous AGN and X-ray binaries.\n===================================================================================================================\n### Received / Accepted\nIntroduction\n============\nIt is widely known that a large fraction of local galaxies contain emission line nuclei that are the result of low nuclear activity. showed that more than 40% of 486 nearby galaxies with *B**T* ≤ 12.5 mag could be considered as active with optical spectra classified as Seyfert nuclei, low-ionization nuclear emission-line regions, or transition objects (objects having intermediate spectra between LINERs and H II nuclei). Pure LINER sources would be the most common component representing 20% of all 486 galaxies. The ionization mechanism responsible for the excitation of emission lines in LINER sources is an ongoing matter of debate and could be explained in terms of: shock heated gas, starburst activity, or a low luminosity active Galactic nucleus (AGN). Many multiwavelength studies were attributed to this subject, looking for a radio, sometimes variable, core or a variable UV core in nearby LINER sources. Nevertheless, the most used tool to search for an active nucleus in a LINER is to look for a hard 2-10 keV unresolved core that could not be due to diffuse emission from shock heated gas or from unusually hot stars. How do LINERs harboring a low luminosity active nucleus compare to luminous Seyfert galaxies and quasars?\n, using high angular resolution multiwavelength observations of 13 LINER sources, demonstrated that the luminosity ratios in different wavebands, mainly UV to X-ray and radio to UV luminosities, follow the same trend as luminous Seyfert galaxies. The authors did not find any sharp change in the spectral energy distribution (SED) of their sample of 13 LINERs compared to more luminous Seyfert and quasar nuclei, suggesting that a thin accretion disk may persist at low accretion rates. Moreover, detected up to 30% flux variations on half a day time-scale in 2 (NGC 3998 and M81) out of 4 LINER and low luminosity AGN sources observed in X-ray with the XRT onboard *Swift*. They combined their X-ray fluxes with simultaneous UV fluxes coming from the UVOT instrument and showed that the SED and the UV to X-ray flux ratios of their 4 sources sample are consistent with those of more luminous sources and that LINERs may have similar accretion and radiative processes at their center compared to luminous Seyfert nuclei.\nOn the other hand, the faintness of LINER sources compared to luminous Seyfert galaxies and quasars has been attributed to a different accretion mechanism owing to some observational contrast between the two classes. No broad nor narrow Fe K*α* emission line at 6.4 keV have been detected in the spectra of the LINER sources with the highest signal to noise ratio, X-ray short time-scale variability has been detected in only a few sources, and the disappearance of the big blue bump in the UV band in the SED of LINER sources, all these signatures could indicate the disappearance of the thin accretion disk at low luminosities and that a different accretion mechanism is responsible for the emission in LINER sources. It has been suggested that when the mass accretion rate falls below a critical value $\\dot{M}\\_{crit}$, the density of the disk could become too low for radiative cooling to be effective. The trapped heat will expand the thin accretion disk into a pressure-supported thick disk with a very low radiative efficiency. Such radiatively inefficient accretion flow (RIAF) models successfully explained the spectral energy distribution of a large number of LINER sources.\nAnother way to assess the geometry of the accretion mode in AGN is to compare them to their less massive counterparts, X-ray binaries (XRBs). showed that the X-ray spectral slope, Γ, of Seyfert 1 galaxies and quasars and the Eddington ratio, *L**b**o**l*/*L**E**d**d*, are positively correlated, similar to XRBs in their high/soft state. Such a behavior could be explained in an AGN accretion mode consistent with an optically thick geometrically thin accretion disk. performed a similar study on a broad sample of LINERs and low luminosity Seyfert galaxies. They found a significant anticorrelation between Γ and the Eddington ratio for the local Seyfert galaxies in their sample analogous to XRBs in the low/hard state where a RIAF mode of accretion takes place. However, no strong correlation was found when considering only the LINER sources in their sample, owing, as suggested by the authors, to heterogeneous fitting models as they have collected their data from different studies. In a separate study, analyzed the X-ray emission of a sample of 107 nearby galaxies including low luminosity Seyferts, LINERs, transitions (nuclei with spectra between Seyferts and LINERs), H II regions, and passive galaxies (lacking optical emission-line activity), *none of which show broad-line components*. Using a Spearman-rank correlation, the authors found an anticorrelation for their sample between Γ and the *L**b**o**l*/*L**E**d**d*. By considering each class separately, a spearman-rank test showed that the anticorrelation persists for the different objects, except for the low luminosity Seyfert galaxies.\nFinally, broad optical emission lines, a characteristic property of classical Seyferts and quasars, are also found in nuclei of much lower luminosities. Thirty three sources out of the 221 nuclei classified as Seyfert, LINER, or transition objects in sample of nearby galaxies show definite detection of a broad H*α* emission, 16 of those ( ∼ 17% of the total pure LINER sources) are LINERs. In this paper, we are aiming for the study of the X-ray characteristics of these LINER 1s observed with the current generation of X-ray telescopes, *XMM-Newton* and *Chandra*. Such a sample insures the responsibility of accretion into a SMBH for the formation of the broad emission lines (given the early type class of this sample where outflows form massive stars and/or supernovae are not expected to be relevant), guarantees the non-existence of large obscuration, and enables X-ray comparison of this class with both XRBs and type 1 AGN. We introduce our sample in § 2, § 3 represents the observations and the data reduction. Temporal and spectral results are given in § 4. In § 5 we discuss the results in the context of LINER 1s-Seyfert-XRB connections, and a conclusion summarizing the main results is given in § 6. We report, in appendix A, some notes on the individual sources and in Appendix B we give spectral results to the surrounding sources around the centers of galaxies observed with *Chandra*. In the remainder of this paper, luminosities are calculated using the distances given in Table [galaxy-param] derived with a Hubble constant *H*0 = 75 km s− 1 Mpc− 1.\nThe sample\n==========\nWe selected objects classified as LINER 1.9 (LINER 1s) sources in showing a definite detection of a broad H*α* emission line. This implies the definite existence of an AGN at the center of all of the sixteen selected galaxies and its responsibility for the excitation of the detected optical emission lines.\n[!th] [galaxy-param]\n| | | | | | |\n| --- | --- | --- | --- | --- | --- |\n| Galaxy Name | R.A. | Dec. | Hubble Type | Distance*a* | log(*M**B**H*)*b* |\n| | | | | (Mpc) | (*M*⊙) |\n| | 00 49 47.8 | +32 16 40 | SB(rs)ab | 62.4 | 8.44 |\n| | 00 57 48.9 | +30 21 09 | E+: | 65.8 | 9.06 |\n| | 08 53 32.7 | +51 18 49 | (R’)SAB(rs)0/a | 17.2 | 6.78 |\n| | 09 19 18.5 | +69 12 12 | SB(r)0+ | 7.48 | 8.15 |\n| | 10 23 27.0 | +19 53 55 | E2:pec | 23.6 | 8.06 |\n| | 11 32 34.8 | +53 04 05 | SB(s)a pec | 17.0 | 7.61 |\n| | 11 57 56.1 | +55 27 13 | SA(r)0? | 14.1 | 9.07 |\n| | 12 09 36.0 | +42 32 03 | SAB(s)0 | 15.9 | 8.18 |\n| | 12 15 05.0 | +33 11 50 | SAB0-: | 15.1 | 7.73 |\n| | 12 20 06.8 | +29 16 51 | E1+ | 16.1 | 8.72 |\n| | 12 50 07.2 | +72 52 28 | (R)SA(rs)ab | 26.1 | 7.27 |\n| | 12 53 29.1 | +02 10 06 | SA(s)a | 16.3 | 7.46 |\n| | 13 10 56.2 | +37 03 33 | SAB(rs)bc | 21.3 | 7.79 |\n*a*Distances adapted from, otherwise from. *b*Black hole mass calculated using updated M-*σ* relation of with stellar velocity dispersion taken from.\nWe excluded three sources from the sample: NGC 3642, NGC 4636 and NGC 1052. NGC 3642 did not have any archived *XMM-Newton* or *Chandra* observations. As for NGC 4636, all of the X-ray archived observations were studied in extreme detail and show a complicated spectrum that requires, for a good spectral parameters measurement, detailed imaging analysis and should be modeled including: sophisticated shock models, temperature and density gradients, and last but not least, steep abundance gradients in the core. Finally, the broad H*α* emission line detected in the spectrum of NGC 1052 is attributed to polarization due to electron scattering within the opening cone of an obscuring torus. NGC 1052 was classified as an obscured AGN showing a large intrinsic absorption in the X-ray spectrum.\nTable [galaxy-param] shows the list of galaxies along with their corresponding right ascension and declination, Hubble type, distance (taken from, otherwise from ), the mass of the black hole derived from the M-*σ* relation where the velocity dispersion is taken from. Multiple snapshot observations (exposure time ≤ 5 ks) were excluded from the analysis either because of high background contamination (NGC 4143, obs.ID: 0150010201), low number of counts detected (NGC 2787, obs. ID: 388), or severe pile-up (NGC 4203 and NGC 4278, obs. IDs:397 and 398, respectively). The final sample consists of 13 LINER 1s with a total of 31 observations summarized in Table [obs-param].\n[!th] [obs-param]\n \n*a*Observations reported for the first time for the LINER 1 nucleus study. *b*Exposure time corrected for solar flare intervals.\nX-ray observations and data reduction\n=====================================\n*Chandra* observations\n----------------------\nAll of the LINER 1s in our sample have at least one *Chandra* observation. Snapshot observations with an exposure time less than 5 ks were performed for eight sources (NGC 266, NGC 315, NGC 3226, NGC 3718, NGC 4143, NGC 4750, NGC 4772, and NGC 5005). Seven sources have observations with a sufficient exposure time for a detailed temporal and spectral study (NGC 315, NGC 2681, NGC 2787, NGC 3226, NGC 3998, NGC 4203, and NGC 4278). All of the *Chandra* observations were obtained with the spectroscopic array where the nucleus was placed on the aim point, except for NGC 3226, of the ACIS-S3 back-illuminated chip. They were taken in either Faint or Very Faint mode to increase their sensitivity. All of the observations are *Chandra* archival data obtained from chaser[1](#fn1). The log of the *Chandra* observations are listed in Table 2.\nAll *Chandra* observations were reduced and analyzed in a systematic, homogeneous way using the CIAO software package version 4.2, *Chandra* Calibration Database, CALDB, version 4.3.1, and the ACIS Extract (AE) software package version 3.175 [2](#fn2). We started by using the level 1 event file produced by the *Chandra* X-ray Center (CXC) to suppress the position randomization applied by the CXC Standard Data Processing when creating a level 2 event file. We also corrected for the effects of charge-transfer inefficiency on event energies and grades. We filtered for bad event grades (only ASCA grades 0, 2, 3, 4 and 6 are accepted) and hot columns to take account of several potential issues such as cosmic rays and bad pixels. Good time intervals, supplied by the pipeline, were applied to the final products.\nThe LINER nucleus source position is determined after running a wavelet transform detection algorithm, the *wavdetect* program within the CIAO data analysis system. This position is then given to the AE software that refines it, extract source photons, construct local backgrounds, extract source, and background spectra, compute redistribution matrix files (RMFs) and auxiliary response files (ARFs), by spawning the *mkarf* and *mkacisrmf* routines of CIAO, and perform spectral grouping and fitting.\nSource events are extracted around the source centroid, inside a polygonal shape of the local PSF, generated by MARX[3](#fn3) at the energy 1.497 keV using the *ae\\_make\\_psf* tool implemented in AE. Background region is defined following the AE procedure. The background region is an annular region centered on the source position where the inner radius delimit 1.1 × 99% encircled energy radius and the outer radius is set such that the background includes between 100 counts and 200 counts (depending on the brightness of the source). This background is obtained from a special image where all events within the ∼ 1.1 × 99% PSF circles of all the sources in the field were excluded (*swiss cheese image*). Background was modeled for snapshot observations.\nPiled-up observations were accounted for by excluding the core of the PSF (see Y10 for more details). We use the tool *dmextract*, called by the AE software, to create spectra over the energy range 0.5–8 keV. We used the tool *ae\\_group\\_spectrum* implemented in AE to group the spectra. Channels between 0.5 and 8 keV are grouped to have a three sigma (3*σ*) signal to noise ratio, which corresponds to a minimum of 20 counts per bin, to enable the use of the *χ*2 statistics in the spectral analysis. The cash statistic (C-stat) is used to derive spectral parameters for snapshot *Chandra* observations with the background being modeled with the *cplinear* background model developed by. The background model is arbitrarily chosen to consist of continuous piecewise-linear (*cplinear*) functions with 2 to 10 vertexes. The model has 2 to 10 parameters representing the X-ray fluxes at the different vertexes. These vertexes are placed on the energy scale so that they divide the energy range into intervals with approximately equal numbers of observed counts in the background spectrum (0.1 to 10 keV). Vertex energies are chosen to coincide with the energies of actual events in the background which helps to prevent the vertex flux from being driven to the hard limit of zero during the fitting process.\n*XMM-Newton* observations\n-------------------------\nThe log of the *XMM-Newton* observations is listed in Table 2. Eight sources were observed at least once with *XMM-Newton* (NGC 315, NGC 2787, NGC 3226, NGC 3718, NGC 3998, NGC 4143, NGC 4278, and NGC 5005) and two have multiple observations (NGC 3226 and NGC 3718). In all of the observations, the EPIC-pn and MOS cameras were operated in Imaging, Prime Full Frame or Large Window Mode (except for the long NGC 3226 observation, where MOS cameras were operating in Small Window Mode[4](#fn4)) using the thin or medium filter. The Reflection Grating Spectra show only few counts for all the different observations and therefore they were not included in our analysis. We did not make use of the optical/UV data taken with the optical/UV monitor (OM) instrument since this paper concentrates on the X-ray characteristics of this sample. A multiwavelength study of our sample will be treated in a forthcoming paper. All data products were obtained from the XMM-Newton Science Archive (XSA)[5](#fn5) and reduced using the Science Analysis System (SAS) version 9.0. Data are selected using event patterns 0–4 and 0–12 for pn and MOS, respectively, during only good X-ray events (“FLAG = 0”). None of the EPIC observations were affected by pile-up, although severe intervals of enhanced solar activity, where the background count rate even exceeds the source count rate, were present during several observations (NGC 315, NGC2787, NGC 3226). In these cases, we reduced the background to 5% by excluding the high background intervals which reduces the observation time usable for spectral analysis, sometimes to less than 30% of the raw exposure time (NGC 315).\n*XMM-Newton* source events of all of the LINER 1s in our sample were extracted from a circle centered at the nucleus using two different radii of 10 and 25[6](#fn6). We compared light curves and spectra of both extraction regions to check if any of the sources (jet emission, diffuse emission, and/or unresolved point-like sources) detected in the *Chandra* image between 10 and 25 around the nucleus contaminate the *XMM-Newton* nucleus emission (see Appendix B and online Fig. 1–4). No change is seen in the light curves, and the fit parameters of the two extracted spectra were consistent, within the error bars. Therefore, and to achieve better statistics and better constrain fit parameters, source events of all of the LINER 1s in our sample observed with *XMM-Newton* were taken from the 25-radius circle centered on the nucleus. We added the spectral contribution of the different sources (jet emission, diffuse emission, and/or unresolved point-like sources), derived from the *Chandra* observation and detected in a 25-radius circle around the nucleus (see Appendix B for more details), to the spectral model used to fit the *XMM-Newton* spectrum. The particular case of NGC 4278 is discussed in detail in Y10. Background events are extracted from a source–free circle with a radius twice of the source on the same CCD. We generated response matrix files using the SAS task *rmfgen*, while ancillary response files were generated using the SAS task *arfgen*. The EPIC spectra were created in the energy range 0.5–10 keV to enable flux and model–parameter comparison with *Chandra* spectra. They were grouped to have a signal to noise ratio of 3 with a minimum of 20 counts per bin to allow the use of the *χ*2 statistic.\nResults\n=======\nLight curves and hardness ratios\n--------------------------------\nTemporal analysis was only done for long exposure observations, not including snapshot *Chandra* observations. Light curves and corresponding hardness ratios, defined as *H**R* = (*H* − *S*)/(*H* + *S*), where *S* is the count rate in the soft 0.5-2 keV band and *H* is the count rate in the hard 2-10 keV band, were extracted for all of the long observations. We corrected the net count rate of the piled-up sources for the excluded fraction of the PSF. *Chandra* and *XMM-Newton* light curves were all binned with a time bin size of 1 ks for a reliable rms variability analysis.\nWe first conducted a Kolmogorov-Smirnov, K-S, test to examine any potential variability within each observation. Based on this test, we do not find short time-scale (hours to days) variability in 5/10 sources (NGC 315, NGC 2681, NGC 3718, NGC 3998, and NGC 5005) with a K-S test probability > 10% that the nuclear emission originates from a constant source. Three sources (NGC 2787, NGC 4143, and NGC 4203) indicate a possible variability with a K-S test probability between 4% and 2% that the nuclear emission originates from a constant source. Two *XMM-Newton* observations of two different sources exhibit significant short time-scale variability, both already reported in the literature, NGC 4278 (obs. ID: 205010101, Y10) and NGC 3226, where the K-S test gives a probability less than 1% that the core emission is originating from a constant source. NGC 4278 shows an increase at the beginning of the observation of 10% on a time-scale of ∼ 1.5 hour, the emission remains constant for the rest of the observation following that hint of variability. As for NGC 3226, variability is clear through the whole observation where a total increase of ∼ 60% is detected between the beginning and the end of the ∼ 100 ks observation. *XMM-Newton* light curve of NGC 3226 is shown in Fig. [light-hardness] and the *XMM-Newton* and *Chandra* light curves of the other sources are given in online Fig. [LCxmmallLINERs] and Fig. [LCchandraallLINERs].\n [light-hardness]\n[!th]\n [varsigma] \n*a*The exposure times in ks used to calculate the value of *σ**N**X**S*2. *b*Value corresponding to only the longest observation of NGC 3226. Including the other *XMM-Newton* observation would result in a *σ**N**X**S*2 upper limit of 0.013.\nTo check more accurately any intrinsic variability amplitude from the different sources, we calculated the normalized excess variance for all of the long observations with the following expression:\n \n$$\\sigma^2\\_{NXS}=\\frac{1}{N\\mu^2}\\sum\\limits\\_{i=1}^{N}[(X\\_i-\\mu)^2-\\sigma\\_{i}^2]$$ \nwhere *N* is the number of bins in a given light curve, *X**i* and *σ**i* are the count rate and uncertainty of each bin, respectively, and *μ* is the arithmetic mean of the counting rates. To enable *σ**N**X**S*2 comparison between the different light curves, the bin size and the light curve segment duration should be taken equally. For that purpose, we first decided to use light curve segments of 20 ks, as usually done for luminous Seyfert galaxies, splitting any longer observations into multiple ones. That limited our sample to 6 sources observed with *Chandra* and 5 observed with *XMM-Newton*, not enough to draw any safe conclusions. Therefore, and owing to the heterogeneous sampling of the observations for this type of study, we decided to use the whole corrected exposure time of all of the long observations. The mean of the *σ**N**X**S*2 is taken for every source with multiple *Chandra* or *XMM-Newton* observations. The time bin size choice of 1 ks for all of the observations was taken to have a good signal to noise ratio with at least 20 counts in each bin and an acceptable number of bins in each light curve.\nEstimating the error on the *σ**N**X**S*2 could be a tricky task. The variability in an AGN light curve depends, on one hand, on the measurement errors of the data (e.g. Poisson noise) and, on the other hand, on the stochastic nature of the process underlying AGN variability ; even if a source is not intrinsically variable the mean and the variance of different light curves based on observations performed at different times will not be identical. We estimated error due to Poisson noise using the equation\n \n$$\\label{errsig}\n\\resizebox{.43\\textwidth}{!}{$err(\\sigma^2\\_{NXS})=\\sqrt{\\left(\\sqrt{\\frac{2}{N}}\\frac{\\overline{\\sigma\\_{err}^{2}}}{\\overline{x}^2}\\right)^2+\\left(\\sqrt{\\frac{\\overline{\\sigma^{2}\\_{err}}}{N}}\\frac{2\\sigma\\_{NXS}}{\\overline{x}}\\right)^2}$}.$$ \n[NXSvsMbh]\nThe uncertainty owing to the red noise process has been presented by to depend on the power-spectrum shape of the source which we do not know a priori. Therefore, estimated the error on the red noise process directly from the data. Our observations are not well sampled to use method to determine any error due to the stochastic nature of the AGN X-ray variability. The error on *σ**N**X**S*2 shown in equation [errsig] was also used to estimate upper limits to the excess variance in the case of non-variability detection whenever *σ**N**X**S*2 is negative or consistent with zero. Only one object, NGC 3226, in our sample shows a clear short time-scale variability during the longest 100 ks observation with *σ**N**X**S*2 = 0.02 ± 0.002 (comparable to the value of 0.014 found by ). Upper limits were obtained for the rest of the sample. Table [varsigma] and Fig. [NXSvsMbh] summarizes the results that are discussed in § [xrayvar].\nX-ray spectral results\n----------------------\nThe spectral analysis was performed using XSPEC version 12.6.0. The photo-electric cross sections and the solar abundances of are used throughout to account for absorption by neutral gas. An absorbed Galactic column density derived for every single source from (obtained with the W3NH tool[7](#fn7)) was applied to the different spectral models. Spectral uncertainties are given using Δ*χ*2 of 2.71, corresponding to 90% confidence for one interesting parameter, and to 95% confidence for upper limits.\n### snapshot observations\nWe began our spectral analysis with the study of the *Chandra* snapshot observations. Table [specfit-param-snapshot] gives the best fit parameters to the snapshot observations of our sample of LINER 1s. Complicated models, like partial covering and/or two power-law components could not be tested due to the low number of counts. A power-law modified by Galactic absorption gave a good fit in the case of NGC 266 and NGC 4143. An additional intrinsic neutral absorption was needed in the remainder of the cases. The photon indices vary between 1.3 ± 0.2 for the hardest spectra to 2.1 ± 0.7 for the softest ones with a mean value of about 1.7. The hydrogen column density of the intrinsic absorber had an upper limit of ∼ 4 × 1021 cm− 2 in the case of NGC 4750 and NGC 5005. A value consistent with 1021 < *N**H* < 1022 cm− 2 was derived for the rest of the snapshot observations. In one case, NGC 3718, we find a somewhat larger column density with *N**H* in the order of ∼ 1022 cm− 2. For NGC 5005, a thermal component with a 0.8− 0.2+ 0.3 keV temperature, was included in the model to take account for some low energy features, most likely due to diffuse emission from hot gas. In order to rigorously confirm the validity of our best fit spectral-parameter values derived using the C-stat and a modeled background, we compared our results to the results derived from fits applying the *χ*2 statistics to all of the snapshots observations. Spectral parameters derived using the C-stat were all in agreement, within the error bars, with the results derived using the *χ*2 statistics; with smaller deviation from the central value of one interesting parameter. We decided, as a consequence, to use the C-stat fits to calculate model fluxes in the soft, 0.5-2 keV, and in the hard, 2-10 keV, bands. Table [specfit-fluxes] gives the corresponding 0.5-2 keV and 2-10 keV observed fluxes and corrected luminosities.\n[!th] [specfit-param-snapshot]\n \n(*a*)The emission measure (EM) of the *mekal* model, EM=∫n*e*n*H*dV.\n### Long-exposure observations\n[!th] [specfit-param]\n \n(L.) represents a linked paramter in the fit. (*a*)The emission measure (EM) of the *mekal* model, EM=∫n*e*n*H*dV.\n[!th] [specfit-fluxes]\n \n(*a*)The power-law component fraction to the 0.5-10 keV corrected luminosity.\n \n [bestfitmod]\nWe then turned to the analysis of observations with relatively long exposure times. We started with a simple absorbed power-law fit to each of the spectra of the different sources, separately. The fits were acceptable for 6/10 sources but residuals at energies less than 2 keV persisted in the other 4 sources (NGC 315, NGC 2681, NGC 4278, and NGC 5005), suggesting the presence of diffuse hot gas.\nIn order to have a better signal to noise ratio and photon statistics, we decided to fit the different *Chandra* and/or *XMM-Newton* spectra of each source simultaneously (the normalizations of the different models in a fit were left free between the different EPIC instruments to take care of any potential cross calibration uncertainties). Simultaneous fitting routines of different observations performed at different times is done for the first time for any sample of LINER sources. For this purpose, whenever a source is observed with both *Chandra* and *XMM-Newton*, we carried out careful imaging analysis of the *Chandra* observations to disentangle the different components (diffuse emission, LMXBs, and/or jet emission) that are blended in one point-like source in the *XMM-Newton* extraction region of 25-radius circle (see Appendix B and online Fig. 1–4). These components, that we assume non-variable, are included in the simultaneous *Chandra*/*XMM-Newton* fit. We do not expect diffuse hot gas to vary on a time-scale of a few years, although off-nuclear point-like sources and jet X-ray emission could exhibit variation on such time-scales. We present in Appendix B the surrounding medium around the different LINER 1s that are observed with both *Chandra* and *XMM-Newton*, and we give spectral results to the different components.\nWe tested multiple models on the data in order to determine the mechanism responsible for the observed X-ray spectra, noticeably: (1) a simple absorbed power-law, (2) same as (1) but including a thermal *mekal* component to take account of any diffuse hot gas features in the soft band, (3) two power-law components with different photon index values with one representing the hard 2-10 keV emission and the other representing a possible 0.5-2 keV soft excess emission commonly seen in the nuclei of luminous galaxies. We investigated spectral variability in a source by permitting one spectral parameter of a given fit to vary independently between different observations. We then used the F-test to evaluate the improvement in the fit where a 99% probability for an improvement to occur by chance is considered valid. The case for NGC 4278 is already analyzed in Y10 and best fit spectral parameters and fluxes are taken from Y10.\nWe find that 6/10 sources are best fit with model (1) with no additional need for any thermal or two power-law components (NGC 2787, NGC 3226, NGC 3718, NGC 3998, NGC 4143, and NGC 4203). The rest of the sources were best fit with model (2) showing features at low energies below 2 keV, indicating emission from diffuse hot gas (NGC 315, NGC 2681, NGC 4278, and NGC 5005). Model (3) did not improve the quality of the fit in any of the cases, giving worse fits at times. We found that the intrinsic hydrogen column density varies significantly in NGC 3226 decreasing from (8.9 ± 0.7) × 1021 cm− 2 to 4.2− 0.3+ 0.2 × 1021 cm− 2. This is clearly seen in the lower panels of Fig. [bestfitmod] where the soft part of the spectrum during the short *XMM-Newton* observation is much more absorbed compared to the long observation. Additionally, the power-law photon index varies in seven sources (NGC 315, NGC 3226, NGC 3718, NGC 3998, NGC 4143, NGC 4278, and NGC 5005) with the most drastic change being the one observed in NGC 315 where Γ increased from 1.5 ± 0.1 during the *Chandra* observation to 2.1− 0.2+ 0.1 during the *XMM-Newton* one (upper panels of Fig. [bestfitmod]). This increase is accompanied by a decrease in the 2-10 keV flux from 9.8 × 10− 13 to 4.6 × 10− 13 erg s− 1. This behavior is typical of X-ray emission originating in a RIAF structure, which is the accretion flow believed to exist at the center of NGC 315, where the Eddington ratio, which is proportional to L2 − 10 *k**e**V*, is inveresly proportional to the photon index Γ. Best fit models and residuals of the other LINER 1s in our sample are shown in online Fig. [bestfitallLINERs].\nThe photon indicies we derived for all of the sources in our sample observed with a relatively long exposure time varied between 1.5 ± 0.3 and 2.4− 0.3+ 0.2 with a mean value of 2.0. Intrinsic column density covered two orders of magnitude, with *N**H* varying between ∼ 1020 cm− 2 for unabsorbed sources, and up to ∼ 1022 cm− 2 for the only mildly absorbed source NGC 3718. The thermal component had a temperature mean value of 0.63 keV, consistent with all LINER-type sources embedded in diffuse emission. Table [specfit-param] gives the best fit model parameters for our sample of LINER 1s with a relatively long exposure time, and Table [specfit-fluxes] gives the 0.5-2 keV and 2-10 keV observed fluxes and corrected luminosities, as well as the corresponding “Eddington ratios”,. The hard, 2-10 keV, luminosity spans three orders of magnitude from 3.2 × 1038 erg s− 1 to 5.4 × 1041 erg s− 1 which resulted in a range from 2.0 × 10− 8 to 2.3 × 10− 5 which is at least one to two orders of magnitude smaller than seen in luminous AGN.\n[gmaVSedd]\n[!t] \n [other-corr-fig]\n[!th]\n [spear-rank] \nWe looked for any sign of narrow Fe K*α* emission line at 6.4 keV in the EPIC-pn *XMM-Newton* observations of our sample of LINER 1s. None of the observations have clear evidence for the line. Good signal to noise ratio around 6 keV is only found in the spectra of 4 sources that enable the estimate of an upper limit: 112 eV for NGC 3718 (obs.ID: 0200430501), 38 eV for NGC 3226 (obs.ID: 0400270101), 33 eV for NGC 3998, and 22 eV for NGC 4278. Even though a hint for an Fe K*α* emission line around 6.4 keV seems to be apparent in the EPIC-pn spectrum of NGC 315, the addition of a gaussian line to the best fit model does not improve the quality of the fit with an F-test probability of 60% for an improvement to occur by chance.\nX-ray correlations\n------------------\nWe looked for any correlations between the X-ray properties, mainly the photon index Γ and the 2-10 keV luminosity, and the LINER 1s intrinsic parameters, black hole mass and the ratio, which could be directly linked to the Eddington ratio *L**b**o**l*/*L**E**d**d* considering that *L**b**o**l* = *c**o**n**s**t*. × *L*2 − 10 *k**e**V*. We investigated the validity of a correlation by fitting the data with a least square fit to a straight line using the equations of and by minimizing the weighted residuals in both parameters *x* and *y*. We assessed the goodness of the fit following the criteria explained in. If a dependent variable *y* is correlated to a variable *x* with a line slope *b*, then the reciprocity in fitting *x* as a function of *y* should lead to a line with a slope *b*ʹ. Therefore, the linear correlation coefficient *r*, $r\\equiv\\sqrt{bb'}$, varies between 0, when there is no correlation, to ± 1, when there is complete correlation. This correlation coefficient *r* cannot be used directly to indicate the degree of correlation, instead a probability should be calculated that a random sample of N uncorrelated data points would yield to a linear-correlation coefficient as large as or larger than *r*.\nWe found a strong anticorrelation between the photon index Γ, and, for our sample. The fit of these two quantities with a least square fit to a straight line resulted in the following equation:\n \nΓ = ( − 0.31 ± 0.06)*l**o**g*(*L*2 − 10 *k**e**V*/*L**E**d**d*) + (0.11 ± 0.40) \nwith a linear-correlation coefficient *r* ≈ − 0.65 and a probability greater than 99.99% that and Γ yield a linear-correlation coefficient ≥ *r*. Additionally, we performed a spearman-rank correlation between Γ and and found that these two quantities are correlated with a probability greater than 99.99% and a spearman-rank correlation coefficient of -0.62. Figure [gmaVSedd] shows the anticorrelation between Γ and, and the least square best fit to a straight line. Snapshots, *Chandra*, and *XMM-Newton* observations are shown in different colours.\nUsing the same criteria as above, we found a positive correlation between the hard X-ray luminosity, *L*2 − 10 *k**e**V*, and, with a linear correlation coefficient *r* ≈ 0.65 and a probability *P* ≥ 99% that these two quantities would yield to a linear-correlation coefficient as large as or larger than *r*. Another strong positive correlation we found is the increase of the 2-10 keV luminosity with increasing BH mass with *r* ≈ 0.7 and a probability greater than 99.9%. We did not find any strong dependence of the spectral slope Γ or on the BH mass with correlation coefficients of 0.26 and -0.11 respectively and a probability ≤ 90% that these quantities are correlated. To strengthen our conclusions we performed a spearman-rank test on the four different correlations and the results are shown in Table [spear-rank]. Our previous results are in agreement with the spearman-rank test, except for a weak anticorrelation emerging between and the BH mass not seen using our principal correlation test. Fig. [other-corr-fig] shows the dependence of the fit parameters to the LINER 1s parameters. We discuss these results in § [accmode] and § [other-corr].\nDiscussion\n==========\nIn the following discussion, we are comparing X-ray timing and spectral results of our sample of LINER 1s to other results derived on broad samples of LINERs (including type 1 and type 2 LINERs, and transition nucleus) and/or low luminosity AGN. We compare our findings, X-ray variability and absence of an Fe K*α* emission line, to type 1 luminous AGN (Seyfert galaxies and quasars). We discuss correlations between the fit parameters, Γ and *L*2 − 10 *k**e**V*, and the intrinsic parameters of our LINER 1 sample, and *M**B**H*, and compare our results to luminous AGN and XRBs in order to find out the accretion mechanism in LINER 1s.\nX-ray variability\n-----------------\nOne of the most important characteristics of AGN is the X-ray flux-variability on different time-scales. An anti-correlation between the variability amplitude, characterized by the normalized excess variance *σ**N**X**S*2, with both the 2-10 keV luminosity and the black hole mass has been established for a considerable number of AGN. Such variability on time-scales of less than a day was never detected for LINER sources in the past, with observations taken with low spatial resolution telescopes, e.g.:*ASCA*,. showed that LINER and low luminosity AGN sources do not follow the same anticorrelation of luminous AGN showing stronger variability with decreasing luminosity. The authors attributed the non-detection of short time-scale variability in LINERs and low luminosity AGN to a bigger X-ray emitting region, e.g. RIAF, compared to luminous AGN.\nThree sources in our sample of LINER 1s (NGC 2787, NGC 4143, and NGC 4203) show hint of inter-day variability with a K-S test probability between 4% and 2% that the X-ray emission originates from a constant source. Two sources in our sample exhibit significant short time-scale variability, both already reported in the literature (NGC 4278: Y10 and NGC 3226: ).\nNGC 4278 exhibited a short time-scale variability (*t* ∼ 1.5 h) during the *XMM-Newton* observation where the X-ray flux level was highest (compared to the other 6 *Chandra* observations, where no short time-scale variability was detected). During this 1.5 h time-scale, the flux of NGC 4278 increased by 10%. Y10 proposed that this variability could be the result of a more compact X-ray emission region, e.g. an accretion disk truncated at a lower radius during the *XMM-Newton* observation compared to the *Chandra* observations.\nOn the other hand, NGC 3226, which is the only source observed for ∼ 100 ks with *XMM-Newton*, shows significant flux variability during the entire observation, increasing almost continuously. reported a *σ**N**X**S*2 ≈ 0.014, comparable to 0.02 reported here, which is a variability amplitude similar to the one observed in more luminous AGN, but on shorter time-scales., assuming a BH mass of 1.4 × 108 M⊙ and an Eddington ratio of 2 × 10− 5, predicted, using relation, a variability amplitude of ∼ 2 − 3 × 10− 4 on a one day time-scale. The discrepancy between the observed and the predicted value of *σ**N**X**S*2 is the fact that the relation is derived for objects in a high/soft state. As we show in paragraph [accmode], LINER sources, in contrast to luminous Seyfert galaxies, could be in a low/hard state similar to XRBs in their low/hard state. The derived for the NGC 3226 longest observation, where the variability was observed, did not increase compared to the other three observations, in fact it decreased from 4.27 × 10− 6 to 2.95 × 10− 6. Therefore, the significant variability seen during the ∼ 100 ks observation cannot be attributed to an increase in $\\dot{m}$ (as also denied by ). showed that the PSD break time-scale of the LLAGN NGC 4258 is greater than 4.5 days at > 90% confidence level. The authors suggested that LLAGN, like XRBs in their low/hard state, might have longer break time-scales compared to luminous AGN and XRBs in the high/soft state. The X-ray variability detected in the case of NGC 3226 could be the result of a break time-scale of ∼ 1 day ( ∼ 10− 5 Hz) in the NGC 3226 PSD, however, this assumption is purely speculative and PSD measurement of NGC 3226, which is not possible with the present 100 ks observation due to a low number of counts for this kind of study, could help confirm or discard this idea. Similar to NGC 3226, variability on short time-scales (half a day to several days) was observed in the LINER 1 source NGC 3998 and in the low luminosity AGN M 81.\n[!t]\n[longtermflux]\nAlthough all of the normalized excess variance derived for our sample consist of upper limits, except for NGC 3226, we show in Fig. [NXSvsMbh] *σ**N**X**S*2 as a function of BH mass. Fig. [NXSvsMbh] rules out any variability on time-scales shorter than 50 ks in our sample of LINER 1s, commonly observed in more luminous AGN, except for NGC 4278 that shows a 10% flux increase in a ∼ 1.5 h time-scale.\nVariability on time-scales of months to years is common in our sample. Seven out of nine sources observed for more than once show variability on months (NGC 5005 and NGC 4278) to years (NGC 315, NGC 3226, NGC 3718, NGC 3998, NGC 4143, and NGC 4278) time-scales[8](#fn8) with a 5 times flux increase in the most variable source, NGC 4278, and a 1.4 times flux increase in the least variable source, NGC 4143. The only two sources that do not show variability on long time-scales are the ones with a very low : NGC 2681 and NGC 2787 (Table [specfit-fluxes]). Both sources were observed twice in less than a 5 months period and therefore, variability on years-timescale cannot be tested. For illustration purposes, we report in Figure [longtermflux] the fluxes derived for the sources observed more than once. All but NGC 2681 and NGC 2787 exhibit months and/or years time-scale variability.\nWith the increasing number of X-ray observations of LINER and low luminosity AGN sources, more sources are revealing short time-scale variability ( ≤ 1 day). This variability is detected with the help of the current generation of X-ray telescopes (*Chandra*, *XMM-Newton*, and *Swift*) which have a good spatial resolution to isolate the nucleus X-ray emission from contaminating sources. Nevertheless, and before any firm conclusions are made about variability from low accretion rate sources, a homogeneous well sampled population of LINERs and/or low luminosity AGN, with long X-ray observations ( ∼ 100 ks), should be available.\nX-ray spectral shape\n--------------------\nMost recently, studied the X-ray nuclear activity of a distance limited sample, D < 15 Mpc, of 187 galaxies. The authors found that the X-ray emission from ∼ 60% of the elliptical and early type spiral galaxies is consistent with nuclear accreting supermassive BH. They fit the spectra of each source with an absorbed power-law and found a photon index Γ ≈ 1.5 − 2 and an intrinsic column density covering almost 4 orders of magnitude from 1020 cm− 2 to 1024 cm− 2. The authors found a luminosity ranging from ∼ 1038 erg s− 1 to ∼ 1042 erg s− 1 corresponding to a L0.3 − 8 *k**e**V*/L*E**d**d* ratio between 10− 4 to 10− 8. Similar results were derived in previous work such as the work of on a sample of 82 nearby LINER sources. The authors found that 60% of the sample shows a hard unresolved X-ray point-like source in the 4.5-8.0 keV band consistent with an AGN assumption. The data were fit mainly by an absorbed power-law, with a mean photon index Γ = 2.1 ± 0.5 and an intrinsic column density ranging from 1020 cm− 2 to 1024 cm− 2, and a thermal component with a mean temperature *k**T* = 0.5 ± 0.3 keV.\nThe results we derive for our homogeneous and optically selected sample of LINER 1s showing definite detection of broad H*α* emission are in agreement with all of the results derived on broad samples of LINERs and low luminosity AGN in the past. X-ray spectra in the whole 0.5-10 keV band of the majority of our sample are well fitted with an absorbed power-law (9/13). This could mean that the X-ray emission in the soft and in the hard band is originating from the same region. In the remaining 4 sources (NGC 315, NGC 2681, NGC 4278, and NGC 5005), we included a thermal component to take into account some low energy residuals, most likely from diffuse hot gas. We found a photon index ranging from 1.3 ± 0.2 for the hardest source to 2.4− 0.3+ 0.2 for the softest one with a mean value of 1.9 ± 0.2 and a dispersion *σ* = 0.3, similar to values reported in and. The absorption column density observed in our sample spans two orders of magnitude from 1020 cm− 2 for unabsorbed sources (e.g. NGC 3998, NGC 4143) to 1022 cm− 2 for the mildly absorbed source NGC 3718. This is consistent with the fact that strong absorption is not expected in objects showing broad optical emission line, e.g. LINER 1s, as found in luminous type 1 AGN. This result was already confirmed by who studied the *ASCA* X-ray observations of 21 LINER and 17 low luminosity Seyfert galaxies. The authors found a discrepancy in the intrinsic absorbing column density between type 1 LINERs, *N**H* < 1022 cm− 2 (except for NGC 1052 which is not included in our sample, see § [sec:sample]), and type 2 LINERs absorbed with a column density consistent with *N**H* a few 1022 cm− 2. The origin of the slight excess absorption in some of the sources in our sample of LINER 1s could be due to either the host galaxy and/or some compton thin material intrinsic to the central engine. We find a mean value to the temperature of the thermal component of 0.63 ± 0.06 keV typical of diffuse hot gas in early type galaxies. Similar *k**T* values were reported in for a sample of 19 LINER sources observed with *Chandra*. The 2-10 keV luminosities of the LINER 1s in our sample span a range from 3.2 × 1038 erg s− 1 to 5.4 × 1041 erg s− 1 which resulted in range from 2.0 × 10− 8 to 2.3 × 10− 5, in agreement with results derived on broad samples of LINERs and low luminosity AGN. These corresponding “Eddington ratios” are at least an order of magnitude smaller than those reported for luminous AGN.\nAlthough the photon indices measured for our sample of LINER 1s are similar to those of type 1 Seyfert galaxies, this does not necessarily mean that the X-ray emission in LINER 1s is originating from an accretion flow similar to that of more luminous galaxies. Other X-ray timing and spectral aspects could help shed light on the accretion mechanism in LINER 1s.\nThe absence of an Fe K*α* line\n------------------------------\nIt is now believed that a narrow emission Fe K*α* line at 6.4 keV is a common feature in the X-ray spectra of Seyfert galaxies. The origin of this neutral narrow emission line is generally attributed to fluorescence originating from parsec-scale distances (torus) to distances closer than the broad line region. An X-ray Baldwin effect was discovered by where the EW of the Fe line decreases with increasing luminosity. suggested that this Baldwin effect observed in X-rays could be the result of a luminosity-dependent covering fraction of the putative torus. The increase in radiation pressure flattens the torus leading to a bigger opening angle for the torus and, hence, smaller covering factor. This effect was later confirmed on bigger samples of radio-quiet and radio-loud AGN (; ;, but also see ). LINER 1s individually studied in the literature with a high signal to noise ratio around 6.4 keV do not show any sign of Fe K*α* emission line with stringent upper limits on the EW: 25 eV for NGC 3998, 35 eV for NGC 3226, and 22 eV for NGC4278 (Y10). Therefore, it appears that the X-ray Baldwin effect do not hold down to very low luminosity AGN.\nWe do not detect any significant Fe K*α* emission line in our sample with upper limits obtained for the sources with the highest signal to noise ratio around 6 keV (38 eV: NGC 3226, 112 eV: NGC 3718, 33 eV: NGC 3998, and 22 eV: NGC 4278). If the broad line region is responsible for the emission of an Fe K*α* line, such as the case for the intermediate Seyfert-LINER source NGC 7213, we would expect to detect it in at least the sources with high signal to noise ratio around 6.4 keV, since our sample consists of LINER 1s showing a definite detection of *broad* H*α* emission. Instead, the disappearance of the torus structure at low Eddington ratios as suggested by and could explain the lack of Fe emission lines in our sample, assuming that the torus is responsible for the formation of the fluorescence Fe line. Interestingly, we find that the highest upper limit on the EW of the Fe K*α* line is found for NGC 3718 which has the highest hydrogen column density (*N**H* ≈ 1022 cm− 2) in our sample of LINER 1s.\nAccretion mode in LINER 1s?\n---------------------------\nThe accretion mechanism responsible for the bulk of energy from radio to X-rays in LINER sources is still poorly understood. recently demonstrated that through local mass loss from evolved stars and Bondi accretion of hot gas, the accretion rate supply needed for the luminosities observed in LINERs and other LLAGN is easily attained. The author argued that the gas supply present at the center of nearby galaxies should generate more active nuclei and the luminosity deficit seen in the nearby universe is the result of a low radiative efficiency. Indeed, radiatively inefficient accretion flow models have been applied to a growing number of LINERs and LLAGN to explain their energy budget and their spectral energy distribution: M 81 and NGC 4579, NGC 4258, NGC 3998, and NGC 1097. fit the SED of a small sample of eight FR I sources with RIAF and/or jet models and confirmed the prediction of that below a critical value of the Eddington ratio ( ≈ 10− 6) the X-ray emission becomes dominated by the jet rather than the RIAF. suggested an even lower value to distinguish between objects in the “low” state, where an outer thin disk persists, and those in the “quiescent” state containing a pure RIAF. Since we are only considering the X-ray properties of this sample in our study, we decided to examine the Γ − relation which was shown to be a good indicator of accretion rate in luminous AGN.\nIn luminous AGN, a positive correlation is found between the hard X-ray power-law slope and the ratio of the 2-10 keV luminosity to the Eddington luminosity. showed that the relation holds for intermediate-mass (105 − 106 M⊙) BHs in active galaxies and showed that the relation extends up to more luminous objects when studying a sample of 21 low-redshift quasars. A viable explanation is that whenever the disk emission increases, the corona, the origin of the hard X-ray emission, cools more efficiently exhibiting a steepening of the hard X-ray spectrum. This relation was examined for LLAGN (local Seyferts and LINERs) by (see also ) who found a significant anticorrelation between the hard X-ray photon index Γ and the Eddington ratio, L*b**o**l*/L*E**d**d*=30 × , for the local Seyfert galaxies in their sample. However, no strong correlation was found when considering only the LINER sources in their sample owing most likely to heterogeneous fitting models as they have collected their data points from different studies. The authors suggested that this anticorrelation found in their sample, which is in contrast to the positive correlation found for more luminous AGN, could signify that LLAGN resemble XRBs in the low/hard state where a similar anticorrelation is found. suggested that the anticorrelation could mean an accretion mode consistent with a RIAF whereas a positive correlation could mean the existence of the classical thin accretion disk. A plausible explanation for the hardening of the spectrum as the accretion rate increases in a RIAF context (as seen in the Γ- anticorrelation) is the increase of the optical depth of the RIAF which leads to an increase in the Compton *y*-parameter resulting in a harder X-ray spectrum.\nOur well defined optically-selected sample of LINER 1s and our homogeneous data analysis techniques allowed us to establish a strong anticorrelation between Γ and for our sample of LINER 1s (see § [correlations]), not seen in the LINER sample of. This strong anticorrelation support the idea suggested by that LLAGN might be similar to XRBs in the low/hard state where the emission is presumably generated in a RIAF structure. predicted such an anticorrelation in the low/hard state for their accretion flow model consisting of an outer-cool optically-thick disk and inner-hot optically-thin RIAF within the framework of disk and corona with mass evaporation. found that their model can reproduce the anticorrelation between the X-ray photon index and the Eddington ratio observed for the X-ray binary.\n[diff-state]\n noted that the transition between the two different accretion modes in their sample of XRBs is different for different sources but roughly converge to the transitional point where Γ = 1.5 ± 0.1 and *l**o**g*(*L**X*(0.5 − 25 keV)/*L**E**d**d*) = − 2.1 ± 0.2. Assuming that the accretion mode in the high luminosity AGN sample of is the standard thin disk and the accretion mode in our sample of LINER 1s is a RIAF, then the transition point between the two accretion modes in AGNs would be at Γ ≈ 1.3− 0.3+ 0.2 and *l**o**g*(*L**b**o**l*/*L**E**d**d*) ≈ − 2.6− 0.8+ 0.7 (Fig. [diff-state]) in good agreement, within the error bars, with the values reported for the XRBs and for the sample of. One should keep in mind that these two values are affected by several parameters, mainly the BH mass and the bolometric luminosity, that are calculated by different means between and this work. Indeed, calculated the BH masses and the Eddington ratios, *L**b**o**l*/*L**E**d**d*, of their sample of luminous AGNs using the *ν**L**ν*(5100Å) and FWHM(H*β*), whereas our calculation of the BH masses of our sample are based on the *M* − *σ* relation and *L**b**o**l*/*L**E**d**d* is calculated using the bolometric correction *L**b**o**l* = 16 *L**X* of.\nWe do not detect any sharp cut below the critical value of of 10− 6 indicating a change in the X-ray emission, becoming dominated by synchrotron emission from a jet. This could be due to selection effects where none of our sources, except for NGC 315, comprises strong jets. Moreover, a transition between a “low” and a “quiescent” state whenever the accretion rate drops below a critical value ($\\dot{m}\\approx10^{-6}$) as suggested most recently by could not be tested since, as stressed by the author, this kind of analysis should be conducted on large samples.\nOther correlations and implications\n-----------------------------------\nWe found a positive correlation between the hard X-ray luminosity, *L*2 − 10 *k**e**V*, and. An increase in the hard X-ray luminosity with increasing is independent of the accretion model and seen in different types of galaxies from transition nuclei to high luminosity Seyfert nuclei and quasars. Another strong positive correlation we found for our sample of LINER 1s is the increase of the 2-10 keV luminosity with increasing BH mass. A similar trend is seen in a sample of 112 early-type galaxies within a distance of 67 Mpc. Therefore, both the BH mass and the (equivalent to the Eddington ratio L*b**o**l*/L*E**d**d*) are driven factors of the hard 2 − 10 keV luminosity. This could be seen in the right upper panel of Fig. [other-corr-fig] in § [correlations] where NGC 2787 with a relatively high BH mass of 1.4 × 108 M⊙ has a low 2-10 keV luminosity as a consequence of a low (2 × 10− 8). We did not find any strong dependence of the spectral slope Γ or the Eddington ratio on the BH mass (although a spearman-rank correlation showed that is weakly anticorrelated to the BH mass). This is in contrast with the anti-correlation between the X-ray spectral slope and the BH mass seen in a sample of low redshift PG quasars. It appears that in our sample of LINER 1s, which have a very low accretion rate, the BH mass – here spanning 2 orders of magnitude in log(*M**B**H*) space – is not the main driver of the X-ray photon index.\nSummary and conclusion\n======================\nIn the present work, we studied the X-ray properties of a sample of nearby LINER galaxies showing definite detection of broad H*α* emission. Such a sample insures the responsibility of accretion into a SMBH for the detection of the broad emission lines, guarantees the non-existence of large obscuration, and enables X-ray comparison of this class with both XRBs and type 1 AGN.\nOnly two sources in our sample exhibit significant hours to days time-scale variability. The NGC 4278 flux increased by a factor of 10% on a ∼ 1.5 hour period, to remain constant for the rest of the observation (see Y10). On the other hand, NGC 3226 shows variability for the whole observation of ∼ 100 ks, increasing continuously. Three other sources show hint of inter-day variability with a K-S test probability between 2% and 4% that the X-ray emission originates from a constant source. Short time-scale variability study from a homogeneous well sampled population of LINERs and/or low luminosity AGN, with long X-ray observations ( ∼ 100 ks) should be conducted before any firm conclusions are made about variability from low accretion rate sources. On the other hand, variability on longer (months to years) time-scales is common in our sample where 7 out of 9 sources exhibit long time-scale variability. The two sources not exhibiting variability are the ones with a very low.\nThe X-ray spectra of our sample of LINER 1s are typical of all types of LINER sources, fit with an absorbed power-law, or a combination of a thermal component and an absorbed power-law. We found a photon index for our sample between 1.3 ± 0.2 for the hardest source and 2.4− 0.3+ 0.2 for the softest one with a mean value of 1.9 ± 0.2 and a dispersion *σ* = 0.3. None of the sources in our sample is heavily absorbed with NGC 3718 having the highest intrinsic hydrogen column density of ∼ 1022 cm− 2. The thermal component had a mean temperature kT ≈ 0.6 keV, typical of other LINER sources embedded in diffuse emission. Our sample spans three orders of magnitude in both luminosity and space, ranging from 1038 to 1041 erg s− 1 and from 10− 8 to 10− 5 respectively, which is at least an order of magnitude smaller than the Eddington ratios observed in luminous AGN (Seyferts and quasars).\nWe do not detect any significant Fe K*α* emission line at 6.4 keV in the spectra of our sample of LINER 1s. We obtained upper limits on the Fe line for the four sources with the highest signal to noise ratio around 6.4 keV (38 eV: NGC3226, 112 eV: NGC 3718, 33 eV: NGC3998, and 22 eV: NGC 4278). The lack of a narrow Fe line could be due to the disappearance of a torus structure in LLAGN and LINER sources. This implies that the X-ray Baldwin effect or the “Iwasawa-Tanigushi” effect of decreasing Fe K*α* EW with increasing 2-10 keV luminosity does not extend down to LINER 1s.\nFinally, we looked for correlations between the X-ray properties and the AGN properties of our LINER 1 sample. We found a strong anticorrelation between the power-law photon index and the Eddington ratio suggesting that LINER 1s differ from more luminous Seyfert and quasar galaxies that show a positive correlation between the photon index and the Eddington ratio. This anticorrelation, established for the first time for LINER 1s, suggest that LINER 1s mode of accretion could be similar to that of XRBs in their low-hard state, and a RIAF could be responsible for the emitting energy from the nucleus. We found that the 2-10 keV luminosity in our sample is positively correlated to two parameters, the BH mass and the Eddington ratio,, confirming the results found for broad samples of LINERs and low luminosity AGN. On the other hand, it appears that in our sample of LINER 1s, which have a very low accretion rate, the BH mass is not the main driver of the X-ray photon index, as the two quantities do not show any strong correlation.\nThis research has made use of the data obtained from the *Chandra Data Archive* and the *Chandra Source Catalog*, and software provided by the *Chandra X-ray Center* (CXC) in the application packages CIAO and Chips. This work is based on observations with *XMM-Newton*, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. G.Y. would like to thank I. Papadakis for helpful and enlightening discussions. The authors would also like to thank the referee for fruitful comments that improved the quality of the manuscript.\nNotes on individual sources\n===========================\n*NGC 266.* studied the 2 ks snapshot observation made with *Chandra* and discussed here. They model the X-ray spectrum with an absorbed power-law and derive a hydrogen column density and a photon index of < 0.82 × 1022 cm− 2 and 1.4, respectively. They find a 2-10 keV flux of 1.6 × 10− 13 erg s− 1 cm− 2. The values of the photon index and the 2-10 keV flux are with a good agreement with the value we report here (within the error bars), but no additional absorption is added to our model.\n*NGC 315.* first suggested the presence of an active galactic nucleus at the center of NGC 315 using *ROSAT* data. This assumption was later confirmed by when studying a 37 ks ASCA observation (see also ). The authors fit the hard 2-10 keV spectrum with a power-law and find a photon index of ∼ 2 and a luminosity of 3.1 × 1041 erg s− 1. A resolved X-ray jet emission was first reported by when studying the *Chandra* snapshot observation (obs. ID: 855). The fit to the jet emission with a power-law gave a photon index of 2.5 ± 0.7 and a 3.5 × 1040 erg s− 1 luminosity. The authors fit the unresolved core emission with a moderately absorbed power-law with an intrinsic hydrogen column density of ∼ 5 × 1021 cm− 2 and a photon index of 1.4 ± 0.4. found similar results when studying the longer *Chandra* observation reporting a harder core spectrum than the jet with photon indicies of ∼ 1.6 and ∼ 2.2 respectively. Moreover,, after solar flare cleaning, studied the only *XMM-Newton* observation and fit the $60\\arcsec$ core spectrum with a combination of a mekal and a power-law. Our analysis of the two *Chandra* observations and the *XMM-Newton* one gives similar results to all of the above studies with a simultaneous fit to the different extracted spectra where a combination of a mekal (*k**T* ≈ 0.5 keV) and a mildly absorbed (*N**h* ≈ 1022 cm− 2) power-law (Γ between 1.5 and 2) were used. The increase in the power-law slope from 1.5 ± 0.1 during the *Chandra* observation to 2.1− 0.2+ 0.1 during the *XMM-Newton* one is accompanied by a decrease in the 2-10 keV flux from 9.8 × 10− 13 to 4.6 × 10− 13 erg s− 1. This behavior (see § [accmode]) is typical of X-ray emission originating in a RIAF structure which is believed to be the accretion mechanism responsible for the bulk of energy from radio to X-rays in NGC 315.\n*NGC 2681.* One of the two observations performed with *Chandra*has been already reported in. The authors fit the 0.5-8 keV spectrum with a combination of a thermal component with *k**T* ≈ 0.7 keV and a power-law with a photon index Γ ≈ 1.6. No intrinsic absorption was required. The same observation was treated in and same results were derived after fitting the spectrum with a *mekal* and a power-law. derived a 2-10 keV flux of ∼ 2 × 10− 13 erg s− 1 cm− 2. We fit the spectrum of the two *Chandra* observations of NGC 2681 simultaneously with a combination of a thermal component and an absorbed power-law. We found similar results to that derived in the previous works (Γ ≈ 1.5 and *k**T* ≈ 0.6 keV) with a 2-10 keV flux of 3 × 10− 13 erg s− 1 cm− 2.\n*NGC 2787.*, after studying a *Chandra* snapshot, gave this source a class III X-ray morphology, showing hard X-ray nucleus embedded in diffuse emission. derived a 2-10 keV flux of about 3 × 10− 14 erg s− 1 cm− 2, after assuming a photon index of 2 and a Galactic absorption. analyzed both *Chandra* and *XMM-Newton* long observations and fit the *Chandra* spectrum with a power-law with a rather soft Γ of 2.3 and fit the *XMM-Newton* spectrum with a combination of two power-laws and a thermal component with a power-law absorption of ∼ 1022 cm− 2. We found a good fit for both *Chandra* and *XMM-Newton* spectra simultaneously with a single absorbed power-law and found little absorption of ∼ 2 × 1021 cm− 2 and a soft power-law photon index, Γ = 2.4, and a 2-10 keV flux of 4 × 10− 14 erg s− 1 cm− 2.\n*NGC 3226.* fit the 0.5-10 keV spectrum extracted from the long *Chandra* observation with an absorbed (*N**H* ≈ 5 × 1021 cm− 2) power-law (Γ ≈ 1.9). fit the spectrum of the 2.5 ks snapshot *Chandra* observation with a moderately absorbed power-law with *N**H* ≈ 1022 cm− 2 and Γ ≈ 2.2. fit the data of the 35 ks *XMM-Newton* observation with a partial covering absorber to a bremsstrahlung and found that the X-ray emitting region, with a temperature *k**T* ≈ 0.9 keV is 90% covered by an absorber with *N**H* ≈ 5 × 1021 cm− 2. studied the ∼ 100 ks *XMM-Newton* observation and fit the spectrum with a partially covered power-law with Γ ≈ 1.9, a covering fraction of 90%, and an intrinsic hydrogen column density of 1021 cm− 2. We fit the spectra of both *XMM-Newton* and the long *Chandra* observations simultaneously with an absorbed power-law and found an intrinsic column density varying between the observations from ∼ 3 × 1021 cm− 2 to ∼ 9 × 1021 cm− 2 and a mean photon index Γ = 1.9.\n*NGC 3718.* studied the snapshot *Chandra* observation and fit the spectrum with an absorbed (*N**H* ≈ 1022 cm− 2) power-law (Γ ≈ 1.5) in excellent agreement with our fit results to the same observation. We studied two *XMM-Newton* observations of NGC 3718, being in the field of view of the observations of the heavily absorbed Seyfert 2 galaxy. We fit the spectra with an absorbed power-law and found an intrinsic hydrogen column density similar to the one derived for the *Chandra* observation of ≈ 1022 cm− 2 but a somewhat softer power-law with Γ ≈ 1.8. This softening is accompanied with a 2-10 keV flux decrease from 3.3 × 10− 12 erg s− 1 cm− 2 to 1.6 × 10− 12 erg s− 1 cm− 2.\n*NGC 3998.* studied the 10 ks *XMM-Newton* observation and fit the spectrum with a slightly absorbed (*N**H* ≈ 1020 cm− 2) power-law (Γ ≈ 1.9). Same results were found for observations made with *BeppoSAX* and *ASCA*. fit the *Chandra* spectrum with a combination of two absorbed power-laws and a thermal component. We fit the *XMM-Newton* and the *Chandra* spectra simultaneously with a mildly absorbed (*N**H* ≈ 1020 cm− 2) power-law and found a varying Γ from 1.8 to 2.1 respectively, occurring with a flux decrease from 1.1 × 10− 11 erg s− 1 cm− 2 to 6.5 × 10− 12 erg s− 1 cm− 2.\n*NGC 4143.* fit the *Chandra* snapshot observation with an absorbed power-law with *N**H* < 1021 cm− 2 and Γ ≈ 1.7. We fit the same snapshot observation with a power-law without a requirement of an intrinsic absorption and found a similar power-law photon index within the error bars, Γ ≈ 1.9. We fit the *XMM-Newton* observation with an absorbed power-law and found a *N**H* = 6 × 1020 cm− 2 and a Γ ≈ 2.2.\n*NGC 4203.* A power-law fit to the *ASCA* spectrum resulted in a Γ ≈ 1.8. gave NGC 4203 a class I X-ray morphology showing dominant X-ray nucleus. We find that the 40 ks *Chandra* spectrum is well fitted with a simple power-law affected by Galactic absorption with Γ ≈ 2.3, softer than the result reported for *ASCA*, most likely due to contamination from X-ray sources in the *ASCA* extraction region of 1.\n*NGC 4278.* See Y10.\n*NGC 4750.*, and according to the only *Chandra* snapshot observation, gave this source a morphological X-ray type II, exhibiting multiple, hard off-nuclear point sources of comparable brightness to the nuclear source. We fit the spectrum of this same observation with an absorbed power-law with *N**H* < 3 × 1021 cm− 2 and Γ = 1.8.\n*NGC 4772.* We fit the spectrum of the only, *Chandra* snapshot, observation with an absorbed (*N**H* ≈ 5 × 1021 cm− 2) power-law (Γ ≈ 1.7).\n*NGC 5005.* fit the *ASCA* spectrum with a combination of an absorbed (*N**H* < 9 × 1021 cm− 2) power-law (Γ ≈ 1) and a thermal component (*k**T* ≈ 0.8). gave NGC 5005 a morphological X-ray type III, showing a hard nuclear point source embedded in diffuse emission. fit the *XMM-Newton* spectrum with a combination of a thermal component with *k**T* ≈ 0.3 keV and an absorbed (*N**H* ≈ 6 × 1021 cm− 2) power-law (Γ ≈ 1.5). We find a hotter thermal component when fitting the same data set with *k**T* ≈ 0.6 keV and a mildly absorbed power-law with *N**H* ≈ 1021 cm− 2 and Γ ≈ 1.7.\nSurrounding sources of the centers of galaxies observed with *Chandra*\n======================================================================\nIn this Appendix, we report the spectral analysis of resolved and/or unresolved off-nucleus sources detected with the *Chandra* telescope, but blended within the central LINER in the *XMM-Newton* extraction region (§ [xmmobs]). Fig. 1-4 show the surrounding sources of these LINER 1s detected with *Chandra* within a 25-radius circle. The surrounding medium of NGC 4278 is already reported in Y10.\n*NGC 315.* This source is the only source in our sample that shows a resolved X-ray jet. We extracted from the longer *Chandra* observation the spectrum of the jet from an ellipse with a semi-major axis of about 11.3 and semi-minor axis of 5.6. The base of the ellipse extends down to the 1.1 × 99% PSF of the central source. We fit the spectrum with a combination of an absorbed power-law and a thermal *mekal* component and found a good fit with a reduced *χ*2 of 0.9 for 42 d.o.f. We find a hydrogen column density upper limit of 2 × 1021 cm− 2 and a photon index Γ = 2.0− 0.2+ 0.4. The thermal component had a temperature of 0.6− 0.9+ 0.5 keV. We find a corrected 0.5-10 keV flux for the jet emission of (1.2 ± 0.1) × 10− 13 erg s− 1 cm− 2, which corresponds to a 0.5-10 keV luminosity of 6 × 1040 erg s− 1, corresponding to 8% of the nuclear flux. The power-law emission contributes to 95% to the total emission of the jet. For the diffuse emission, we extracted the spectrum from an annulus with inner circle delimited by 1.1 × 99% PSF of the central source and an outer radius of 25 excluding the jet extraction region. The same model fit to the jet gave a good fit with a reduced *χ*2 of 1.3 for 61 d.o.f. We find an intrinsic absorption to the powerlaw component *N**H* = 6− 10+ 12 × 1021 cm− 2 and a photon index Γ = 1.7 ± 0.8, possibly representing emission from unresolved X-ray binaries. The thermal component has a temperature similar to the one derived for the jet emission with *k**T* = 0.6 ± 0.2 keV. We found a corrected 0.5-10 keV flux of 2.9− 0.3+ 0.2 × 10− 13 erg s− 1 cm− 2 with the power-law contributing only to 30% of the total emission. This corresponds to a luminosity of ∼ 1041 erg s− 1 which is ∼ 14% of the total core luminosity.\n*NGC 2787.* An X-ray source south-east of the nucleus of NGC 2787 at a distance less than 10 is present. We fit the spectrum of this source with an absorbed power-law. We used the same redshift and Galactic absorption as for the NGC 2787 nucleus, assuming that this X-ray source is located in NGC 2787 and not a background quasar. The fit is acceptable with a reduced *χ*2 of 1.0 for 19 d.o.f. We found a power-law photon index Γ = 1.5− 0.4+ 0.5, typical of X-ray binaries in nearby galaxies, and an upper limit on the intrinsic hydrogen column density of 2 × 1021 cm− 2. We derived a 0.5-10 keV corrected flux of 7 ± 1 × 10− 14 erg s− 1 cm− 2, which resulted in a luminosity of, adapting the NGC 2787 distance of 7.48 Mpc, 5 ± 1 × 1038 erg s− 1. This luminosity is close to the NGC 2787 core luminosity of ∼ 9 × 1038 erg s− 1. Such a source could be a luminous low mass X-ray binary (LMXB) similar to some seen in early type galaxies. The rest of the medium in an annulus of outer radius 25 around NGC 2787 is formed by six other X-ray sources, much fainter than the closest one to the center. We could not perform spectral analysis on the different sources aside, so we fit the spectrum of the six sources simultaneously with a power-law only affected by Galactic absorption. We found a photon index of ∼ 2 for the six sources and a total corrected 0.5-10 keV flux of 8− 2+ 1 × 10− 15 erg s− 1 cm− 2, resulting in a 0.5-10 keV luminosity of 5− 2+ 1 × 1037 erg s− 1, corresponding to 5% of the nucleus luminosity, when adapting the NGC 2787 distance of 7.48 Mpc.\n*NGC 3226.* Two X-ray sources are within a ∼ 12 distance from the nucleus of NGC 3226 (source 1: CXOU J102334.1+195347, source 2: CXOU J102326.7+195407). Both sources were reported in. Based on the hardness ratio between the counts in the 0.3-2 keV band and the counts in the 2-10 keV band, the authors estimated the sources to have a flux of a few 10− 14 erg s− 1 cm− 2, and therefore a luminosity between a few times 1038 erg s− 1 to a few times 1039 erg s− 1. We fit the spectrum of both sources with an absorbed power-law, using the cash statistic due to a low number of counts. We find that source 1 and source 2 have an intrinsic absorption upper limit of 6 × 1022 cm− 2 and 5 × 1021 cm− 2, respectively. The photon index we found for source 2 is typical, within the error bars, of accreting objects with a Γ = 1.2− 0.6+ 0.9. On the other hand, we found a harder spectrum for source 1 with a Γ = 0.2 ± 1.3. The corrected 0.5-10 keV flux we derive for source 1 and source 2 are ∼ 4 × 10− 14 erg s− 1 cm− 2 and ∼ 2 × 10− 14 erg s− 1 cm− 2, respectively. This implies, assuming the distance of NGC 3226 to both sources, a corrected 0.5-10 keV luminosity of 3 × 1039 erg s− 1 and 2 × 1039 erg s− 1 for source 1 and source 2 respectively. Both luminosities are well beyond the luminosity of a typical neutron star LMXB of ∼ 3 × 1038 erg s− 1, and hence could be BHs greater than or equal to a few solar masses, very young supernovae, or microquasars.\n*NGC 3998.* Only one source is detected within a 25 circle around NGC 3998 in the *Chandra* image. We fit the source with an absorbed power-law, using the cash statistic. We found an upper limit on the intrinsic column density of 1021 cm− 1 and a power-law photon index of 1.4− 0.5+ 0.9. We derived a corrected 0.5-10 keV flux of 3 ± 1 × 10− 14 erg s− 1 cm− 2, which corresponds to a luminosity of 7 ± 1 × 1038 erg s− 1, adapting the NGC 3998 distance of 14.1 Mpc. This corresponds to only 0.2% of the total core luminosity of NGC 3998 and match the luminosity of XRBs in nearby galaxies.\n114 natexlab#1#1\nAlonso-Herrero, A., Rieke, M. J., Rieke, G. H., & Shields, J. C. 2000,, 530, 688\nArnaud, K. A. 1996, in ASP Conf. Ser. 101: Astronomical Data Analysis Software and Systems V, 17\nAwaki, H., Terashima, Y., Hayashida, K., & Sakano, M. 2001,, 53, 647\nBaldi, A., Forman, W., Jones, C., et al. 2009,, 707, 1034\nBarth, A. J., Filippenko, A. V., & Moran, E. C. 1999,, 515, L61\nBevington, P. R. & Robinson, D. K. 2003, Data reduction and error analysis for the physical sciences, ed. Bevington, P. R. & Robinson, D. K.\nBianchi, S., Guainazzi, M., Matt, G., & Fonseca Bonilla, N. 2007,, 467, L19\nBianchi, S., La Franca, F., Matt, G., et al. 2008,, 389, L52\nBinder, B., Markowitz, A., & Rothschild, R. E. 2009,, 691, 431\nBroos, P. S., Townsley, L. K., Feigelson, E. D., et al. 2010,, 714, 1582\nChaudhary, P., Brusa, M., Hasinger, G., Merloni, A., & Comastri, A. 2010,, 518, A58+\nConstantin, A., Green, P., Aldcroft, T., et al. 2009,, 705, 1336\nCroston, J. H., Hardcastle, M. J., Birkinshaw, M., Worrall, D. M., & Laing, R. A. 2008,, 386, 1709\nDopita, M. A. & Sutherland, R. S. 1995,, 455, 468\nDudik, R. P., Satyapal, S., Gliozzi, M., & Sambruna, R. M. 2005,, 620, 113\nFabbiano, G. 1989,, 27, 87\nFabbiano, G. 2006,, 44, 323\nFlohic, H. M. L. G., Eracleous, M., Chartas, G., Shields, J. C., & Moran, E. C. 2006,, 647, 140\nFreeman, P. E., Kashyap, V., Rosner, R., & Lamb, D. Q. 2002,, 138, 185\nGammie, C. F., Narayan, R., & Blandford, R. 1999,, 516, 177\nGeorge, I. M., Mushotzky, R. F., Yaqoob, T., et al. 2001,, 559, 167\nGondoin, P., Orr, A., & Siddiqui, H. 2004,, 420, 905\nGonzález-Martín, O., Masegosa, J., Márquez, I., Guainazzi, M., & Jiménez-Bailón, E. 2009,, 506, 1107\nGonzález-Martín, O., Masegosa, J., Márquez, I., Guerrero, M. A., & Dultzin-Hacyan, D. 2006,, 460, 45\nGraham, A. W., Onken, C. A., Athanassoula, E., & Combes, F. 2010, ArXiv e-prints\nGreene, J. E. & Ho, L. C. 2007,, 667, 131\nGreene, J. E. & Ho, L. C. 2007,, 656, 84\nGu, M. & Cao, X. 2009,, 399, 349\nGuainazzi, M. & Antonelli, L. A. 1999,, 304, L15\nHarris, D. E. 2003,, 47, 617\nHeckman, T. M. 1980,, 87, 152\nHo, L. C. 1999,, 516, 672\nHo, L. C. 2008,, 46, 475\nHo, L. C. 2009,, 699, 626\nHo, L. C., Feigelson, E. D., Townsley, L. K., et al. 2001,, 549, L51\nHo, L. C., Filippenko, A. V., & Sargent, W. L. 1995,, 98, 477\nHo, L. C., Filippenko, A. V., & Sargent, W. L. W. 1997,, 112, 315\nHo, L. C., Filippenko, A. V., Sargent, W. L. W., & Peng, C. Y. 1997,, 112, 391\nHo, L. C., Greene, J. E., Filippenko, A. V., & Sargent, W. L. W. 2009,, 183, 1\nIrwin, J. A., Athey, A. E., & Bregman, J. N. 2003,, 587, 356\nIshisaki, Y., Makishima, K., Iyomoto, N., et al. 1996,, 48, 237\nIwasawa, K. & Taniguchi, Y. 1993,, 413, L15\nIyomoto, N. & Makishima, K. 2001,, 321, 767\nIyomoto, N., Makishima, K., Matsushita, K., et al. 1998,, 503, 168\nJiménez-Bailón, E., Piconcelli, E., Guainazzi, M., et al. 2005,, 435, 449\nJones, C., Forman, W., Vikhlinin, A., et al. 2002,, 567, L115\nKalberla, P. M. W., Burton, W. B., Hartmann, D., et al. 2005,, 440, 775\nKomossa, S., Böhringer, H., & Huchra, J. P. 1999,, 349, 88\nKonigl, A. & Kartje, J. F. 1994,, 434, 446\nLiu, B. F., Mineshige, S., Meyer, F., Meyer-Hofmeister, E., & Kawaguchi, T. 2002,, 575, 117\nLobban, A. P., Reeves, J. N., Porquet, D., et al. 2010,, 408, 551\nMaoz, D. 2007,, 377, 1696\nMaoz, D., Nagar, N. M., Falcke, H., & Wilson, A. S. 2005,, 625, 699\nMarkowitz, A. & Uttley, P. 2005,, 625, L39\nMason, K. O., Breeveld, A., Much, R., et al. 2001,, 365, L36\nMatsumoto, Y., Fukazawa, Y., Nakazawa, K., Iyomoto, N., & Makishima, K. 2001,, 53, 475\nMcHardy, I. M., Koerding, E., Knigge, C., Uttley, P., & Fender, R. P. 2006,, 444, 730\nMewe, R., Gronenschild, E. H. B. M., & van den Oord, G. H. J. 1985,, 62, 197\nNagar, N. M., Falcke, H., & Wilson, A. S. 2005,, 435, 521\nNandra, K., George, I. M., Mushotzky, R. F., Turner, T. J., & Yaqoob, T. 1997,, 476, 70\nNandra, K., George, I. M., Mushotzky, R. F., Turner, T. J., & Yaqoob, T. 1997,, 477, 602\nNandra, K., George, I. M., Mushotzky, R. F., Turner, T. J., & Yaqoob, T. 1997,, 488, L91+\nNandra, K., O’Neill, P. M., George, I. M., & Reeves, J. N. 2007,, 382, 194\nNarayan, R. & McClintock, J. E. 2008,, 51, 733\nNemmen, R. S., Storchi-Bergmann, T., Eracleous, M., & Yuan, F. 2010, ArXiv e-prints\nNemmen, R. S., Storchi-Bergmann, T., Yuan, F., et al. 2006,, 643, 652\nOhto, A., Kawano, N., & Fukazawa, Y. 2003,, 55, 819\nO’Neill, P. M., Nandra, K., Papadakis, I. E., & Turner, T. J. 2005,, 358, 1405\nO’Sullivan, E., Vrtilek, J. M., & Kempner, J. C. 2005,, 624, L77\nPage, K. L., O’Brien, P. T., Reeves, J. N., & Turner, M. J. L. 2004,, 347, 316\nPapadakis, I. E. 2004,, 348, 207\nPellegrini, S. 2010,, 717, 640\nPellegrini, S., Cappi, M., Bassani, L., della Ceca, R., & Palumbo, G. G. C. 2000,, 360, 878\nPian, E., Romano, P., Maoz, D., et al. 2010,, 401, 677\nPorquet, D., Reeves, J. N., O’Brien, P., & Brinkmann, W. 2004,, 422, 85\nPtak, A., Serlemitsos, P., Yaqoob, T., & Mushotzky, R. 1999,, 120, 179\nPtak, A., Terashima, Y., Ho, L. C., & Quataert, E. 2004,, 606, 173\nPtak, A., Yaqoob, T., Mushotzky, R., Serlemitsos, P., & Griffiths, R. 1998,, 501, L37+\nQiao, E. & Liu, B. F. 2010,, 62, 661\nQuataert, E. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 224, Probing the Physics of Active Galactic Nuclei, ed. B. M. Peterson, R. W. Pogge, & R. S. Polidan, 71–+\nQuataert, E., Di Matteo, T., Narayan, R., & Ho, L. C. 1999,, 525, L89\nSatyapal, S., Dudik, R. P., O’Halloran, B., & Gliozzi, M. 2005,, 633, 86\nShakura, N. I. & Sunyaev, R. A. 1973,, 24, 337\nShemmer, O., Brandt, W. N., Netzer, H., Maiolino, R., & Kaspi, S. 2006,, 646, L29\nShemmer, O., Brandt, W. N., Netzer, H., Maiolino, R., & Kaspi, S. 2008,, 682, 81\nShu, X. W., Yaqoob, T., & Wang, J. X. 2010,, 187, 581\nSobolewska, M. A. & Papadakis, I. E. 2009,, 399, 1597\nStr\" uder, L., Briel, U., Dennerl, K., et al. 2001,, 365, L18\nTerashima, Y., Ho, L. C., & Ptak, A. F. 2000,, 539, 161\nTerashima, Y., Iyomoto, N., Ho, L. C., & Ptak, A. F. 2002,, 139, 1\nTerashima, Y. & Wilson, A. S. 2003,, 583, 145\nTerlevich, R. & Melnick, J. 1985,, 213, 841\nTonry, J. L., Dressler, A., Blakeslee, J. P., et al. 2001,, 546, 681\nTremaine, S., Gebhardt, K., Bender, R., et al. 2002,, 574, 740\nTully, R. B. 1988, Nearby galaxies catalog, ed. Tully, R. B.\nTurner, M. J. L., Abbey, A., Arnaud, M., et al. 2001,, 365, L27\nTurner, T. J., George, I. M., Nandra, K., & Turcan, D. 1999,, 524, 667\nVaughan, S., Edelson, R., Warwick, R. S., & Uttley, P. 2003,, 345, 1271\nWang, J., Watarai, K., & Mineshige, S. 2004,, 607, L107\nWeisskopf, M. C., Brinkman, B., Canizares, C., et al. 2002,, 114, 1\nWilms, J., Allen, A., & McCray, R. 2000,, 542, 914\nWorrall, D. M. & Birkinshaw, M. 1994,, 427, 134\nWorrall, D. M., Birkinshaw, M., & Hardcastle, M. J. 2003,, 343, L73\nWorrall, D. M., Birkinshaw, M., Laing, R. A., Cotton, W. D., & Bridle, A. H. 2007,, 380, 2\nWu, Q. & Gu, M. 2008,, 682, 212\nWu, Q., Yuan, F., & Cao, X. 2007,, 669, 96\nXu, H., Gu, J., Gu, L., et al. 2010, Research in Astronomy and Astrophysics, 10, 220\nXu, H., Kahn, S. M., Peterson, J. R., et al. 2002,, 579, 600\nYamaoka, K., Uzawa, M., Arai, M., Yamazaki, T., & Yoshida, A. 2005, Chinese Journal of Astronomy and Astrophysics Supplement, 5, 273\nYork, D. 1966, Canadian Journal of Physics, 44, 1079\nYounes, G., Porquet, D., Sabra, B., et al. 2010,, 517, A33+\nYuan, F. & Cui, W. 2005,, 629, 408\nYuan, F., Taam, R. E., Misra, R., Wu, X., & Xue, Y. 2007,, 658, 282\nZhang, W. M., Soria, R., Zhang, S. N., Swartz, D. A., & Liu, J. F. 2009,, 699, 281\n---\n1. [chaser]http://cda.harvard.edu/chaser/Descriptions.[↩](#fnref1)\n2. [AEnote] The *ACIS Extract* software package and User’s Guide are available at http://www.astro.psu.edu/xray/acis/acis\\_analysis.html.[↩](#fnref2)\n3. http://space.mit.edu/ASC/MARX/[↩](#fnref3)\n4. NGC 3226 is off axis during the long observation and hence is not detected with the MOS cameras when operating in a Small Window Mode[↩](#fnref4)\n5. http://xmm.esac.esa.int/xsa/index.shtml[↩](#fnref5)\n6. The lower 10 limit was taken so to encircle at least 50% of the EPIC *XMM-Newton* PSF.[↩](#fnref6)\n7. http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl[↩](#fnref7)\n8. All of the sources reported to show variability on years time-scale and not on months time-scale lack inter-year observations and therefore, monthly variability cannot be tested.[↩](#fnref8)\n\nStudy of LINER sources with broad H*α* emission.X-ray properties and comparison to luminous AGN and X-ray binaries.\n===================================================================================================================\n### Received / Accepted\nIntroduction\n============\nIt is widely known that a large fraction of local galaxies contain emission line nuclei that are the result of low nuclear activity. showed that more than 40% of 486 nearby galaxies with *B**T* ≤ 12.5 mag could be considered as active with optical spectra classified as Seyfert nuclei, low-ionization nuclear emission-line regions, or transition objects (objects having intermediate spectra between LINERs and H II nuclei). Pure LINER sources would be the most common component representing 20% of all 486 galaxies. The ionization mechanism responsible for the excitation of emission lines in LINER sources is an ongoing matter of debate and could be explained in terms of: shock heated gas, starburst activity, or a low luminosity active Galactic nucleus (AGN). Many multiwavelength studies were attributed to this subject, looking for a radio, sometimes variable, core or a variable UV core in nearby LINER sources. Nevertheless, the most used tool to search for an active nucleus in a LINER is to look for a hard 2-10 keV unresolved core that could not be due to diffuse emission from shock heated gas or from unusually hot stars. How do LINERs harboring a low luminosity active nucleus compare to luminous Seyfert galaxies and quasars?\n, using high angular resolution multiwavelength observations of 13 LINER sources, demonstrated that the luminosity ratios in different wavebands, mainly UV to X-ray and radio to UV luminosities, follow the same trend as luminous Seyfert galaxies. The authors did not find any sharp change in the spectral energy distribution (SED) of their sample of 13 LINERs compared to more luminous Seyfert and quasar nuclei, suggesting that a thin accretion disk may persist at low accretion rates. Moreover, detected up to 30% flux variations on half a day time-scale in 2 (NGC 3998 and M81) out of 4 LINER and low luminosity AGN sources observed in X-ray with the XRT onboard *Swift*. They combined their X-ray fluxes with simultaneous UV fluxes coming from the UVOT instrument and showed that the SED and the UV to X-ray flux ratios of their 4 sources sample are consistent with those of more luminous sources and that LINERs may have similar accretion and radiative processes at their center compared to luminous Seyfert nuclei.\nOn the other hand, the faintness of LINER sources compared to luminous Seyfert galaxies and quasars has been attributed to a different accretion mechanism owing to some observational contrast between the two classes. No broad nor narrow Fe K*α* emission line at 6.4 keV have been detected in the spectra of the LINER sources with the highest signal to noise ratio, X-ray short time-scale variability has been detected in only a few sources, and the disappearance of the big blue bump in the UV band in the SED of LINER sources, all these signatures could indicate the disappearance of the thin accretion disk at low luminosities and that a different accretion mechanism is responsible for the emission in LINER sources. It has been suggested that when the mass accretion rate falls below a critical value $\\dot{M}\\_{crit}$, the density of the disk could become too low for radiative cooling to be effective. The trapped heat will expand the thin accretion disk into a pressure-supported thick disk with a very low radiative efficiency. Such radiatively inefficient accretion flow (RIAF) models successfully explained the spectral energy distribution of a large number of LINER sources.\nAnother way to assess the geometry of the accretion mode in AGN is to compare them to their less massive counterparts, X-ray binaries (XRBs). showed that the X-ray spectral slope, Γ, of Seyfert 1 galaxies and quasars and the Eddington ratio, *L**b**o**l*/*L**E**d**d*, are positively correlated, similar to XRBs in their high/soft state. Such a behavior could be explained in an AGN accretion mode consistent with an optically thick geometrically thin accretion disk. performed a similar study on a broad sample of LINERs and low luminosity Seyfert galaxies. They found a significant anticorrelation between Γ and the Eddington ratio for the local Seyfert galaxies in their sample analogous to XRBs in the low/hard state where a RIAF mode of accretion takes place. However, no strong correlation was found when considering only the LINER sources in their sample, owing, as suggested by the authors, to heterogeneous fitting models as they have collected their data from different studies. In a separate study, analyzed the X-ray emission of a sample of 107 nearby galaxies including low luminosity Seyferts, LINERs, transitions (nuclei with spectra between Seyferts and LINERs), H II regions, and passive galaxies (lacking optical emission-line activity), *none of which show broad-line components*. Using a Spearman-rank correlation, the authors found an anticorrelation for their sample between Γ and the *L**b**o**l*/*L**E**d**d*. By considering each class separately, a spearman-rank test showed that the anticorrelation persists for the different objects, except for the low luminosity Seyfert galaxies.\nFinally, broad optical emission lines, a characteristic property of classical Seyferts and quasars, are also found in nuclei of much lower luminosities. Thirty three sources out of the 221 nuclei classified as Seyfert, LINER, or transition objects in sample of nearby galaxies show definite detection of a broad H*α* emission, 16 of those ( ∼ 17% of the total pure LINER sources) are LINERs. In this paper, we are aiming for the study of the X-ray characteristics of these LINER 1s observed with the current generation of X-ray telescopes, *XMM-Newton* and *Chandra*. Such a sample insures the responsibility of accretion into a SMBH for the formation of the broad emission lines (given the early type class of this sample where outflows form massive stars and/or supernovae are not expected to be relevant), guarantees the non-existence of large obscuration, and enables X-ray comparison of this class with both XRBs and type 1 AGN. We introduce our sample in § 2, § 3 represents the observations and the data reduction. Temporal and spectral results are given in § 4. In § 5 we discuss the results in the context of LINER 1s-Seyfert-XRB connections, and a conclusion summarizing the main results is given in § 6. We report, in appendix A, some notes on the individual sources and in Appendix B we give spectral results to the surrounding sources around the centers of galaxies observed with *Chandra*. In the remainder of this paper, luminosities are calculated using the distances given in Table [galaxy-param] derived with a Hubble constant *H*0 = 75 km s− 1 Mpc− 1.\nThe sample\n==========\nWe selected objects classified as LINER 1.9 (LINER 1s) sources in showing a definite detection of a broad H*α* emission line. This implies the definite existence of an AGN at the center of all of the sixteen selected galaxies and its responsibility for the excitation of the detected optical emission lines.\n[!th] [galaxy-param]\n| | | | | | |\n| --- | --- | --- | --- | --- | --- |\n| Galaxy Name | R.A. | Dec. | Hubble Type | Distance*a* | log(*M**B**H*)*b* |\n| | | | | (Mpc) | (*M*⊙) |\n| | 00 49 47.8 | +32 16 40 | SB(rs)ab | 62.4 | 8.44 |\n| | 00 57 48.9 | +30 21 09 | E+: | 65.8 | 9.06 |\n| | 08 53 32.7 | +51 18 49 | (R’)SAB(rs)0/a | 17.2 | 6.78 |\n| | 09 19 18.5 | +69 12 12 | SB(r)0+ | 7.48 | 8.15 |\n| | 10 23 27.0 | +19 53 55 | E2:pec | 23.6 | 8.06 |\n| | 11 32 34.8 | +53 04 05 | SB(s)a pec | 17.0 | 7.61 |\n| | 11 57 56.1 | +55 27 13 | SA(r)0? | 14.1 | 9.07 |\n| | 12 09 36.0 | +42 32 03 | SAB(s)0 | 15.9 | 8.18 |\n| | 12 15 05.0 | +33 11 50 | SAB0-: | 15.1 | 7.73 |\n| | 12 20 06.8 | +29 16 51 | E1+ | 16.1 | 8.72 |\n| | 12 50 07.2 | +72 52 28 | (R)SA(rs)ab | 26.1 | 7.27 |\n| | 12 53 29.1 | +02 10 06 | SA(s)a | 16.3 | 7.46 |\n| | 13 10 56.2 | +37 03 33 | SAB(rs)bc | 21.3 | 7.79 |\n*a*Distances adapted from, otherwise from. *b*Black hole mass calculated using updated M-*σ* relation of with stellar velocity dispersion taken from.\nWe excluded three sources from the sample: NGC 3642, NGC 4636 and NGC 1052. NGC 3642 did not have any archived *XMM-Newton* or *Chandra* observations. As for NGC 4636, all of the X-ray archived observations were studied in extreme detail and show a complicated spectrum that requires, for a good spectral parameters measurement, detailed imaging analysis and should be modeled including: sophisticated shock models, temperature and density gradients, and last but not least, steep abundance gradients in the core. Finally, the broad H*α* emission line detected in the spectrum of NGC 1052 is attributed to polarization due to electron scattering within the opening cone of an obscuring torus. NGC 1052 was classified as an obscured AGN showing a large intrinsic absorption in the X-ray spectrum.\nTable [galaxy-param] shows the list of galaxies along with their corresponding right ascension and declination, Hubble type, distance (taken from, otherwise from ), the mass of the black hole derived from the M-*σ* relation where the velocity dispersion is taken from. Multiple snapshot observations (exposure time ≤ 5 ks) were excluded from the analysis either because of high background contamination (NGC 4143, obs.ID: 0150010201), low number of counts detected (NGC 2787, obs. ID: 388), or severe pile-up (NGC 4203 and NGC 4278, obs. IDs:397 and 398, respectively). The final sample consists of 13 LINER 1s with a total of 31 observations summarized in Table [obs-param].\n[!th] [obs-param]\n \n*a*Observations reported for the first time for the LINER 1 nucleus study. *b*Exposure time corrected for solar flare intervals.\nX-ray observations and data reduction\n=====================================\n*Chandra* observations\n----------------------\nAll of the LINER 1s in our sample have at least one *Chandra* observation. Snapshot observations with an exposure time less than 5 ks were performed for eight sources (NGC 266, NGC 315, NGC 3226, NGC 3718, NGC 4143, NGC 4750, NGC 4772, and NGC 5005). Seven sources have observations with a sufficient exposure time for a detailed temporal and spectral study (NGC 315, NGC 2681, NGC 2787, NGC 3226, NGC 3998, NGC 4203, and NGC 4278). All of the *Chandra* observations were obtained with the spectroscopic array where the nucleus was placed on the aim point, except for NGC 3226, of the ACIS-S3 back-illuminated chip. They were taken in either Faint or Very Faint mode to increase their sensitivity. All of the observations are *Chandra* archival data obtained from chaser[1](#fn1). The log of the *Chandra* observations are listed in Table 2.\nAll *Chandra* observations were reduced and analyzed in a systematic, homogeneous way using the CIAO software package version 4.2, *Chandra* Calibration Database, CALDB, version 4.3.1, and the ACIS Extract (AE) software package version 3.175 [2](#fn2). We started by using the level 1 event file produced by the *Chandra* X-ray Center (CXC) to suppress the position randomization applied by the CXC Standard Data Processing when creating a level 2 event file. We also corrected for the effects of charge-transfer inefficiency on event energies and grades. We filtered for bad event grades (only ASCA grades 0, 2, 3, 4 and 6 are accepted) and hot columns to take account of several potential issues such as cosmic rays and bad pixels. Good time intervals, supplied by the pipeline, were applied to the final products.\nThe LINER nucleus source position is determined after running a wavelet transform detection algorithm, the *wavdetect* program within the CIAO data analysis system. This position is then given to the AE software that refines it, extract source photons, construct local backgrounds, extract source, and background spectra, compute redistribution matrix files (RMFs) and auxiliary response files (ARFs), by spawning the *mkarf* and *mkacisrmf* routines of CIAO, and perform spectral grouping and fitting.\nSource events are extracted around the source centroid, inside a polygonal shape of the local PSF, generated by MARX[3](#fn3) at the energy 1.497 keV using the *ae\\_make\\_psf* tool implemented in AE. Background region is defined following the AE procedure. The background region is an annular region centered on the source position where the inner radius delimit 1.1 × 99% encircled energy radius and the outer radius is set such that the background includes between 100 counts and 200 counts (depending on the brightness of the source). This background is obtained from a special image where all events within the ∼ 1.1 × 99% PSF circles of all the sources in the field were excluded (*swiss cheese image*). Background was modeled for snapshot observations.\nPiled-up observations were accounted for by excluding the core of the PSF (see Y10 for more details). We use the tool *dmextract*, called by the AE software, to create spectra over the energy range 0.5–8 keV. We used the tool *ae\\_group\\_spectrum* implemented in AE to group the spectra. Channels between 0.5 and 8 keV are grouped to have a three sigma (3*σ*) signal to noise ratio, which corresponds to a minimum of 20 counts per bin, to enable the use of the *χ*2 statistics in the spectral analysis. The cash statistic (C-stat) is used to derive spectral parameters for snapshot *Chandra* observations with the background being modeled with the *cplinear* background model developed by. The background model is arbitrarily chosen to consist of continuous piecewise-linear (*cplinear*) functions with 2 to 10 vertexes. The model has 2 to 10 parameters representing the X-ray fluxes at the different vertexes. These vertexes are placed on the energy scale so that they divide the energy range into intervals with approximately equal numbers of observed counts in the background spectrum (0.1 to 10 keV). Vertex energies are chosen to coincide with the energies of actual events in the background which helps to prevent the vertex flux from being driven to the hard limit of zero during the fitting process.\n*XMM-Newton* observations\n-------------------------\nThe log of the *XMM-Newton* observations is listed in Table 2. Eight sources were observed at least once with *XMM-Newton* (NGC 315, NGC 2787, NGC 3226, NGC 3718, NGC 3998, NGC 4143, NGC 4278, and NGC 5005) and two have multiple observations (NGC 3226 and NGC 3718). In all of the observations, the EPIC-pn and MOS cameras were operated in Imaging, Prime Full Frame or Large Window Mode (except for the long NGC 3226 observation, where MOS cameras were operating in Small Window Mode[4](#fn4)) using the thin or medium filter. The Reflection Grating Spectra show only few counts for all the different observations and therefore they were not included in our analysis. We did not make use of the optical/UV data taken with the optical/UV monitor (OM) instrument since this paper concentrates on the X-ray characteristics of this sample. A multiwavelength study of our sample will be treated in a forthcoming paper. All data products were obtained from the XMM-Newton Science Archive (XSA)[5](#fn5) and reduced using the Science Analysis System (SAS) version 9.0. Data are selected using event patterns 0–4 and 0–12 for pn and MOS, respectively, during only good X-ray events (“FLAG = 0”). None of the EPIC observations were affected by pile-up, although severe intervals of enhanced solar activity, where the background count rate even exceeds the source count rate, were present during several observations (NGC 315, NGC2787, NGC 3226). In these cases, we reduced the background to 5% by excluding the high background intervals which reduces the observation time usable for spectral analysis, sometimes to less than 30% of the raw exposure time (NGC 315).\n*XMM-Newton* source events of all of the LINER 1s in our sample were extracted from a circle centered at the nucleus using two different radii of 10 and 25[6](#fn6). We compared light curves and spectra of both extraction regions to check if any of the sources (jet emission, diffuse emission, and/or unresolved point-like sources) detected in the *Chandra* image between 10 and 25 around the nucleus contaminate the *XMM-Newton* nucleus emission (see Appendix B and online Fig. 1–4). No change is seen in the light curves, and the fit parameters of the two extracted spectra were consistent, within the error bars. Therefore, and to achieve better statistics and better constrain fit parameters, source events of all of the LINER 1s in our sample observed with *XMM-Newton* were taken from the 25-radius circle centered on the nucleus. We added the spectral contribution of the different sources (jet emission, diffuse emission, and/or unresolved point-like sources), derived from the *Chandra* observation and detected in a 25-radius circle around the nucleus (see Appendix B for more details), to the spectral model used to fit the *XMM-Newton* spectrum. The particular case of NGC 4278 is discussed in detail in Y10. Background events are extracted from a source–free circle with a radius twice of the source on the same CCD. We generated response matrix files using the SAS task *rmfgen*, while ancillary response files were generated using the SAS task *arfgen*. The EPIC spectra were created in the energy range 0.5–10 keV to enable flux and model–parameter comparison with *Chandra* spectra. They were grouped to have a signal to noise ratio of 3 with a minimum of 20 counts per bin to allow the use of the *χ*2 statistic.\nResults\n=======\nLight curves and hardness ratios\n--------------------------------\nTemporal analysis was only done for long exposure observations, not including snapshot *Chandra* observations. Light curves and corresponding hardness ratios, defined as *H**R* = (*H* − *S*)/(*H* + *S*), where *S* is the count rate in the soft 0.5-2 keV band and *H* is the count rate in the hard 2-10 keV band, were extracted for all of the long observations. We corrected the net count rate of the piled-up sources for the excluded fraction of the PSF. *Chandra* and *XMM-Newton* light curves were all binned with a time bin size of 1 ks for a reliable rms variability analysis.\nWe first conducted a Kolmogorov-Smirnov, K-S, test to examine any potential variability within each observation. Based on this test, we do not find short time-scale (hours to days) variability in 5/10 sources (NGC 315, NGC 2681, NGC 3718, NGC 3998, and NGC 5005) with a K-S test probability > 10% that the nuclear emission originates from a constant source. Three sources (NGC 2787, NGC 4143, and NGC 4203) indicate a possible variability with a K-S test probability between 4% and 2% that the nuclear emission originates from a constant source. Two *XMM-Newton* observations of two different sources exhibit significant short time-scale variability, both already reported in the literature, NGC 4278 (obs. ID: 205010101, Y10) and NGC 3226, where the K-S test gives a probability less than 1% that the core emission is originating from a constant source. NGC 4278 shows an increase at the beginning of the observation of 10% on a time-scale of ∼ 1.5 hour, the emission remains constant for the rest of the observation following that hint of variability. As for NGC 3226, variability is clear through the whole observation where a total increase of ∼ 60% is detected between the beginning and the end of the ∼ 100 ks observation. *XMM-Newton* light curve of NGC 3226 is shown in Fig. [light-hardness] and the *XMM-Newton* and *Chandra* light curves of the other sources are given in online Fig. [LCxmmallLINERs] and Fig. [LCchandraallLINERs].\n [light-hardness]\n[!th]\n [varsigma] \n*a*The exposure times in ks used to calculate the value of *σ**N**X**S*2. *b*Value corresponding to only the longest observation of NGC 3226. Including the other *XMM-Newton* observation would result in a *σ**N**X**S*2 upper limit of 0.013.\nTo check more accurately any intrinsic variability amplitude from the different sources, we calculated the normalized excess variance for all of the long observations with the following expression:\n \n$$\\sigma^2\\_{NXS}=\\frac{1}{N\\mu^2}\\sum\\limits\\_{i=1}^{N}[(X\\_i-\\mu)^2-\\sigma\\_{i}^2]$$ \nwhere *N* is the number of bins in a given light curve, *X**i* and *σ**i* are the count rate and uncertainty of each bin, respectively, and *μ* is the arithmetic mean of the counting rates. To enable *σ**N**X**S*2 comparison between the different light curves, the bin size and the light curve segment duration should be taken equally. For that purpose, we first decided to use light curve segments of 20 ks, as usually done for luminous Seyfert galaxies, splitting any longer observations into multiple ones. That limited our sample to 6 sources observed with *Chandra* and 5 observed with *XMM-Newton*, not enough to draw any safe conclusions. Therefore, and owing to the heterogeneous sampling of the observations for this type of study, we decided to use the whole corrected exposure time of all of the long observations. The mean of the *σ**N**X**S*2 is taken for every source with multiple *Chandra* or *XMM-Newton* observations. The time bin size choice of 1 ks for all of the observations was taken to have a good signal to noise ratio with at least 20 counts in each bin and an acceptable number of bins in each light curve.\nEstimating the error on the *σ**N**X**S*2 could be a tricky task. The variability in an AGN light curve depends, on one hand, on the measurement errors of the data (e.g. Poisson noise) and, on the other hand, on the stochastic nature of the process underlying AGN variability ; even if a source is not intrinsically variable the mean and the variance of different light curves based on observations performed at different times will not be identical. We estimated error due to Poisson noise using the equation\n \n$$\\label{errsig}\n\\resizebox{.43\\textwidth}{!}{$err(\\sigma^2\\_{NXS})=\\sqrt{\\left(\\sqrt{\\frac{2}{N}}\\frac{\\overline{\\sigma\\_{err}^{2}}}{\\overline{x}^2}\\right)^2+\\left(\\sqrt{\\frac{\\overline{\\sigma^{2}\\_{err}}}{N}}\\frac{2\\sigma\\_{NXS}}{\\overline{x}}\\right)^2}$}.$$ \n[NXSvsMbh]\nThe uncertainty owing to the red noise process has been presented by to depend on the power-spectrum shape of the source which we do not know a priori. Therefore, estimated the error on the red noise process directly from the data. Our observations are not well sampled to use method to determine any error due to the stochastic nature of the AGN X-ray variability. The error on *σ**N**X**S*2 shown in equation [errsig] was also used to estimate upper limits to the excess variance in the case of non-variability detection whenever *σ**N**X**S*2 is negative or consistent with zero. Only one object, NGC 3226, in our sample shows a clear short time-scale variability during the longest 100 ks observation with *σ**N**X**S*2 = 0.02 ± 0.002 (comparable to the value of 0.014 found by ). Upper limits were obtained for the rest of the sample. Table [varsigma] and Fig. [NXSvsMbh] summarizes the results that are discussed in § [xrayvar].\nX-ray spectral results\n----------------------\nThe spectral analysis was performed using XSPEC version 12.6.0. The photo-electric cross sections and the solar abundances of are used throughout to account for absorption by neutral gas. An absorbed Galactic column density derived for every single source from (obtained with the W3NH tool[7](#fn7)) was applied to the different spectral models. Spectral uncertainties are given using Δ*χ*2 of 2.71, corresponding to 90% confidence for one interesting parameter, and to 95% confidence for upper limits.\n### snapshot observations\nWe began our spectral analysis with the study of the *Chandra* snapshot observations. Table [specfit-param-snapshot] gives the best fit parameters to the snapshot observations of our sample of LINER 1s. Complicated models, like partial covering and/or two power-law components could not be tested due to the low number of counts. A power-law modified by Galactic absorption gave a good fit in the case of NGC 266 and NGC 4143. An additional intrinsic neutral absorption was needed in the remainder of the cases. The photon indices vary between 1.3 ± 0.2 for the hardest spectra to 2.1 ± 0.7 for the softest ones with a mean value of about 1.7. The hydrogen column density of the intrinsic absorber had an upper limit of ∼ 4 × 1021 cm− 2 in the case of NGC 4750 and NGC 5005. A value consistent with 1021 < *N**H* < 1022 cm− 2 was derived for the rest of the snapshot observations. In one case, NGC 3718, we find a somewhat larger column density with *N**H* in the order of ∼ 1022 cm− 2. For NGC 5005, a thermal component with a 0.8− 0.2+ 0.3 keV temperature, was included in the model to take account for some low energy features, most likely due to diffuse emission from hot gas. In order to rigorously confirm the validity of our best fit spectral-parameter values derived using the C-stat and a modeled background, we compared our results to the results derived from fits applying the *χ*2 statistics to all of the snapshots observations. Spectral parameters derived using the C-stat were all in agreement, within the error bars, with the results derived using the *χ*2 statistics; with smaller deviation from the central value of one interesting parameter. We decided, as a consequence, to use the C-stat fits to calculate model fluxes in the soft, 0.5-2 keV, and in the hard, 2-10 keV, bands. Table [specfit-fluxes] gives the corresponding 0.5-2 keV and 2-10 keV observed fluxes and corrected luminosities.\n[!th] [specfit-param-snapshot]\n \n(*a*)The emission measure (EM) of the *mekal* model, EM=∫n*e*n*H*dV.\n### Long-exposure observations\n[!th] [specfit-param]\n \n(L.) represents a linked paramter in the fit. (*a*)The emission measure (EM) of the *mekal* model, EM=∫n*e*n*H*dV.\n[!th] [specfit-fluxes]\n \n(*a*)The power-law component fraction to the 0.5-10 keV corrected luminosity.\n \n [bestfitmod]\nWe then turned to the analysis of observations with relatively long exposure times. We started with a simple absorbed power-law fit to each of the spectra of the different sources, separately. The fits were acceptable for 6/10 sources but residuals at energies less than 2 keV persisted in the other 4 sources (NGC 315, NGC 2681, NGC 4278, and NGC 5005), suggesting the presence of diffuse hot gas.\nIn order to have a better signal to noise ratio and photon statistics, we decided to fit the different *Chandra* and/or *XMM-Newton* spectra of each source simultaneously (the normalizations of the different models in a fit were left free between the different EPIC instruments to take care of any potential cross calibration uncertainties). Simultaneous fitting routines of different observations performed at different times is done for the first time for any sample of LINER sources. For this purpose, whenever a source is observed with both *Chandra* and *XMM-Newton*, we carried out careful imaging analysis of the *Chandra* observations to disentangle the different components (diffuse emission, LMXBs, and/or jet emission) that are blended in one point-like source in the *XMM-Newton* extraction region of 25-radius circle (see Appendix B and online Fig. 1–4). These components, that we assume non-variable, are included in the simultaneous *Chandra*/*XMM-Newton* fit. We do not expect diffuse hot gas to vary on a time-scale of a few years, although off-nuclear point-like sources and jet X-ray emission could exhibit variation on such time-scales. We present in Appendix B the surrounding medium around the different LINER 1s that are observed with both *Chandra* and *XMM-Newton*, and we give spectral results to the different components.\nWe tested multiple models on the data in order to determine the mechanism responsible for the observed X-ray spectra, noticeably: (1) a simple absorbed power-law, (2) same as (1) but including a thermal *mekal* component to take account of any diffuse hot gas features in the soft band, (3) two power-law components with different photon index values with one representing the hard 2-10 keV emission and the other representing a possible 0.5-2 keV soft excess emission commonly seen in the nuclei of luminous galaxies. We investigated spectral variability in a source by permitting one spectral parameter of a given fit to vary independently between different observations. We then used the F-test to evaluate the improvement in the fit where a 99% probability for an improvement to occur by chance is considered valid. The case for NGC 4278 is already analyzed in Y10 and best fit spectral parameters and fluxes are taken from Y10.\nWe find that 6/10 sources are best fit with model (1) with no additional need for any thermal or two power-law components (NGC 2787, NGC 3226, NGC 3718, NGC 3998, NGC 4143, and NGC 4203). The rest of the sources were best fit with model (2) showing features at low energies below 2 keV, indicating emission from diffuse hot gas (NGC 315, NGC 2681, NGC 4278, and NGC 5005). Model (3) did not improve the quality of the fit in any of the cases, giving worse fits at times. We found that the intrinsic hydrogen column density varies significantly in NGC 3226 decreasing from (8.9 ± 0.7) × 1021 cm− 2 to 4.2− 0.3+ 0.2 × 1021 cm− 2. This is clearly seen in the lower panels of Fig. [bestfitmod] where the soft part of the spectrum during the short *XMM-Newton* observation is much more absorbed compared to the long observation. Additionally, the power-law photon index varies in seven sources (NGC 315, NGC 3226, NGC 3718, NGC 3998, NGC 4143, NGC 4278, and NGC 5005) with the most drastic change being the one observed in NGC 315 where Γ increased from 1.5 ± 0.1 during the *Chandra* observation to 2.1− 0.2+ 0.1 during the *XMM-Newton* one (upper panels of Fig. [bestfitmod]). This increase is accompanied by a decrease in the 2-10 keV flux from 9.8 × 10− 13 to 4.6 × 10− 13 erg s− 1. This behavior is typical of X-ray emission originating in a RIAF structure, which is the accretion flow believed to exist at the center of NGC 315, where the Eddington ratio, which is proportional to L2 − 10 *k**e**V*, is inveresly proportional to the photon index Γ. Best fit models and residuals of the other LINER 1s in our sample are shown in online Fig. [bestfitallLINERs].\nThe photon indicies we derived for all of the sources in our sample observed with a relatively long exposure time varied between 1.5 ± 0.3 and 2.4− 0.3+ 0.2 with a mean value of 2.0. Intrinsic column density covered two orders of magnitude, with *N**H* varying between ∼ 1020 cm− 2 for unabsorbed sources, and up to ∼ 1022 cm− 2 for the only mildly absorbed source NGC 3718. The thermal component had a temperature mean value of 0.63 keV, consistent with all LINER-type sources embedded in diffuse emission. Table [specfit-param] gives the best fit model parameters for our sample of LINER 1s with a relatively long exposure time, and Table [specfit-fluxes] gives the 0.5-2 keV and 2-10 keV observed fluxes and corrected luminosities, as well as the corresponding “Eddington ratios”,. The hard, 2-10 keV, luminosity spans three orders of magnitude from 3.2 × 1038 erg s− 1 to 5.4 × 1041 erg s− 1 which resulted in a range from 2.0 × 10− 8 to 2.3 × 10− 5 which is at least one to two orders of magnitude smaller than seen in luminous AGN.\n[gmaVSedd]\n[!t] \n [other-corr-fig]\n[!th]\n [spear-rank] \nWe looked for any sign of narrow Fe K*α* emission line at 6.4 keV in the EPIC-p"},"data_id":{"kind":"string","value":"arxiv_0002464"}}},{"rowIdx":598,"cells":{"content":{"kind":"string","value":"One-way communication complexity and non-adaptive decision trees[3](#fn3)\n=========================================================================\nWe study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. More generally, we show the following when the gadget is Inner Product on 2*b* input bits for all *b* ≥ 2, denoted IP.\n* If *f* is a total Boolean function that depends on all of its *n* input bits, then the bounded-error one-way quantum communication complexity of *f* ∘ IP equals Ω(*n*(*b* − 1)).\n* If *f* is a *partial* Boolean function, then the deterministic one-way communication complexity of *f* ∘ IP is at least Ω(*b* ⋅ Ddt→(*f*)), where Ddt→(*f*) denotes non-adaptive decision tree complexity of *f*.\nTo prove our quantum lower bound, we first show a lower bound on the VC-dimension of *f* ∘ IP. We then appeal to a result of Klauck [STOC’00], which immediately yields our quantum lower bound. Our deterministic lower bound relies on a combinatorial result independently proven by Ahlswede and Khachatrian [Adv. Appl. Math.’98], and Frankl and Tokushige [Comb.’99].\nIt is known due to a result of Montanaro and Osborne [arXiv’09] that the deterministic one-way communication complexity of *f* ∘ XOR *equals* the non-adaptive parity decision tree complexity of *f*. In contrast, we show the following when the inner gadget is the AND function on 2 input bits.\n* There exists a function for which even the *quantum* non-adaptive AND decision tree complexity of *f* is exponentially large in the deterministic one-way communication complexity of *f* ∘ AND.\n* However, for symmetric functions *f*, the non-adaptive AND decision tree complexity of *f* is at most quadratic in the (even two-way) communication complexity of *f* ∘ AND.\nIn view of the first bullet, a lower bound on non-adaptive AND decision tree complexity of *f* *does not* lift to a lower bound on one-way communication complexity of *f* ∘ AND. The proof of the first bullet above uses the well-studied *Odd-Max-Bit* function. For the second bullet, we first observe a connection between the one-way communication complexity of *f* and the *Möbius sparsity* of *f*, and then give a lower bound on the Möbius sparsity of symmetric functions. An upper bound on the non-adaptive AND decision tree complexity of symmetric functions follows implicitly from prior work on combinatorial group testing; for the sake of completeness, we include a proof of this result.\nIt is well known that the rank of the communication matrix of a function *F* is an upper bound on its deterministic one-way communication complexity. This bound is known to be tight for some *F*. However, in our final result we show that this is not the case when *F* = *f* ∘ AND. More precisely we show that for all *f*, the deterministic one-way communication complexity of *F* = *f* ∘ AND is at most (rank(*M**F*))(1 − Ω(1)), where *M**F* denotes the communication matrix of *F*.\nIntroduction\n============\nComposed functions are important objects of study in analysis of Boolean functions and computational complexity. For Boolean functions *f* : {0, 1}*n* → {0, 1} and *g* : {0, 1}*m* → {0, 1}, their composition *f* ∘ *g* : ({0, 1}*m*)*n* → {0, 1} is defined as follows: *f* ∘ *g*(*x*1, …, *x**n*) : = *f*(*g*(*x*1), …, *g*(*x**n*)). In other words, *f* ∘ *g* is the function obtained by first computing *g* on *n* disjoint inputs of *m* bits each, and then computing *f* on the *n* resultant bits. Composed functions have been extensively looked at in the complexity theory literature, with respect to various complexity measures .\nOf particular interest to us is the case when *g* is a communication problem (also referred to as “gadget”). More precisely, let *g* : {0, 1}*b* × {0, 1}*b* → {0, 1} and *f* : {0, 1}*n* → {0, 1} be Boolean functions. Consider the following communication problem: Alice has input *x* = (*x*1, …, *x**n*) and Bob has input *y* = (*y*1…, *y**n*) where *x**i*, *y**i* ∈ {0, 1}*b* for all *i* ∈ [*n*]. Their goal is to compute *f* ∘ *g*((*x*1, *y*1), …, (*x**n*, *y**n*)) using as little communication as possible. A natural protocol is the following: Alice and Bob jointly simulate an efficient query algorithm for *f*, using an optimal communication protocol for *g* to answer each query. Lifting theorems are statements that say this naive protocol is essentially optimal. Such theorems enable us to prove lower bounds on the rich model of communication complexity by proving feasibly easier-to-prove lower bounds in the query complexity (decision tree) model. Various lifting theorems have been proved in the literature .\nIn this work we are interested in the *one-way* communication complexity of composed functions. Here, a natural protocol is for Alice and Bob to simulate a *non-adaptive* decision tree for the outer function, using an optimal *one-way* communication protocol for the inner function. Thus, the one-way communication complexity of *f* ∘ *g* is at most the non-adaptive decision tree complexity of *f* times the one-way communication complexity of *g*.\nLifting theorems in the one-way model are less studied than in the two-way model. Montanaro and Osborne observed that the deterministic one-way communication complexity of *f* ∘ XOR *equals* the non-adaptive parity decision tree complexity of *f*. Thus, non-adaptive parity decision tree complexity lifts “perfectly” to deterministic communication complexity with the XOR gadget. Kannan et al. showed that under uniformly distributed inputs, bounded-error non-adaptive parity decision tree complexity lifts to one-way bounded-error distributional communication complexity with the XOR gadget. Hosseini, Lovett and Yaroslavtsev showed that randomized non-adaptive parity decision tree complexity lifts to randomized communication complexity with the XOR gadget in the one-way broadcasting model with Θ(*n*) players.\nWe explore the tightness of the naive communication upper bound for two different choices of the gadget *g*: the Inner Product function, and the two-input AND function. For each choice of *g*, we compare the one-way communication complexity of *f* ∘ *g* with an appropriate type of non-adaptive decision tree complexity of *f*. Below, we motivate and state our results for each choice of the gadget. Formal definitions of the measures considered in this section can be found in Section [sec: prelims].\nInner Product Gadget\n--------------------\nLet Qcc, *ɛ*→( ⋅ ), Rcc, *ɛ*→( ⋅ ) and Dcc→( ⋅ ) denote quantum *ɛ*-error, randomized *ɛ*-error and deterministic one-way communication complexity, respectively. When we allow the parties to share an arbitrary input-independent entangled state in the beginning of the protocol, denote the one-way quantum *ɛ*-error communication complexity by Qcc, *ɛ*\\* , →( ⋅ ). Let Qdt→( ⋅ ) and Ddt→( ⋅ ) denote bounded-error quantum non-adaptive decision tree complexity and deterministic non-adaptive decision tree complexity, respectively. For an integer *b* > 0, let IP : {0, 1}*b* × {0, 1}*b* → {0, 1} denote the *Inner Product Modulo 2* function, that outputs the parity of the bitwise AND of two *b*-bit input strings. Our first result shows that if *f* is a total function that depends on all of its input bits, the quantum (and hence, randomized) bounded-error one-way communication complexity of *f* ∘ IP is Ω(*n*(*b* − 1)). Let Hbin( ⋅ ) denote the binary entropy function. If *ɛ* = 1/2 − Ω(1), then 1 − Hbin(*ɛ*) = Ω(1).\n[thm:main] Let *f* : {0, 1}*n* → {0, 1} be a total Boolean function that depends on all its inputs (i.e., it is not a junta on a strict subset of its inputs), and let *ɛ* ∈ (0, 1/2). Let IP : {0, 1}*b* × {0, 1}*b* → {0, 1} denote the Inner Product function on 2*b* input bits for *b* ≥ 1. Then Qcc, *ɛ*→(*f* ∘ IP) ≥ (1 − Hbin(*ɛ*))*n*(*b* − 1) and Qcc, *ɛ*\\* , →(*f* ∘ IP) ≥ (1 − Hbin(*ɛ*))*n*(*b* − 1)/2.\nIn an earlier manuscript , the second author proved a lower bound of (1 − Hbin(*ɛ*))*n*(*b* − 1) on a weaker complexity measure, namely Rcc, *ɛ*→(*F*), via information-theoretic tools. Kundu subsequently observed that a quantum lower bound can also be obtained by additionally using Holevo’s theorem. They also suggested to the second author via private communication that one might be able to recover these bounds using a result of Klauck . This is indeed the approach we take, and we thank them for suggesting this and pointing out the reference.\nIn order to prove Theorem [thm:main], we appeal to a result of Klauck , who showed that the one-way *ɛ*-error quantum communication complexity of a function *F* is at least (1 − Hbin(*ɛ*)) ⋅ VC(*F*), where VC(*F*) denotes the VC-dimension of *F* (see Definition [defn: VC]). In the case when the parties can share an arbitrary entangled state in the beginning of a protocol, Klauck showed a lower bound of (1 − Hbin(*ɛ*)) ⋅ VC(*F*)/2. We exhibit a set of inputs that witnesses the fact that VC(*f* ∘ IP) ≥ *n*(*b* − 1). Note that Theorem [thm:main] is useful only when *b* > 1. Indeed, no non-trivial lifting statement is true for *b* = 1 when *f* is the AND function on *n* bits, since in this case, *f* ∘ IP = AND2*n*, whose one-way communication complexity is 1.\nOur second result with the Inner Product gadget relates the deterministic one-way communication complexity of *f* ∘ IP to the deterministic non-adaptive decision tree complexity of *f*, where *f* is an arbitrary *partial* Boolean function.\n[thm:detlift] Let S ⊆ {0, 1}*n* be arbitrary, and *f* : S → {0, 1} be a *partial* Boolean function. Let *b* ≥ 2 and IP : {0, 1}*b* × {0, 1}*b* → {0, 1}. Then Dcc→(*f* ∘ IP) = Ω(*b* ⋅ Ddt→(*f*)).\nGiven a protocol Π, our proof extracts a set of variables of cardinality linear in the complexity of Π, whose values always determine the value of *f*. The following claim which follows directly from a result due to Ahlswede and Khachatrian and independently Frankl and Tokushige, is a crucial ingredient in our proof.\n[thm: packing] Let *q* ≥ 3 and 1 ≤ *d* ≤ *n*/3. Let A ⊆ [*q*]*n* be such that for all *x*(1) = (*x*1(1), …, *x**n*(1)), *x*(2) = (*x*1(2), …, *x**n*(2)) ∈ A, ∣{*i* ∈ [*n*] ∣ *x**i*(1) = *x**i*(2)}∣ ≥ *d*. Then, $|{\\mathcal{A}}| < q^{n-\\frac{d}{10}}$.\nWe prove Theorem [thm: packing] in Appendix [erdos]. Theorem [thm: packing] admits simple proofs when *q* is large compared to *n*. See for a proof when *q* is a prime power, and *q* ≥ *n*. Their proof is based on polynomials over finite fields. We give a different proof for all $q > (\\frac{en}{d})^2$ in Appendix [erdos] (for every *δ* > 0, the proof can be extended to work for all *q* = Ω(*n*/*d*)1 + *δ*). However, such a statement that only holds for large *q* will only enable us to prove a lifting theorem for a gadget of size *b* = Ω(log*n*). To prove Theorem [thm:detlift] for constant-sized gadgets we need to set *q* to *O*(1).\nAn analogous lifting theorem for deterministic one-way protocols for *total* outer functions follows as a special case of both Theorem [thm:main] and Theorem [thm:detlift]. However, the statement admits a simple and direct proof based on a fooling set argument.\nTheorem [thm:main] and Theorem [thm:detlift] give lower bounds even when the gadget is the Inner Product function on 4 input bits (and lower bounds do not hold for the Inner Product gadget with fewer inputs). It is worth mentioning here that prior works that consider lifting theorems with the Inner Product gadget , albeit in the two-way model of communication complexity, require a super-constant gadget size.\nAND Gadget\n----------\nInteractive communication complexity of functions of the form *f* ∘ AND have gained a recent interest . In order to state and motivate our results regarding when the inner gadget is the 2-bit AND function, we first discuss some known results in the case when the inner gadget is the 2-bit XOR function.\nConsider non-adaptive decision trees, where the trees are allowed to query arbitrary parities of the input variables. Denote the minimum cost (number of parity queries) of such a tree computing a Boolean function *f*, by NAPDT(*f*). An efficient non-adaptive parity decision tree for *f* can easily be simulated to obtain an efficient deterministic one-way communication protocol for *f* ∘ XOR. Thus, Dcc→(*f* ∘ XOR) ≤ NAPDT(*f*). Montanaro and Osborne observed that this inequality is, in fact, tight for all Boolean functions. More precisely,\n[][claim: foxor oneway equals napdt] For all Boolean functions *f* : {0, 1}*n* → {0, 1}, Dcc→(*f* ∘ XOR) = NAPDT(*f*).\nIf the inner gadget were AND instead of XOR, then the natural analogous decision tree model to consider would be non-adaptive decision trees that have query access to arbitrary ANDs of subsets of inputs. Denote the minimum cost (number of AND queries) of such a tree computing a Boolean function *f* by NAADT(*f*). Clearly, Dcc→(*f* ∘ AND) is bounded from above by NAADT(*f*), since a non-adaptive AND decision tree can be easily simulated to give a one-way communication protocol for *f* ∘ AND of the same complexity. Thus, Dcc→(*f* ∘ AND) ≤ NAADT(*f*). On the other hand, one can show that Dcc→(*f* ∘ AND) ≥ log(NAADT(*f*)) (see Claim [claim: doneway vs naandt]). Thus\n \nlog(NAADT(*f*)) ≤ Dcc→(*f* ∘ AND) ≤ NAADT(*f*). \nWe explore if an analogous statement to Claim [claim: foxor oneway equals napdt] holds true if the inner function were AND instead of XOR. That is, is the second inequality in Equation always tight?\nWe give a negative answer in a very strong sense and exhibit a function for which the first inequality is tight (up to an additive constant). We show that there is an exponential separation between these measures even if one allows the decision trees to have *quantum* query access to ANDs of subsets of input variables. It is worth noting that, in contrast, if one is given quantum query access to *parities* (in place of ANDs) of subsets of input variables, then one can completely recover an *n*-bit string using just 1 query , rendering this model trivial. Let QNAADT(*f*) denote the bounded-error quantum non-adaptive AND decision tree complexity of *f*.\n[thm: intro omb separation] There exists a function *f* : {0, 1}*n* → {0, 1} such that QNAADT(*f*) = Ω(2Dcc→(*f* ∘ AND)).\nThe function *f* we use to witness the bound in Theorem [thm: intro omb separation] is a modification of the well-studied *Odd-Max-Bit* function, which we denote OMB*n*. This function outputs 1 if and only if the maximum index of the input string that contains a 0, is odd (see Definition [defn: omb]). A ⌈log(*n* + 1)⌉-cost one-way communication protocol is easy to show, since Alice can simply send Bob the maximum index where her input is 0 (if it exists), and Bob can use this along with his input to conclude the parity of the maximum index where the bitwise AND of their inputs is 0. A crucial property that we use to show a lower bound of Ω(*n*) on QNAADT(OMB*n*) is that OMB*n* has large *alternating number*, that is, there is a monotone path on the Boolean hypercube from 0*n* to 1*n* on which the value of OMB*n* flips many times.\nTheorem [thm: intro omb separation] implies that, in contrast to the lifting theorem with the XOR gadget (Claim [claim: foxor oneway equals napdt]), the measure of non-adaptive AND decision tree complexity *does not* lift to a one-way communication lower bound for *f* ∘ AND. However we show that a statement analogous to Claim [claim: foxor oneway equals napdt] does hold true for symmetric functions *f*, albeit with a quadratic factor, even when the measure is two-way communication complexity, denoted Dcc( ⋅ ).\n[thm: intro symmetric naandt communication] Let *f* : {0, 1}*n* → {0, 1} be a symmetric function. Then NAADT(*f*) = *O*(Dcc(*f* ∘ AND)2).\nIn fact we prove a stronger bound in which Dcc(*f* ∘ AND) above is replaced by logrank(*M**f* ∘ AND), where *M**f* ∘ AND denotes the communication matrix of *f* ∘ AND. That is, we show that for symmetric functions *f*,\n \nNAADT(*f*) = *O*(log2rank(*M**f* ∘ AND)). \nSince it is well known (Equation ) that the communication complexity of a function is at least as large as the logarithm of the rank of its communication matrix, this implies Theorem [thm: intro symmetric naandt communication]. There have been multiple works (see, for example, and the references therein) studying the communication complexity of AND functions in connection with the log-rank conjecture which states that the communication complexity is bounded from above by a polynomial in the logarithm of the rank of the communication matrix. Among other things, Buhrman and de Wolf observed that the log-rank conjecture holds for symmetric functions composed with AND. In particular, they showed that if *f* is symmetric, then Dcc(*f* ∘ AND) = *O*(logrank(*M**f* ∘ AND))). Most recently, Knop et al. showed that ${{\\mathsf{D}}\\_{{\\mathrm{cc}}}}(f \\circ {\\mathsf{AND}}) = O(\\textnormal{poly}(\\log {\\mathrm{rank}}(M\\_{f \\circ {\\mathsf{AND}}}), \\log n)$ for all Boolean functions *f* : {0, 1}*n* → {0, 1}, nearly resolving the log-rank conjecture for AND functions.\nWhile we have a quadratically worse dependence in the RHS of Equation as compared to the above-mentioned bound for symmetric functions due to Buhrman and de Wolf, our upper bound is on a complexity measure that can be exponentially larger than communication complexity in general (Theorem [thm: intro omb separation]).\nBuhrman and de Wolf showed a lower bound on logrank(*M**f* ∘ AND) for symmetric functions *f*. An upper bound on NAADT(*f*) implicitly follows from prior work on group testing , but we provide a self-contained probabilistic proof for completeness. Combining these two results yields Equation , and hence Theorem [thm: intro symmetric naandt communication].\nSuitable analogues of Theorem [thm: intro omb separation] and Theorem [thm: intro symmetric naandt communication] can be easily seen to hold when the inner gadget is OR instead of AND. In this case, the relevant decision tree model is non-adaptive OR decision trees. Interestingly, these decision trees are studied in the seemingly different context of *non-adaptive group testing algorithms*. Non-adaptive group testing is an active area of research (see, for, example, and the references therein), and has additionally gained significant interest of late in view of the ongoing pandemic (see, for example, ).\nOur final result regarding the AND gadget deals with the relationship between one-way communication complexity and rank of the underlying communication matrix. It is easy to show that for functions *F* : {0, 1}*m* × {0, 1}*n* → {0, 1},\n \nlogrank(*M**F*) ≤ Dcc→(*F*) ≤ rank(*M**F*), \nwhere *M**F* denotes the communication matrix of *F* and is defined by *M**F*(*x*, *y*) = *F*(*x*, *y*), and rank( ⋅ ) denotes real rank. The first bound can be seen to be tight for functions with maximal rank, for example the Equality function. The second inequality is tight, for example, for the Addressing function on (log*n* + *n*) input bits (see Definition [defn: addressing]) where Alice receives *n* target bits and Bob receives log*n* addressing bits. Sanyal showed that the upper bound can be improved for functions of the form *F* = *f* ∘ XOR. More precisely they showed that for all Boolean functions *f* : {0, 1}*n* → {0, 1},\n \n$${{\\mathsf{D}}\\_{{\\mathrm{cc}}}^\\rightarrow}(f \\circ {\\mathsf{XOR}}) \\leq O\\left(\\sqrt{{\\mathrm{rank}}(M\\_{f \\circ {\\mathsf{XOR}}})} \\log {\\mathrm{rank}}(M\\_{f \\circ {\\mathsf{XOR}}})\\right),$$ \nand moreover this bound is tight up to the logarithmic factor on the RHS, when *f* is the Addressing function. We show that the same bound does not hold when the XOR gadget is replaced by AND. We show in Corollary [cor: doneway vs rank addressing] that when *f* is the Addressing function, then\n \nDcc→(*f* ∘ AND) ≥ rank(*M**f* ∘ AND)log32 ≈ rank(*M**f* ∘ AND)0.63, \nThus it is plausible that the upper bound in terms of rank from Equation might be tight for some function of the form *f* ∘ AND. We show that this is not the case.\n[thm: 1way sublinear rank] Let *f* : {0, 1}*n* : {0, 1} be a Boolean function. Then, \nDcc→(*f* ∘ AND) ≤ (rank(*M**f* ∘ AND))(1 − Ω(1)). \nWe show that Dcc→(*f* ∘ AND) is equal to the logarithm of a measure that we define in this work: the *Möbius pattern complexity* of *f*, which is the total number of distinct evaluations of the monomials in the Möbius expansion of *f* (see Section [subsec: expansions] for a formal definition of Möbius expansion).\n[Möbius pattern complexity][defn: patcomp] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function, and let $f = \\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S$ be its Möbius expansion. For an input *x* ∈ {0, 1}*n*, define the *pattern of *x** to be (AND*S*(*x*))*S* ∈ S*f*. Define the *Möbius pattern complexity* of *f*, denoted PatM(*f*), by ${\\mathsf{Pat}^{\\mathsf{M}}}(f) := \\abs{{\\left\\{P \\in {\\{0, 1\\}}^{{\\mathcal{S}}\\_f} : P = {\\left({\\mathsf{AND}}\\_S(x)\\right)}\\_{S \\in {\\mathcal{S}}\\_f}~\\textnormal{for some}~x \\in {\\{0, 1\\}}^n\\right\\}}}$.\nWhen clear from context, we refer to the Möbius pattern complexity of *f* just as the pattern complexity of *f*.\nAll of our results involving bounds for Dcc→(*f* ∘ AND) use the above-mentioned equivalence between it and log(PatM(*f*)) (see Claim [claim: doneway equals log pattern]). We unravel interesting mathematical structure in the Möbius supports of Boolean functions, and use them to bound their pattern complexity. We hope that pattern complexity will prove useful in future research.\nOrganization\n------------\nWe introduce the necessary preliminaries in Section [sec: prelims]. In Section [sec: IP composition] we prove our results regarding the Inner Product gadget (Theorem [thm:main] and Theorem [thm:detlift]). In Section [sec: naandt is not communication] we prove our results regarding the AND gadget (Theorem [thm: intro omb separation] and Theorem [thm: intro symmetric naandt communication]). In Appendix [app: addressing] we show some results regarding the Addressing function. We provide a combinatorial proof missing from the main text in Appendix [erdos].\nPreliminaries\n=============\nAll logarithms in this paper are taken base 2. We use the notation [*n*] to denote the set {1, …, *n*}. We often identify subsets of [*n*] with their corresponding characteristic vectors in {0, 1}*n*. The view we take will be clear from context. Let S ⊆ {0, 1}*n* be an arbitrary subset of the Boolean hypercube, and let *f* : S → {0, 1} be a partial Boolean function. If S = {0, 1}*n*, then *f* is said to be a total Boolean function. When not explicitly mentioned otherwise, we assume Boolean functions to be total.\n[Binary entropy] For *p* ∈ (0, 1), the binary entropy of *p*, Hbin(*p*), is defined to be the Shannon entropy of a random variable taking two distinct values with probabilities *p* and 1 − *p*. \n$${\\mathbb{H}}\\_{\\mathsf{bin}}(p):=p \\log \\frac{1}{p} + (1-p) \\log \\frac{1}{1-p}.$$ \nWe define the Inner Product Modulo 2 function on 2*b* input bits, denoted IP (we drop the dependence of IP on *b* for convenience; the value of *b* will be clear from context).\n[Inner Product Modulo 2][defn: ip] For an integer *b* > 0, define the *Inner Product Modulo 2* function, denoted IP : {0, 1}*b* × {0, 1}*b* → {0, 1} by IP(*x*1, …, *x**b*, *y*1, …, *y**b*) = ⊕ *i* ∈ [*b*](AND(*x**i*, *y**i*)).\nIf *f* is a partial function, so is *f* ∘ IP. We also define the following functions.\n[Odd-Max-Bit][defn: omb] Define the *Odd-Max-Bit* function,[4](#fn4) denoted OMB*n* : {0, 1}*n* → {0, 1}, by OMB*n*(*x*) = 1 if max{*i* ∈ [*n*] : *x**i* = 0} is odd, and OMB*n*(*x*) = 0 otherwise. Define OMB*n*(1*n*) = 0.\n[Addressing][defn: addressing] For an integer *n* ≥ 2 that is a power of 2, define the *Addressing* function, denoted ADDR*n* : {0, 1}log*n* + *n* → {0, 1}, by \nADDR*n*(*x*, *y*) = *y*bin(*x*), \n where bin(*x*) denotes the integer in [*n*] whose binary representation is *x*. We refer to the *x*-variables as *addressing variables* and the *y*-variables as *target variables*.\nDecision Trees and Their Variants\n---------------------------------\nFor a partial Boolean function *f* : S → {0, 1}, the deterministic non-adaptive query complexity (alternatively the non-adaptive decision tree complexity) Ddt→(*f*) is the minimum integer *k* such that the following is true: there exist *k* indices *i*1, …, *i**k* ∈ [*n*], such that for every Boolean assignment *a**i*1, …, *a**i**k* to the input variables *x**i*1, …, *x**i**k*, *f* is constant on S ∩ {*x* ∈ {0, 1}*n* ∣ ∀*j* = 1, …, *k*, *x**i**j* = *a**i**j*}. Equivalently Ddt→(*f*) is the minimum number of variables such that *f* can be expressed as a function of these variables. It is easy to see that if *f* is a total function that depends on all input variables, then Ddt→(*f*) = *n*.\n[Non-adaptive parity decision tree complexity][defn: napdt] Define the *non-adaptive parity decision tree complexity* of *f* : {0, 1}*n* → {0, 1}, denoted by NAPDT(*f*), to be the minimum number of parities such that *f* can be expressed as a function of these parities. In other words, the non-adaptive parity decision tree complexity of *f* equals the minimal number *k* for which there exists S = {{*S*1, …, *S**k*} : *S**i* ⊆ [*n*] for all *i* ∈ [*k*]} such that the function value *f*(*x*) is determined by the values { ⊕ *j* ∈ *S**i**x**j* : *i* ∈ [*k*]} for all *x* ∈ {0, 1}*n*.\n[Non-adaptive AND decision tree complexity][defn: naandt] Define the *non-adaptive AND decision tree complexity* of *f* : {0, 1}*n* → {0, 1}, denoted by NAADT(*f*), to be the minimum number of monomials such that *f* can be expressed as a function of these monomials. In other words, the non-adaptive AND decision tree complexity of *f* equals the minimal number *k* for which there exists S = {{*S*1, …, *S**k*} : *S**i* ⊆ [*n*] for all *i* ∈ [*k*]} such that the function value *f*(*x*) is determined by the values {AND*S**i*(*x*) : *i* ∈ [*k*]} for all *x* ∈ {0, 1}*n*. We refer to such a set S as an *NAADT basis* for *f*.\n[Randomized non-adaptive AND decision tree complexity][defn: rnaandts] A randomized non-adaptive AND decision tree *T* computing *f* is a distribution over non-adaptive AND decision trees with the property that Pr[*T*(*x*) = *f*(*x*) ≥ 2/3] for all *x* ∈ {0, 1}*n*. The cost of *T* is the maximum cost of a non-adaptive AND decision tree in its support. Define the *randomized non-adaptive AND decision tree complexity* of *f* : {0, 1}*n* → {0, 1}, denoted by RNAADT(*f*), to be the minimum cost of a randomized non-adaptive AND decision tree that computes *f*.\nWe refer the reader to for the basics of quantum computing.\n[Quantum non-adaptive AND decision tree complexity][def: qnaandt] A quantum non-adaptive AND decision tree of cost *c* is a query algorithm that works with a state space ∣*S*1, …, *S**c*⟩∣*b*⟩∣*w*⟩, where each *S**j* ⊆ [*n*], *b* ∈ {0, 1}*c* and the last register captures a workspace of an arbitrary dimension. It is specified by a starting state ∣*ψ*⟩ and a projective measurement {Π, *I* − Π}. For an input *x* ∈ {0, 1}*n*, the action of the non-adaptive query oracle *O**x*⊗ *c* is captured by its action on the basis states, described below. \n*O**x*⊗ *c*∣*S*1, …, *S**c*⟩∣*b*1, …, *b**c*⟩∣*w*⟩ ↦ ∣*S*1, …, *S**c*⟩∣*b*1 ⊕ AND*S*1(*x*), …, *b**c* ⊕ AND*S**c*(*x*)⟩∣*w*⟩. \n We use *O**x* to refer to this oracle since *c* is already unambiguously determined by the state space. The algorithm accepts *x* with probability ∥Π*O**x*∣*ψ*⟩∥2.\nDefine the *quantum non-adaptive AND decision tree complexity* of *f* : {0, 1}*n* → {0, 1}, denoted by QNAADT(*f*), to be the minimum cost of a quantum non-adaptive AND decision tree that outputs the correct value of *f*(*x*) with probability at least 2/3 for all *x* ∈ {0, 1}*n*.\nThe quantum non-adaptive query complexity, denoted Qdt→, is defined similarly, the only difference being that the sets *S*1, …, *S**c* are restricted to be singletons. Montanaro observed that Qdt→(*f*) = Ω(*n*) for all total Boolean functions *f* : {0, 1}*n* → {0, 1} that depend on all input bits. Our proofs of Theorem [thm: quantum NAANDT OMB] and Claim [claim: addr qnaandt large] use ideas from their proof.\nWe require the following well-known fact about quantum algorithms.\n[claim: final states far] Let A be a two-outcome quantum algorithm computing a (possibly partial) Boolean function *f* : S → {0, 1} to error *ɛ*, and let {Π, *I* − Π} be the final measurement it performs. Let ∣*ψ**z*⟩ denote the final state of the algorithm on input *z* ∈ S before measurement. Then for any *x* ∈ *f*− 1(1) and *y* ∈ *f*− 1(0), \n$$\\|{|\\psi\\_x\\rangle} - {|\\psi\\_y\\rangle}\\|^2 \\geq 2 - 4\\sqrt{{\\varepsilon}(1 - {\\varepsilon})}.$$ \nWe provide a proof below for completeness.\nThe correctness of the algorithm guarantees that ∥Π∣*ψ**x*⟩∥2 ≥ 1 − *ɛ* and ∥Π∣*ψ**y*⟩∥2 ≤ *ɛ*. Thus we have\n \n$$\\begin{aligned}\n \\|{|\\psi\\_x\\rangle} - {|\\psi\\_y\\rangle}\\|^2 & = \\|\\Pi({|\\psi\\_x\\rangle} - {|\\psi\\_y\\rangle})\\|^2 + \\|(I - \\Pi)({|\\psi\\_x\\rangle} - {|\\psi\\_y\\rangle})\\|^2 \\tag\\*{since $\\Pi$ and $I - \\Pi$ are orthogonal projectors}\\\\\n & \\geq {\\left(\\|\\Pi{|\\psi\\_x\\rangle}\\| - \\|\\Pi{|\\psi\\_y\\rangle}\\|\\right)}^2 + {\\left(\\|(I - \\Pi){|\\psi\\_x\\rangle}\\| - \\|(I - \\Pi){|\\psi\\_y\\rangle}\\|\\right)}^2 \\tag\\*{by the triangle inequality}\\\\\n & \\geq 2(\\sqrt{1 - {\\varepsilon}} - \\sqrt{{\\varepsilon}})^2 \\tag\\*{by assumption}\\\\\n & = 2 - 4\\sqrt{{\\varepsilon}(1 - {\\varepsilon})}.\\end{aligned}$$ \nMöbius and Fourier Expansions of Boolean Functions\n--------------------------------------------------\nEvery Boolean function *f* : {0, 1}*n* → {0, 1} has a unique expansion as $f = \\sum\\_{S \\subseteq [n]}{\\widetilde}{f}(S) {\\mathsf{AND}}\\_S$, where AND*S* denotes the AND of the input variables in *S* and each ${\\widetilde}{f}(S)$ is a real number. We refer to the functions AND*S* as *monomials*, the expansion as the *Möbius expansion* of *f*, and the real coefficients ${\\widetilde}{f}(S)$ as the Möbius coefficients of *f*. It is known that the Möbius coefficients can be expressed as ${\\widetilde}{f}(S) = \\sum\\_{X \\subseteq S}(-1)^{|S \\setminus X|}f(X)$. Define the *Möbius support* of *f*, denoted S*f*, to be the set ${\\mathcal{S}}\\_f := {\\left\\{S \\subseteq [n] : {\\widetilde}{f}(S) \\neq 0\\right\\}}$. Define the *Möbius sparsity* of *f*, denoted spar(*f*), to be spar(*f*) : = |S*f*|.\nEvery Boolean function *f* : {0, 1}*n* → R can also be uniquely written as $f= \\sum\\limits\\_{S \\subseteq [n]}\\widehat{f}(S)(-1)^{\\oplus\\_{j \\in S}x\\_j}$. This representation is called the *Fourier expansion* of *f* and the real values *f̂*(*S*) are called the Fourier coefficients of *f*. The Fourier sparsity of *f* is defined to be number of non-zero Fourier coefficients of *f*. Sanyal showed the following relationship between non-adaptive parity decision complexity of a Boolean function and its Fourier sparsity.\n[][thm: san19] Let *f* : {0, 1}*n* → { − 1, 1} be a Boolean function with Fourier sparsity *r*. Then ${\\mathsf{NAPDT}}(f) = O(\\sqrt{r} \\log r)$.\nThis theorem is tight up to the logarithmic factor, witnessed by the Addressing function. We now note some simple observations relating the non-adaptive AND decision tree complexity of Boolean functions and their Möbius expansions.\n[claim: naandt basis monomial product] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function and let S = {*S*1, …, *S**k*} be a NAADT basis for *f*. Then, every monomial in the Möbius support of *f* equals ∏*i* ∈ *T*AND*S**i*, for some *T* ⊆ [*k*].\nSince S is an NAADT basis for *f*, the values of {AND*S**i* : *i* ∈ [*k*]} determine the value of *f*. That is, we can express *f* as \n$$f = \\sum\\_{T \\subseteq [k]}b\\_T \\prod\\_{i \\in T}{\\mathsf{AND}}\\_{S\\_i}\\prod\\_{j \\notin T}(1 - {\\mathsf{AND}}\\_{S\\_j}),$$ \n for some values of *b**T* ∈ {0, 1}. Expanding this expression only yields monomials that are products of AND*S**i*’s from S. The claim now follows since the Möbius expansion of a Boolean function is unique.\n[claim: trivial bounds] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function with Möbius sparsity *r*. Then log*r* ≤ NAADT(*f*) ≤ *r*.\nThe upper bound NAADT(*f*) ≤ *r* follows from the fact that knowing the values of all ANDs in the Möbius support of *f* immediately yields the value of *f* by plugging these values in the Möbius expansion of *f*. That is, the Möbius support of *f* acts as an NAADT basis for *f*.\nFor the lower bound, let NAADT(*f*) = *k*, and let S = {*S*1, …, *S**k*} be an NAADT basis for *f*. Claim [claim: naandt basis monomial product] implies that every monomial in the Möbius expansion of *f* is a product of some of these AND*S**i*’s. Thus, the Möbius sparsity of *f* is at most 2*k*, yielding the required lower bound.\nCommunication Complexity\n------------------------\nThe standard model of two-party communication complexity was introduced by Yao . In this model, there are two parties, say Alice and Bob, each with inputs *x*, *y* ∈ {0, 1}*n*. They wish to jointly compute a function *F*(*x*, *y*) of their inputs for some function *F* : U → {0, 1} that is known to them, where U is a subset of {0, 1}*n* × {0, 1}*n*. They use a communication protocol agreed upon in advance. The cost of the protocol is the number of bits exchanged in the worst case (over all inputs). Alice and Bob are required to output the correct answer for all inputs (*x*, *y*) ∈ U. The communication complexity of *F* is the best cost of a protocol that computes *F*, and we denote it by Dcc(*F*). See, for example, , for an introduction to communication complexity.\nIn a deterministic one-way communication protocol, Alice sends a message *m*(*x*) to Bob. Then Bob outputs a bit depending on *m*(*x*) and *y*. The complexity of the protocol is the maximum number of bits a message contains for any input *x* to Alice. In a randomized one-way protocol, the parties share some common random bits R. Alice’s message is a function of *x* and R. Bob’s output is a function of *m*(*x*), *y* and R. The protocol Π is said to compute *F* with error *ɛ* ∈ (0, 1/2) if for every (*x*, *y*) ∈ U, the probability over R of the event that Bob’s output equals *F*(*x*, *y*) is at least 1 − *ɛ*. The cost of the protocol is the maximum number of bits contained in Alice’s message for any *x* and R. In the one-way quantum model, Alice sends Bob a quantum message, after which Bob performs a projective measurement and outputs the measurement outcome. Depending on the model of interest, Alice and Bob may or may not share an arbitrary input-independent entangled state for free. We refer the reader to for an introduction to quantum communication complexity. As in the randomized setting, a protocol Π computes *F* with error *ɛ* if Pr[Π(*x*, *y*) ≠ *f*(*x*, *y*)] ≤ *ɛ* for all (*x*, *y*) ∈ U.\nThe deterministic (*ɛ*-error randomized, *ɛ*-error quantum, *ɛ*-error quantum with entanglement, respectively) one-way communication complexity of *F*, denoted by Dcc→( ⋅ ) (Rcc, *ɛ*→( ⋅ ), Qcc, *ɛ*→( ⋅ ), Qcc, *ɛ*\\* , →( ⋅ ), respectively), is the minimum cost of any deterministic (*ɛ*-error randomized, *ɛ*-error quantum, *ɛ*-error quantum with entanglement, respectively) one-way communication protocol for *F*.\nTotal functions *F* whose domain is {0, 1}*n* × {0, 1}*n* induce a communication matrix *M**F* whose rows and columns are indexed by strings in {0, 1}*n*, and the (*x*, *y*)’th entry equals *F*(*x*, *y*). It is known that\n \n$$\\label{eqn: logrank bounds}\n\\log{\\mathrm{rank}}(M\\_F) \\leq {{\\mathsf{D}}\\_{{\\mathrm{cc}}}}(F) \\leq O(\\sqrt{{\\mathrm{rank}}(M\\_F)} \\log{\\mathrm{rank}}(M\\_F)),$$ \nwhere rank( ⋅ ) denotes real rank. The first inequality is well known (see, for instance ), and the second inequality was shown by Lovett . One of the most famous conjectures in communication complexity is the log-rank conjecture, due to Lovász and Saks , that proposes that the communication complexity of any Boolean function is polylogarithmic in its rank, i.e. the first inequality in Equation is always tight up to a polynomial dependence.\nBuhrman and de Wolf observed that the Möbius sparsity of a Boolean function *f* equals the rank of the communication matrix of *f* ∘ AND. That is, for all Boolean functions *f* : {0, 1}*n* → {0, 1},\n \nspar(*f*) = rank(*M**f* ∘ AND). \nIn view of the first inequality in Equation , this yields\n \nDcc(*f* ∘ AND) ≥ log(spar(*f*)). \nWe require the definition of the Vapnik-Chervonenkis (VC) dimension .\n[VC-dimension][defn: VC] Consider a function *F* : {0, 1}*n* × {0, 1}*n* → {0, 1}. A subset of columns *C* of *M**F* is said to be *shattered* if all of the 2∣*C*∣ patterns of 0’s and 1’s are attained by some row of *M**F* when restricted to the columns *C*. The *VC-dimension* of a function *F* : {0, 1}*n* × {0, 1}*n*, denoted VC(*F*), is the maximum size of a shattered subset of columns of *M**F*.\nKlauck showed that the one-way quantum communication complexity of a function *F* is bounded below by the VC-dimension of *F*.\n[][thm: klauck] Let *F* : {0, 1}*n* × {0, 1}*n* → {0, 1} be a Boolean function. Then, Qcc, *ɛ*→(*F*) ≥ (1 − Hbin(*ɛ*))VC(*F*) and Qcc, *ɛ*\\* , →(*F*) ≥ (1 − Hbin(*ɛ*))VC(*F*)/2.\nComposition with Inner Product\n==============================\nIn this section we prove Theorem [thm:main] and Theorem [thm:detlift], which are our results regarding the quantum and deterministic one-way communication complexities, respectively, of functions composed with a small Inner Product gadget.\nQuantum Complexity\n------------------\n[Proof of Theorem [thm:main]] By Theorem [thm: klauck], it suffices to show that VC(*f* ∘ IP) ≥ *n*(*b* − 1). Since *f* is a function that depends on all its input variables, the following holds. For each index *i* ∈ [*n*], there exist inputs *z*(*i*, 0) = *z*1(*i*), …, *z**i* − 1(*i*), 0, *z**i* + 1(*i*), …, *z**n*(*i*) and *z*(*i*, 1) = *z*1(*i*), …, *z**i* − 1(*i*), 1, *z**i* + 1(*i*), …, *z**n*(*i*) such that *f*(*z*(*i*, 0)) = *v**i* and *f*(*z*(*i*, 1)) = 1 − *v**i*. That is, *z*(*i*, 0) and *z*(*i*, 1) have different function values, but differ only on the *i*’th bit.\nFor each *i* ∈ [*n*] and *j* ∈ {2, 3, …, *b*}, define a string *y*(*i*, *j*) ∈ {0, 1}*n**b* as follows. For all *k* ∈ [*n*] and ℓ ∈ [*b*], \n$$y^{(i, j)}\\_{k, \\ell} = \\begin{cases}\nz^{(i)}\\_k & \\text{if}~k \\neq i~\\text{and}~\\ell = 1\\\\\n1 & \\text{if}~k = i~\\text{and}~\\ell = j\\\\\n0 & \\text{otherwise}.\n\\end{cases}$$ \n That is, for *k* ≠ *i*, the *k*’th block of *y*(*i*, *j*) is (*z**k*(*i*), 0*b* − 1), and the *i*’th block of *y*(*i*, *j*) is (0*j* − 1, 1, 0*b* − *j*). Consider the set of *n*(*b* − 1)-many columns of *M**f* ∘ IP, one for each *y*(*i*, *j*). We now show that this set of columns is shattered. Consider an arbitrary string *c* = *c*1, 2, …, *c*1, *b*, …, *c**n*, 2, …, *c**n*, *b* ∈ {0, 1}*n*(*b* − 1). We now show the existence of a row that yields this string on restriction to the columns described above. Define a string *x* ∈ {0, 1}*n**b* as follows. For all *i* ∈ [*n*] and *j* ∈ [*b*], *x**i*, 1 = 1 and\n \n$$\\begin{aligned}\nx\\_{i, j} & = \\begin{cases}\nc\\_{i, j} & \\text{if}~v\\_i = 0\\\\\n1 - c\\_{i, j} & \\text{if}~v\\_i = 1.\n\\end{cases}\\end{aligned}$$ \nThat is, the first element of each block of *x* is 1, and the remaining part of any block, say the *i*’th block, equals either the string *c**i*, 2, …, *c**i*, *b* or its bitwise negation, depending on the value of *v**i*.\nTo complete the proof, we claim that the row of *M**f* ∘ IP corresponding to this string *x* equals the string *c* when restricted to the columns {*y*(*i*, *j*)}*i* ∈ [*n*], *j* ∈ {2, 3, …, *b*}. To see this, fix *i* ∈ [*n*] and *j* ∈ {2, 3, …, *b*} and consider *M**f* ∘ IP(*x*, *y*(*i*, *j*)). Next, for each *k* ∈ [*n*] with *k* ≠ *i*, the inner product of the *k*’th block of *x* with the *k*’th block of *y* equals *z**k*(*i*), since *x**k*, 1 = 1 and the first element of the *k*’th block of *y*(*i*, *j*) equals *z**k*(*i*), and all other elements of the block are 0 by definition. In the *i*’th block of *y*(*i*, *j*), only the *j*’th element is non-zero, and equals 1 by definition. Moreover, *x**i*, *j* = *c**i*, *j* if *v**i* = 0, and equals 1 − *c**i*, *j* otherwise. Hence, the inner products of the *i*’th blocks of *x* and *y*(*i*, *j*) equals *c**i*, *j* if *v**i* = 0, and equals 1 − *c**i*, *j* otherwise. Thus, the string obtained on taking the block-wise inner product of *x* and *y*(*i*, *j*) equals *z*1(*i*), …, *z**i* − 1(*i*), *c**i*, *j*, *z**i* + 1(*i*), …, *z**n*(*i*) if *v**i* = 0 and *z*1(*i*), …, *z**i* − 1(*i*), 1 − *c**i*, *j*, *z**i* + 1(*i*), …, *z**n*(*i*) if *v**i* = 1. By our definitions of *z*(*i*, 0), *z*(*i*, 1) and *v**i* for each *i* ∈ [*n*], it follows that the value of *f* when applied to either of these inputs equals *c**i*, *j*. This concludes the proof.\nDeterministic Complexity\n------------------------\nWe now prove Theorem [thm:detlift], which gives a lower bound on the deterministic one-way communication complexity of *f* ∘ IP for partial functions *f*. A crucial ingredient of our proof is Theorem [thm: packing], which we prove in Appendix [erdos]. Now we proceed to the proof of Theorem [thm:detlift].\n[Proof of Theorem [thm:detlift]] Let *q* : = 2*b* − 1 and let Π be an optimal one-way deterministic protocol for *f* ∘ IP of complexity Dcc→(*f* ∘ IP) = : *c*log*q*. The theorem is trivially true if *c* ≥ *n*/30 since Ddt→(*f*) ≤ *n*. In the remainder of the proof we assume that *c* < *n*/30. Π induces a partition of {0, 1}*n**b* into at most *q**c* parts; each part corresponds to a distinct message. There are (2*b* − 1)*n* = *q**n* inputs (*x*1, …, *x**n*) to Alice such that for each *i*, *x**i* ≠ 0*b*. Let Z be the set of those inputs. Identify Z with [*q*]*n*. By the pigeon-hole principle there exists one part P in the partition induced by Π that contains at least *q**n* − *c* strings in Z. We now invoke Theorem [thm: packing] with *d* set to 10*c*. This is applicable since *d* ≤ *n*/3 and the assumption *b* ≥ 2 implies that *q* ≥ 3. Theorem [thm: packing] implies that there are two strings *x*(1) = (*x*1(1), …, *x**n*(1)), *x*(2) = (*x*1(2), …, *x**n*(2)) ∈ P ∩ Z such that ∣{*i* ∈ [*n*] ∣ *x**i*(1) = *x**i*(2)}∣ < 10*c*. Let I : = {*i* ∈ [*n*] ∣ *x**i*(1) = *x**i*(2)}. Let *z* = (*z*1, …, *z**n*) denote a generic input to *f*. We claim that for each Boolean assignment (*a**i*)*i* ∈ I to the variables in I, *f* is constant on S ∩ {*z* : ∀*i* ∈ I, *z**i* = *a**i*}. This will prove the theorem, since querying the variables {*z**i* ∣ *i* ∈ I} determines *f*; thus Ddt→(*f*) ≤ ∣I∣ < 10*c*. Towards a contradiction, assume that there exist *z*(1), *z*(2) ∈ S ∩ {*z* : ∀*i* ∈ I, *z**i* = *a**i*} such that *f*(*z*(1)) ≠ *f*(*z*(2)). We will construct a string *y* = (*y*1, …, *y**n*) ∈ {0, 1}*n**b* in the following way:\n*i* ∈ I\n: Choose *y**i* such that IP(*y**i*, *x**i*(1)) = IP(*y**i*, *x**i*(2)) = *a**i*.\n$i \\notin {\\mathcal{I}}$\n: Choose *y**i* such that IP(*y**i*, *x**i*(1)) = *z**i*(1) and IP(*y**i*, *x**i*(2)) = *z**i*(2).\nNote that we can always choose a *y* as above since for each *i* ∈ [*n*], *x**i*(1), *x**i*(2) ≠ 0*b*, and for each $i \\notin {\\mathcal{I}}$, *x**i*(1) ≠ *x**i*(2). By the above construction, *f* ∘ IP(*x*(1), *y*) = *f*(*z*(1)) and *f* ∘ IP(*x*(2), *y*) = *f*(*z*(2)). Since by assumption *f*(*z*(1)) ≠ *f*(*z*(2)), we have that *f* ∘ IP(*x*(1), *y*) ≠ *f* ∘ IP(*x*(2), *y*). But since Alice sends the same message on inputs *x*(1) and *x*(2), Π produces the same output on (*x*(1), *y*) and (*x*(2), *y*). This contradicts the correctness of Π.\nIt can be seen that the proof of Theorem [thm:detlift] also works when the inner gadget *g* : {0, 1}*b*1 × {0, 1}*b*2 → {0, 1} satisfies the following general property: There exists a subset *X* of {0, 1}*b*1 (Alice’s input in the gadget) such that:\n* ∣*X*∣ ≥ 3,\n* for all *x*1 ≠ *x*2 ∈ *X* and all *b*1, *b*2 ∈ {0, 1}, there exists *y* ∈ {0, 1}*b*2 such that *g*(*x*1, *y*) = *b*1 and *g*(*x*2, *y*) = *b*2.\nThis is satisfied, for example, for the Addressing function on {0, 1}log*b* + *b* when *b* ≥ 4 (see Definition [defn: addressing]). For *g* = IP*b*, the set *X* equals {0, 1}*b* \\ {0*b*}.\nComposition with AND\n====================\nWe first investigate the relationship between non-adaptive AND decision tree complexity and Möbius sparsity of Boolean functions. Recall that Claim [claim: trivial bounds] shows that for all Boolean functions *f* : {0, 1}*n* → {0, 1}, logspar(*f*) ≤ NAADT(*f*) ≤ spar(*f*). A natural question to ask is whether both of the bounds are tight, i.e. are there Boolean functions witnessing tightness of each bound? The first bound is trivially tight for any Boolean function with full Möbius sparsity, for example, the NOR function: querying all the input bits (which is querying *n* many ANDs) immediately yields the value of the function, and its Möbius sparsity can be shown to be 2*n*. One might expect that the upper bound is not tight in view of Theorem [thm: san19]. The Addressing function witnesses tightness of the quadratic gap in Theorem [thm: san19]. This gives rise to the natural question of whether an analogous bound holds true in the Möbius-world: is it true for all Boolean functions *f* that ${\\mathsf{NAADT}}(f) = \\widetilde{O}(\\sqrt{{\\mathrm{spar}}(f)})$? Interestingly we show in Appendix [app: addressing] that the Addressing function already gives a negative answer to this question. In this section we first observe that a stronger separation holds, and there exists a function (OMB*n*) for which the second inequality in Claim [claim: trivial bounds] is in fact an equality. We then use this same function to prove Theorem [thm: intro omb separation], which gives a maximal separation between QNAADT(*f*) and Dcc→(*f* ∘ AND2). Finally, we prove Theorem [thm: intro symmetric naandt communication], which says that NAADT(*f*) is at most quadratically large in Dcc(*f* ∘ AND) for symmetric *f*.\nPattern Complexity and One-Way Communication Complexity\n-------------------------------------------------------\nIn this section we observe that the logarithm of the pattern complexity, PatM(*f*), of a Boolean function *f* equals the deterministic one-way communication complexity of *f* ∘ AND. We also give bounds on NAADT(*f*) in terms of PatM(*f*). As a consequence we also show that Dcc→(*f* ∘ AND) ≥ log(NAADT(*f*)).\n[claim: doneway equals log pattern] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function. Then Dcc→(*f* ∘ AND) = ⌈log(PatM(*f*))⌉.\nWrite the Möbius expansion of *f* as\n \n$$\\begin{aligned}\n\\label{eqn: mobius expansion of f}\n f = \\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S.\\end{aligned}$$ \nSay PatM(*f*) = *k*. We first show that Dcc→(*f* ∘ AND) ≤ ⌈log*k*⌉ by exhibiting a one-way protocol of cost ⌈log*k*⌉. Alice computes the pattern of *x* and sends Bob the pattern using ⌈log*k*⌉ bits of communication. Bob now knows the values of {AND*S*(*x*) : *S* ∈ S*f*}. Since Bob can compute {AND*S*(*y*) : *S* ∈ S*f*} without any communication, he can now compute the value of *f* ∘ AND(*x*, *y*) using the formula \n$$(f \\circ {\\mathsf{AND}})(x,y) = \\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x){\\mathsf{AND}}\\_S(y).$$ \nIt remains to show that Dcc→(*f* ∘ AND) ≥ ⌈log*k*⌉. Let Dcc→(*f* ∘ AND) = *d*. Thus there are at most 2*d* messages that Alice can send Bob. We show that any two inputs *x*, *x*ʹ ∈ {0, 1}*n* for which Alice sends the same message have the same pattern, which would prove 2*d* ≥ *k*, and prove the claim since *d* must be an integer.\nLet *x*, *x*ʹ be 2 inputs to Alice for which her message to Bob is *m*. We have\n \n$$\\begin{aligned}\n (f \\circ {\\mathsf{AND}})(x,y) & = \\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x){\\mathsf{AND}}\\_S(y)\\\\\n (f \\circ {\\mathsf{AND}})(x',y) & = \\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x'){\\mathsf{AND}}\\_S(y)\\end{aligned}$$ \nSince *m* and *y* completely determine the value of the function, we must have\n \n$$\\begin{aligned}\n\\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x){\\mathsf{AND}}\\_S(y) = \\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x'){\\mathsf{AND}}\\_S(y) \\qquad\\text{for all $y \\in {\\{0, 1\\}}^n$.}\\end{aligned}$$ \nDefine the functions *g**x*, *g**x*ʹ : {0, 1}*n* → {0, 1} by\n \n$$\\begin{aligned}\n g\\_x(y) & = \\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x){\\mathsf{AND}}\\_S(y)\\\\\n g\\_{x'}(y) & = \\sum\\_{S \\in {\\mathcal{S}}\\_f}{\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x'){\\mathsf{AND}}\\_S(y).\\end{aligned}$$ \nThus by uniqueness of the Möbius expansion of Boolean functions, *g**x* = *g**x*ʹ as functions of *y*. This implies ${\\widetilde}{g\\_x}(S) = {\\widetilde}{g\\_{x'}}(S)$ for all *S* ∈ S*f*. Since ${\\widetilde}{g\\_x}(S) = {\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x)$ and ${\\widetilde}{g\\_{x'}}(S) = {\\widetilde}{f}(S){\\mathsf{AND}}\\_S(x')$ for all *S* ∈ S*f*,\n \n$$\\begin{aligned}\n{\\mathsf{AND}}\\_S(x) = {\\mathsf{AND}}\\_S(x') \\qquad \\textnormal{for all}~S \\in {\\mathcal{S}}\\_f,\\end{aligned}$$ \nThis shows that the pattern induced by *x* and the pattern induced by *x*ʹ are the same, concluding the proof.\nNext we show that the pattern complexity of *f* is bounded below by the Möbius sparsity of *f*.\n[claim: patcomp at least spar] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function. Then PatM(*f*) ≥ spar(*f*).\nRecall that S*f* denotes the Möbius support of *f*. For each *S* ∈ S*f*, define the input *x**S* to be the *n*-bit characteristic vector of the set *S*. We now show that each of these inputs induces a different pattern for *f*. Let *S*1 ≠ *S*2 ∈ S*f*, with ∣*S*1∣ ≥ ∣*S*2∣. Since they are different sets, there must be an index *j* ∈ *S*1 such that $j \\notin S\\_2$. Note that AND*S*1(*x**S*1) = 1. On the other hand *x**j**S*2 = 0 implies AND*S*1(*x**S*2) = 0. Hence *x**S*1 and *x**S*2 induce different patterns. Since ${\\mathrm{spar}}(f) = \\abs{{\\mathcal{S}}\\_f}$, this completes the proof.\nFrom Claim [claim: trivial bounds] we know that spar(*f*) ≥ NAADT(*f*) and from Claim [claim: doneway equals log pattern] we know that Dcc→(*f* ∘ AND) = ⌈log(PatM(*f*))⌉. Along with Claim [claim: patcomp at least spar], these imply the following claim.\n[claim: doneway vs naandt] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function. Then ⌈log(NAADT(*f*))⌉ ≤ Dcc→(*f* ∘ AND) ≤ NAADT(*f*).\nFor the upper bound on Dcc→(*f* ∘ AND), let S = {*S*1, …, *S**k*} be an NAADT basis for *f*. By Claim [claim: naandt basis monomial product], every monomial in the Möbius support of *f* is a product of some of these AND*S**i*’s. Since there are at most 2*k* possible values for {AND*S**i*(*x*) : *i* ∈ [*k*]} and since these completely determine the pattern of *x* for any given *x* ∈ {0, 1}*n*, we have \nPatM(*f*) ≤ 2NAADT(*f*), \n which proves the required upper bound in view of Claim [claim: doneway equals log pattern].\nFor the lower bound, we have \nDcc→(*f* ∘ AND) = ⌈log(PatM(*f*))⌉ ≥ ⌈log(spar(*f*))⌉ ≥ ⌈log(NAADT(*f*))⌉, \n where the equality follows from Claim [claim: doneway equals log pattern], the first inequality follows from Claim [claim: patcomp at least spar] and the last inequality follows from Claim [claim: trivial bounds].\nThe pattern complexity of *f* is trivially at most 2spar(*f*) since each pattern is a spar(*f*)-bit string. Interestingly we show that there is no function for which this bound is tight.\n[claim: patcomp is never full] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function. Then PatM(*f*) ≤ 2(1 − Ω(1))spar(*f*).\nIn fact, we obtain an explicit constant that yields an upper bound of 2((log6)/3)spar(*f*) + 1 ≈ 20.86spar(*f*) + 1. Our proof of Claim [claim: patcomp is never full] relies on the following observation about the structure of the Möbius support of any Boolean function.\n[clm: mobius support structure] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function with Möbius support S*f*. For any two distinct sets *S*, *T* ∈ S*f* there exists a set of ‘partners’ *p*({*S*, *T*}) ⊆ S*f* such that\n* *p*({*S*, *T*}) ≠ {*S*, *T*},\n* $\\abs{p({\\left\\{S, T\\right\\}})} = 2$ if $S \\cup T \\notin {\\mathcal{S}}\\_f$ and $\\abs{p({\\left\\{S, T\\right\\}})} = 1$ if *S* ∪ *T* ∈ S*f*, and\n* ⋃*U* ∈ *p*({*S*, *T*})*U* = *S* ∪ *T*.\nLet $\\sum\\_{S \\in {\\mathcal{S}}\\_f} {\\widetilde}{f}(S) {\\mathsf{AND}}\\_{S}$ be the Möbius expansion of *f*. Since *f* has range {0, 1}, we know that *f* = *f*2. However, \n$$f^2 = \\left( \\sum\\_{S \\in {\\mathcal{S}}\\_f} {\\widetilde}{f}(S) {\\mathsf{AND}}\\_{S} \\right) \\left( \\sum\\_{T \\in {\\mathcal{S}}\\_f} {\\widetilde}{f}(T) {\\mathsf{AND}}\\_{T} \\right) = \\sum\\_{W \\subseteq [n]} \\left( \\sum\\_{S,T \\subseteq [n] : S \\cup T = W} {\\widetilde}{f}(S) {\\widetilde}{f}(T) \\right) {\\mathsf{AND}}\\_W.$$ \n Since the Möbius expansion of *f* is unique, we can compare the two expansions to see that for all sets *W* ⊆ [*n*],\n \n$$\\label{eqn: mobius titsworth}\n {\\widetilde}{f}(W) = \\sum\\_{S,T \\subseteq [n] : S \\cup T = W} {\\widetilde}{f}(S) {\\widetilde}{f}(T).$$ \nAs a consequence we have the following structure. Let *S* ≠ *T* ∈ S*f* such that $S \\cup T \\notin {\\mathcal{S}}\\_f$. Since ${\\widetilde}{f}(S \\cup T) = 0$, the summation corresponding to *W* = *S* ∪ *T* in Equation must have at least one non-zero summand apart from ${\\widetilde}{f}(S) {\\widetilde}{f}(T)$. Hence there must exist *U* ≠ *V* ∈ S*f* such that {*S*, *T*} ≠ {*U*, *V*} and *U* ∪ *V* = *S* ∪ *T*. We choose an arbitrary such pair {*U*, *V*} and define *p*({*S*, *T*}) = {*U*, *V*}. For *S*, *T* ∈ S*f* such that *S* ∪ *T* ∈ S*f*, let *p*({*S*, *T*}) be defined as {*S* ∪ *T*}. It clearly satisfies the necessary conditions.\n[obs: mobius partners patterns] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function with Möbius support S*f*. For any two distinct sets *S*, *T* ∈ S*f*, let *p*({*S*, *T*}) ⊆ S*f* be as in Claim [clm: mobius support structure]. Then for any pattern *P* ∈ {0, 1}S*f*, \n*P**S* ⋅ *P**T* = ∏*W* ∈ *p*({*S*, *T*})*P**W*. \nLet *P* be a pattern in {0, 1}S*f*. There must exist an *x* ∈ {0, 1}*n* such that for all sets *W* ∈ S*f*, *P**W* = AND*W*(*x*). Since *S* ∪ *T* = ⋃*W* ∈ *p*({*S*, *T*})*W*, we have *P**S* ⋅ *P**T* = AND*S* ∪ *T*(*x*) = ∏*W* ∈ *p*({*S*, *T*})*P**W*.\n[Proof of Claim [claim: patcomp is never full]] We analyze the pattern complexity of *f* in iterations. To define these iterations, we define a sequence of subsets of S*f*, described in Algorithm [algo: iteration generation].\n[H][algo: iteration generation] **Initialize** T0 ← ∅, *i* ← 0. \n Set num\\\\_iterations ← *i*. \nSet Tnum\\\\_iterations + 1 ← S*f*.\nFor *i* ∈ {0, …, num\\\\_iterations + 1}, define the partial patterns \n$$\\mathcal{P}\\_i := {\\left\\{P \\in {\\{0, 1\\}}^{\\mathcal{T}\\_i} : P = {\\left({\\mathsf{AND}}\\_S(x)\\right)}\\_{S \\in \\mathcal{T}\\_i}~\\textnormal{for some}~x \\in {\\{0, 1\\}}^n\\right\\}}.$$ \n We will now prove that\n \n$$\\label{eqn: partial patterns bound}\n \\forall j \\in {\\left\\{0,\\dots,\\mathsf{num\\\\_iterations}\\right\\}},~ \\abs{\\mathcal{P}\\_j} \\leq 2^{\\frac{\\log 6}{3}\\abs{\\mathcal{T}\\_j}}.$$ \nWe prove this by induction. In the remainder of this proof, let $\\alpha := \\frac{\\log 6}{3} \\approx 0.86$. Equation is true when *j* = 0 since both sides are 1. Now let *i* > 0 and assume as our induction hypothesis that Equation is true when *j* = *i* − 1. As our inductive step, we will prove that for every partial pattern *P* ∈ P*i* − 1, the number of partial patterns *Q* ∈ P*i* that extend *P* (in the sense that *Q* restricted to indices in T*i* − 1 is equal to *P*) is at most $2^{\\alpha(\\abs{\\mathcal{T}\\_i} - \\abs{\\mathcal{T}\\_{i-1}})}$. Since every partial pattern in P*i* is an extension of a partial pattern in P*i* − 1, this would imply that $\\abs{\\mathcal{P}\\_i} \\leq 2^{\\alpha(\\abs{\\mathcal{T}\\_i} - \\abs{\\mathcal{T}\\_{i-1}})}\\abs{\\mathcal{P}\\_{i-1}}$. Along with our induction hypothesis, this will prove Equation for *j* = *i*, and hence for all *j*.\nOur proof of the inductive step will involve some case analysis. Fix a partial pattern *P* ∈ T*i* − 1. Let *S*, *T* be the sets chosen when constructing T*i* from T*i* − 1. We know from Observation [obs: mobius partners patterns] that any partial pattern *Q* ∈ P*i* must satisfy *Q**S* ⋅ *Q**T* = ∏*W* ∈ *p*({*S*, *T*})*Q**W*. We split the analysis into cases based on whether or not *S* ∪ *T* ∈ S*f* and also based on the number of sets in T*i* \\ T*i* − 1.\nWe now show the inductive step, that is, for all *i* ∈ {1, …, num\\\\_iterations}, the number of extensions of any partial pattern *P* ∈ P*i* − 1 to a partial pattern *Q* in P*i* is at most $2^{\\alpha(\\abs{\\mathcal{T}\\_i} - \\abs{\\mathcal{T}\\_{i-1}})}$.\n* Suppose $S \\cup T \\notin {\\mathcal{S}}\\_f$. In this case we know $\\abs{p({\\left\\{S,T\\right\\}})} = 2$. Let *p*({*S*, *T*}) = {*U*, *V*}. Consider the set W : = T*i* \\ T*i* − 1 = ({*U*, *V*} ∪ {*S*, *T*}) \\ T*i* − 1.\n\t+ If $\\abs{\\mathcal{W}} = 2$: This can happen either because {*U*, *V*} ⊆ T*i* − 1 or because one of *U* and *V* is in T*i* − 1 and the other is in {*S*, *T*}. We analyze these subcases below.\n\t\n\t\n\t\t- If {*U*, *V*} ⊆ T*i* − 1, then *P* already determines *Q**U* and *Q**V*, and hence *Q**U* ⋅ *Q**V*. One of the following two cases hold. The third implication in both of the cases below use Observation [obs: mobius partners patterns].\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t (P\\_{U}, P\\_{V}) = (1, 1) & \\implies (Q\\_{U}, Q\\_{V}) = (1, 1) \\implies Q\\_{U} \\cdot Q\\_{V} = 1 \\\\\n\t\t & \\implies Q\\_{S} \\cdot Q\\_{T} = 1 \\implies (Q\\_{S}, Q\\_{T}) = (1, 1)\\\\\n\t\t & \\textnormal{or}\\\\\n\t\t (P\\_{U}, P\\_{V}) \\in {\\left\\{(0,0), (0,1), (1,0)\\right\\}} & \\implies (Q\\_{U}, Q\\_{V}) \\in {\\left\\{(0,0), (0,1), (1,0)\\right\\}} \\\\ & \\implies Q\\_{U} \\cdot Q\\_{V} = 0 \\implies Q\\_{S} \\cdot Q\\_{T} = 0 \\\\ \n\t\t & \\implies (Q\\_{S}, Q\\_{T}) \\in {\\left\\{(0,0), (0,1), (1,0)\\right\\}}.\n\t\t \\end{aligned}$$\n\t\t- One of *U* or *V* is in T*i* − 1 and the other is in {*S*, *T*}. Without loss of generality assume *U* ∈ T*i* − 1 (which implies *Q*(*U*) = *P*(*U*)) and *V* = *T*. Thus *S* ∪ *T* = *U* ∪ *T*, and Observation [obs: mobius partners patterns] implies *P**U* ⋅ *Q**T* = *Q**S* ⋅ *Q**T*. If *P**U* = 0 then\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t P\\_{U} \\cdot Q\\_{T} = 0 \\implies Q\\_{S} \\cdot Q\\_{T} = 0 \\implies (Q\\_{S}, Q\\_{T}) \\in {\\left\\{(0,0), (0,1), (1,0)\\right\\}}.\n\t\t \\end{aligned}$$ \n\t\t\n\t\t\n\t\t\n\t\tIf *P**U* = 1 then\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t Q\\_T = Q\\_S \\cdot Q\\_T \\implies (Q\\_{S}, Q\\_{T}) \\in {\\left\\{(0, 0), (1,0), (1,1)\\right\\}}.\n\t\t \\end{aligned}$$In both of the above subcases, *P* has at most 3 extensions in P*i*. Note that 3 < 22*α* ≈ 3.3, so our inductive step holds in this case.\n\t+ If $\\abs{\\mathcal{W}} = 3$: Without loss of generality, let W = {*S*, *T*, *U*}. This case can arise either because *V* ∈ T*i* − 1 or because *V* ∈ {*S*, *T*}. Again, we analyze both subcases.\n\t\n\t\n\t\t- If *V* ∈ T*i* − 1, then *Q**V* = *P**V* and thus Observation [obs: mobius partners patterns] implies *Q**U* ⋅ *P**V* = *Q**S* ⋅ *Q**T*. If *P**V* = 0 then\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t &Q\\_{U} \\cdot P\\_{V} = 0 \\implies Q\\_S \\cdot Q\\_T = 0\\\\\n\t\t \\implies &(Q\\_{S}, Q\\_{T}, Q\\_{U}) \\in {\\left\\{(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1)\\right\\}}.\n\t\t \\end{aligned}$$ \n\t\t\n\t\t\n\t\t\n\t\tIf *P**V* = 1 then\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t Q\\_{U} = Q\\_S \\cdot Q\\_T \\implies (Q\\_{S}, Q\\_{T}, Q\\_{U}) \\in {\\left\\{(0,0,0), (0,1,0), (1,0,0), (1,1,1)\\right\\}}.\n\t\t \\end{aligned}$$\n\t\t- If *V* ∈ {*S*, *T*} (without loss of generality assume *V* = *T*), then Observation [obs: mobius partners patterns] implies that\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t &Q\\_{U} \\cdot Q\\_T = Q\\_S \\cdot Q\\_T \\\\ \\implies &(Q\\_{S}, Q\\_{T}, Q\\_{U}) \\in {\\left\\{(0,0,0), (0,0,1), (0,1,0), (1,0,0), (1,0,1), (1,1,1)\\right\\}}.\n\t\t \\end{aligned}$$In both of these subcases, *P* had at most 6 extensions in P*i*. Since 23*α* = 6, the inductive step holds in this case.\n\t+ If $\\abs{\\mathcal{W}} = 4$, Observation [obs: mobius partners patterns] implies that\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\t &Q\\_{U} \\cdot Q\\_{V} = Q\\_{S} \\cdot Q\\_{T} \\\\ \\implies & (Q\\_{S},Q\\_{T},Q\\_{U},Q\\_{V}) \\in {\\left({\\left\\{(0,0),(0,1),(1,0)\\right\\}} \\times {\\left\\{(0,0),(0,1),(1,0)\\right\\}}\\right)} \\cup {\\left\\{(1,1,1,1)\\right\\}}.\n\t \\end{aligned}$$ \n\t\n\t\n\t\n\tThus there are 10 possibilities for (*Q**S*, *Q**T*, *Q**U*, *Q**V*), so *P* has at most 10 extensions in P*i*. Since 10 < 24*α* ≈ 10.9, the induction step holds in this case as well.\n* Suppose *S* ∪ *T* ∈ S*f*. In this case *p*({*S*, *T*}) = {*S* ∪ *T*}. Again, consider the set W : = T*i* \\ T*i* − 1 = ({*S* ∪ *T*} ∪ {*S*, *T*}) \\ T*i* − 1.\n\t+ If $\\abs{\\mathcal{W}} = 2$: This can arise either because *S* ∪ *T* ∈ T*i* − 1 or because *S* ∪ *T* ∈ {*S*, *T*}. We analyze both of these subcases below.\n\t\n\t\n\t\t- If *S* ∪ *T* ∈ T*i* − 1, Observation [obs: mobius partners patterns] implies that *P**S* ∪ *T* = *Q**S* ⋅ *Q**T*. If *P**S* ∪ *T* = 1, then\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t Q\\_S \\cdot Q\\_T = 1 \\implies (Q\\_S,Q\\_T) = (1,1).\n\t\t \\end{aligned}$$ \n\t\t\n\t\t\n\t\t\n\t\tIf *P**S* ∪ *T* = 0, then\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t Q\\_S \\cdot Q\\_T = 0 \\implies (Q\\_S,Q\\_T) \\in {\\left\\{(0,0),(0,1),(1,0)\\right\\}}.\n\t\t \\end{aligned}$$\n\t\t- If *S* ∪ *T* ∈ {*S*, *T*} (without loss of generality assume *S* ∪ *T* = *S*), then Observation [obs: mobius partners patterns] implies that *Q**S* = *Q**S* ⋅ *Q**T*. Therefore\n\t\t\n\t\t\n\t\t \n\t\t$$\\begin{aligned}\n\t\t (Q\\_S,Q\\_T) \\in {\\left\\{(0,0),(0,1),(1,1)\\right\\}}.\n\t\t \\end{aligned}$$In both of the above subcases, *P* has at most 3 extensions in P*i*. Since 3 < 22*α* ≈ 3.3, the inductive step holds in this case.\n\t+ If $\\abs{\\mathcal{W}} = 3$, Observation [obs: mobius partners patterns] implies\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\t Q\\_{S \\cup T} = Q\\_S \\cdot Q\\_T \\implies (Q\\_S,Q\\_T,Q\\_{S \\cup T}) \\in {\\left\\{(0,0,0),(0,1,0),(1,0,0),(1,1,1)\\right\\}}.\n\t \\end{aligned}$$ \n\t\n\t\n\t\n\tIn this case *P* has at most 4 extensions in P*i*. Since 4 < 23*α* = 6, the inductive step holds here as well.\nSince the inductive step holds in all the cases, we conclude that Equation is true. Finally, note that the while loop in Algorithm [algo: iteration generation] quits when $\\abs{\\mathcal{T}\\_i} \\geq {\\mathrm{spar}}(f)-1$. If it quits with $\\abs{\\mathcal{T}\\_i} = {\\mathrm{spar}}(f)$, then Equation implies that PatM(*f*) ≤ 2*α* ⋅ spar(*f*). If it quits with $\\abs{\\mathcal{T}\\_i} = {\\mathrm{spar}}(f)-1$, then each of the partial patterns in P*i* can have at most two extensions to actual patterns of *f*. Hence even in this case PatM(*f*) ≤ 2*α* ⋅ spar(*f*) + 1.\nThe analogous relation to Claim [claim: patcomp is never full] in the Fourier-world has been nearly determined by Sanyal ; their main result (Theorem [thm: san19]) essentially shows that the Fourier analog of pattern complexity of a Boolean function is at most exponential in the square root of its Fourier sparsity. This is a stronger bound than that in Claim [claim: patcomp is never full], but the same bound cannot hold in the Möbius-world since the Addressing function witnesses PatM(ADDR*n*) ≥ 2spar(ADDR*n*)log32 (see Claims [claim: addr sparsity] and [claim: addr patcomp lower bound]). Nevertheless we conjecture that a stronger bound is possible.\n[conj: patcomp ub] Let *f* : {0, 1}*n* → {0, 1} be a Boolean function. Then PatM(*f*) ≤ 2(spar(*f*)1 − Ω(1)).\nConjecture [conj: patcomp ub] would strengthen Theorem [thm: 1way sublinear rank], showing that Dcc→(*f* ∘ AND) = rank(*M**f* ∘ AND)1 − Ω(1).\n[Proof of Theorem [thm: 1way sublinear rank]] We have \nDcc→(*f* ∘ AND) = ⌈log(PatM(*f*))⌉ ≤ (1 − Ω(1))spar(*f*) ≤ (1 − Ω(1))rank(*M**f* ∘ AND), \n where the equality holds by Claim [claim: doneway equals log pattern], the first inequality follows from Claim [claim: patcomp is never full] and the last inequality holds by Equation .\nOur results regarding the one-way communication complexity of *f* ∘ AND use the Booleanness of *f* to bring out mathematical insights about the dependencies of monomials in the Möbius support of *f*. These dependencies enable us to establish interesting bounds on the pattern complexity of *f*. We hope that pattern complexity will find more use in future research.\nDeterministic Complexity\n------------------------\nWe prove in this section that the non-adaptive AND decision tree complexity of OMB*n* is maximal whereas the one-way communication complexity of OMB*n* ∘ AND is small.\n[claim: max naandt OMB] Let *n* be a positive integer. Then NAADT(OMB*n*) = *n*. Moreover, spar(OMB*n*) = *n* if *n* is even, and spar(OMB*n*) = *n* + 1 if *n* is odd.\n[Proof of Claim [claim: max naandt OMB]] Write the polynomial representation of OMB*n* as OMB*n*(*x*) = \n \n$$\\begin{aligned}\n\\label{eqn: OMB and expansion}\n(1-x\\_n)\\cdot 0 + x\\_n(1-x\\_{n-1}) \\cdot 1 + x\\_nx\\_{n-1}{\\mathsf{OMB}}\\_{n-2}(x\\_1, \\dots, x\\_{n-2}) & \\quad \\text{if $n$ is even, or}\\\\\n(1-x\\_n)\\cdot 1 + x\\_n(1-x\\_{n-1}) \\cdot 0 + x\\_nx\\_{n-1}{\\mathsf{OMB}}\\_{n-2}(x\\_1, \\dots, x\\_{n-2}) & \\quad \\text{if $n$ is odd}.\\end{aligned}$$ \nThe Möbius support of OMB*n* equals {{*j*, …, *n*} : *j* ≤ *n*} ∪ {∅} if *n* is odd, and {{*j*, …, *n*} : *j* ≤ *n*} if *n* is even. Thus spar(OMB*n*) = *n* + 1 if *n* is odd, and equals *n* if *n* is even.\nWe now show that the NAADT(OMB*n*) = *n*. Let S denote a NAADT basis for OMB*n*. By Claim [claim: naandt basis monomial product], any monomial in the Möbius expansion of OMB*n* can be expressed as a product of some ANDs from S. Thus, {*n*} must participate in S since it appears in its Möbius support. Next, since {*n* − 1, *n*} appears in the support as well, either {*n* − 1, *n*} or {*n* − 1} must appear in S. Continuing iteratively, we conclude that for all *i* ∈ [*n*], there must exist a set in S that contains *i*, but does not contain any *j* for *j* < *i*. This implies that ∣S∣ ≥ *n*. Equality holds since NAADT(*f*) ≤ *n* for any Boolean function *f* : {0, 1}*n* → {0, 1}.\nThus OMB*n* witnesses that non-adaptive AND decision tree complexity can be as large as sparsity. We remark here that OMB*n* admits a simple (adaptive) AND-decision tree that makes *O*(log*n*) AND-queries in the worst case. This uses a binary search using AND-queries to determine the right-most index where a 0 is present. One might expect that a result similar to Claim [claim: foxor oneway equals napdt] holds when the inner function is AND instead of XOR. That is, it is plausible that the deterministic one-way communication complexity of *f* ∘ AND equals the non-adaptive AND decision tree complexity of *f*. We show that this is not true, and exhibit an exponential separation between Dcc→(OMB*n* ∘ AND) and NAADT(OMB*n*).\n[claim: doneway omb] Let *n* be a positive integer. Then Dcc→(OMB*n* ∘ AND) = ⌈log(*n* + 1)⌉.\nBy expanding Equation we have that the Möbius support of OMB*n* equals the set S = {{*n*}, {*n* − 1, *n*}, …, {*n*, *n* − 1, …, 1}} if *n* is an even integer, and equals the set S = {∅, {*n*}, {*n* − 1, *n*}, …, {*n*, *n* − 1, …, 1}} if *n* is an odd integer. It is easy to verify that the only possible Möbius patterns attainable (ignoring the empty set since it always evaluates to 1) are 1*i*0*n* − *i*, for *i* ∈ {0, 1, …, *n*}. Moreover, all of these patterns are attainable: the pattern 1*i*0*n* − *i* is attained by the input string 0*n* − *i*1*i*. Thus PatM(OMB*n*) = *n* + 1. Claim [claim: doneway equals log pattern] implies Dcc→(OMB*n* ∘ AND) = ⌈log(*n* + 1)⌉.\nWe obtain our main result of this section, which follows from Claim [claim: max naandt OMB] and Claim [claim: doneway omb].\n[thm: omb and exponential separation] Let *n* be a positive integer. Then NAADT(OMB*n*) = *n* and Dcc→(OMB*n* ∘ AND) = ⌈log(*n* + 1)⌉.\nQuantum Complexity\n------------------\nWe prove that even the quantum non-adaptive AND decision tree complexity of OMB*n* is Ω(*n*). In view of the small one-way communication complexity of OMB*n* ∘ AND from Claim [claim: doneway omb], Theorem [thm: intro omb separation] then follows.\n[thm: quantum NAANDT OMB] Let *n* be a positive integer. Then QNAADT(OMB*n*) = Ω(*n*).\nThe fact that the non-adaptive AND decision tree complexity (in fact even the non-adaptive *monotone* decision tree complexity, where the tree is allowed to make arbitrary monotone queries rather than just ANDs) of OMB*n* equals Ω(*n*) already follows by a recent result of Amireddy, Jayasurya and Sarma . They show that any function with alternating number *k* (which is the largest number of switches of the function’s value along a monotone path from 0*n* to 1*n*) must have non-adaptive monotone decision tree complexity at least *k*. In Theorem [thm: quantum NAANDT OMB], we show that even the *quantum* non-adaptive AND decision tree complexity of OMB*n* is Ω(*n*). In fact our proof can easily be adapted to show that any function with alternating number *k* must have quantum non-adaptive monotone decision tree complexity Ω(*k*), thus generalizing the above-mentioned result .\nBefore we prove this theorem, we introduce an auxiliary function and state some properties of it that are of use to us.\n[defn: omb’] Let *n* be a positive integer. Let us define the set S ⊂ {0, 1}*n* to be S = {*x* ∈ {0, 1}*n* : *x* = 0*i*1*n* − *i* for some *i* ∈ [*n*]}. Define the partial function OMBʹ*n* : S → {0, 1} by OMBʹ*n*(*x*) = OMB*n*(*x*).\n[claim: omb’ reduction] Let *n* be a positive integer. Then RNAADT(OMBʹ*n*) = Rdt→(OMBʹ*n*) and QNAADT(OMBʹ*n*) = Qdt→(OMBʹ*n*).\nThe inequalities RNAADT(OMBʹ*n*) ≤ Rdt→(OMBʹ*n*) and QNAADT(OMBʹ*n*) ≤ Qdt→(OMBʹ*n*) are clear by definitions. In the other direction, consider a (randomized or quantum) non-adaptive AND decision tree computing OMBʹ*n*. Let *S* ⊆ [*n*] be a subset of variables whose AND is queried (possibly as part of a superposition) in this tree. Let *S* = {*i*1, …, *i**k*}, where *i*1 < *i*2 < ⋯ < *i**k*. Observe that AND*S*(*x*) = *x**i*1 for all *x* in the domain of OMBʹ*n*. Thus the query to AND*S* can be replaced by a query to the *i*1’th variable. Hence any (randomized or quantum) non-adaptive AND decision tree can be converted to a (randomized or quantum) non-adaptive decision tree without changing its correctness or complexity. Thus RNAADT(OMBʹ*n*) ≥ Rdt→(OMBʹ*n*) and QNAADT(OMBʹ*n*) ≥ Qdt→(OMBʹ*n*), proving the claim.\nWe require the following result, which follows implicitly from a result of Montanaro .\n[thm: quantum dt lower bound] Let S ⊆ {0, 1}*n*, *I* ⊆ [*n*] and *f* : S → {0, 1} be such that for all *i* ∈ *I* there exists *x* ∈ S such that *f*(*x* ⊕ *e**i*) = 1 − *f*(*x*). Then ${{\\mathsf{Q}}\\_{{\\mathrm{dt}}}^\\rightarrow}(f) = \\Omega(\\abs{I})$.\nWe provide a proof below for completeness.\nConsider a *T*-query non-adaptive quantum query algorithm that computes *f* to error at most *ɛ* = 1/3. The algorithm works with a state space ∣*a*1, …, *a**T*⟩∣*b*⟩∣*w*⟩, where each *a**j* ∈ [*n*], *b* ∈ {0, 1}*T* and the last register captures the workspace. The algorithm does the following on input *x* ∈ *S*:\n* It starts in an input-independent state, say ∑*a*1, …, *a**T*, *b*, *w**α**a*1, …, *a**T*, *b*, *w*∣*a*1, …, *a**T*⟩∣*b*⟩∣*w*⟩,\n* applies *T* non-adaptive queries, that act on the basis states as follows: \n*O**x* : ∣*a*1, …, *a**T*⟩∣*b*1, …, *b**T*⟩∣*w*⟩ ↦ ∣*a*1, …, *a**T*⟩∣*b*1 ⊕ *x**a*1, …, *b**T* ⊕ *x**a**T*⟩∣*w*⟩.\n* It then applies a two-outcome projective measurement {Π, *I* − Π} on the resulting state *O**x*∣*ψ*⟩ and outputs a value depending on the measurement outcome.\nLet ∣*ψ**x*⟩ denote the state of the algorithm on input *x* ∈ {0, 1}*n* just before measurement.\nFix *i* ∈ *I*. By assumption, there exists *x* ∈ {0, 1}*n* such that *f*(*x*) = 1 − *f*(*x* ⊕ *e**i*). We define \n*J**i* : = {(*a*1, …, *a**T*, *b*, *w*) ∣ *i* ∈ {*a*1, …, *a**T*}} \n to be (the indices of) those basis states that query *x**i*. Write the initial state of our algorithm as ∣*ψ*⟩ = ∣*ϕ*1*i*⟩ + ∣*ϕ*2*i*⟩, where ∣*ϕ*1*i*⟩ = ∑*s* ∈ *J**i**α**s*∣*s*⟩. Thus,\n \n$$\\begin{aligned}\n 0.114 \\approx 2 - 4\\sqrt{2}/3 & \\leq \\|O\\_x {|\\psi\\rangle} - O\\_{x \\oplus e\\_i} {|\\psi\\rangle}\\|^2 \\tag\\*{by Claim~\\ref{claim: final states far}}\\\\\n & = \\| O\\_x {|\\phi\\_{1i}\\rangle} + O\\_x {|\\phi\\_{2i}\\rangle} - O\\_{x \\oplus e\\_i} {|\\phi\\_{1i}\\rangle} - O\\_{x \\oplus e\\_i} {|\\phi\\_{2i}\\rangle} \\|^2\\\\\n & = \\| O\\_x {|\\phi\\_{1i}\\rangle} - O\\_{x \\oplus e\\_i} {|\\phi\\_{1i}\\rangle} \\|^2 \\tag\\*{since $O\\_{x}$ and $O\\_{x \\oplus e\\_i}$ have the same action on ${|\\phi\\_{2i}\\rangle}$}\\\\\n & \\leq (\\|O\\_x {|\\phi\\_{1i}\\rangle}\\| + \\|O\\_{x \\oplus e\\_i} {|\\phi\\_{1i}\\rangle}\\|)^2 \\tag\\*{by the triangle inequality}\\\\\n & \\leq \\left(2\\cdot\\sqrt{\\sum\\_{(a\\_1, \\dots, a\\_T, b, w) \\in J\\_i}\\abs{\\alpha\\_{a\\_1, \\dots, a\\_T, b, w}}^2}\\right)^2.\\end{aligned}$$ \nSumming over all *i* ∈ *I*, this implies\n \n$$\\begin{aligned}\n \\Omega(\\abs{I}) \\leq \\sum\\_{i \\in [n]}\\sum\\_{(a\\_1, \\dots, a\\_T,b,w) \\in J\\_i} \\abs{\\alpha\\_{a\\_1, \\dots, a\\_T, b, w}}^2 \\leq T \\sum\\_{a\\_1, \\dots, a\\_T,b,w} \\abs{\\alpha\\_{a\\_1, \\dots, a\\_T, b, w}}^2 \\leq T,\\end{aligned}$$ \nwhere the second inequality follows because each $\\abs{\\alpha\\_{a\\_1, \\dots, a\\_T, b, w}}^2$ can appear in the summation for at most *T* values of *i*, and the last inequality follows since the sum of squares of the moduli of the amplitudes must equal 1. The theorem now follows.\n[Proof of Theorem [thm: quantum NAANDT OMB]] Clearly QNAADT(OMB*n*) ≥ QNAADT(OMBʹ*n*). Claim [claim: omb’ reduction] implies that QNAADT(OMBʹ*n*) = Qdt→(OMBʹ*n*). Recall that the domain of OMBʹ*n* consists only of the inputs {*x* ∈ {0, 1}*n* : *x* = 0*i*1*n* − *i* for some *i* ∈ [*n*]}. By definition, OMBʹ*n*(0*i*1*n* − *i*) ≠ OMBʹ*n*(0*i* − 11*n* − *i* + 1) for all *i* ∈ [*n*]. Thus Theorem [thm: quantum dt lower bound] is applicable with *I* = [*n*] and *f* = OMBʹ*n*. Combining the above, we have QNAADT(OMB*n*) ≥ QNAADT(OMBʹ*n*) = Qdt→(OMBʹ*n*) = Ω(*n*).\n[Proof of Theorem [thm: intro omb separation]] It follows from Claim [claim: doneway omb] and Theorem [thm: quantum NAANDT OMB].\nSymmetric Functions\n-------------------\nIn this section we show that symmetric functions *f* admit efficient non-adaptive AND decision trees in terms of the deterministic (even two-way) communication complexity of *f* ∘ AND. We require the following bounds on the Möbius sparsity of symmetric functions, due to Buhrman and de Wolf . For a non-constant symmetric function *f* : {0, 1}*n* → {0, 1}, define the following measure which captures the smallest Hamming weight inputs before which *f* is not a constant: ${\\mathsf{switch}}(f) := \\min{\\left\\{k : f ~\\textnormal{is a constant on all}~x~\\textnormal{such that}~|x| < n-k\\right\\}}$.\n[][claim: symmetric sparsity] Let *n* be sufficiently large, let *f* : {0, 1}*n* → {0, 1} be a symmetric Boolean function, and let *k* : = switch(*f*). Then $\\log {\\mathrm{spar}}(f) \\geq \\frac{1}{2}\\log\\left(\\sum\\_{i = n - k}^n \\binom{n}{i}\\right)$.\nUpper bounds on the non-adaptive AND decision tree complexity of symmetric functions follow from known results in the non-adaptive group testing literature. To the best of our knowledge, the following upper bounds were first shown (formulated differently) by Dyachkov and Rykov . Also see and the references therein.\n[thm: upper bound naandt in terms of switch symmetric] Let *f* : {0, 1}*n* → {0, 1} be a symmetric Boolean function with switch(*f*) = *k* < *n*/2. Then ${\\mathsf{NAADT}}(f) = O\\left(\\log^2 \\binom{n}{k}\\right)$.\nWe give a self-contained proof of Theorem [thm: upper bound naandt in terms of switch symmetric] for clarity and completeness. The proof is via the probabilistic method. We construct a random family of $O\\left(\\log^2 \\binom{n}{k}\\right)$ many ANDs and argue that with non-zero probability, their evaluations on any input determine the function’s value.\nWe require the following intermediate claim.\n[claim: intermediate probabilistic claim] Let *n* be a positive integer, and let 1 ≤ *k* < *n*/2 be an integer. Then, there exists a collection X of $O\\left(\\log^2 \\binom{n}{k}\\right)$ many subsets of [*n*] satisfying the following.\n \n$$\\label{eqn: properties of X}\n\\forall i\\_1, \\dots, i\\_{k+1} \\in [n], j \\in [k+1], \\exists X \\in {\\mathcal{X}}~\\text{such that}~ i\\_j \\in X, i\\_{\\ell} \\notin X~\\text{for all}~\\ell \\neq j.$$ \nConsider a random set *X* ⊆ [*n*] chosen as follows: For each index *i* ∈ [*n*] independently, include *i* in *X* with probability 1/(2*k*). Pick *w* many sets (where *w* is a parameter that we fix later) independently using the above sampling process, giving the multiset of sets X = {*X*1, …, *X**w*}.\nFor fixed *i*1, …, *i**k* + 1 ∈ [*n*], *j* ∈ [*k* + 1] and *t* ∈ [*w*],\n \n$$\\label{eqn: single k+1tuple prob bound}\n\\Pr\\_{X\\_t}[i\\_{j} \\in X\\_t ~\\text{and}~i\\_{\\ell} \\notin X\\_t ~\\text{for all}~{\\ell} \\neq j] = \\frac{1}{2k} \\cdot {\\left(1 - \\frac{1}{2k}\\right)}^k \\geq \\frac{1}{2k\\cdot e},$$ \nwhere the last inequality uses the fact that *k* ≥ 1 and the standard inequality that 1 − *x* ≥ *e*− 2*x* for all 0 ≤ *x* ≤ 1/2. Thus Equation implies that for fixed *i*1, …, *i**k* + 1 ∈ [*n*] and *j* ∈ [*k* + 1],\n \n$$\\Pr\\_{{\\mathcal{X}}}[\\nexists X \\in {\\mathcal{X}}: i\\_{j} \\in X ~\\text{and}~i\\_{\\ell} \\notin X ~\\text{for all}~{\\ell} \\neq j] \\leq {\\left(1 - \\frac{1}{2k \\cdot e}\\right)}^w \\leq \\exp(-w/(2ke)).$$ \nBy a union bound over these “bad events” for all *i*1, …, *i**k* + 1 ∈ [*n*] and *j* ∈ [*k* + 1], we conclude that\n \n$$\\begin{aligned}\n& \\Pr\\_{{\\mathcal{X}}}[\\forall i\\_1, \\dots, i\\_{k+1} \\in [n] ~\\text{and}~ j \\in [k+1],~\\exists X \\in {\\mathcal{X}}: i\\_{j} \\in X ~\\text{and}~i\\_{\\ell} \\notin X ~\\text{for all}~{\\ell} \\neq j]\\nonumber\\\\ \n& \\geq 1 - \\binom{n}{k+1} \\cdot (k+1) \\cdot \\exp(-w/(2ke)). \\label{eqn: union bound prob RHS}\\end{aligned}$$ \nWe want to choose *w* such that this probability is greater than 0. Thus we require\n \n$$\\begin{aligned}\n1 & > \\binom{n}{k+1} \\cdot (k+1) \\cdot \\exp(-w/(2ke))\\\\\n\\iff \\exp(w/(2ke)) & > (k+1) \\cdot \\binom{n}{k + 1}\\\\\n\\iff w & > 2ke{\\left(\\log (k+1) + \\log\\binom{n}{k+1}\\right)}.\\end{aligned}$$ \nSince $\\binom{n}{j+1} \\geq n > j+1$ for all *j* ∈ {1, 2, 3, …, *n*/2} and *n* > 2, and since $\\log\\binom{n}{j} \\geq j\\log(n/j) \\geq j$ for all *j* ∈ {1, 2, …, *n*/2}, it suffices to choose\n \n$$\\label{eqn: w choice}\nw \\geq 2e\\log\\binom{n}{k}{\\left(2\\log \\binom{n}{k+1}\\right)}.$$ \nBy standard binomial inequalities we have $\\log\\binom{n}{k+1} \\leq (k+1)\\log(ne/(k+1))$, and $\\log\\binom{n}{k} > k\\log(n/k)$. Next, since *k* + 1 ≤ 2*k* for *k* ≥ 1 and *n**e*/(*k* + 1) < *n*3/*k*3 for *k* ∈ {1, 2, …, *n*/2}, Equation implies that it suffices to choose \n$$w \\geq 2e\\log\\binom{n}{k}{\\left(12\\log\\binom{n}{k}\\right)}.$$ \n For this choice of *w*, the RHS of Equation is strictly positive. This proves the claim.\n[Proof of Theorem [thm: upper bound naandt in terms of switch symmetric]] Let *f* be a symmetric function with switch(*f*) = *k* < *n*/2, and let X be as in Claim [claim: intermediate probabilistic claim] with $|{\\mathcal{X}}| = O{\\left(\\log^2\\binom{n}{k}\\right)}$. We now show how X yields a NAADT for *f*. Without loss of generality assume that *f*(*x*) = 0 for all ∣*x*∣ < *n* − *k* (if not, output 1 in place of 0 in the **Output** step of Algorithm [algo: thn-k naandt] below).\n[H][algo: thn-k naandt] **Input:** *x* ∈ {0, 1}*n* \n1. Let X be as obtained from Claim [claim: intermediate probabilistic claim].\n2. Query {AND*X*(*x*) : *X* ∈ X} to obtain a string *P**x* ∈ {0, 1}∣X∣.\n**Output:** *f*(*y*) if *P**x* = *P**y* for some *y* with ∣*y*∣ ≥ *n* − *k*, and 0 otherwise.\nWe show below that the following holds: *P**x* ≠ *P**y* for all *x* ≠ *y* ∈ {0, 1}*n* such that ∣*y*∣ ≥ *n* − *k*. This would show correctness of the algorithm as follows:\n* If *P**x* = *P**y* for some ∣*y*∣ ≥ *n* − *k*, then *x* must equal *y* by the above. In this case we output the correct value since we have learned *x*.\n* If *P**x* ≠ *P**y* for any ∣*y*∣ ≥ *n* − *k*, then ∣*x*∣ < *n* − *k*. Since *f* evaluates to 0 on all such inputs, we output the correct value in this case.\nLet *x* ≠ *y* ∈ {0, 1}*n* be two strings such that ∣*y*∣ ≥ *n* − *k*. Without loss of generality assume ∣*y*∣ ≥ ∣*x*∣ (else swap the roles of *x* and *y* above). Let *I**x*, *I**y* ⊆ [*n*] denote the sets of indices where *x* and *y* take value 0, respectively. By assumption, *x* ≠ *y* and ∣*I**x*∣ ≥ ∣*I**y*∣. Thus there exists an index *i**x* ∈ *I**x* \\ *I**y*.\nSince ∣*I**y*∣ ≤ *k*, by Claim [claim: intermediate probabilistic claim] there exists *X* ∈ X such that *i**x* ∈ *X* and *X* ∩ *I**y* = ∅. Thus, for this *X* we have \nAND*X*(*x*) = 0, AND*X*(*y*) = 1. \n Hence *P**x* ≠ *P**y*, which proves the correctness of the algorithm and yields the theorem.\nThe proof above in fact yields a NAADT of cost $O\\left(\\log^2 \\binom{n}{k}\\right)$ for any function *f* : {0, 1}*n* → {0, 1} for which *f* is a constant on inputs of Hamming weight less than *n* − *k* for some *k* < *n*/2 (in particular, *f* need not be symmetric on inputs of larger Hamming weight).\nWe are now ready to prove Theorem [thm: intro symmetric naandt communication].\n[Proof of Theorem [thm: intro symmetric naandt communication]] If switch(*f*) ≥ *n*/2, then Claim [claim: symmetric sparsity] implies that spar(*f*) = 2Ω(*n*). Equation implies that Dcc(*f* ∘ AND) = Ω(*n*). Thus, a trivial NAADT of cost *n* witnesses NAADT(*f*) = *O*(Dcc(*f* ∘ AND)) in this case.\nHence, we may assume switch(*f*) = *k* < *n*/2. We have \n$${\\mathsf{NAADT}}(f) = O\\left(\\log^2 \\binom{n}{k}\\right) = O(\\log^2({\\mathrm{spar}}(f))) = O({{\\mathsf{D}}\\_{{\\mathrm{cc}}}}(f \\circ {\\mathsf{AND}})^2),$$ \n where the first equality follows from Theorem [thm: upper bound naandt in terms of switch symmetric], the second from Claim [claim: symmetric sparsity], and the third from Equation .\nAcknowledgements\n================\nSwagato Sanyal thanks Prahladh Harsha and Jaikumar Radhakrishnan for pointing out the reference , and Srijita Kundu for pointing out the reference . N.S.M. thanks Ronald de Wolf for useful discussions. We thank Justin Thaler for pointing out a bug from an earlier version of the paper, and Arnab Maiti for helpful comments. We thank anonymous reviewers for helpful comments and suggestions, including an improvement of the value *b* from 4 to 2 in Theorem [thm:detlift].\nAddressing Function\n===================\nRecall that Theorem [thm: san19] shows that for all Boolean functions *f*, we have ${\\mathsf{NAPDT}}(f) = O(\\sqrt{r} \\log r)$, where *r* denotes the Fourier sparsity of *f*. Moreover a quadratic separation is witnessed by the Addressing function. We showed in Claim [claim: max naandt OMB] that such an upper bound does not hold in the Möbius-world, and NAADT(OMB*n*) ∈ {spar(OMB*n*), spar(OMB*n*) − 1}. In this section we show that the Addressing function already witnesses that a separation similar to that in Theorem [thm: san19] cannot hold in the Möbius-world (even on allowing randomization in the decision trees). While the bound we obtain here is weaker than that in Claim [claim: max naandt OMB], it is interesting that the Addressing function, which witnesses a quadratic separation in the Fourier-world, no longer does so in the Möbius-world.\nThe following is our main claim of this section.\n[claim: addressing naandt] Let *n* ≥ 2 be a positive integer that is a power of 2. Then, \n$${\\mathsf{QNAADT}}({\\mathsf{ADDR}}\\_n) = \\Theta({\\mathrm{spar}}({\\mathsf{ADDR}}\\_n)^{\\frac{1}{\\log 3}}).$$ \nTo show this, we show a lower bound on the quantum non-adaptive AND decision tree complexity of the Addressing function, and we compute its sparsity exactly.\n[claim: addr qnaandt large] For an integer *n* ≥ 2 that is a power of 2, \nQNAADT(ADDR*n*) = Θ(*n*). \n[claim: addr sparsity] For an integer *n* ≥ 2 that is a power of 2, \nspar(ADDR*n*) = *n*log3. \nClaim [claim: addressing naandt] follows immediately from the two above claims. We now prove these claims.\n[Proof of Claim [claim: addr qnaandt large]] Let T be a quantum non-adaptive AND query algorithm of cost *c* that computes ADDR*n* to error at most *ɛ* = 1/3. An input to ADDR*n* consists of log*n* addressing bits and *n* target bits. Let *I*1 be the set of indices of the addressing bits and *I*2 be the set of indices of the target bits. The algorithm works with a state space ∣*S*1, …, *S**c*⟩∣*b*⟩∣*w*⟩, where each *S**j* ⊆ *I*1 ∪ *I*2, *b* ∈ {0, 1}*c* and the last register captures the workspace. The algorithm does the following on input *z* ∈ {0, 1}log*n* + *n*:\n* It starts in an input-independent state, say ∑*S*1, …, *S**c*, *b*, *w**α**S*1, …, *S**c*, *b*, *w*∣*S*1, …, *S**c*⟩∣*b*⟩∣*w*⟩,\n* applies *c* non-adaptive queries, that act on the basis states as follows: \n*O**z* : ∣*S*1, …, *S**c*⟩∣*b*1, …, *b**c*⟩∣*w*⟩ ↦ ∣*S*1, …, *S**c*⟩∣*b*1 ⊕ AND*S*1(*z*), …, *b**c* ⊕ AND*S**c*(*z*)⟩∣*w*⟩.\n* It then applies a two-outcome projective measurement {Π, *I* − Π} on the resulting state *O**z*∣*ψ*⟩ and outputs a value depending on the measurement outcome.\nFix any *i* ∈ {0, 1}log*n* and consider the inputs *z**i"},"data_id":{"kind":"string","value":"arxiv_0002465"}}},{"rowIdx":599,"cells":{"content":{"kind":"string","value":"Reasoning about Discrete and ContinuousNoisy Sensors and Effectors in Dynamical Systems\n=======================================================================================\nAmong the many approaches for reasoning about degrees of belief in the presence of noisy sensing and acting, the logical account proposed by Bacchus, Halpern, and Levesque is perhaps the most expressive. While their formalism is quite general, it is restricted to fluents whose values are drawn from discrete finite domains, as opposed to the continuous domains seen in many robotic applications. In this work, we show how this limitation in that approach can be lifted. By dealing seamlessly with both discrete distributions and continuous densities within a rich theory of action, we provide a very general logical specification of how belief should change after acting and sensing in complex noisy domains.\nKnowledge representation Reasoning about action Reasoning about knowledge Reasoning about uncertainty Cognitive robotics\nIntroduction\n============\n> \n> *On numerous occasions it has been suggested that the formalism [the situation calculus] take uncertainty into account by attaching probabilities to its sentences. We agree that the formalism will eventually have to allow statements about the probabilities of events, but attaching probabilities to all statements has the following objections:* \n> \n> \n> 1. It is not clear how to attach probabilities to statements containing quantifiers in a way that corresponds to the amount of conviction people have.\n> 2. The information necessary to assign numerical probabilities is not ordinarily available. Therefore, a formalism that required numerical probabilities would be epistemologically inadequate.\n> \n> \n> − McCarthy and Hayes.\n> \n> \n> \nMuch of high-level AI research is concerned with the behaviour of some putative agent, such as an autonomous robot, operating in an environment. Broadly speaking, an intelligent agent interacting with a dynamic and incompletely known world grapples with two special sorts of reasoning problems. First, because the world is *dynamic*, it will need to reason about change: how its actions affect the state of the world. Pushing an object on a table, for example, may cause it to fall on the floor, where it will remain unless picked up. Second, because the world is incompletely known, the agent will need to make do with partial specifications about what is true. As a result, the agent will often need to augment what it believes about the world by performing perceptual actions, using sensors of one form or another.\nFor many AI applications, and robotics in particular, these reasoning problems are more involved. Here, it is not enough to deal with incomplete knowledge, where some formula *ϕ* might be unknown. One must also know which of *ϕ* or ¬*ϕ* is the more *likely*, and by how much. In addition, both the sensors and the effectors that the agent uses to modify its world are often subject to uncertainty in that they are *noisy.*\nTo see a very simple example, imagine a robot moving towards a wall as shown in Figure [fig:robot], and a certain distance *h* from it. Suppose the robot can move towards and away from the wall, and it is equipped with a distance sensor aimed at the wall. Here, the robot may not know the true value of *h* but may believe that it takes values from some set, say {2, …, 11}. If the sensor is noisy, a reading of, say, 5 units, does not guarantee that the agent is actually 5 units from the wall, although it should serve to increase the agent’s degree of belief in that fact. Analogously, if the robot intends to move by 1 unit and the effector is noisy, it may end up moving by 0.9 units, which the agent does not get to observe. Be that as it may, the robot’s degree of belief that it is closer to the wall should increase.\nWhile many proposals have appeared in the literature to address such concerns (*cf.*penultimate section), very few are embedded in a general theory of action whilst supporting features like disjunction and quantification. For example, graphical models such as Bayesian networks can represent and reason about the probabilistic dependencies between random variables, and how that might change over time. However, it lacks first-order features and a rich account of actions. Relational graphical models, including Markov logic networks, borrow devices from first-order logic to allow the succinct modelling of relational dependencies, but ultimately they are purely syntactic extensions to graphical models, and do not attempt to address the deeper issues pertaining to the specification of probabilities in the presence of logical connectives and quantifiers. Building on first-order accounts of probabilistic reasoning, perhaps the most general formalism for dealing with *degrees of belief* in formulas, and in particular, with how degrees of belief should evolve in the presence of noisy sensing and acting is the account proposed by Bacchus, Halpern, and Levesque, henceforth BHL. Among its many properties, the BHL model shows precisely how beliefs can be made less certain by acting with noisy effectors, but made more certain by sensing (even when the sensors themselves are noisy).\nThe main advantage of a logical account like BHL is that it allows a specification of belief that can be partial or incomplete, in keeping with whatever information is available about the application domain. It does not require specifying a prior distribution over some random variables from which posterior distributions are then calculated, as in Kalman filters, for example. Nor does it require specifying the conditional independences among random variables and how these dependencies change as the result of actions, as in the temporal extensions to Bayesian networks. In the BHL model, some logical constraints are imposed on the initial state of belief. These constraints may be compatible with one or very many initial distributions and sets of independence assumptions. All the properties of belief will then follow at a corresponding level of specificity.\n [fig:robot]\nSubjective uncertainty is captured in the BHL account using a possible-world model of belief. In classical possible-world semantics, a formula *ϕ* is believed to be true when *ϕ* holds in all possible worlds that are deemed accessible. In BHL, the degree of belief in *ϕ* is defined as a normalized sum over the possible worlds where *ϕ* is true of some nonnegative *weights* associated with those worlds. (Inaccessible worlds are assigned a weight of zero.) To reason about belief change, the BHL model is then embedded in a rich theory of action and sensing provided by the situation calculus. The BHL account provides axioms in the situation calculus regarding how the weight associated with a possible world changes as the result of acting and sensing. The properties of belief and belief change then emerge as a direct logical consequence of the initial constraints and these changes in weights.\nFor example, suppose *h* is a fluent representing the robot’s horizontal distance to the wall in Figure [fig:robot]. The fluent *h* would have different values in different possible worlds. In a BHL specification, each of these worlds might be given an initial weight. For example, a uniform distribution might give an equal weight of.1 to ten possible worlds where *h* ∈ {2, 3, …, 11}. The degree of belief in a formula like (*h* < 9) is then defined as a sum of the weights, and would lead here to a value of.7. The theory of action would then specify how these weights change as the result of acting (such as moving away or towards the wall) and sensing (such as obtaining a reading from a sonar aimed at the wall). Naturally, the logical language permits weaker specifications, involving disjunctions and quantifiers, and the appropriate behavior would still emerge.\nWhile this model of belief is widely applicable, it does have one serious drawback: it is ultimately based on the addition of weights and is therefore restricted to fluents having discrete finite values. This is in stark contrast to robotics and machine learning applications, where event and observation variables are characterized by continuous distributions, or perhaps combinations of discrete and continuous ones. There is no way to say in BHL that the initial value of *h* is any real number drawn from a uniform distribution on the interval [2, 12]. One would again expect the belief in (*h* < 9) to be.7, but instead of being the result of *summing* weights, it must now be the result of *integrating* densities over a suitable space of values, something quite beyond the BHL approach.\nSo, on the one hand, the BHL account and others like it can be seen as general formal theories that attempt to address important philosophical problems such as those raised by McCarthy and Hayes above. But on the other, a serious criticism levelled at this line of work, and indeed at much of the work in reasoning about action, is that the theory is far removed from the kind of probabilistic uncertainty and noise seen in typical robotic applications.\nThe goal of this work is to show how with minimal additional assumptions this serious limitation of BHL can be lifted.[1](#fn1) By lifting this limitation, one obtains, for the first time, a logical language for representing real-world robotic specifications without any modifications, but also extend beyond it by means of the logical features of the underlying framework. In particular, we present a *formal specification* of the degrees of belief in formulas with real-valued fluents (and other fluents too), and how belief changes as the result of acting and sensing. Our account will retain the advantages of BHL but work seamlessly with discrete probability distributions, probability densities, and perhaps most significantly, with difficult combinations of the two. More broadly, we believe the model of belief proposed in this work provides the necessary bridge between logic-based reasoning modules, on the one hand, and probabilistic specifications as seen in real-world data-intensive applications, on the other.\nThe paper is organized as follows. We first review the formal preliminaries, the BHL model in particular, and introduce definitions for modeling continuous probability distributions. We then show how the definition of belief in BHL can be reformulated as a different summation, which then provides sufficient foundation for our extension to continuous domains. We then discuss how this model can be extended for noisy acting, and combinations of discrete and continuous properties. We conclude after discussing related work.\nA Theory of Action\n==================\nOur account is formulated in the language of the situation calculus, as developed in. The situation calculus is a special-purpose knowledge representation formalism for reasoning about dynamical systems. Informally, the formalism is best understood by arranging the world in terms of three kinds of things: *situations*, *actions* and *objects*. Situations represent “snapshots,” and can be viewed as possible histories. A set of initial situations correspond to the ways the world can be prior to the occurrence of actions. The result of doing an action, then, leads to a successor (non-initial) situation. Naturally, dynamic worlds change the properties of objects, which are captured using predicates and functions whose last argument is always a situation, called *fluents*.\nThe Logical Language\n--------------------\nFormally, the language L of the situation calculus is a many-sorted dialect of predicate calculus, with sorts for *actions*, *situations* and *objects* (for everything else). (We do not review standard predicate logic here; see, for example,. We further assume familiarity with the notions of *models*, *structures*, *satisfaction* and *entailment*.) In full length, let L include:\n* logical connectives ¬, ∀, ∧ , = , with other connectives such as ⊃ understood for their usual abbreviations;\n* an infinite supply of *variables* of each sort;\n* an infinite supply of *constant* symbols of the sort object;\n* for each *k* ≥ 1, *object* *function* symbols *g*1, *g*2, … of type (*action* ∪ *object*)*k* → *object*;\n* for each *k* ≥ 0, *action function* symbols *A*1, *A*2, … of type (*action* ∪ *object*)*k* → *action*;\n* a special *situation* *function* symbol *do*: *action* × *situation* → *situation*;\n* a special predicate symbol *Poss*: *action* × *situation*;[2](#fn2)\n* for each *k* ≥ 0, *fluent function* symbols *f*1, *f*2, … of type (*action* ∪ *object*)*k* × *situation* → *object*;\n* a special constant *S* 0 to represent the actual initial situation.\nTo reiterate, apart from some syntactic particulars, the logical basis for the situation calculus is the regular (many-sorted) predicate calculus.[3](#fn3) So, terms and well-formed formulas are defined inductively, as usual, respecting sorts. See, for example, for an exposition.\nWe follow some conventions in the ways we use Latin and Greek alphabets: *a* for both terms and variables of the action sort (the context would make this clear); *s* for terms and variables of the situation sort (the context would make this clear); and finally, *x*, *u*, *v*, *z*, *n*, and *y* to range over variables of the object sort. We let *ϕ* and *ψ* range over formulas, and Σ over sets of formulas. (These may be further decorated using superscripts or subscripts.)\nWe sometimes suppress the situation term in a formula *ϕ*, or use a distinguished variable *now*, to denote the current situation. Either way, we let *ϕ*[*t*] denote the formula with the restored situation term *t*.\nWe will often use the the usual “case” notation with curly braces as a convenient abbreviation for a logical formula: \n$$z = \\begin{cases}\n t\\_1 & \\mbox{if\\,\\,} \\psi\\\\\n t\\_2 & \\mbox{otherwise}\n \\end{cases}\n \\,{\\doteq}\\,\\,\\, (\\psi \\supset z = t\\_1) \\land (\\lnot\\psi \\supset z = t\\_2).$$ \nFinally, for convenience, we often introduce formula and term abbreviations that are meant to expand as L-formulas. For example, we might introduce a new formula *A* by *A* ≐ *ϕ*, where *ϕ* ∈ L. Then any expression *E*(*A*) containing *A* is assumed to mean *E*(*ϕ*). Analogously, if we introduce a new term *t* by *t* = *u* ≐ *ϕ*(*u*) then any expression *E*(*t*) is assumed to mean ∃*u*(*E*(*u*) ∧ *ϕ*(*u*)).\nDynamic worlds are enabled by performing actions, and in the language, this is realized using the *do* operator. That is, the result of doing an action *a* at situation *s* is the situation *d**o*(*a*, *s*). Functional fluents, which take situations as arguments, may then have different values at different situations, thereby capturing *changing properties* of the world. As noted, the constant *S* 0 is assumed to give the actual initial state of the domain, but the agent may consider others possible that capture the beliefs and ignorance of the agent. In general, we say a situation is an *initial* one when it is a situation without a *predecessor*: \n*Init*(*s*) ≐ ¬∃*a*, *s*ʹ. *s* = *d**o*(*a*, *s*ʹ). \n The picture that emerges is that situations can be structured as a set of trees, each rooted at an initial situation and whose edges are actions. More conventions: we use *ι* to range over such initial situations only, and let *α* denote sequences of action terms or variables, and freely use this with *d**o*, that is, if $\\alpha = [a{\\_}1, \\ldots, a{\\_}n]$ then *d**o*(*α*, *s*) stands for $do(a{\\_}n, do(\\ldots, do(a{\\_}1, s)\\ldots ))$.\nDomains are modeled in the situation calculus as *axioms*. A set of L-sentences specify the actions available, what they depend on, and the ways they affect the world. Specifically, these axioms are given in the form of a *basic action theory*, reviewed below.\nBasic Action Theories\n---------------------\nIn general, a basic action theory $ {{\\cal D}}$ is a set of sentences consisting of (free variables understood as universally quantified from the outside):\n* an *initial theory* $ {{\\cal D}}\\_0 $ that describes what is true initially;\n* *precondition* *axioms*, of the form ${\\textit{Poss}}(A(\\vec x,s)) \\equiv \\beta(\\vec x, s)$ that describe the conditions under which actions are executable;\n* *successor state axioms*, of the form $f(\\vec x, do(a,s)) = u \\equiv \\gamma{\\_}f (u, \\vec x, s)$, that describe the changes to fluent values after doing actions;\n* domain-agnostic *foundational* *axioms*, such as a second-order induction axiom for the space of situations and unique name axioms for actions, the details of which need not concern us here.\nThe formulation of successor state axioms, in particular, incorporates Reiter’s monotonic solution to the frame problem.\nAn agent reasons about actions by means of entailments of a basic action theory $ {{\\cal D}}$, for which standard first-order (Tarskian) models suffice (although see below). A fundamental task in reasoning about action is that of *projection*, where we test which properties hold after actions. Formally, suppose *ϕ* is a situation-suppressed formula or uses the special symbol ${{\\textit}{now}}.$ Given a sequence of actions $a{\\_}1$ through $a{\\_}n$, we are often interested in asking whether *ϕ* holds after these: \n$${{\\cal D}}\\models \\phi[do([a{\\_}1, \\ldots, a{\\_}n], {S\\_{\\!\\textrm{0}}})]?$$ \nThis concludes our review of the basic features of the language. In the subsequent sections, we will discuss how the formalism is first extended for knowledge, and then, degrees of belief against discrete probability distributions and beyond. To prepare for that, the rest of the section will introduce three logical constructions that will be used in our work. First, we will define a class of Tarskian structures. Second, we define a logical term standing for summation in the usual mathematical sense, that is to be understood as an abbreviation for a formula involving second-order quantification. Third, we analogously define a logical term standing for integration in the usual mathematical sense.\nR-interpretations\n-----------------\nFor our purposes, the notion of entailment will be assumed wrt a class of Tarskian structures that we call R-interpretations. See for a review of Tarskian structures; we assume some familiarity with the underlying notions. Below, for any L-term *t* and L-interpretation *M*, we use *t**M* to mean the domain element that *t* references. If *t* has a free variable *x* and *y* is any L-term, then we write *t**y**x* to mean the L-term obtained by replacing *x* in *t* with *y*. Finally, for any variable map *μ* and variable *X*, we use *μ**Y**X* to mean a variable map that is exactly like *μ* except that for the variable *X* it takes the value *Y*.\nBy an R-interpretation we mean any L-structure where = , < , > , 0, 1, + , × , /, − , *e*, *π*, exponentiation and logarithms have their usual interpretations.\n(That is, ``1 + 0 = 1“ is true in all R-interpretations, if ``*x* > *y*” is true then ``¬(*y* > *x*)\" is true, and so on.) So, henceforth, when we write Σ ⊨ *ϕ*, we mean that in all R-interpretations where Σ is true, so is *ϕ*.[4](#fn4)\nNatural numbers can be defined in terms of a predicate by appealing to R-interpretations. Let \n*Natural*(*x*) ≐ ∀*P*[(*P*(0) ∧ ∀*x*(*P*(*x*) ⊃ *P*(*x* + 1))) ⊃ *P*(*x*)]. \n Then:\n[thm:real interpretations natural domain] Let *M* be any R-interpretation, and *c* a constant symbol of L. Then, *M* ⊨ *Natural*(*c*) iff $c\\su M \\in {{\\mathbb{N}}}$.\nThe proof relies on two lemmas. First, we argue that the set of natural numbers satisfies the antecedent *A* ≐ *P*(0) ∧ ∀*x*(*P*(*x*) ⊃ *P*(*x* + 1)):\n[lem:nat ante] For every R-interpretation *M* and map *μ*, $M, \\mu \\su P{\\_}{{\\mathbb{N}}}\\models A.$\nSince ``0“ and ``+” have their usual interpretations, and 0 ∈ N, $M, \\mu\\su P{\\_}{{\\mathbb{N}}}\\models P(0).$ Suppose *k* ∈ N, and so, $M, {\\mu \\su {P~ x} {\\_}{{{\\mathbb{N}}}~ k}} \\models P(x)$. Since N includes the successors of all its elements, $M, {\\mu \\su {P~ x} {\\_}{{{\\mathbb{N}}}~ k}} \\models P(x+1).$\nNext, we argue that every set satisfying the antecedent includes the set of natural numbers:\n[lem:nat subset] For every *M* and *μ* as above, and for any set *Q*, if $M, \\mu\\su P{\\_}Q \\models A$ then N is a subset of *Q*.\nSuppose $M, \\mu\\su P{\\_}Q \\models A$. We prove the claim by induction on natural numbers. For the base case, we consider 0 ∈ N, and by assumption $M, \\mu\\su P{\\_}Q \\models P(0)$, and so 0 is in the set *Q*. For the hypothesis, assume for any *k* ∈ N, *k* is also in *Q*. That is, $M, \\mu\\su {P~ x}{\\_}{Q~ k} \\models P(x)$. Of course, the successor of *k* is a natural number and by assumption, $M, \\mu\\su {P~ x}{\\_}{Q~ k} \\models P(x+1)$. Thus, starting from 0, every natural number and its successor is in *Q*, and so *Q* must include N.\nSo, the main claim is as follows:\nSuppose *M* ⊨ *Natural*(*c*), that is, *M* ⊨ ∀*P*(*A* ⊃ *P*(*c*)). Since N is a subset of the domain by assumption, for any map *μ*, $M, \\mu\\su P{\\_}{{\\mathbb{N}}}\\models A\\supset P(c)$. By Lemma [lem:nat ante], $M, \\mu\\su P{\\_}{{\\mathbb{N}}}\\models A$, and so $M, \\mu\\su P{\\_}{{\\mathbb{N}}}\\models P(c)$. Hence $c\\su M\\in {{\\mathbb{N}}}$.\nSuppose instead $c\\su M\\in {{\\mathbb{N}}}$. Suppose for any set *Q* and any map *μ*, $M, \\mu\\su P{\\_}Q \\models A$. By Lemma [lem:nat subset], N is a subset of *Q* and so $c\\su M$ is in *Q*, that is, $M, \\mu\\su P{\\_}Q \\models P(c)$. Therefore, $M, \\mu\\su P {\\_}{{\\mathbb{N}}}\\models A\\supset P(c)$. Since this holds for any set *Q*, *M* ⊨ ∀*P*[*A* ⊃ *P*(*c*)], that is, *M* ⊨ *Natural*(*c*).\nSummation\n---------\nHere we show how finite summations can be characterized as an abbreviation for an L-term using second-order quantification.\nLet *f* be any L-function from N to R. Let ${\\textsc{SUM}}(f,n)$, standing for the sum of the values of *f* for the argument 1 through *n*, be defined as an abbreviation: \n$$\\begin{array}{l}\n {\\textsc{SUM}}(f,n) = z \\doteq {\\exists}g[g(1) = f(1) ~\\land \\\\ \\hspace{3cm}~~ g(n) = z ~\\land \\\\ \\hspace{3cm}~~ \\forall i(~1\\leq i{<}n \\supset g(i+1) = g(i) + f(i+1)~)].\n\\end{array}$$ \n The variable *i* is understood to be chosen here not to conflict with any of the variables in *n* and *z*. The function *g* from N to R is assumed to not conflict with *f*. (That is, the logical terms are distinct.)\nThis can then be argued to correspond to summations in the usual mathematical sense as follows:\nLet *f* be a function symbol of L from N to R, *c* be a term, and *n* be a constant symbol of L. Let *M* be any R-interpretation. Then, \n$$\\textrm{if } \\sum{\\_}{i=1}\\su {n\\su M} f\\su M (i) = c\\su M \\textrm{ then } M\\models {\\textsc{SUM}}(f,n) = c.$$ \nWe prove by induction on $n\\su M$. For the base case, suppose $n\\su M =1$. Then, the antecedent would give us $f\\su M(1)$. As for the consequent, clearly $M \\models {\\textsc{SUM}}(f,n) = f(1),$ and so the base case holds.\nAssume the hypothesis holds for $n\\su M$, and we prove the case for $(n\\su M + 1)$. (Note that owing to ``1“ and ``+” having their usual interpretations, $(n+1)\\su M = n\\su M +1$.) So suppose \n$$\\sum {\\_}{i=1} \\su {{n\\su M} +1} f\\su M (i) = c\\su M.$$ \n On expanding the summation expression, we obtain: \n$$\\sum{\\_}{i=1}\\su {n\\su M} f\\su M (i) = c\\su M - f\\su M (n\\su M + 1)$$ \n By induction hypothesis, $ M \\models {\\textsc{SUM}}(f,n) = c - f(n+1) $, and so, by definition, $ M\\models {\\textsc{SUM}}(f,n+1) = c. $\nHenceforth, we write: \n$${\\displaystyle}\\sum{\\_}{i=1} \\su n t$$ \n to mean the logical formula ${\\textsc{SUM}}(t, n)$ for a logical term *t*. Here, *i* is assumed to not conflict with any of the variables in *n* and *t*.[5](#fn5)\nIt is worth noting that this logical formula can be applied to summation expressions where the arguments to the terms are not restricted to natural numbers, but taken from any finite set. For example, suppose *H* is any finite set of terms {*h*1, …, *h**n*}. We can then use terms such as: \n$${\\displaystyle}\\sum \\_{h\\in H} t(h)$$ \n standing for an abbreviation, similar to ${\\textsc{SUM}}(t,n)$: let *g* be a function, and let *g*(*i*) = *t*(*h**i*). Then clearly the above sum defines the same number as ∑*i* = 1*n**g*. In the sequel, we sometimes sum over a finite set of situations, or a finite vector of values, which is then understood as an abbreviation in this sense.\nIntegration\n-----------\nFinally, we characterize integrals as logical terms.[6](#fn6) We present this first for a continuous real-valued function of *one* variable, and then discuss its (straightforward) extension to the many-variable case.\nWe begin by introducing a notation for limits to positive infinity. For any logical term *t* and variable *x*, we introduce a term characterized as follows: \n$${\\textsc{LIM}[x,t]} = z \\,\\,{\\doteq}\\,\\,\n\\forall{u}(u>0 \\supset \\exists{m}\\,\\forall{n}(n>m \\supset {\\left| z-t^x\\_n\\right|}<u)).$$ \n The variables *u*, *m*, and *n* are understood to be chosen here not to conflict with any of the variables in *x*, *t*, and *z*. The abbreviation can be argued to correspond to the limit of a function at infinity in the usual sense:\n[lem:limits] Let *g* be a function symbol of L standing for a function from R to R, and let *c* be a constant symbol of L. Let *M* be any R-interpretation of L. Then we have the following: \n$$\\mbox{If}\\,\n\\lim\\_{x\\to{\\infty}} g\\su M (x) \\,=\\, c^M\n\\,\\,\\,\\,\\mbox{then}\\,\\,\\,\nM \\,\\models\\, (c\\,=\\, {\\textsc{LIM}[x,g]}).$$ \nSuppose the limit holds. Then, by the standard notion of limits, for every real number *ε* > 0, there is a natural number *j* such that for all natural numbers *k* > *j*: \n$$| g\\su M (k) - c\\su M | {<}\\epsilon.$$ \n Suppose $ M \\not\\models {\\textsc{LIM}[x,g]} = c. $ Then there is some *r* > 0 such that $ M, \\mu \\su u {\\_}r \\not\\models \\exists m \\forall n (n{>}m \\supset | c - g(n)| {<}u). $ By Theorem [thm:real interpretations natural domain], the domain of *M* includes N, and so let *m* and *n* be variables that are mapped to *j* and *k* respectively. Then, given that *M* interprets arithmetic symbols in the usual way, the claim $ M, \\mu \\su {u~m~n} {\\_}{r~j~k} \\not\\models (n{>}m) \\supset (| c - g(n) | {<}u) $ is a contradiction.\nHenceforth, we write: \nlim*x* → ∞*t* \n to mean the logical formula $ {\\textsc{LIM}[x,t]}. $\nNext, for any variable *x* and terms *a*, *b*, and *t*, we introduce a term ${\\textsc{INT}[x,a,b,t]}$ denoting the definite integral of *t* over *x* from *a* to *b*: \n$${\\textsc{INT}[x,a,b,t]} \\,\\,{\\doteq}\\,\\,\n\\lim\\_{n\\to{\\infty}}\\,h\\cdot \\sum\\_{i=1}^n\\, t^x\\_{(a+h\\cdot{i})}$$ \n where *h* stands for (*b* − *a*)/*n*. The variables are chosen not to conflict with any of the other variables. We now show:\n[lem:definite integral] Let *g* be a function symbol of L standing for a function from R to R, and let *a*, *b*, *c* be constant symbols of L. Let *M* be any R-interpretation of L. Then we have the following: \n$$\\mbox{If}\\,\n\\int\\_{a\\su M}^{b\\su M}\\! g^M(x)\\,dx \\,=\\, c^M\n\\,\\,\\,\\,\\mbox{then}\\,\\,\\,\nM \\,\\models\\, (c={\\textsc{INT}[x,a,b,g]}).$$ \nSuppose $\\int{\\_}{a\\su M} \\su {b\\su M} g\\su M (x) dx = c\\su M$, that is, $g\\su M$ is integrable. By definition, then: \n$$\\lim{\\_}{k\\to{\\infty}} h\\cdot \\sum{\\_}{i=1} \\su {k} g\\su M({a\\su M} + h\\cdot i) = c\\su M$$ \n where $h = (b\\su M - a\\su M)/k$. By Lemma [lem:limits], we have \n$$M \\models \\lim{\\_}{n\\to {\\infty}} \\,h\\cdot \\sum\\_{i=1}^n\\, g^x\\_{(a+h\\cdot{i})} = c$$ \n where *h* = (*b* − *a*)/*n*, that is, $ M \\models {\\textsc{INT}[x,a,b,g]} = c. $\nFinally, we define the definite integral of *t* over all real values of *x* by the following: \n$$\\int\\_{x}\\!{t} \\,\\,{\\doteq}\\,\\,\n\\lim\\_{u\\to{\\infty}} \\, \\lim\\_{v\\to{\\infty}} \\, {\\textsc{INT}[x,-u,v,t]}.$$ \n The main result for this logical abbreviation is the following:\n[thm:singleint] Let *g* be a function symbol of L standing for a function from R to R, and let *c* be a constant symbol of L. Let *M* be any R-interpretation of L. Then we have the following: \nIf ∫− ∞∞ *g**M*(*x*) *d**x* = *c**M* then *M* ⊨ (*c* = ∫*x* *g*(*x*)). \nSuppose $\\int{\\_}{-{\\infty}} \\su {\\infty}g\\su M (x) dx = c\\su M$, that is, $g\\su M$ is integrable. By definition, \n$$\\lim{\\_}{u\\to{\\infty}} \\lim{\\_}{v\\to{\\infty}} \\int{\\_}{-u}\\su {v} g\\su M (x) = c.$$ \n By Lemma [lem:limits] and Lemma [lem:definite integral], we obtain: \n$$M \\models \\lim{\\_}{u\\to{\\infty}} \\lim{\\_}{v\\to{\\infty}} {\\textsc{INT}[x,-u,v,g]} = c.$$ \n That is, $ M\\models \\int {\\_}x g(x) = c. $\nThe characterization of integrals for a many-variable function *f*, from ${{\\mathbb{R}}}\\su k$ to R, is then an easy exercise. For variables $x{\\_}1, \\ldots, x{\\_}k$ and terms $a{\\_}1, \\ldots, a{\\_}k$, $b{\\_}1, \\ldots, b{\\_}k$, and *t*, we introduce a term ${\\mbox{MINT}}[x{\\_}1, \\ldots, x{\\_}k, a{\\_}1, \\ldots, a{\\_}k, b{\\_}1, \\ldots, b{\\_}k, t]$ denoting the definite integral of *t* from $(a{\\_}1, \\ldots, a{\\_}k) \\in {{\\mathbb{R}}}\\su k$ to $(b{\\_}1, \\ldots, b{\\_}k)\\in {{\\mathbb{R}}}\\su k$: \n$${\\mbox{MINT}}[x{\\_}1, \\ldots, x{\\_}k, a{\\_}1, \\ldots, a{\\_}k, b{\\_}1, \\ldots, b{\\_}k, t] \\,\\,{\\doteq}\\,\\,\n\\lim\\_{n{\\_}1\\to{\\infty}}\\ldots \\lim\\_{n{\\_}k\\to{\\infty}}\\,h{\\_}1 \\cdots h{\\_}k \\cdot \\sum\\_{i{{\\_}1}=1}^{n{\\_}1}\\,\\ldots \\sum\\_{{i{\\_}k}=1}^{n{\\_}k}\\, t^{x{\\_}1, \\ldots, x{\\_}k}\\_{(a{\\_}1+h{\\_}1 \\cdot{i{\\_}1}), \\ldots, (a{\\_}k + h{\\_}k\\cdot {i{\\_}k})}$$ \n where $h{\\_}j$ stands for $(b{\\_}j-a{\\_}j)/{n{\\_}j}$. The variables are chosen not to conflict with any of the other variables. Finally, we define the definite integral of *t* over all real values of $x{\\_}1, \\ldots, x{\\_}k$ by the following: \n$$\\int\\_{x{\\_}1, \\ldots, x{\\_}k}\\!{t} \\,\\,{\\doteq}\\,\\,\n\\lim\\_{u{\\_}1\\to{\\infty}}\\ldots \\lim\\_{u{\\_}k\\to{\\infty}} \\, \\lim\\_{v{\\_}1\\to{\\infty}}\\ldots \\lim\\_{v{\\_}k\\to{\\infty}} \\, {\\mbox{MINT}}[x{\\_}1, \\ldots, x{\\_}k, -u{\\_}1, \\ldots, -u{\\_}k, v{\\_}1, \\ldots, v{\\_}k, t].$$ \n From this we get, as a corollary to Theorem [thm:singleint]:\nLet *g* be a function symbol of L standing for a function from ${{\\mathbb{R}}}\\su k$ to R, and let *c* be a constant symbol of L. Let *M* be any R-interpretation of L. Then we have the following: \n$$\\mbox{If}\\,\n\\int\\_{-{\\infty}}^{{\\infty}}\\ldots\\int\\_{-{\\infty}}^{{\\infty}}\\! g^M(x{\\_}1, \\ldots, x{\\_}k)\\,d x{\\_}1\\ldots d x{\\_}k \\,=\\, c^M\n\\,\\,\\,\\,\\mbox{then}\\,\\,\\,\nM \\,\\models\\, (c=\\!\\!\\int\\_{x{\\_}1, \\ldots, x{\\_}k}\\!{g(x{\\_}1, \\ldots, x{\\_}k)}).$$ \nA Theory of Knowledge\n=====================\nKnowledge\n---------\nAn early treatment of *knowledge* in the situation calculus is due to Moore. The classical *possible-world* interpretation for knowledge is based on the notion that there many different ways the world can be, where each world stands for a complete state of affairs. Some of these are considered possible by a putative agent, and they determine what the agent knows and does not know. Moore’s observation was that situations can be viewed as possible worlds. (Different from standard modal logics, however, worlds are reified as part of the syntax, but this is a minor technicality.) A special binary fluent *K*, taking two situation arguments, determines the accessibility relation between worlds: *K*(*s*ʹ, *s*) says that when the agent is at *s*, he considers *s*ʹ possible. (Also different from standard modal logics, we note that the order of the terms in the accessibility relation is reversed.) *Knowledge*, then, is simply truth at accessible worlds:\n**(Knowledge.)** Let *ϕ* be any situation-suppressed formula. The agent *knowing* *ϕ* at situation *s*, written *Knows*(*ϕ*, *s*), is the following abbreviation: \n*Knows*(*ϕ*, *s*) ≐ ∀*s*ʹ. *K*(*s*ʹ, *s*) ⊃ *ϕ*[*s*ʹ]. \nIn English: if in every situation *s*ʹ that is considered possible at *s* the formula *ϕ* holds, then the agent knows *ϕ* at *s*.[7](#fn7)\nMoore’s account has been adapted to the arrangement of dynamic laws via basic action theories by Scherl and Levesque. In this scheme, the initial theory is assumed to specify the agent’s initial beliefs. For example, *Knows*(*velocity*(*obj5*) = 50, *S* 0) expands to ∀*s*ʹ. *K*(*s*ʹ, *S* 0) ⊃ *velocity(obj5,**s*ʹ) = 50; that is, every accessible world initially agrees that the velocity of object 5 is 50, which means that the agent knows that the velocity of object 5 is 50. In addition, ¬*Knows*(*velocity*(*obj6*) = 60, *S* 0) and *Knows*(*f* = 0 ∨ *f* = 1, *S* 0) say that initially, the agent does not know that the velocity of object 6 is 60, and knows that the functional fluent *f* takes a value of either 0 or 1. Equivalently, one specifies an initial constraint on *K*; for example: [eq:categorical knowledge] K(, S) (f = 1 f =0)[] characterizes the agent’s knowledge about the fluent *f* taking a value of either 0 or 1.\nSensing\n-------\nTo specify the behavior of *K* at non-initial situations, Scherl and Levesque provide a *successor state axiom* for *K*. This axiom, intuitively, tests whether situations are to remain accessible as actions occur. Without going into full details, assume the provision of domain-specific *sensing* axioms, also part of the basic action theory $ {{\\cal D}}$. For example, \n$${\\textit}{SF}({\\textit}{sensetrue-f},s) \\equiv f(s) = 1.$$ \n formalizes the sensing outcome for an action to check whether *f* has value 1 in situation *s*. Here $ {\\textit}{SF} $ is a special L-predicate, similar to *Poss*. Then, for *s*ʹ to be accessible from *s*, we need the following successor state axiom to be included in $ {{\\cal D}}$: \n$$K(s', do(a,s)) \\equiv \\exists s''[K(s'', s) \\land s' = do(a,s'') \\land Poss(a, s'') \\land ({\\textit}{SF}(a,s'') \\equiv {\\textit}{SF}(a,s))].$$ \n This says that if *s*ʺ is the predecessor of *s*ʹ, such that *s*ʺ was considered possible at *s*, then *s*ʹ would be considered possible from *d**o*(*a*, *s*) contingent on sensing outcomes.\nTo appreciate how this axiom works in dynamical domains, assume that for actions without any sensing aspect, such as an action that toggles the value of *f*, one simply lets ${\\textit}{SF}$ be vacuously true: \n$${\\textit}{SF}({\\textit}{toggle-f},s) \\equiv {{\\textit{true}}}.$$ \n The idea, then, is that the accessibility between situations would not change as physical actions occur. However, as the agent operates in an environment where it senses various properties, those situations that are *incompatible* with the real world regarding the sensed value will be deemed *impossible* after such sensing actions. This leads to a notion of knowledge *expansion*,[8](#fn8) as the agent becomes more certain about the true nature of the world. See Figure [fig:kssa] for an illustration of the axiom: we imagine three situations *s*, *s*ʹ and *s*ʺ that are epistemically related prior to any sensing (that is, *s*ʹ and *s*ʺ are possible worlds when the agent is at *s*). These situations disagree on the value of *f*, and consequently, the agent does *not* know *f*’s value. After executing a sensing action for the truth of *f*, however, *d**o*(\\it sensetrue-f, *s*ʺ) is not epistemically related to *d**o*(\\it sensetrue-f, *s*). The upshot is that the agent knows the value of *f* at *d**o*(\\it sensetrue-f, *s*) and believes that this value is 1.\n [fig:kssa]\nDegrees of Belief and Likelihood\n--------------------------------\nThe Scherl and Levesque scheme, however, lacks constructs to quantify the agent’s uncertainty. One measure to quantify uncertainty is with *degrees of belief.* What is also lacking in their scheme is the ability to formalize the *probabilistic noise* in effectors and sensors, as seen in many real-world robotics applications. These limitations, to discrete approximations, were addressed by BHL.\nThe BHL scheme builds on Scherl and Levesque’s ideas, especially regarding how accessibility relations between worlds vary as a result of actions. In fact, the reader may observe many parallels between the two extensions. BHL’s remarkably simple proposal consists of introducing two new distinguished fluents, *p* and *l*, in addition to *K*. We present a simpler alternative involving only *p* and *l*.\nAs a simple running example, imagine a robot moving towards a wall, as shown in Figure [fig:robot]. Its distance to the wall is given by a functional fluent *h*, and it is assumed to be equipped with a sonar sensor that measures how far the robot is from the wall. In other words, ideally, the sensor’s reading would correspond to the actual value of *h*.\nThe *p* fluent determines a probability distribution on situations, by associating situations with *weights*. More precisely, the term *p*(*s*ʹ, *s*) denotes the relative *weight* accorded to situation *s*ʹ when the agent happens to be in situation *s*. Of course, *p* can be seen as a companion to *K*. As one would for *K*, the properties of *p* in initial states, which vary from domain to domain, are specified with axioms as part of $\n{{\\cal D}}\\_0. $ For example, [eq:uniform-1]\nl p(,S) = u ((*h*() = 2 *h*() = 3) u =.5) \n (*h*() 2 *h*() 3 u=0 ).\nsays that those initial situations where *h* has the integer values 2 or 3 obtain a weight of.5. All other situations, then, obtain 0 weight. We expect, of course, that weights are nonnegative, and that non-initial situations are given a weight of 0 initially. The following nonnegative constraint, also part of $ {{\\cal D}}\\_0 $, ensures this:\n \n$$\\label{eq:p initial constraint}\n{\\forall}{\\iota},s.\\, p(s,{\\iota}) \\geq 0 \\,\\land\\, (p(s,{\\iota}) {>}0 \\supset\n{\\textit{Init}}(s)) \n\\tag{P1}$$ \nNote that this is a stipulation about initial situations *ι* only. But BHL provide a *successor state axiom* for *p*, to be listed shortly, that ensures that this constraint holds in all situations.\nNext, the term *l*(*a*, *s*) is intended to denote the likelihood of action *a* occurring in situation *s*. Among other things, *l* can be used to model noisy sensors. This is perhaps best demonstrated using an example. Imagine a sonar aimed at the wall, which gives a reading for the true value of *h*. Supposing the sonar’s readings are subject to additive Gaussian noise.[9](#fn9) If now a reading of *z* were observed on the sonar, intuitively, those situations where *h* = *z* should be considered more probable than those where *h* ≠ *z*.[10](#fn10) This occurrence is captured using likelihoods in the formalism. Basically, if *sonar*(*z*) is the sonar sensing action with *z* being the value read, we specify a likelihood axiom describing its error profile as follows:\n \n$$\\label{eq:sonar error profile simple} \n\\begin{array}{l}\nl({\\textit{sonar}}(z),s) = u \\equiv (z\\ge0 \\land u={{\\cal N}}(z-{\\textit{h}}(s);\\mu,\\sigma^2)) \n~\\lor \\\\ \\qquad \\mbox{} \\qquad \\mbox{} \\qquad \\mbox{} \\qquad \\mbox{} (z<0 \\land u=0). \n\\end{array}$$ \nThis stipulates that the difference between a nonnegative reading of *z* and the true value *h* is normally distributed with a variance of *σ*2 and mean of *μ*.[11](#fn11)\nClearly, the error profile of various hardware devices is application dependent, and it is this profile that is modeled as shown above using *l*. Notice, for example, when *μ* = 0, which indicates that the sensor has no systematic bias, then *l*(*sonar*(5), *s*) will be higher when *h*(*s*) = 5 than when *h*(*s*) = 25. Roughly, then, the idea is that after an observation, the weights on situations would get redistributed based on their compatibility with the observed value.\nOne may contrast such likelihood specifications to (trivial) ones for deterministic physical actions,[12](#fn12) such as an action *move(z)* of moving towards the wall by precisely *z* units. For such actions, we simply write \n$$l({\\textit}{move}(z),s) = u \\equiv u = 1$$ \n in which case the *p* value of *s* is the same as that for $ do({\\textit}{move(3)},s) $. Thus, this is a form of *imaging*, where the weights of worlds are simply ``transferred\" to their successors.\nFormally, we add *action likelihood* axioms to $ {{\\cal D}}$:\nAction likelihood axioms for each action type *A* are sentences of the form: \n$$l(A(\\vec x),s) = u \\equiv \\psi\\_A(\\vec x, u, s).$$ \n Here $ \\psi\\_A(\\vec x, u, s) $ is any formula characterizing the conditions under which action $ A(\\vec x) $ has likelihood *u* in *s*.\nIn general, likelihood axioms can depend on any number of features of the world besides the fluent that the sensor is measuring. For example, imagine that the sonar’s accuracy depends on the room temperature. We could then specify an error profile as follows:\n \n$$\\label{eq:sonar error profile complicated}\n \\begin{array}{l}\n l({\\textit{sonar}}(z), s) = u \\equiv \\\\\n \\qquad \\mbox{} \\qquad \\mbox{} \\qquad \\mbox{} (z\\geq 0 \\land \\mathit{temp}(s) \\geq 0 \\land u = {{\\cal N}}(z - {\\textit{h}}(s);\\mu, 1)) ~\\lor \\\\\n \\qquad \\mbox{} \\qquad \\mbox{} \\qquad \\mbox{} (z \\geq 0 \\land \\mathit{temp}(s) < 0 \\land u = {{\\cal N}}(z-{\\textit{h}}(s);\\mu,16)) ~\\lor \\\\\n \\qquad \\mbox{} \\qquad \\mbox{} \\qquad \\mbox{} (z <0 \\land u = 0). \n \\end{array}$$ \nThat is, the sonar’s accuracy worsens severely when the temperature drops below 0, as seen by the larger variance.\nHaving introduced the new fluents, we are now ready to provide the successor state axiom for *p*, which is analogous to the one for *K*:\n \n$$\\label{eq:p ssa}\n \\begin{array}{l}\n p(s',do(a,s)) = u \\,\\,\\equiv\\\\\n \\quad \\quad \\exists s''~[s' = do(a,s'') \\land {\\textit{Poss}}(a,s'') \\,\\land\\\\\n \\quad\\quad\\quad\\quad\\quad u = p(s'',s) \\times l(a,s'')] \\\\\n \\quad \\lor\\,\\, \\lnot\\exists s''~[s' = do(a,s'') \\land {\\textit{Poss}}(a,s'')] \n \\land u = 0. \n \\end{array}\n \\tag{P2}$$ \nThis says that the weight of situations *s*ʹ relative to *d**o*(*a*, *s*) is the weight of their predecessors *s*ʺ times the likelihood of *a* contingent on the successful execution of *a* at *s*ʺ.\n [fig:sensing]\nTo see an application of this axiom using the specifications and, consider two situations *s* and *s*ʹ associated with the same weight initially, as shown in Figure [fig:sensing]. These situations have *h* values 2 and 3 respectively. Suppose the robot obtains a reading of 2 on the sensor. Given a sensor with mean *μ* = 0 and variance $ \\sigma\\su 2 = 1, $ the likelihood axiom is such that the weight of the successor of *s* is higher than that of *s*ʹ because the *h* value at *s* coincides with the sensor reading. More precisely, the weight for the successor of *s* is given by the prior weight.5 multiplied by the likelihood factor $ {{\\cal N}}(z-h(s);\\mu,\\sigma\\su 2) = {{\\cal N}}(2-2;0,1) = {{\\cal N}}(0,0,1). $ The weight for the successor of *s*ʹ is obtained analogously.\nInterestingly, by means of the above axiom, if predecessors are not *epistemically* related, which is another way of saying that *p*(*s*ʺ, *s*) is 0, then their successors will also not be. Similarly, when *a* is not executable at *s*ʺ, its successor is no longer accessible from *d**o*(*a*, *s*). One other consequence of and is that (*p*(*s*ʹ, *s*) > 0) will be true only when *s*ʹ and *s* share the same history of actions. This is because insists that initial situations are only epistemically related to other initial ones, and respects this relation over actions.\nWe are now prepared to define the *degree of belief* in a formula *ϕ* at a situation *s*, written *Bel*(*ϕ*, *s*). (Henceforth, whenever a formula *ϕ* appears in the context of *Bel*, we assume that it is either situation-suppressed, or it only mentions the situation term *now*.) Intuitively, this is simply the *weight* of *accessible situations*. Formally:\n[defn:belief] **(Degrees of belief.)** Suppose *ϕ* is any situation-suppressed L-formula. Then the *degree of belief* in *ϕ* is an abbreviation for: \n$${\\textit{Bel}}(\\phi,s) \\,{\\doteq}\\,\n \\frac{1}{\\gamma}\\!\\sum\\_{\\{s':\\phi[s'\\!]\\}} p(s'\\!,s),$$ \n where *γ*, the normalization factor, is understood (throughout) as the same expression as the numerator but with *ϕ* replaced by *true*. So, here, *γ* is ∑*s*ʹ*p*(*s*ʹ, *s*).\nNote that we do not have to insist that *s*ʹ and *s* share histories since *p*(*s*ʹ, *s*) will be 0 otherwise, as discussed above. The summation term in this logical formula is not a new logical symbol, but simply an abbreviation for a second-order formula, by way of Section [sub:summation].\nDiscussion\n----------\nLet us conclude this section by remarking that since *p* is a fluent, the syntax of basic action theories allows us to express probabilistic knowledge in a very general way, quite beyond standard probabilistic formalisms. For example, to represent categorical uncertainty like, we would do: [eq:uniform-3] p(,S) =\n1 & (h=2h=3)[] \n0 &\nIn English: all initial situations where the value of *h* is either 2 or 3 are considered epistemically possible, and accorded a weight of 1. All other initial situations are accorded a weight of 0. Observe, however, that this *p*-specification does not say which value of *h* is more likely. Thus, unlike standard probabilistic frameworks, we do not assume that it is always possible to find a single probability distribution for the robot to use.\nIt is also possible to handle partial specifications. Let us contrast with the following initial axiom for *p*: [eq:uniform-2] p(,S) =\n.1 & h() { 1,2,…,10 } \n0 &\nThen, letting a basic action theory include the sentence: \n$$\\eqref{eq:uniform-1} \\lor \\eqref{eq:uniform-2}$$ \n means that the robot believes *h* is uniformly distributed on {2, 3} or on {1, …, 10} without being able to say which. To reiterate, we do not assume that it is always possible to find a single probability distribution for the robot to use.\nOf course, a much weaker specification is possible by replacing one of these probabilistic alternatives with categorical ones. For example, suppose the basic action theory were to instead include: \n$$\\eqref{eq:uniform-3} \\lor \\eqref{eq:uniform-2}.$$ \n In this case, the agent believes that *h* may take a value from {2, 3} or is uniformly distributed on {1, 2, …, 10} without being able to say which.\nIn sum, the framework allows one to freely combine categorical and probabilistic specifications, leading to a very general model of belief. For simplicity of presentation, we consider a fairly simple set of specifications in this article, and refer readers to for more involved ones.\nBelief Reformulated\n===================\nThe definition for degrees of belief, as given by BHL, is intuitive and simple. It is closely fashioned after the semantics for belief in modal probability logics, where the probabilities of formulas is calculated from the weights of possible worlds satisfying the formula. Unfortunately, this definition is not easily amenable to generalizations. Notice, for example, that *Bel* is well-defined only when the sum over all those situations *s*ʹ such that *ϕ*[*s*ʹ] holds is finite. This immediately precludes domains that involve an infinite set of situations agreeing on a formula. Moreover, the definition does not have an obvious analogue for continuous probability distributions. Observe that such an analogue would involve *integrating* over the space of situations, which makes little sense. Indeed, it is not certain what the space of situations would look like in general, but even if this was fixed, how such a thing can be tinkered with so as to obtain an appropriate notion of *integration* is far from obvious.\nTherefore, what we propose is to shift the calculating of probabilities from situations to *fluent values*, that is, to the well-understood domain of numbers. The current section is an exploration of this idea. What we will show in this section is that Definition [defn:belief] can be reformulated as a summation over numeric indices. That will allow, among other things, a seamless generalization from summation to integration, which is to be the topic of the next section.\nTo prepare for that, in addition to the usual case notation used, for example, in, we will introduce another kind of conditional term for convenience. This involves a quantifier and a default value of 0, like in formula. If *z* is a variable, *ψ* is a formula and *t* is a term, we use ⟨*z*.*ψ* → *t*⟩ as a logical term characterized as follows:\n \n$$\\begin{aligned}\n\\lefteqn{{{\\langle}z.\\psi\\to t {\\rangle}} = u \\,\\,{\\doteq}}\\\\[-\\smallskipamount]\n& & [(\\exists\n z \\psi) \\supset \\forall z(\\psi \\supset u=t)] \n \\land [(\\neg\\exists z\n \\psi) \\supset u=0)].\\end{aligned}$$ \nThe notation says that when ∃*z**ψ* is true, the value of the term is *t*; otherwise, the value is 0. When *t* uses *z* (the usual case), this will be most useful if there is a *unique* *z* that satisfies *ψ*.\nReturning to the task at hand, we will now need a way to enumerate the primitive fluent terms of the language. Intuitively, these correspond to the probabilistic variables in the language. Perhaps the simplest way is to assume there are *n* fluents *f*1, *f*2, …, *f**n* in L which take no arguments other than the situation argument,[13](#fn13) and that they take their values from some finite sets. We can then rephrase Definition [defn:belief] as follows:\n[defn:bel with two sums] Suppose *ϕ* is as before. Let *Bel*(*ϕ*, *s*) be an abbreviation for: \n$$\\frac{1}{\\gamma}\n \\sum\\_{\\vec{x}} \\sum\\_{s'}\n \\left\\{\n \\begin{array}{ll}\n p(s'\\!,s) & \\mbox{if\\, ${\\bigwedge f\\_i(s')=x\\_i \\,\\land\\, \n \\phi[s'\\!]}$}\\\\ 0\n & \\mbox{otherwise}\n \\end{array}\n \\right.$$ \n where *γ* is the numerator but with *ϕ* replaced by *true*, as usual.\n(For readability, we often drop the index variables in sums and connectives when the context makes it clear: in this case, *i* ranges over the set {1, …, *n*}, that is, the indices of the fluents in L.) Definition [defn:bel with two sums] suggests that for each possible value of the fluents, we are to sum over all possible situations and for each one, if the fluents have those values and *ϕ* holds, then the *p* value is to be used, and 0 otherwise. Roughly speaking, if one were to group situations satisfying ⋀*f**i*(*s*) = *x**i* into sets for every possible vector $ \\vec x, $ the union of these sets would give the space of situations. Our claim about the relationship between the two abbreviations can be made precise as follows:\n[thm:belief two sums] Let $ {{\\cal D}}$ be any basic action theory and *ϕ* any L-formula. Then the abbreviations for *Bel*(*ϕ*, *s*) from Definition [defn:belief] and [defn:bel with two sums] define the same number.\nFor the proof, we focus solely on the numerators of the two abbreviations. That is,\n \n$$\\label{eq:N1T1}\n \\sum\\_{\\{s':\\phi[s'\\!]\\}} p(s'\\!,s)\n \\tag{$\\dag$}$$ \non the one hand, and\n \n$$\\label{eq:N2T1}\n \\sum\\_{\\vec{x}} \\sum\\_{s'}\n \\left\\{\n \\begin{array}{ll}\n p(s'\\!,s) & \\mbox{if\\, ${\\bigwedge f\\_i(s')=x\\_i \\,\\land\\, \n \\phi[s'\\!]}$}\\\\ 0\n & \\mbox{otherwise}\n \\end{array}\n \\right. \n\\tag{$\\ddag$}$$ \non the other. We show that these expressions define the same number. With this, the case for the denominators follows trivially, since *true* is a special case of *ϕ*. Then, the claim is proven.\nLet *S* be a set such that *s*ʹ ∈ *S* iff *p*(*s*ʹ, *s*) > 0. (That is, for any ground situation term *s*, this is the set of all ground situation terms *s*ʹ such that *p*(*s*ʹ, *s*) > 0.) Then, let *T* ⊆ *S* be the set such that *s*ʹ ∈ *T* iff *ϕ*[*s*ʹ]. Intuitively, *T* is the set of all situations that are epistemically related to *s*, but where *ϕ* holds. It is easy to see that \n$$\\eqref{eq:N1T1} = \\sum\\_{s' \\in T} p(s',s).$$ \nSuppose now $f{\\_}i$ ranges over ${\\left\\{ c{\\_}{1i}, \\ldots, c{\\_}{ki} \\right\\}} $, and so by extension, suppose $\\vec f = \\langle f{\\_}1, \\ldots, f{\\_}n {\\rangle}$ ranges over $ \\{ {\\vec {c\\_1 {}}}, {\\vec {c\\_2 {}}}, \\ldots, {\\vec {c\\_k {}}} \\}. $ For any $ \\vec {c\\_{j} {}} $ in that set, let *T**j* ⊆ *T* be a set such that *s*ʹ ∈ *T**j* iff ⋀*f**i*(*s*ʹ) = *c**j**i*. That is, *T**j* identifies those situations from *T* where fluents satisfy the vector of values $ \\vec {c\\_{j} {}}. $ Observe that \n$$\\eqref{eq:N2T1} = \\sum\\_{j=1}^k \\sum\\_{s' \\in T\\_j} p(s',s).$$ \nBut, of course, *T* = ⋃*T**j*. Therefore, \n∑*j* = 1*k*∑*s*ʹ ∈ *T**j**p*(*s*ʹ, *s*) = ∑*s*ʹ ∈ *T**p*(*s*ʹ, *s*). \n Therefore, and define the same number.\nBe that as it may, Definition [defn:bel with two sums] still involves summations over situations. To arrive at a definition that eschews the summing of situations, we start with the case of initial situations. In this matter, we will be insisting on a precise space of initial situations. For this, let us recall the axiomatization of the situation calculus presented in for multiple initial situations, which includes a sentence saying there is precisely one initial situation for *any* possible vector of fluent values. This can be written as follows:\n \n$$\\label{eq:stara}\n[\\forall{\\vec{x}}\\,\\exists{{\\iota}}\\!\\bigwedge f\\_i({\\iota}) = x\\_i]\n\\,\\land\\,\n[{\\forall}{\\iota}, {\\iota}'\\!.\\!\\bigwedge f\\_i({\\iota}) = f\\_i({\\iota}') \n \\supset {\\iota}= {\\iota}']\n\\tag{{\\textrm{\\upshape{P3}}}}$$ \n(Recall that *i* ranges over the indices of the fluents in L, that is, {1, …, *n*}.) Under the assumption , we can rewrite Definition [defn:belief] for *s* = *S* 0 as\n \n$$\\label{eq:belief-ini}\n{\\textit{Bel}}(\\phi,{S\\_{\\!\\textrm{0}}}) {\\doteq}\\frac{1}{\\gamma}\\!\n \\sum\\_{\\vec{x}}\n {{\\langle}{\\iota}.\\bigwedge\\! f\\_i({\\iota})=x\\_i \\land \\phi[{\\iota}]\\to p({\\iota},{S\\_{\\!\\textrm{0}}}) {\\rangle}}\n\\tag{B0}$$ \nThe two abbreviations, in fact, are equivalent:\nLet $ {{\\cal D}}$ be any basic action theory, *ϕ* any L-formula, and suppose $ {{\\cal D}}\\_0 $ includes. Then the abbreviations for *Bel*(*ϕ*, *S* 0) in Definition [defn:belief] and define the same number.\nAs in Theorem [thm:belief two sums], we focus on the numerators for the two abbreviations. That is,\n \n$$\\label{eq:N1T2}\n \\sum\\_{\\{s':\\phi[s'\\!]\\}} p(s'\\!,{S\\_{\\!\\textrm{0}}})\n \\tag{$\\dag$}$$ \non the one hand, and\n \n$$\\label{eq:N2T2}\n \\sum\\_{\\vec{x}}\n {{\\langle}{\\iota}.\\bigwedge\\! f\\_i({\\iota})=x\\_i \\land \\phi[{\\iota}]\\to p({\\iota},{S\\_{\\!\\textrm{0}}}) {\\rangle}}\n \\tag{$\\ddag$}$$ \non the other. We show that these expressions define the same number. The denominators represent a special case, and so the claim will follow.\nLet *S* be the set of initial situations. Suppose $ f{\\_}i $ ranges over $ {\\left\\{ c{\\_}i, c'{\\_}i, \\ldots, c''{\\_}i \\right\\}} $. By way of, for any vector of values $ \\langle c{\\_}1, \\ldots, c{\\_}n \\rangle $ for the vector of fluents $ \\langle f{\\_}1, \\ldots, f{\\_}n \\rangle $, there exists a (unique) situation *s* ∈ *S* such that ⋀*f**i*(*s*) = *c**i*. Let *T* ⊆ *S* be such that *s* ∈ *T* iff *ϕ*[*s*]. It is easy to see that \n$$\\eqref{eq:N2T2} = \\sum\\_{s' \\in T} p(s', {S\\_{\\!\\textrm{0}}}).$$ \n Since ensures that *p*(*s*ʹ, *S* 0) > 0 only if *s*ʹ is an initial situation, we get that \n$$\\eqref{eq:N1T2} = \\sum\\_{s' \\in T} p(s', {S\\_{\\!\\textrm{0}}}).$$ \n Therefore and define the same number.\nThis shows that for *S* 0, summing over possible worlds can be replaced by summing over fluent values.\nUnfortunately, is only geared for initial situations. For non-initial situations, the assumption that no two agree on all fluent values is untenable. To see why, imagine an action ${{\\textit}{move}}(z)$ that moves the robot *z* units to the left (towards the wall) but that the motion stops if the robot hits the wall. The successor state axiom for fluent *h*, then, might be like this:\n \n$$\\label{eq:xpos ssa} \n \\begin{array}{l}\n {\\textit{h}}(do(a,s)) = u \\equiv \\\\\n \\qquad \\mbox{} \\lnot\\exists{z}(a={{\\textit}{move}}(z)) \\land u = {\\textit{h}}(s) \n \\,\\,\\,\\lor\\\\ \n \\qquad \\mbox{} \\exists{z}(a = {{\\textit}{move}}(z) \\land u = \\max(0,{\\textit{h}}(s) - z)).\n \\end{array}$$ \nIn this case, if we have two initial situations that are identical except that *h* = 3 in one and *h* = 4 in the other, then the two distinct successor situations that result from doing ${{\\textit}{move}}(4)$ would agree on all fluents (since both would have *h* = 0). Ergo, we cannot sum over fluent values for non-initial situations unless we are prepared to count some fluent values more than once.\nIt turns out there is a simple way to circumvent this issue by appealing to Reiter’s solution to the frame problem. Indeed, Reiter’s solution gives us a way of computing what holds in non-initial situations in terms of what holds in initial ones, which can be used for computing belief at arbitrary successors of *S* 0. More precisely,\n[defn:discrete belief] **(Degrees of belief (reformulated).)** Let *ϕ* be any L-formula. Given any sequence of ground action terms *α* = [*a*1, …, *a**k*], let \n$${\\textit{Bel}}(\\phi, s) \\doteq \\displaystyle \\frac{1}{\\gamma}\\sum\\_{\\vec{x}} {\\textit{P}}(\\vec x, \\phi, s)$$ \n where if *s* = *d**o*(*α*, *S* 0) then \n$${\\textit{P}}(\\vec x, \\phi, do(\\alpha,{S\\_{\\!\\textrm{0}}})) \\doteq {\\langle}{\\iota}.\\,\n {\\bigwedge\\! f\\_i({\\iota})=x\\_i \\,\\land\\, \\phi[do(\\alpha,{\\iota})]}\\,\\to \n {p(do(\\alpha,{\\iota}),do(\\alpha,{S\\_{\\!\\textrm{0}}}))}\\,{\\rangle}.$$ \n(As before, *i* ranges over the indices of the fluents in L.) To say more about how (and why) this definition works, we first note that by and, *p* will be 0 unless its two arguments share the same history. So the *s*ʹ argument of *p* in Definition [defn:belief] is *expanded* and written as *d**o*(*α*, *ι*) in Definition [defn:discrete belief]. By ranging over all fluent values, we range over all initial *ι* as before, but without ever having to deal with fluent values in non-initial situations. Of course, we test that the *ϕ* holds and use the *p* weight in the appropriate non-initial situation. In particular, owing to *p*’s successor state axiom, the weight for non-initial situations accounts for the likelihood of actions executed in the history. We now establish the following result:\nLet $ {{\\cal D}}$ be any basic action theory with initially, *ϕ* any L-formula, and *α* any sequence of ground actions terms. Then the abbreviations for *Bel*(*ϕ*, *d**o*(*α*, *S* 0)) in Definition [defn:belief] and Definition [defn:discrete belief] define the same number.\nAs in Theorem [thm:belief two sums], we will focus on the numerators of the two abbreviations. That is,\n \n$$\\label{eq:dag}\n \\sum\\_{\\{s':\\phi[s'\\!]\\}} p(s'\\!,do(\\alpha,{S\\_{\\!\\textrm{0}}}))\n \\tag{$\\dag$}$$ \non the one hand, and\n \n$$\\label{eq:ddag}\n \\sum\\_{\\vec{x}} {\\langle}{\\iota}.\\,\n {\\bigwedge\\! f\\_i({\\iota})=x\\_i \\,\\land\\, \\phi[do(\\alpha,{\\iota})]}\\,\\to \n {p(do(\\alpha,{\\iota}),do(\\alpha,{S\\_{\\!\\textrm{0}}}))}\\,{\\rangle}.\n \\tag{$\\ddag$}$$ \non the other. We show that these expressions define the same number. The denominators represent a special case, and so the claim will follow.\nLet *S* be the set of initial situations, as determined by. Let *S*\\* = {*d**o*(*α*, *s*ʹ): *s*ʹ ∈ *S*}. Let *T* ⊆ *S*\\* such that *s*ʺ ∈ *T* iff *ϕ*[*s*ʺ]. It is easy to see that \n$$\\eqref{eq:ddag} = \\sum\\_{ s' \\in T} p(s', do(\\alpha, {S\\_{\\!\\textrm{0}}})).$$ \nOn the other hand, by and, *p*(*s*, *d**o*(*α*, *S* 0)) = 0 for all *s* ∉ *S*\\*. This means that \n$$\\eqref{eq:dag} = \\sum\\_{s' \\in T} p(s', do(\\alpha, {S\\_{\\!\\textrm{0}}})).$$ \n Therefore and define the same number.\nThus, by incorporating a simple constraint on initial situations, we now have a notion of belief that does not require summing over situations.\nReaders may notice that our reformulation only applies when we are given an explicit sequence *α* of actions, including the sensing ones. But this is just what we would expect to be given for the *projection problem*, where we are interested in inferring whether a formula holds after an action sequence. In fact, we can use regression on the *ϕ* and the *p* to reduce the belief formula from Definition [defn:discrete belief] to a formula involving initial situations only. See for work in this direction.\nFrom Weights to Densities\n=========================\nThe framework presented so far is *fully* *discrete*, which is to say that fluents, sensors and effectors are characterized by finite values and finite outcomes. Belief in *ϕ*, in particular, is the summing over a finite set of situations where *ϕ* holds. We now generalize this framework. We structure our work by first focusing on *fully continuous domains*, which is to say that fluents, sensors and effectors are characterized by values and outcomes ranging over R. This section, in particular, explores the very first installment: effectors are assumed to be deterministic, but *sensors* have continuous noisy error profiles. The next section, then, allows both effectors and sensors to have continuous noisy profiles. Further generalizations are deferred to Section [sec:generalizations].\nLet us begin by observing that the uncountable nature of continuous domains precludes summing over possible situations. In this section, we present a new formalization of belief in terms of *integrating* over fluent values. This, in particular, is made possible by the developments in the preceding section.\nAllowing real-valued fluents implies that there will be uncountably many initial situations. Imagine, for example, the scenario from Figure [fig:robot], and that the fluent *h* can now be any nonnegative real number. Then for any nonnegative real *x* there will be an initial situation where (*h* = *x*) is true. Suppose further that ${{\\cal D}}\\_0$ includes:\n \n$$\\label{eq:uniform 2 12 continuous}\n {p}({\\iota},{S\\_{\\!\\textrm{0}}}) = \\begin{cases}\n.1 & \\text{if}~2 \\leq {\\textit{h}}({\\iota}) \\leq 12 \\\\ \n 0 & \\text{otherwise}\n \\end{cases}$$ \nwhich says that the true value of *h* initially is drawn from a uniform distribution on the interval [2,12]. Then there are uncountably many situations where *p* is non-zero initially. So the *p* fluent now needs to be understood as a *density*, not as a weight. (That is, we now interpret *p*(*s*ʹ, *s*) as the *density* of *s*ʹ when the agent is in *s*.) In particular, for any *x*, we would expect the initial degree of belief in the formula (*h* = *x*) to be 0, but in (*h* ≤ 12) to be 1.\nWhen actions enter the picture, even if deterministic, there is more to be said. Numerous subtleties arise with *p* in non-initial situations. For example, if the robot were to do a ${{\\textit}{move}}(4)$ there would be an uncountable number of situations agreeing on *h* = 0: namely, those where 2 ≤ *h* ≤ 4 was true initially. In a sense, the point *h* = 0 now has *weight*, and the degree of belief in *h* = 0 should be.2. On the other hand, the other points *h* ∈ (0, 8] should retain their densities. That is, belief in *h* ≤ 2 should be.4 but belief in *h* = 2 should still be 0. In effect, we have moved from a continuous to a *mixed* distribution on *h*. Of course, a subsequent rightward motion will retain this mixed density. For example, if the robot were to now move away by 4 units, the belief in *h* = 4 would then be.2.\nTo address the concern of belief change in continuous domains, we now present a generalization to BHL. One of the advantages of our approach is that we will not need to specify how to handle changing densities and distributions like the ones above. These will emerge as side-effects, that is, shifting density changes will be *entailed* by the action theory.\nFor our formulation of belief, we first observe that we have fluents *f*1, …, *f**n* in L as before, that take no argument other than the situation term but which now take their values from R. Then:\n[defn:continuous belief] **(Degrees of belief (continuous noisy sensors).)** Let *ϕ* be any situation-suppressed L-formula, and *α* = [*a*1, …, *a**k*] any ground sequence of action terms. The *degree of belief* in *ϕ* at *s* is an abbreviation: \n$${\\textit{Bel}}(\\phi, s) \\doteq \\displaystyle\\frac{1}{\\gamma}{\\int\\_{\\vec x}}{\\textit{P}}(\\vec x, \\phi, s)$$ \n where, as in Definition [defn:discrete belief], if *s* = *d**o*(*α*, *S* 0) then \n$$P(\\vec x, \\phi, do(\\alpha,{S\\_{\\!\\textrm{0}}})) \\doteq {\\langle}{\\iota}.\\,\n {\\bigwedge\\! f\\_i({\\iota})=x\\_i \\,\\land\\, \\phi[do(\\alpha,{\\iota})]}\\,\\to \n {p(do(\\alpha,{\\iota}),do(\\alpha,{S\\_{\\!\\textrm{0}}}))}\\,{\\rangle}.$$ \nThat is, the belief in *ϕ* is obtained by ranging over all possible fluent values, and integrating the densities of situations where *ϕ* holds. If we were to compare the above definition to Definition [defn:discrete belief], we see that we have simply shifted from summing over finite domains to integrating over reals. In fact, we could read *P* as the *(unnormalized) density* associated with a situation satisfying *ϕ*. As discussed, by insisting on an explicit world history, the *ι* need only range over initial situations, giving us the exact correspondence with fluent values.\nThis completes our new definition of belief. To summarize, our extension to the BHL scheme is defined using a few convenient abbreviations, such as for *Bel* and mathematical integration, and where an action theory consists of:\n1. $ {{\\cal D}}\\_0 $ (with ) as usual, but now also including ;\n2. precondition axioms as usual;\n3. successor state axioms, including one for *p*, namely, as usual;\n4. foundational domain-independent axioms as usual; and\n5. action likelihood axioms, one for each action type.\nNote that, apart from and *Bel*’s new abbreviation, we carry over precisely the same components as would BHL. By and large, the extension, thus, retains the simplicity of their proposal, and comes with minor additions. We will show that it has reasonable properties using an example and its connection to Bayesian conditioning below.\nIn the sequel, we assume, without explicitly mentioning so, that basic action theories include the sentences, and.\nBayesian Conditioning\n---------------------\nWe now explicate the relationship between our definition for *Bel* and *Bayesian conditioning*. Bayesian conditioning is a standard model for belief change wrt noisy sensing and it rests on two significant assumptions. First, sensors do not physically change the world, and second, conditioning on a random variable *f* is the same as conditioning on the event of observing *f*.\nIn general, in the language of the situation calculus, there need not be a distinction between sensing actions and physical actions. In that case, the agent’s beliefs are affected by the sensed value as well as any other physical changes that the action might enable to adequately capture the “total evidence” requirement of Bayesian conditioning.\nThe second assumption expects that sensors only depend on the true value for the fluent. For example, in the formulation of the sonar’s error profile is determined solely by *h*. But to suggest that the error profile might depend on other factors about the environment, as formulated by for example, goes beyond this simplified view. In fact, here, the agent also learns about the room temperature, apart from sensing the value of *h*.\nThus, our theory of action admits a view of dynamical systems far richer than the standard setting where Bayesian conditioning is applied. Be that as it may, when a similar set of assumptions are imposed as axioms in an action theory, we obtain a sensor fusion model identical to Bayesian conditioning. This connection was demonstrated in BHL for the discrete case. We prove the property formally for continuous variables below.\nWe begin by stipulating that actions are either of the *physical* type or of the *sensing* type, the latter being the kind that do not change the value of any fluent, that is, such actions do not appear in the successor state axioms for any fluent. Now, if $ {\\textit}{obs}(z) $ senses the true value of fluent *f*, then assume the sensor error model to be: \n$$l({\\textit}{obs}(z),s)=u\n\\,\\,\\equiv\\,\\, u={\\textit}{Err}(z,f(s))$$ \n where $ {\\textit}{Err}(u\\_1,u\\_2) $ is some expression with only two free variables, both numeric. This captures the notion established above: the error model of a sensor measuring *f* depends only on the true value of *f*, and is independent of other factors. Finally, for simplicity, assume *obs(z)* is always executable: \n*Poss*(*obs(z)*, *s*) ≡ *true*. \n Then we obtain:\n[thm:bayes theorem] Suppose $ {{\\cal D}}$ is any basic action theory with likelihood and precondition axioms for *obs(z)* as above, *ϕ* is any L-formula mentioning only *f*, and *u* ∈ {*x*1, …, *x**n*} that *f* takes a value from. Then we obtain*:* \n$$\\begin{array}{l}\n {{\\cal D}}\\models {\\textit{Bel}}(\\phi, do({\\textit}{obs}(z),{S\\_{\\!\\textrm{0}}})) = \\displaystyle\\frac{ \\int\\_{\\vec x } \n [{\\textit{P}}(\\vec x, \\phi \\land f = u, {S\\_{\\!\\textrm{0}}}) \\times {\\textit}{Err}(z,u)]}{\n \\int\\_{\\vec x } \n [{\\textit{P}}(\\vec x, f = u, {S\\_{\\!\\textrm{0}}}) \\times {\\textit}{Err}(z,u)] }\n \\end{array}$$ \nThat is, the *posterior belief* in *ϕ* is obtained from the *prior density* and the error likelihoods for all points where *ϕ* holds given that *z* is observed, normalized over all points. The proof for the theorem is as follows. \nWithout loss of generality, assume *f* takes the value *x*, and the remaining fluents are *f*1, …*f**n* that take values from *x*1, …, *x**n*. Let *a* denote *obs(z)*. From Definition [defn:continuous belief], $ {\\textit{Bel}}(\\phi, do({\\textit}{obs(z)},{S\\_{\\!\\textrm{0}}})) $ is an abbreviation for: \n$$\\begin{array}{l}\n {{\\displaystyle}}\\frac{1}{\\gamma} \\displaystyle \\int\\_{\\vec x} \\underline{{\\textit{P}}(\\vec x, \\phi, do(a,{S\\_{\\!\\textrm{0}}}))} = \\hfill \\text{(a)} \\\\[3ex]\n{{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {{\\langle}{\\iota}.f({\\iota}) = x \\land \\bigwedge\\_{i=1}^n f\\_i({\\iota}) = x\\_i \\land \\underline{\\phi[do(a,{\\iota})]}\\to p(do(a,{\\iota}), do(a, {S\\_{\\!\\textrm{0}}})) {\\rangle}} = \\hfill \\text{(b)} \\\\[3ex] \n{{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {{\\langle}{\\iota}.f({\\iota}) = x \\land \\bigwedge f\\_i({\\iota}) = x\\_i \\land \\underline{\\phi[{\\iota}]}\\to {p(do(a,{\\iota}), do(a, {S\\_{\\!\\textrm{0}}}))} {\\rangle}} = \\hfill \\text{(c)} \\\\[3ex] \n{{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {{\\langle}{\\iota}.f({\\iota}) = x \\land \\bigwedge f\\_i({\\iota}) = x\\_i \\land {(\\phi \\land f = x)[{\\iota}]}\\to \\underline{p(do(a,{\\iota}), do(a, {S\\_{\\!\\textrm{0}}}))} {\\rangle}} = \\hfill \\hspace{7em} \\text{(d)} \\\\[3ex] \n{{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {{\\langle}{\\iota}.f({\\iota}) = x \\land \\bigwedge f\\_i({\\iota}) = x\\_i \\land (\\phi\\land f=x)[{\\iota}]\\to \\underline{p({\\iota},{S\\_{\\!\\textrm{0}}})\\times {\\textit}{Err}(z,x)} {\\rangle}} = \\hfill \\text{(e)} \\\\[3ex] \n{{\\displaystyle}}\\frac{1}{\\gamma} \\fullx Err(z,x) \\times \\underline{{{\\langle}{\\iota}.f({\\iota}) = x \\land \\bigwedge f\\_i({\\iota}) = x\\_i \\land (\\phi\\land f=x)[{\\iota}]\\to {p({\\iota},{S\\_{\\!\\textrm{0}}})} {\\rangle}}} = \\hfill \\text{(f)} \\\\[3ex] \n{{\\displaystyle}}\\frac{1}{\\gamma} \\fullx Err(z,x) \\times {\\textit{P}}(\\vec x, \\phi \\land f = x, {S\\_{\\!\\textrm{0}}}).\n \\end{array}$$ \n The arguments underline those parts of the sentences that are being reduced. Step (a) expands *P*. In step (b), owing to the fact that sensing actions do not change fluent values (by our first assumption tailored to Bayesian conditioning), *ϕ*[*d**o*(*a*, *ι*)] is equivalent to *ϕ*[*ι*]. In step (c), we observe that what appears to the left of the right-arrow in step (b) is equivalent to one where *ϕ*[*ι*] is replaced by (*ϕ* ∧ *f* = *x*)[*ι*]. (The main reason for introducing the formula *f* = *x* is to allow us to identify those situations where *f* takes the value *x* which are all to be multiplied by the error likelihood for observing *z* when the true value is *x*.) In step (d), we use and the fact that *a* is always executable to replace *p*(*d**o*(*a*, *ι*), *d**o*(*a*, *S* 0)) by *p*(*ι*, *S* 0) × *l*(*a*, *ι*) = *p*(*ι*, *S* 0) × *Err*(*z*, *x*). Now note that the term appearing in the context of an integral is suggesting that if there is an initial situation where a particular condition holds, then a certain value is returned, and otherwise 0 is returned. This allows us to place the term *E**r**r*(*z*, *x*) on the outside in step (f), giving us the required numerator that appears in the claim. The expansion of the denominator is analogous with *true* being a special case for *ϕ*, and so we are done.\nThe usual case for posteriors are formulas such as *b* ≤ *f* ≤ *c*, which is estimated from the prior and error likelihoods for all points in the range [*b*, *c*], as demonstrated by the following consequence:\nSuppose *D* is any basic action theory with likelihood and precondition axioms for *obs(z)* as above, *f* is any L-fluent, and *u* is a variable from $ \\vec x = {\\langle}x\\_1, \\ldots, x\\_m {\\rangle}$ that *f* takes a value from. Then we obtain*:* \n$$\\begin{array}{l}\n {{\\cal D}}\\models {\\textit{Bel}}( a\\leq f \\leq b, do({\\textit}{obs}(z),{S\\_{\\!\\textrm{0}}})) = \\displaystyle\\frac{ \\int\\_{\\vec x } \n [{\\textit{P}}(\\vec x, f = u \\land a \\leq u \\leq b, {S\\_{\\!\\textrm{0}}}) \\times {\\textit}{Err}(z,u)]}{\n \\int\\_{\\vec x } \n [{\\textit{P}}(\\vec x, f = u, {S\\_{\\!\\textrm{0}}}) \\times {\\textit}{Err}(z,u)] }\n \\end{array}$$ \nMore generally however, and unlike many probabilistic formalisms, we are able to reason about any logical property *ϕ* of the random variable *f* being measured.\nExample\n-------\nUsing an example, we demonstrate the formalism and Theorem [thm:bayes theorem] in particular. To reason about the beliefs of our robot, let us build a simple basic action theory ${{\\cal D}}$. We extend the setting from Figure [fig:robot] to a 2-dimensional grid, as shown in Figure [fig:robot-v]. As before, let *h* be the fluent denoting its horizontal position (that is, its distance to the wall), and let the robot’s vertical position be given by a fluent *v*. The components of $ {{\\cal D}}$ are as below.\n [fig:robot-v]\n* Imagine a *p* of the form:\n \n$$\\label{eq:initial p spec}\n p({\\iota},{S\\_{\\!\\textrm{0}}}) = \\begin{cases}\n.1 \\times {{\\cal N}}({\\textit{v}}({\\iota});0,16) & \\text{if } 2\\leq {\\textit{h}}({\\iota}) \\leq 12 \\\\\n 0 & \\text{otherwise}\n \\end{cases}$$ \nThis says that the value of *v* is normally distributed about the horizontal axis with variance 16, and independently, that the value of *h* is uniformly distributed between 2 and 12.\nNote also that initial beliefs can be specified for ${{\\cal D}}\\_0$ using *Bel* directly. For example, to express that the true value of *h* is believed to be uniformly distributed on the interval [2, 12] we might equivalently include the following theory in $ {{\\cal D}}\\_0 $: \n{*Bel*(2 ≤ *h* ≤ 12, *S* 0) = .1, *Bel*(*h* ≤ 2 ∨ *h* ≥ 12, *S* 0) = 0}, \n and analogously for the fluent *v*.\nFor this example, a simple distribution has been chosen for illustrative purposes. In general, recall from Section [sub:discussion] that the *p*-specification does not require the variables to be independent, nor does it have to mention all variables.\n* For simplicity, let us assume that actions are always executable, *i.e.,* that $ {{\\cal D}}$ includes\n \n*Poss*(*a*, *s*) ≡ *true* \nfor all actions *a*. For this example, we assume three action types: action ${{\\textit}{move}}(z)$ that moves the robot *z* units towards the wall, action *up*(*z*) that moves the robot *z* units away from the horizontal axis, and action *sonar*(*z*) that gives a reading of *z* for the distance between the robot and the wall.\n* The successor state axiom for *h* is as in, and the one for *v* is as follows:\n \n$$\\label{eq:ypos ssa}\n\\begin{array}{l}\n {\\textit{v}}(do(a,s)) = u \\equiv \\lnot\\exists{z} (a={\\textit{up}}(z)) \\land \nu ={\\textit{v}}(s) ~\\lor \\\\\n\\qquad \\mbox{}\\qquad \\mbox{}\\qquad \\mbox{}\\qquad \\mbox{} \\exists{z} (a = {\\textit{up}}(z) \\land u = {\\textit{v}}(s) + z).\n\\end{array}$$\n* For the sensor device, suppose its error model is given as follows:\n \n$$\\label{eq:sonar error model}\n l({\\textit{sonar}}(z),s) = u \\equiv (z\\geq 0\\land u = {{\\cal N}}(z-{\\textit{h}}(s);0,4)) \\lor (z<0 \\land u=0).$$ \nThe error model says that for nonnegative *z* readings, the difference between the reading and the true value is normally distributed with mean 0 (which indicates that there is no systematic bias) and variance 4.[14](#fn14)\nFor the remaining (physical) actions, we let\n \n$$\\label{eq:likelihood deterministic physical actions}\n l({{\\textit}{move}}(z),s) = 1,~~ l({\\textit{up}}(z),s) =1$$ \nsince they are assumed to be deterministic for this section.\nThen we obtain:\n[thm:example] Let $ {{\\cal D}}$ be a basic action theory that is the union of $ \\{ \\eqref{eq:initial p spec}, \\eqref{eq:poss always true}, \\eqref{eq:xpos ssa}, \\eqref{eq:ypos ssa}, \\eqref{eq:sonar error model}, \\eqref{eq:likelihood deterministic physical actions} \\} $. Then the following are logical entailments of $ {{\\cal D}}$:\n1. [i:point 4 has 0 weight] *Bel*([*h* = 3 ∨ *h* = 4 ∨ *h* = 7], *S* 0) = 0.\nTo see how this follows, let us begin by expanding *Bel*([*h* = 3 ∨ *h* = 4 ∨ *h* = 7], *S* 0): \n$${{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {\\textit{P}}(\\vec x, {\\textit{h}}= 3 \\lor {\\textit{h}}= 4 \\lor {\\textit{h}}= 5, {S\\_{\\!\\textrm{0}}}). \n\\tag{a}$$ \n For the rest of the section, let *h* take its value from $ \\xo $ and *v* take its value from $ \\xt. $ By means of, there is exactly one situation for any set of values for *x*1 and *x*2. The *P* term for any such situation, however, is 0 unless *h* = 3 ∨ *h* = 4 ∨ *h* = 5 holds at the situation. Thus, (a) basically simplifies to: \n$${{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases}\n.1 \\times {{\\cal N}}(\\xt;0,16) & \\text{if \\( \\xo\\in \\{3,4,5\\} \\)} \\\\ \n 0 & \\text{otherwise}\n \\end{cases} \\hspace{3em} = 0.$$ \nIn effect, although we are integrating a function *δ*(*x*1, *x*2) over all real values, *δ*(*x*1, *x*2) = 0 unless *x*1 ∈ {3, 4, 7}.\n2. [i:initial belief about h] *Bel*(*h* ≤ 9, *S* 0) = .7.\nWe might contrast this with the previous property in that for any given value for $ \\xo $ and $ \\xt $, the *P* term is 0 when $ \\xo >9. $ When $ \\xo\\leq 9, $ however, the *p* value for the situation is obtained from the specification given by. That is, we have: \n$$\\begin{array}{l}\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases} \\text{Initial specification given by \\eqref{eq:initial p spec}} & \\text{if \\( {\\textit{h}}\\leq 9 \\)} \\\\\n 0 & \\text{otherwise} \n \\end{cases} \\hspace{9em} \\hfill = \\\\[3ex]\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases}.1 \\times {{\\cal N}}(\\xt; 0, 16) & \\text{if \\({\\exists}{\\iota}.~{\\textit{h}}({\\iota}) = \\xo, \\xo \\in [2,12] \\) and \\( {\\textit{h}}({\\iota}) \\leq 9 \\)} \\\\\n 0 & \\text{otherwise}\n \\end{cases} \n \\hfill =\\\\[3ex]\n {{\\displaystyle}}\\frac{1}{\\gamma} \\int\\_{{\\mathbb{R}}}\\int\\_2^9.1 \\times {{\\cal N}}(x\\_2;0,16) ~{\\textrm{d}{x}}\\_1~\n {\\textrm{d}{x}}\\_2 \n \\end{array}$$ \n The numerator evaluates to.7, and the denominator to 1.\n3. *Bel*(*h* > 7*v*, *S* 0) ≈ .6.\nBeliefs about any mathematical expression involving the random variables, even when that does not correspond to well known density functions, are entailed. To evaluate this one, for example, observe that we have \n$$\\begin{array}{l}\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases}.1\\times {{\\cal N}}(\\xt;0,16) & \\text{if \\( \\xo \\in [2,12] \\) and \\( \\xo > 7 \\xt \\)} \\\\\n 0 & \\text{otherwise}\n \\end{cases} \\hspace{3em} = \\\\[3ex] \n {{\\displaystyle}}\\frac{1}{\\gamma} \\int\\_2^{12} \\int\\_{-{\\infty}}^{\\xo /7}.1\\times {{\\cal N}}(\\xt;0,16) {\\textrm{d}{\\xo}}.\n\\end{array}$$\n4. [i:weight for 0] $ {\\textit{Bel}}({\\textit{h}}= 0, do({{\\textit}{move}}(4),{S\\_{\\!\\textrm{0}}})) =.2 $.\nHere a *continuous* distribution evolves into a *mixed* distribution. This results from $ {\\textit{Bel}}({\\textit{h}}= 0, do({{\\textit}{move}}(4),{S\\_{\\!\\textrm{0}}})) $ first expanding as: \n$${{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {\\textit{P}}(\\vec x, {\\textit{h}}=0, do({{\\textit}{move}}(4),{S\\_{\\!\\textrm{0}}})) \\tag{a}$$ \n The *P* term, then, simplifies to: \n$${{\\langle}{\\iota}.\\bigwedge f({\\iota}) = x \\land ({\\textit{h}}= 0)[do({{\\textit}{move}}(4),{\\iota})]\\to p({\\iota},{S\\_{\\!\\textrm{0}}}) {\\rangle}} \\tag{b}$$ \n That is, since $ {{\\textit}{move}}(z) $ has no error component, $ l({{\\textit}{move}}(z),s) = 1 $ for any *s* in accordance with $ {{\\cal D}}. $ Therefore, *p*(*d**o*(*a*, *ι*), *d**o*(*a*, *S* 0)) = *p*(*ι*, *S* 0). Now (b) says that for every possible value for *h* and *v*, if there is an initial situation where *h* = 0 holds after moving leftwards, then its *p* value is to be considered. Note that for any initial situation *s* where *h*(*s*) ∈ [2, 4], we get $ {\\textit{h}}(do({{\\textit}{move}}(4),s)) = 0 $ by. This leaves us with: \n$$\\begin{array}{l} \n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases}\n.1 \\times {{\\cal N}}(\\xt;0,16) & \\text{if \\( {\\exists}{\\iota}.~{\\textit{h}}({\\iota}) =\\xo, \\xo\\in [2,12], {\\textit{h}}({\\iota}) \\in [2,4] \\)} \\\\ \n 0 & \\text{otherwise}\n \\end{cases} \\qquad \\mbox{} \\qquad \\mbox{} = \\\\[3ex]\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases}\n.1 \\times {{\\cal N}}(\\xt;0,16) & \\text{if \\( \\xo \\in [2,4] \\)} \\\\ \n 0 & \\text{otherwise}\n \\end{cases} \n \\end{array} \\tag{c}$$ \n We can show that *γ* = 1, which means (c) =.2. This change in beliefs is shown in Figure [fig:xposmixed].\n5. $ {\\textit{Bel}}({\\textit{h}}\\leq 3, do({{\\textit}{move}}(4),{S\\_{\\!\\textrm{0}}})) =.5 $.\n*Bel*’s definition is amenable to a set of *h* values, where one value has a weight of.2, and all the other real values have a uniformly distributed density of.1.\n6. ${\\textit{Bel}}([\\exists{a,s}.~now\\!=\\!do(a,s) \\land {\\textit{h}}(s)\\!>\\!1],\ndo({{\\textit}{move}}(4),{S\\_{\\!\\textrm{0}}})) = 1$.\nIt is possible to refer to earlier or later situations using *n**o**w* as the current situation. This says that after moving, there is full belief that (*h* > 1) held before the action.\n7. [i:moving towards moving away] $ {\\textit{Bel}}(h=4,do([{{\\textit}{move}}(4),{{\\textit}{move}}(-4)],{S\\_{\\!\\textrm{0}}})) =.2 $\n$ {\\textit{Bel}}({\\textit{h}}=4,do([{{\\textit}{move}}(-4),{{\\textit}{move}}(4)],{S\\_{\\!\\textrm{0}}})) = 0 $.\nThe point *h* = 4 has 0 weight initially, as shown in item [i:point 4 has 0 weight]. Roughly, if the agent were to move leftwards first then many points would “collapse”, as shown in item [i:weight for 0]. The point would then obtain a *h* value of 0, and have a weight of.2. The weight is then retained on moving away by 4 units, where the point once again gets *h* value 4. On the other hand, if this entire phenomena were reversed then none of these features are observed because the collapsing does not occur and the entire space remains fully continuous.\n8. $ {\\textit{Bel}}(-1 \\leq {\\textit{v}}\\leq 1, do({{\\textit}{move}}(4),{S\\_{\\!\\textrm{0}}})) = {\\textit{Bel}}(-1 \\leq {\\textit{v}}\\leq\n 1, {S\\_{\\!\\textrm{0}}}) = \\smallint\\_{-1}^1 {{\\cal N}}(x\\_2;0,16) {\\textrm{d}{x}}\\_2 $.\nOwing to Reiter’s solution to the frame problem, belief in *v* is unaffected by a lateral motion. That is, a leftwards motion does not change *v* in accordance with. As per, the initial belief in *v* ∈ [ − 1, 1] is the area between [ − 1, 1] bounded by the specified Gaussian.\n9. *Bel*(*v* ≤ 7, *d**o*(*up*(2.5), *S* 0)) = *Bel*(*v* ≤ 4.5, *S* 0).\nAfter the action *up*(2.5), the Gaussian for *v*’s value has its mean “shifted” by 2.5 because the density associated with *v* = *x*2 initially is now associated with *v* = *x*2 + 2.5. Intuitively, we have: \n$$\\begin{array}{l}\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases}\n.1 \\times {{\\cal N}}(\\xt;0,16) & \\text{if \\( {\\exists}{\\iota}.{\\textit{v}}({\\iota}) = \\xt \\) and \\( ({\\textit{v}}\\leq 7)[do({\\textit{up}}(2.5),{\\iota})] \\)} \\\\ \n 0 & \\text{otherwise}\n \\end{cases} \\qquad \\mbox{} \\qquad \\mbox{} = \\\\[3ex]\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases}\n.1 \\times {{\\cal N}}(\\xt;0,16) & \\text{if \\( {\\exists}{\\iota}.{\\textit{v}}({\\iota}) = \\xt \\) and \\( {\\textit{v}}({\\iota}) \\leq 4.5 \\)} \\\\ \n 0 & \\text{otherwise}\n \\end{cases} \\hfill = \\\\[3ex]\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\begin{cases}\n.1 \\times {{\\cal N}}(\\xt;0,16) & \\text{if \\( \\xt \\leq 4.5 \\)} \\\\ \n 0 & \\text{otherwise}\n \\end{cases}\n \\end{array}$$\n10. [i:first reading] *Bel*(*h* ≤ 9, *d**o*(*sonar*(5), *S* 0)) ≈ .97.\n*Bel*(*h* ≤ 9, *d**o*([*sonar*(5), *sonar*(5)], *S* 0) ≈ .99.\nCompared to item [i:initial belief about h], belief in *h* ≤ 9 is sharpened by obtaining a reading of 5 on the sonar, and sharpened to almost certainty on a second reading of 5. This is because the *p* function, according to, incorporates the likelihood of each *sonar*(5) action. More precisely, the belief term in the first entailment simplifies to: \n$${{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {{\\langle}{\\iota}.{\\textit{h}}({\\iota}) = \\xo \\land {\\textit{v}}({\\iota}) =\\xt \\land {\\textit{h}}({\\iota}) \\leq 9\\to p({\\iota},{S\\_{\\!\\textrm{0}}}) \\times {{\\cal N}}(5-\\xo;0,4) {\\rangle}} \\tag{a}$$ \n Note that we have replaced (*h* ≤ 9)[*d**o*(*sonar*(5), *S* 0)] by (*h* ≤ 9)[*ι*] since *sonar*(*z*) does not affect *h*. From (a), we get \n$${{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {{\\cal N}}(5-\\xo;0,4)\\times {{\\langle}{\\iota}.{\\textit{h}}({\\iota}) = \\xo \\land {\\textit{v}}({\\iota}) =\\xt \\land {\\textit{h}}({\\iota}) \\leq 9\\to p({\\iota},{S\\_{\\!\\textrm{0}}}) {\\rangle}} \\tag{b}$$ \n We know from that those initial situations where *h* ≤ 2 have *p* values 0. Therefore, from (b), we get: \n$$\\begin{array}{l}\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx {{\\cal N}}(5-\\xo;0,4)\\times \\begin{cases}\n.1 \\times {{\\cal N}}(\\xt;0,16) & \\text{if \\( \\xo \\in [2,9] \\)} \\\\ \n 0 & \\textrm{otherwise}\n \\end{cases} \\hspace{3em} =\\\\[3ex]\n {{\\displaystyle}}\\frac{1}{\\gamma} \\int\\_{{\\mathbb{R}}}\\int\\_2^9 {{\\cal N}}(5-\\xo;0,4)\\times.1 \\times {{\\cal N}}(\\xt;0,16) {\\textrm{d}{\\xo}} {\\textrm{d}{\\xt}}\n\\end{array}$$ \n After a second reading of 5 from the sonar, the expansion for belief is analogous, except that the function to be integrated gets multiplied by a second ${{\\cal N}}(5-x\\_1;0,4)$ term. It is then not hard to see that belief sharpens significantly with this multiplicand. The agent’s changing densities are shown in Figure [fig:sonargraph].\n [fig:xposmixed]\n [fig:sonargraph]\nNoisy Acting\n============\nIn the presentation so far, we assumed physical actions to be deterministic. By this we mean that when a physical action *a* occurs, it is clear to us (as modelers) but also the agent how the world has changed on *a*. Of course, in realistic domains, especially robotic applications, this is *not* the usual case. In this section, in a domain that has continuous fluents, we show how our current account of belief can be extended to reason with sensors as well as effectors that are noisy.\nIn line with the rest of this work, effector noise is given a quantitative account. Let us first reflect on what is expected with noisy acting. When an agent senses, as in the case of *sonar*(*z*), the argument for this action is not chosen by the agent. That is, the world decides what *z* should be, and based on this reading of *z*, the agent comes to certain conclusions about its own state. The noise factor, then, simply addresses the phenomena that the number *z* returned may differ from the true value of whatever fluent the sensor is measuring.\nNoisy acting diverges from that picture in the following sense. The agent *intends* to do action *a*, but what actually occurs is *a*ʹ that is possibly different from *a*. For example, the agent may want to move 3 units, but, unbeknownst to the agent, it may move by 3.042 units. The agent, of course, does not observe this outcome. Nevertheless, provided the agent has an account of its effector’s inaccuracies, it is reasonable for the agent to believe that it is in fact closer to the wall, even if it may not be able to precisely tell by how much. Intuitively, the result of a nondeterministic action is that any number of successor situations might be obtained, which are all indistinguishable in the agent’s perspective (until sensing is performed). Depending on the likelihoods of the action’s potential outcomes, some of these successor situations are considered more probable than others. The agent’s belief about what holds then must incorporate these relative likelihoods. So, in our view, nondeterminism is really an *epistemic* notion.\nNoisy Action Types\n------------------\nFollowing, perhaps the simplest extension to make all this precise is to assume that deterministic actions such as $ {{\\textit}{move}}(x) $ now have companion action types $ {{\\textit}{move}}(x,y) $ in L. The intuition is that *x* represents the *nominal* value, which is the number of units that the agent intends to move, while *y* represents the actual distance moved. The actual value of *y* in any ground action, of course, is not observable for the agent. This simple idea will need three adjustments to our account:\n1. successor state axioms need to be built using these new action types;\n2. the formalism must allow the modeler to formalize that certain outcomes are more likely than others, that is, noisy actions may be associated with a probabilistic account of the various outcomes; and\n3. the notion of belief must incorporate the nominal value, the range of possible outcomes and their likelihoods.\nFirst, we address successor state axioms. These are now specified as usual, but using the second argument, which is the actual outcome, rather than the nominal value, which is ignored. For example, for the fluent *h*, instead of, we will now have:\n \n$$\\label{eq:xpos ssa noisy} \n\\begin{array}{l}\n{\\textit{h}}(do(a, s)) = u \\equiv \\exists x, y[a = \\move(x,y) \\land u = \\max(0, {\\textit{h}}- y)] ~\\lor \\\\ \n\\qquad \\mbox{} \\qquad \\mbox{} \\qquad \\mbox{}~~~~ \\neg \\exists x, y[a = \\move(x,y)] \\land u = {\\textit{h}}(s).\n\\end{array}$$ \nThe reason for this modification is obvious. If *y* is the actual outcome then the fluent change should be contingent on this value rather than what was intended. It is important to note that no adjustment to the existing (Reiter’s) solution to the frame problem is necessary.\nThe approach\n-------------\nThe foremost issue now is to use the above idea to allow for more than one possible successor situation. Clearly, we do not want the agent to control the actual outcome in general. So the approach taken by BHL is to think of picking the second argument as a *nondeterministic* program. Briefly, is an agent programming proposal where one is allowed to formulate complex actions that denote sequential and nondeterministic executions of actions, among others, and is essentially a basic action theory. Given the action $ \\move(x,y) $, for example, the program $ \\textsc{Move(x)} $ might stand for the abbreviation $ \\pi y.~\\move(x,y) $, which corresponds to a ground action $ \\move(x,n) $ where *n* is chosen nondeterministically. For our purposes, we would then imagine that the agent executes programs.\nThere are some advantages to this approach: namely, we only have to look at the logical entailments, including ones mentioning *Bel*, of such programs. Since traces of these programs account for many potential outcomes, *Bel* does the right thing and accommodates all of these when considering knowledge. But the disadvantage is that the resulting formal specification turns out be unnecessarily complex, at least as far as projection is concerned.\nFor projection tasks, we show that we can settle on a simpler alternative, one that does not appeal to. Like BHL, we assume that the world is deterministic, where the result of doing a ground action leads to a distinct successor. Roughly, the intuition then is that when a noisy action is performed, the various outcomes of the action as well as the potential successor situations that are obtained wrt these are treated at the level of belief.\nAlternate Action Axioms\n-----------------------\nInspired by, our approach is based on the introduction of a distinguished predicate $\\alt.$ The idea is this: if $ \\alt(a, a', \\vec z) $ holds for ground action $ a = A(\\vec c) $ then we understand this to mean that the agent believes that any instance of $ a' = A(\\vec z) $ might have been executed instead of *a*. Here, $ \\vec z $ denotes the range of the arguments for potential outcomes.\nTo see how that gets used with the new action types such as $ {{\\textit}{move}}(x,y) $, consider the ground action $ \\move(3,3.1) $. So, the agent intends to move by 3 units but what has actually occurred is a move by 3.1 units. Since the agent does not observe the latter argument, from its perspective, what occurred could have been a move by 2.9 units, but also perhaps (although less likely) a move by 9 units. Thus, the ground actions $ \\move(3,2.9) $ and $ \\move(3,9) $ are $ \\alt $-related to $ \\move(3,3.1). $ (The likelihoods for these may vary, of course.) In logical terms, we might have have an axiom of the following form in the background theory:\n \n$$\\label{eq:alt move action}\n \\alt(\\move(x,y),a', z) \\equiv a' = \\move(x, z).$$ \nThis to be read as saying that $ \\move(3,z) $ for every *z* ∈ R are alternatives to $ \\move(3,3.1) $. If we required that *z* is only a certain range from 3, for example, we might have: \n$$\\alt(\\move(x,y),a', z) \\equiv a' = \\move(x,z) \\land | x-z| \\leq c$$ \n where *c* bounds the magnitude of the maximal possible error. On the other hand, for actions such as *sonar*(*z*), which do not have any alternatives, we simply write:\n \n$$\\label{eq:alt sonar}\n \\alt({\\textit{sonar}}(x), a', z ) \\equiv a' = {\\textit{sonar}}(z) \\land z=x.$$ \nThis says that *sonar*(*x*) is only $ \\alt $-related to itself.\nWith this simple technical device, one can now additionally constrain the likelihood of various outcomes using *l*. For example: [eq:simple noisy move] l((x,y), s) = u u = N(y-x;, 2) says that the difference between nominal value and the actual value is normally distributed with mean *μ* and variance *σ*2. This essentially corresponds to the standard additive Gaussian noise model in robotics.\nTo see an example of how, say, and work together with the successor state axiom for *p*, consider three situations *s*, *s*ʹ and *s*ʺ associated with the same density, as shown in Figure [fig:situations]. Suppose their *h* values are 6, 6.1 and 5.9 respectively. After attempting to move 3 units, the action $ {{\\textit}{move}}(3,z) $ for any *z* ∈ R may have occurred. So, for each of the three situations, we explore successors from different values for *z*. Assume the motion effector is defined by a mean *μ* = 0 and variance $ \\sigma\\su 2 = 1. $ Then, the *p*-value of the situation $ do({{\\textit}{move}}(3,5.7),s') $, for example, is obtained from the *p*-value for *s*ʹ multiplied by the likelihood of $ {{\\textit}{move}}(3,5.7) $, which is $ {{\\cal N}}(5.7-3;0,1) = {{\\cal N}}(2.7;0,1) $. Thus, the successor situation $ do({{\\textit}{move}}(3,5.7),s') $ is much less likely than the successor situation $ do({{\\textit}{move}}(3,3),s) $, as should be the case.\n [fig:situations]\nIn general, we define *alternate actions* axioms that are to be a part of the basic action theory henceforth:\nLet $ A(\\vec x, \\vec y) $ be any action. *Alternate actions* *axioms* are sentences of the form: \n$$\\alt(A(\\vec x, \\vec y), a', \\vec z) \\equiv a' = A(\\vec x, \\vec z) \\land \\psi(\\vec x, \\vec y, \\vec z)$$ \n where *ψ* is a formula that characterizes the relationship between the nominal and true values.\nThe one limitation with this definition is that only actions of the same *type*, *i.e.,* built from the same function symbol, are alternatives to each other. This does not allow, for example, situations where the agent intends a physical move, but instead unlocks the door. Nevertheless, this definition is not unreasonable because noisy actions in robotic applications typically involve additive noise. Moreover, this limitation only assists us in arriving at a simple and familiar definition for belief. A more involved definition would allow for other variants.\nA Definition for Belief\n-----------------------\nWe have thus far successfully augmented successor state axioms and extended the formalism for modeling noisy actions. The final question, then, is how can the outcomes of a noisy action, and their likelihoods, be accounted for? Indeed, a formula might not only be true as a result of the actions intended, but also as a result of those that were not.\nConsider the simple case of deterministic actions, where the density associated with *s* is simply transferred to *d**o*(*a*, *s*). This is an instance of Lewis’s *imaging*. In contrast, if *a* and *a*ʹ are $\\alt$-related, then the result of doing *a* at *s* would lead to successor situations *d**o*(*a*, *s*) and *d**o*(*a*ʹ, *s*). Moreover, unlike noisy sensors, *a* and *a*ʹ may affect fluent values in different ways, which is certainly the case with $ \\move(3,3.074) $ and $ \\move(3,3) $ on the fluent *h*. Thus, the idea then is that when reasoning about the agent’s beliefs about *ϕ*, one would need to integrate over the densities of all those potential successors where *ϕ* would hold.\nTo make this precise, let us first consider the result of doing a single action *a* at *S* 0. The *degree of belief* in *ϕ* after doing *a* is now an abbreviation for: \n$${{\\displaystyle}}{\\textit{Bel}}(\\phi, do(a, {S\\_{\\!\\textrm{0}}})) {\\doteq}\\frac{1}{\\gamma} \\fullx \\int\\_z P(\\vec x, z, \\phi, do(a,{S\\_{\\!\\textrm{0}}}) )$$ \n where \n$$P(\\vec x, z, \\phi, do(a,{S\\_{\\!\\textrm{0}}})) \\doteq {{\\langle}{{\\iota}, {b}}.\\bigwedge f\\_i({\\iota}) = x\\_i \\land \\alt(a, b, z) \\land \\phi[do(b,{\\iota})]\\to p(do(b, {\\iota}), do(a, {S\\_{\\!\\textrm{0}}}))~ {\\rangle}}.$$ \n (As before, the *i* ranges over the indices of the fluents in L, that is, {1, …, *n*}.) The intuition is this. Recall that by integrating over $ \\vec x $, all possible initial situations are considered by $ f{\\_}i ({\\iota}) = x{\\_}i. $ Analogously, by integrating over *z*, all possible action outcomes are considered by $ \\alt(a, b, z). $ Supposing *a* = *A*(*x*, *y*), for each outcome *b* = *A*(*x*, *z*),[15](#fn15) we test whether *ϕ* holds at the resulting situation *d**o*(*b*, *ι*) as before, and use its *p*-value. Here, this *p*-value is given by *p*(*d**o*(*b*, *ι*), *d**o*(*a*, *S* 0)), where the first argument is the successor of interest *d**o*(*b*, *ι*) and the second is the real world *d**o*(*a*, *S* 0).\nThe generalization, then, for a sequence of actions is as follows:\n[defn:belief continuous effector sensor] **(Degrees of belief (continuous noisy effectors and sensors.))** Suppose *ϕ* is any L-formula. Then the *degree of belief* in *ϕ* at *s*, written *Bel*(*ϕ*, *s*), is defined as an abbreviation: \n$${\\textit{Bel}}(\\phi,s) \\doteq \\begin{array}{l}\n {{\\displaystyle}}\\frac{1}{\\gamma} \\fullx \\int\\_{\\vec z} P(\\vec x, \\vec z, \\phi,s)\n \\end{array}$$ \n where, if *s* = *d**o*([*a*1, …, *a**k*], *S* 0), then \n$$\\begin{array}{l}\n P(\\vec x, \\vec z, \\phi,s) \\doteq \\\\ \\quad\\quad {\\langle}{\\iota}, {b\\_1, \\ldots, b\\_k}.~\\bigwedge f\\_i({\\iota}) = x\\_i \\land \\bigwedge \\alt(a\\_j, b\\_j, z\\_j) \\land \\phi[do([b\\_1, \\ldots, b\\_k], {\\iota})] \\\\ \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad \\rightarrow p(do([b\\_1, \\ldots, b\\_k],{\\iota}), do([a\\_1, \\ldots, a\\_k], {S\\_{\\!\\textrm{0}}})) ~{\\rangle}.\n\\end{array}$$ \n(Here, *i* ranges over {1, …, *n*} as before, and *j* ranges over the indices of the ground actions {1, …, *k*}.) That is, given any sequence, for all possible $ \\vec z $ values, we consider alternate sequences of ground action terms and integrate the densities of successor situations that satisfy *ϕ*, using the appropriate *p*-value.\nExample\n-------\nLet us now build a simple example with noisy actions. Consider the robot scenario in Figure [fig:robot]. Suppose the basic action theory $ {{\\cal D}}$ includes the foundational axioms, and the following components.\nThe initial theory $ {{\\cal D}}\\_0 $ includes the following *p* specification:\n \n$$\\label{eq:p uniform 10 12}\n p({\\iota},{S\\_{\\!\\textrm{0}}}) = \\begin{cases}\n.5 & \\text{ if \\( 10\\leq {\\textit{h}}({\\iota}) \\leq 12 \\)} \\\\ \n 0 & \\text{ otherwise}\n \\end{cases}$$ \nFor simplicity, let *h* be the only fluent in the domain, and assume that actions are always executable. The successor state axiom for the fluent *h* is. For *p*, it is the usual one, *viz.*.\nWe imagine two actions in this domain, one of which is the noisy move $ \\move(x,y) $ and a sonar sensing action *sonar*(*z*). For the alternate actions axioms, let us use and.\nFinally, we specify the likelihood axioms. Let the sonar’s error profile be l(*sonar*(z),s) = u (z0u = N(z-*h*(s);0,.25)) (z<0 u=0). Readers may note that this sonar is more accurate than the one characterized by, as it has a smaller variance. Regarding the likelihood axiom for $ \\move(x,y) $, let that be:\n \n$$\\label{eq:error profile noisy move}\n l(\\move(x,y), s) = u \\equiv u = {{\\cal N}}(y-x; 0, 1).$$ \nThis completes the specification of $ {{\\cal D}}. $\n[thm:continuous noisy effector and continuous noisy sensor] The following are entailments of $ {{\\cal D}}$.\n [fig:uniform-gaussian]\n1. $ {\\textit{Bel}}({\\textit{h}}\\geq 11, do(\\move(-2,-2.01), {S\\_{\\!\\textrm{0}}})) \\approx.95 $\nWe first observe that for calculating the degrees of belief, we have to consider all those successors of initial situations wrt $ \\move(-2, z) $ for every *z*, where *ϕ* holds. By, the *p* value for such situations is the initial *p* value times the likelihood of $ \\move(-2, z) $, which is $ {{\\cal N}}(z+2; 0, 1) $ by. Therefore, we get \n$$\\begin{array}{l}\n{{\\displaystyle}}\\frac{1}{\\gamma} \\int\\_x \\int\\_z \\begin{cases}\n.5\\times {{\\cal N}}(z+2;0,1) & \\text{ if \\( {\\exists}{\\iota}. \\) \\( x \\in [10,12] \\), \\( {\\textit{h}}({\\iota}) =x \\) and \\( ({\\textit{h}}\\geq 11)[do(\\move(2,z),{\\iota})] \\) } \\\\ \n 0 & \\text{otherwise}\n \\end{cases} \\qquad \\mbox{} \\hfill = \\\\[3ex]\n {{\\displaystyle}}\\frac{1}{\\gamma} \\int\\_x \\int\\_z \\begin{cases}\n.5\\times {{\\cal N}}(z+2;0,1) & \\text{if \\( {\\exists}{\\iota}. \\) $ x\\in [10,12]$, \\( {\\textit{h}}({\\iota}) = x \\) and \\( {\\textit{h}}({\\iota}) - z \\geq 11 \\)} \\\\ \n 0 & \\text{otherwise}\n \\end{cases} \\hfill = \\\\[3ex]\n {{\\displaystyle}}\\frac{1}{\\gamma} {{\\int\\_{-\\infty}^{\\infty}}}\\int\\_{10}^{12} \\begin{cases}\n.5\\times {{\\cal N}}(z+2;0,1) & \\text{if \\( x\\in [10,12] \\) and \\( x-z \\geq 11 \\)} \\\\\n 0 & \\text{otherwise}\n \\end{cases}\n \\end{array}$$ \n It is not hard to see that had the action been deterministic, the degree of belief in *h* ≥ 11 after moving away by 2 units should have been precisely 1. In Figure [fig:uniform-gaussian], we see the effect of this move, where the range of *h* values with non-zero densities extends considerably more than 2 units.\n2. $ {\\textit{Bel}}({\\textit{h}}\\geq 10, do([\\move(-2,-2.01), \\move(2,2.9)], {S\\_{\\!\\textrm{0}}})) \\approx.74 $\nThe argument proceeds in a manner identical to the previous demonstration. The density function is further multiplied by a factor of $ {{\\cal N}}(u-2; 0, 1) $, from and. More precisely, we have \n$$\\begin{array}{l} \n {{\\displaystyle}}\\frac{1}{\\gamma} \\int\\_x \\int\\_z \\int\\_u \\begin{cases}.5\\times{{\\cal N}}(z+2;0,1)\\times{{\\cal N}}(u-2;0,1) & \\text{if \\( {\\exists}{\\iota}. \\) \\( x\\in [10,12] \\), \\( {\\textit{h}}({\\iota})=x \\) and \\( {\\textit{h}}({\\iota}) - z- u \\geq 10 \\)} \\\\\n 0 & \\textrm{otherwise}\n \\end{cases} \\hspace{1cm} \\hfill = \\\\[3ex]\n{{\\displaystyle}}\\frac{1}{\\gamma} \\int\\_z \\int\\_u \\int\\_{10}^{12} \\begin{cases}\n.5\\times{{\\cal N}}(z+2;0,1)\\times{{\\cal N}}(u-2;0,1) & \\text{if \\( x\\in [10,12] \\) and \\( x - z - u \\geq 10 \\)} \\\\ \n 0 & \\text{otherwise}\n\\end{cases} \\end{array}$$ \n If the action were deterministic, yet again the degree of belief about *h* ≥ 10 would be 1 after the intended actions. That is, the robot moved away by 2 units and then moved towards the wall by another 2 units, which means that *h*’s current value should have been precisely what the initial value was.\nSee Figure [fig:uniform-gaussian] for the resulting density change. Intuitively, the resulting density changes as effectuated by the moves degrades the agent’s confidence considerably. In Figure [fig:uniform-gaussian], for example, we see that in contrast to a single noisy move, the range of *h* values considered possible has extended further, leading to a wide curve.\n3. ${\\textit{Bel}}({\\textit{h}}\\geq 11, do([\\move(-2, -2.01), {{\\textit{sonar}}}(11.5)], {S\\_{\\!\\textrm{0}}}) \\approx.94 $\nThis demonstrates the result of a sensing action after a noisy move. Using arguments analogous to those in the previous item, it is not hard to see that we have: \n$$\\begin{array}{l}\n{{\\displaystyle}}\\frac{1}{\\gamma} \\int\\_x \\int\\_ z \\begin{cases}\n.5\\times {{\\cal N}}(z+2; 0, 1) \\times {{\\cal N}}(x-z-11.5;0,.25) & \\text{if \\( {\\exists}{\\iota}.~x\\in [10,12], {\\textit{h}}({\\iota}) = x \\) and \\( \n {\\textit{h}}({\\iota}) - z \\geq 11 \\)} \\\\ \n 0 & \\text{otherwise}\n \\end{cases}\n \\end{array}$$\n4. ${\\textit{Bel}}({\\textit{h}}\\geq 11, do([\\move(-2, -2.01), {{\\textit{sonar}}}(11.5),"},"data_id":{"kind":"string","value":"arxiv_0002471"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":5,"numItemsPerPage":100,"numTotalItems":46503,"offset":500,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6Im1sR1pFYktIVHRVSGRhX1RRdEczQ0N1S3k5ME9Qd25DNHY5elRjM3FVYzgifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4Mzc4ODY3NCwianRpIjoiMTI4NjY4NDgtNTVlMi00YWQ2LTgxZDgtNGJlNjQ2NDdkMWYwIiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODM3OTIyNzQsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.LutUEElNdpOBFKpuOCU5q4oMpNg-wgsyiOXtQ1saUTSIPSnYDUsEcJDZqG7emg0FUpdmta4WgTEd7Y7_9g6gAw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
|---|---|
MIR selected red QSOs are unobscured, 37–40% are obscured by the torus, and the remaining 23–25% are obscured, but not by the torus. These values constrain the covering factor of the torus at these extreme luminosities to be ∼ 37–40%, and the half torus opening angle, measured from the torus axis, to be ∼ 67. This i... | arxiv_0002065 |
The Cox-Voinov law for traveling waves in the partial wetting regime
====================================================================
[1](#fn1)
We consider the thin-film equation ∂*t**h* + ∂*y*(*m*(*h*)∂*y*3*h*) = 0 in {*h* > 0} with partial-wetting boundary conditions and inhomogeneous mobility of the form *m*(*h*... | arxiv_0002066 |
ae in our Galaxy, M31 and LMC, respectively, which have reached absolute visual magnitude $M\_{\rm V} \lesssim -9.0$, suggests luminosities highly above the Eddington limit.
The super-Eddington state of novae was investigated by, who suggested a decrease of the effective opacity of the inhomogeneous atmosphere of novae... | arxiv_0002073 |
alous self-bound regions of the host galaxy from a past merger the host galaxy experienced. We excluded these galaxies as potential progenitors by removing galaxies with dark matter content < 107 $\textup{M}\_\odot$.
Because of the presence of diffuse stars and the possibility of particles being miss-assigned to the wr... | arxiv_0002076 |
Scale Dependence of Multiplier Distributions for Particle Concentration, Enstrophy and Dissipation in the Inertial Range of Homogeneous Turbulence
==================================================================================================================================================
### 12 December 2016 (Subm... | arxiv_0002077 |
regime the deviations from GR increase for smaller mass ratios, which corresponds with our analysis of the dependencies of the tidal and GB contributions in Fig. [fig:tidGBcontr]. In the QD regime, the net phase differences from GR are small. However for a coupling of *ε* = 0.0015 and mass ratio of *q* = 0.2 the phase... | arxiv_0002085 |
k^2$
*AMOS-CD* & $ \frac{64}{3}N^4$ & 96*N*5 & $ \frac{32N\_1N\_2}{3}\sum\_{i=1}^2 N\_k^2$ & $ \frac{96N\_1N\_2N\_3}{3}\sum\_{i=1}^3 N\_k^2$
*AOS-CD Banded* & 200*N*2 & 300*N*3 & 200*N*1*N*2 & 300*N*1*N*2*N*3
*AMOS-CD Banded* & 398*N*2 & 1195*N*3 & 398*N*1*N*2 & 1195*N*1*N*2*N*3
[tab:speedups]
[!htb]![image]... | arxiv_0002086 |
}}}\left( x \right)dx} = \frac{{\pi {{10}^{ - \frac{{2{P\_L}\left( {{d\_0}} \right)}}{{10n}}}}}}{{\left| {{{\cal V}\_{{d\_1}}}} \right|\Gamma \left( \xi \right)}} \left. {\left[ {g{{\left( x \right)}^{ - \frac{2}{n}}}\mathcal{G} \left( {\xi,\frac{z}{{\delta g\left( x \right)}}} \right) - {{\left( {\frac{z}{\delta }} \r... | arxiv_0002087 |
caging and release of the masses following launch and the micro-Newton thrusters. As much as possible the nominal LISA systems and hardware are being used. LISA Pathfinder is currently scheduled for launch in mid 2013, after which it will orbit the L1 point, with a 180-day mission plan.
Other missions
--------------
L... | arxiv_0002088 |
�= 0 or *m* ≥ 2 are always damped.
2. Growing modes exist where *θ* < *π*/3 or *θ* > 2*π*/3. They do not exist at the equator.
3. Modes only grow in the presence of finite (but not too large) magnetic diffusivity, and the fastest growing modes have growth rates [taylergamma] ~kr2 ~.
These results confirm the findings o... | arxiv_0002089 |
A graph convolutional autoencoder approach to model order reduction for parametrized PDEs
=========================================================================================
The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM). In the reduc... | arxiv_0002097 |
& & & & & &
& Magpl & & 12.58 & 12.22 & 11.44 & 10.87 & 10.51 & 9.31 & & 12.72 & 12.80 & 12.84 & 12.60 & 12.10 & 11.58
2*M*J & *A*band & & 26.33 & 16.84 & 10.60 & 5.84 & 4.57 & 8.24 & & 26.33 & 16.84 & 10.60 & 5.84 & 4.57 & 8.24
& Magexpected & & 38.91 & 29.06 & 22.04 & 16.71 & 15.08 & 17.55 & & 39.06 & 29.65 & ... | arxiv_0002099 |
Statistical analysis of *m/n* = 2/1 locked and quasi-stationary modes with rotating precursors at DIII-D
========================================================================================================
=4
A database has been developed to study the evolution, the nonlinear effects on equilibria, and the disrupti... | arxiv_0002102 |
Noncommutative solutions to Zamolodchikov’s tetrahedron equation and matrix six-factorisation problems
======================================================================================================
It is known that the local Yang–Baxter equation is a generator of potential solutions to Zamolodchikov’s tetrahedr... | arxiv_0002105 |
Decompositional Minimisation of Monolithic Processes
====================================================
Compositional minimisation can be an effective technique to reduce the state space explosion problem. This technique considers a parallel composition of several processes. In its simplest form, each sequential proc... | arxiv_0002111 |
A LOFAR study of non-merging massive galaxy clusters
====================================================
### Received...; accepted...
Introduction
============
Diffuse radio emission in galaxy clusters is caused by relativistic electrons that emit synchrotron radiation in the intra-cluster magnetic field (see and ref... | arxiv_0002117 |
Strength of the non-locality of two-qubit entangled states and its applications
===============================================================================
[theorem]Lemma [theorem]Corollary [theorem]Proposition [theorem]Conjecture
[theorem]Definition
Non-locality is a feature of quantum mechanics that cannot be exp... | arxiv_0002126 |
Long-term monitoring in IC4665: Fast rotation and weak variability in very low mass objects
===========================================================================================
### Accepted. Received.
[firstpage]
We present the combined results of three photometric monitoring campaigns targeting very low mass (V... | arxiv_0002133 |
Fitting functions for dark matter density profiles
==================================================
[firstpage]
We present a unified parameterization of the fitting functions suitable for density profiles of dark matter haloes or elliptical galaxies. A notable feature is that the classical Einasto profile appears nat... | arxiv_0002135 |
Controllability of the Jaynes-Cummings-Hubbard model
====================================================
In quantum control theory, the fundamental issue of controllability covers the questions whether and under which conditions a system can be steered from one pure state into another by suitably tuned time evolution ... | arxiv_0002138 |
Spin-dependent sum rules connecting real and virtual Compton scattering verified
================================================================================
We present a detailed derivation of the two sum rules relating the spin polarizabilities measured in real, virtual, and doubly-virtual Compton scattering. For... | arxiv_0002151 |
either encounters the interior of one of the polygons or follows for some time one of the sides of a polygon *R**j*1 *on the left* and for some time one of the sides of a polygon *R**j*2 (possibly the same) *on the right*.
Our main result is the following
[theorem.1] The damping *a* stabilizes uniformly the wave equat... | arxiv_0002157 |
The local-triviality dimension of actionsof compact quantum groups
==================================================================
We define the local-triviality dimension for actions of compact quantum groups on unital C\*-algebras. The resulting compact quantum principal bundle is said to be locally trivial when t... | arxiv_0002162 |
*π*1/2 will only be a local minima. If we have a kink and anti-kink pair separated by a length *l*, the potential energy linearly increases as *V*(*l*) = *l*/2 leading to charge confinement.
[tb] [fig:toysolve]
Below, we neglect spatial and quantum fluctuations, and simply treat it as a problem of classical dynamics ... | arxiv_0002173 |
and related topics Mount Holyoke, 1994*, ed. C. Dong and G. Mason, Contemporary Math., Vol. 193, Amer. Math. Soc., Providence, 1996, 203–236.
R. McRae, On the tensor structure of modules for compact orbifold vertex operator algebras, *Math. Z.* (2019), https://doi.org/10.1007/s00209-019-02445-z
R. McRae, Twisted modul... | arxiv_0002175 |
80.31 & 16.63 & 52720.0728 & 8.58 & 5.38 & 54906.1494 & -3.53 & 3.78
54189.9051 & -108.21 & 1.75 & 51648.8687 & 79.03 & 13.64 & 53095.8583 & -117.18 & 6.85 & 54951.9511 & -21.41 & 1.94
54192.9248 & -84.69 & 1.44 & 51650.8991 & 101.74 & 6.12 & 53095.8797 & -109.81 & 10.61 & 54951.9651 & -21.32 & 1.90
54192.9353 & ... | arxiv_0002176 |
**Colliding and Fixed Target Mode in a Single Experiment–A Novel Approach to Study the Matter under New Extreme Conditions**
Oleksandr V. Vitiuk1,, Valery M. Pugatch 2, Kyrill A. Bugaev 3, 4, Nazar S. Yakovenko 3,
Pavlo P. Panasiuk 3, Elizaveta S. Zherebtsova 5, 6, Vasyl M. Dobishuk 2, Sergiy B. Chernyshenko2,
Bo... | arxiv_0002177 |
----------
The presented agents differ between each other in the input game state (Json string or screen capture), the amount of learning time, the algorithm used. Additionally, some of the agents have been tested on a different set of games and sometimes using different game length (i.e., maximal number of game ticks ... | arxiv_0002179 |
Improved and Scalable Online Learning ofSpatial Concepts and Language Models with Mapping [1](#fn1)
====================================================================================================
### Received: date / Accepted: date
We propose a novel online learning algorithm, called SpCoSLAM 2.0, for spatial con... | arxiv_0002181 |
Time Step Rescaling Recovers Continuous-Time Dynamical Properties for Discrete-Time Langevin Integration of Nonequilibrium Systems
==================================================================================================================================
When simulating molecular systems using deterministic equa... | arxiv_0002183 |
On Klein’s *So-called Non-Euclidean geometry*
=============================================
In two papers titled *On the so-called non-Euclidean geometry, I* and *II* ( and ), Felix Klein proposed a construction of the spaces of constant curvature -1, 0 and and 1 (that is, hyperbolic, Euclidean and spherical geometry) ... | arxiv_0002188 |
Boltzmann hierarchies for self-interacting warm dark matter scenarios
=====================================================================
Introduction
============
In the standard cosmological paradigm, ΛCDM, Dark Matter (DM) has long been a necessary ingredient in the Big Bang model of the universe and in understand... | arxiv_0002189 |
On quaternionic contact Fefferman spaces
========================================
We investigate the Fefferman spaces of conformal type which are induced, via parabolic geometry, by the quaternionic contact (qc) manifolds introduced by O.Biquard. Equivalent characterizations of these spaces are proved: as conformal man... | arxiv_0002201 |
**Theory on Covariate-Adaptive Randomized Clinical Trials: Efficiency, Selection bias and Randomization Methods\***
[1](#fn1)
*School of Mathematical Sciences, Zhejiang University, Hangzhou 310027*
(Email:stazlx@zju.edu.cn)
The theocratical properties of the power of the conventional testing hypotheses and the ... | arxiv_0002209 |
Runge-Kutta convolution quadrature and FEM-BEM coupling for the time-dependent linear Schrödinger equation
==========================================================================================================
We propose a numerical scheme to solve the time-dependent linear Schrödinger equation. The discretization ... | arxiv_0002217 |
Degeneration of curves on some polarized toric surfaces
=======================================================
We address the following question: *Given a polarized toric surface $(S,{{\mathcal L}})$, and a general integral curve *C* of geometric genus *g* in the linear system $|{{\mathcal L}}|$, do there exist degene... | arxiv_0002229 |
*i*, we must also account for *Gaia*’s measurement uncertainty. Thus, **v**obs, *i* is related to its actual velocity by the equation
$$\mathbf{v}\_{\mathrm{obs},i} = \mathbf{v}\_i + \boldsymbol{\varepsilon}\_i,$$
where measurement errors $\boldsymbol{\varepsilon}\_i$ are drawn from Gaussian distributions with cov... | arxiv_0002232 |
High-precision stellar abundances of the elements
=================================================
Methods and applications
========================
### Received: date / Accepted: date
Efficient spectrographs at large telescopes have made it possible to obtain high-resolution spectra of stars with high signal-to-noise... | arxiv_0002233 |
**Gelfand–Tsetlin-type weight bases for all special linear Lie algebra
representations corresponding to skew Schur functions**
Robert G. Donnelly[1](#fn1)
and Molly W. Dunkum[2](#fn2)
*Dedicated to Robert A. (Bob) Proctor, in honor of his 65\tiny th birthday.*
We generalize the famous weight basis constructions of th... | arxiv_0002234 |
*ρ* distribution ∣*D**ρ**R*∣2 in Eq. ([eq:modCFrhodilep]), while using the invariant mass dependent version of the Cooper-Frye (CF) integral, namely ∫*d*3Σ*μ**p**μ* *n**ρ*(*M*). Note that apart from the invariant mass dependence, *n**ρ* is otherwise the same as in Ref. . The second option uses the vacuum description f... | arxiv_0002240 |
A novel scheme for the wave function renormalization of the composite operators
===============================================================================
We propose a novel renormalization scheme for the hadronic operators. The renormalization factor of the operator in this scheme is normalized by the correlation... | arxiv_0002243 |
Factorization for groomed jet substructure beyond the next-to-leading logarithm
================================================================================
Introduction
============
The high luminosity proton collisions at the Large Hadron Collider (LHC) enable an unprecedented sensitivity to rare and high scale ... | arxiv_0002247 |
_{\rm OBD} = -|\Gamma|\sum\_{i,j}
\left(\mathbf{S}\_{i}\cdot\mathbf{S}\_{j}\right)^{2},
\label{eqn:collinear}$$
that selects the most collinear spin configurations among the classical ground states (see, for example, Refs. []). The usual argument for the selection of such states is that collinear spin configurations... | arxiv_0002249 |
7748 & 0.8305 & 81.82 & 0.0 & 3.9841974 & 54967.88329 & 1.342 & 1.04 & 0.191& B
4386047 & 0.1716 & 0.3599 & 0.0753 & 87.62 & 0.002 & 2.90069 & 55001.19665 & 1.162 & 0.418 & - & B
4484356 & 0.3102 & 0.528 & 0.8263 & 79.62 & 0.0 & 1.1441593 & 54965.01868 & 0.949 & 0.501 & 0.572& B
4670267 & 0.4928 & 0.6643 & 0.8283... | arxiv_0002255 |
A Hybrid High-Order method for highly oscillatory elliptic problems
===================================================================
We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomi... | arxiv_0002257 |
The viscous overstability, nonlinear wavetrains, and finescale structure in dense planetary rings
=================================================================================================
,
This paper addresses the fine-scale axisymmetric structure exhibited in Saturn’s A and B-rings. We aim to explain both the... | arxiv_0002258 |
Observing Strategy for the SDSS-IV/MaNGA IFU Galaxy Survey
==========================================================
MaNGA (Mapping Nearby Galaxies at Apache Point Observatory) is an integral-field spectroscopic survey that is one of three core programs in the fourth-generation Sloan Digital Sky Survey (SDSS-IV). MaNG... | arxiv_0002262 |
The nuclear region of NGC 613. I - Multiwavelength analysis
===========================================================
### Accepted 2019 December 28. Received 2019 November 22; in original form 2019 August 16.
[firstpage]
In this paper, we report a detailed study with a variety of data from optical, near-infrared, X-r... | arxiv_0002263 |
**New thought on Matsumura-Nishida theory in the *L**p*-*L**q* maximal regularity framework**
=============================================================================================
This paper is devoted to proving the global well-posedness of initial-boundary value problem for Navier-Stokes equations describing ... | arxiv_0002267 |
Unified modelling of epidemics by coupled dynamics via Monte-Carlo Markov Chain algorithms.
===========================================================================================
Introduction
============
The Covid-19 pandemic has given rise to manifold new studies which aim at understanding the dynamics of the di... | arxiv_0002269 |
in each collision is chosen randomly. For shattering, the fragment mass distribution is determined following using the tensile strength appropriate for compact grains [in their notation, we adopt *Q*D⋆ = 8.9 × 109 cm2 s− 2 (valid for graphite; for silicate, *Q*D⋆ = 4.3 × 1010 cm2 s− 2)]. The maximum and minimum masse... | arxiv_0002272 |
order 2 × 10− 4.
[H]
|c|c|c|c|c| Function & Step & cEP & tMP & **cMP**
⟨*x*⟩ & 0.1 & 0.38 & **0.11** & **0.11**
& 0.05 & 0.19 & **2 × 10− 2** & **2 × 10− 2**
⟨∣*f*∣⟩ & 0.1 & 3.7 × 10− 5 & 0.3 & **1.2 × 10− 5**
& 0.05 & 1.8 × 10− 7 & 0.03 & **6 × 10− 16**
Catenoid
--------
The catenoid is an hyperbola rotate... | arxiv_0002275 |
ization of the following expression:
$$\chi^2=\sum\_{x,y}{\frac{1}{\sigma\_{x,y}^2}}[\{(R(x,y) \otimes
K(u,v)\} + B(x,y) -S(x,y)]^2,$$
where *σ**x*, *y*2 is the variance of a Gaussian distribution used to approximate the images Poisson statistics, S(x,y) is a single image, R(x,y) is the reference frame and B(x,... | arxiv_0002276 |
A heterogeneous group CNN for image super-resolution
====================================================
Convolutional neural networks (CNNs) have obtained remarkable performance via deep architectures. However, these CNNs often achieve poor robustness for image super-resolution (SR) under complex scenes. In this pape... | arxiv_0002280 |
Nonequilibrium transport in mesoscopic multi-terminal SNS Josephson junctions
=============================================================================
We report the results of several nonequilibrium experiments performed on superconducting/normal/superconducting (S/N/S) Josephson junctions containing either one or... | arxiv_0002288 |
regime. The radio data presented here were taken at eight frequencies at three different radio observatories but, for four of these frequency bands the radio coverage was very sparse. The fluxes and their associated standard errors are given in janskys, so they have already been normalized for the bandwidth of their r... | arxiv_0002298 |
for communications among processes. The MPI library on Garnet derives from the Argonne National Laboratory MPICH, which implements the MPI-3.0 standard. The code was compiled in double precision using the Intel Fortran Compiler XE version 14.0.2.144 with the `-fast` optimization level. The code performance is affected... | arxiv_0002299 |
Multiscale prediction of microstructure length scales in metallic alloy casting
===============================================================================
Microstructural length scales, such as dendritic spacings in cast metallic alloys, play an essential role in the properties of structural components. Therefore,... | arxiv_0002301 |
Chemical Power for Swarms of Microscopic Robots in Blood Vessels
================================================================
Microscopic robots in the bloodstream could obtain power from fuel cells using glucose and oxygen. Previous studies of small numbers of such robots operating near each other showed how robot... | arxiv_0002305 |
The vanishing volume of *D* = 4 superspace
==========================================
CBPF-NF-011/11
CPHT-RR045.0511
IHES/P/11/14
Imperial/TP/11/KSS/02
IPHT-T-11/124
KCL-MTH-11-11
NSF-KITP-11-069
Introduction
============
The problem of ultraviolet divergences in supergravity has attracted the attention of ... | arxiv_0002309 |
to the real mass with an error proportional to *N*− 1/2, where *N* is the sample number. Moreover, the information from every tracer particle is equally weighted, contrary to what happens for the virial theorem case.
Previous studies successfully used these mass estimator methods to “weigh” M31; and more recently, W10... | arxiv_0002312 |
Multiple stellar populations in the globular cluster NGC 1851[1](#fn1)[2](#fn2)
===============================================================================
Introduction
============
Spectroscopy (starting several decades ago) and more recently photometry (in particular in the near ultraviolet window) have shown tha... | arxiv_0002313 |
Fluid-fluid demixing and density anomaly in a ternary mixture of hard spheres
=============================================================================
We report the grand-canonical solution of a ternary mixture of discrete hard spheres defined on a Husimi lattice built with cubes, which provides a mean-field appro... | arxiv_0002317 |
Competitive Allocation of a Mixed Manna
=======================================
We study the fair division problem of allocating a mixed manna under additively separable piecewise linear concave (SPLC) utilities. A mixed manna contains goods that everyone likes and bads that everyone dislikes, as well as items that som... | arxiv_0002323 |
central charge. We have seen that the loop model can only be interpreted as a Potts model when *L* is even. Since (*l*1, *l*2) must necessarily have the same parity as *L*, we shall henceforth set *l*1 = 2*n*1 and *l*2 = 2*n*2. As far as the two-colored loop model is concerned, one could also consider the “twisted sec... | arxiv_0002326 |
A distance limited sample of massive molecular outflows[2](#fn2)
================================================================
### Accepted 2015 July 16. Received 2015 July 16; in original form 2015 Februray 23
[firstpage]
We have observed 99 mid-infrared-bright, massive young stellar objects and compact Hii regions... | arxiv_0002327 |
kind of actors neither can influence nor can be influenced by other members of the society. The collective decision-making model includes a procedure to reach an individual final decision, from a given initial decision of the participants. Finally, a global decision is taken by applying the single majority rule to the... | arxiv_0002328 |
:dNdzhigh]), shows that structural evolution in the IGM must also play a role.
### Characteristic sizes and spatial correlations
Multiple QSOs in proximity to each other on the sky provide an invaluable means for estimating the physical sizes of the regions in the IGM that give rise to the absorption features. Especial... | arxiv_0002332 |
Critical Balance and the Physics of MHD Turbulence
==================================================
A discussion of the advantages and limitations of the concept of critical balance, as employed in turbulence phenomenologies, is presented. The incompressible magnetohydrodynamic (MHD) case is a particular focus. The d... | arxiv_0002347 |
Coherent state coding approaches the capacity of non-Gaussian bosonic channels
==============================================================================
The additivity problem asks if the use of entanglement can boost the information-carrying capacity of a given channel beyond what is achievable by coding with sim... | arxiv_0002350 |
Efficient and Robust Semi-supervised Estimation of ATE with Partially Annotated Treatment and Response
======================================================================================================
A notable challenge of leveraging Electronic Health Records (EHR) for treatment effect assessment is the lack of p... | arxiv_0002351 |
Un théorème A de Quillen pourles.9ptcatégories strictes II :la preuve.9ptcatégorique
======================================================================================
Cet article est le second d’une série de deux articles consacrés à une généralisation du théorème A de Quillen aux -catégories strictes. Dans le pre... | arxiv_0002355 |
Testing Properties of Multiple Distributions with Few Samples
=============================================================
We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from *s* distributions, *p*1, *... | arxiv_0002360 |
that the notion of *simulation* in Definition [def:subdynamics] is rather restrictive. We should keep in mind that in general a *d*-dimensional subdynamics of a minimal Z*d* + 1-shift is not necessarily minimal.
Theorem [thm-main] implies the following rather surprising corollary: a quasiperiodic Z2-SFT can have a hig... | arxiv_0002363 |
Modelling and Verifying BDI Agents with Bigraphs
================================================
The Belief-Desire-Intention (BDI) architecture is a popular framework for rational agents; most verification approaches are based on reasoning about implementations of BDI programming languages. We investigate an alternati... | arxiv_0002365 |
Multiobjective Evolutionary Pruning of Deep Neural Networks with Transfer Learning for improving their Performance and Robustness
=================================================================================================================================
Evolutionary Computation algorithms have been used to solve ... | arxiv_0002369 |
Visualising Basins of Attraction for the Cross-Entropy and the Squared Error Neural Network Loss Functions
==========================================================================================================
Quantification of the stationary points and the associated basins of attraction of neural network loss sur... | arxiv_0002370 |
#1(see Fig. #1.)
1.4cm
***E*11-extended spacetime and gauged supergravities**
Fabio Riccioni and Peter West
*Department of Mathematics*
King’s College London
Strand London WC2R 2LS
UK
1.5cm
addtoresetequationsection
Introduction
============
One of the most surprising discoveries in the development of superg... | arxiv_0002383 |
nm, and *d* = 5 nm. The results are obtained with the red detuning *δ*K ∼ *δ*NV − Δm and initial state ∣*e*, 0, 0⟩.
Fig. [fig:f3] (a) shows, without mechanical amplification, the population of the mechanical mode can be neglected, and the dominant term is the spin-mangon coupling. As the center-of-mass motion is modul... | arxiv_0002392 |
Learning Asymmetric and Local Features in Multi-Dimensional Data through Wavelets with Recursive Partitioning
=============================================================================================================
learning of asymmetric and local features in images and other data observed on multi-dimensional gri... | arxiv_0002396 |
partial M/\partial
\rho\_c\right|\_{\rm tp} > 0$. We believe this reflects closely the properties of the BNS-merger remnant, which has a larger amount of angular momentum in the outer layers and a slowly rotating core, so that the reduced centrifugal support leads also to larger central rest-mass densities. On the basi... | arxiv_0002402 |
Accretion disc cooling and narrow absorption lines in the tidal disruption event AT2019dsg
==========================================================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
We present the results of a large multi-wavelength follow-up campaign of ... | arxiv_0002411 |
The Rank-One Theorem on $\rm RCD$ spaces
========================================
We extend Alberti’s Rank-One Theorem to RCD(*K*, *N*) metric measure spaces.
Introduction
============
The Rank-One Theorem in the Euclidean setting
---------------------------------------------
Let Ω be an open subset of $\mathbb R^n$, a... | arxiv_0002412 |
Energetics of collapsible channel flow with a nonlinear fluid-beam model
========================================================================
We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strai... | arxiv_0002417 |
the outflow direction (Fig. [case] (d)). There are clear peak offsets between the blueshifted and redshifted emission in the P–V diagrams perpendicular to the outflow direction (Fig. [case] (e) & (f)). In Case 3, the moment 1 map shows that the direction of the velocity gradient is from northeast to southwest (Fig. [c... | arxiv_0002422 |
Two-sided cartesian fibrations of synthetic (∞, 1)-categories
=============================================================
Within the framework of Riehl–Shulman’s synthetic (∞, 1)-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness ... | arxiv_0002425 |
Cosmology in nonlocal gravity
=============================
Introduction
============
Most often a prominent goal in the quantum gravity research is to achieve a short distance and time scale modification of Einstein’s general theory of relativity (GR) and to find its ultraviolet (UV) completion via demanding Unitarity... | arxiv_0002426 |
Modelling dust evolution in galaxies with a multiphase, inhomogeneous ISM
=========================================================================
[2]
We develop a model of dust evolution in a multiphase, inhomogeneous ISM including dust growth and destruction processes. The physical conditions for grain evolution are... | arxiv_0002428 |
'}\left[\frac{c\_{mn,K'-K}(c^\dagger )\_{nm,-K'}}{(\omega-\omega\_1) + K'\omega\_{21}+(\epsilon\_{mn}/\hbar+i V\_{mn}) }\right. \nonumber\\
&+\left.\frac{(c^\dagger )\_{mn,-K'}c\_{nm,K'-K}}{-(\omega-\omega\_1+K\omega\_{21})+(K-K')\omega\_{21}+(\epsilon\_{mn}/\hbar-i V\_{mn})} \right] \end{aligned}$$
$$\begin{align... | arxiv_0002435 |
A Noncommutative Gauss Map
==========================
The aim of this paper is to transfer the Gauss map, which is a Bernoulli shift for continued fractions, to the noncommutative setting. We feel that a natural place for such a map to act is on the AF algebra A considered separately by F. Boca and D. Mundici. The cent... | arxiv_0002436 |
Two-way sparsity for time-varying networks, with applications in genomics
=========================================================================
,
We propose a novel way of modelling time-varying networks, by inducing two-way sparsity on local models of node connectivity. This two-way sparsity separately promotes sp... | arxiv_0002437 |
Quantifying EPR: the resource theory of nonclassicality of common-cause assemblages
===================================================================================
Einstein-Podolsky-Rosen (EPR) steering is often (implicitly or explicitly) taken to be evidence for spooky action-at-a-distance. An alternative perspect... | arxiv_0002439 |
In this paper, we consider the evaluation of the effective action for photons coupled to charged scalar fields in the framework of a (2 + 1)-dimensional noncommutative spacetime. In order to determine the noncommutative Maxwell Lagrangian density, we follow a perturbative approach, by integrating out the charged scalar... | arxiv_0002442 |
Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order
=======================================================================================================
We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth orde... | arxiv_0002444 |
fragmentation brings in the membership disclosure protection. In general, the fragmentation process has the potential to be extended for other privacy models, because of its meta-approach, which is more easily generalizable. This might eventually provide unprecedented flexibility in using the fragmentation method as a... | arxiv_0002453 |
provides insight and instruction into more recent changes to the article template.
The “`acmart`” document class can be used to prepare articles for any ACM publication — conference or journal, and for any stage of publication, from review to final “camera-ready” copy, to the author’s own version, with *very* few chan... | arxiv_0002458 |
already defined in 1951, it was not until 1980 that the origin of the observed overabundances was understood as the result of mass transfer in a binary system. The polluting heavy elements were formerly produced within an asymptotic giant branch (AGB) companion, which is now a very faint white dwarf (WD) that cannot ... | arxiv_0002461 |
Study of LINER sources with broad H*α* emission.X-ray properties and comparison to luminous AGN and X-ray binaries.
===================================================================================================================
### Received / Accepted
Introduction
============
It is widely known that a large fracti... | arxiv_0002464 |
One-way communication complexity and non-adaptive decision trees[3](#fn3)
=========================================================================
We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function... | arxiv_0002465 |
Reasoning about Discrete and ContinuousNoisy Sensors and Effectors in Dynamical Systems
=======================================================================================
Among the many approaches for reasoning about degrees of belief in the presence of noisy sensing and acting, the logical account proposed by Bac... | arxiv_0002471 |
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