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samples/texts/1239855/page_1.md

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+Gap solitons in a two-channel microresonator structure
+
+Suresh Pereira and J. E. Sipe
+
+Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
+
+John E. Heebner and Robert W. Boyd
+
+Institute of Optics, University of Rochester, Rochester, New York 14627
+
+Received October 5, 2001
+
+We show that, when two channel waveguides are coupled by a sequence of periodically spaced microresonators, the group-velocity dispersion is low in the vicinity of the gap associated with the resonant frequency of the resonators. This low dispersion permits the excitation of a gap soliton with much lower energy than in a gap of similar width caused by Bragg reflection. © 2002 Optical Society of America
+
+OCIS codes: 190.5530, 190.4390.
+
+We consider nonlinear optical propagation in two
+channel waveguides coupled by periodically spaced
+microresonators [Fig. 1(a)]; we call such a device a
+two-channel, side-coupled, integrated, spaced sequence
+of resonators (SCISSOR).¹ The nonlinear properties
+of a similar structure with only one channel guide
+were studied previously,¹,² as were the linear proper-
+ties of the two-channel structure.³,⁴ In a bottom (top)
+mode, light traveling in the forward direction in the
+bottom (top) channel is coupled via the resonator to
+light traveling in the backward direction in the top
+(bottom) channel. Two types of gap open in the
+dispersion relation: Bragg gaps associated with the
+resonator spacing, *d*, and resonator gaps associated
+with *ρ*, the radius of the resonators. We show that,
+for a Kerr nonlinear SCISSOR structure, the propaga-
+tion of optical pulses is well described by a nonlinear
+Schrödinger equation (NLSE). The NLSE supports
+soliton solutions, and we find that much less energy is
+required for exciting a gap soliton in a resonator gap
+than in a Bragg gap with the same gap width.
+
+sociated with the waveguide; we ignore any small fre-
+quency dependence of n<sub>eff</sub>. Without loss of generality,
+we assume that light is traveling forward (backward)
+in the bottom (top) channel. At the coupling points we
+use the model<sup>1</sup>
+
+$$
+\begin{bmatrix} q(0_+) \\ l(a_+) \end{bmatrix} = \begin{bmatrix} \sigma_b & i\kappa_b \\ i\kappa_b & \sigma_b \end{bmatrix} \begin{bmatrix} q(0_-) \\ l(a_-) \end{bmatrix}, \quad (1)
+$$
+
+where $a_{\pm} = a \pm (\delta a)$ and $0_{\pm} = \pm \delta \theta$, where $\delta a$ and $\delta \theta$ are infinitesimal quantities. A similar expression is used for the top channel coupling point. To conserve energy, the coupling coefficients, $\sigma_b$, $\sigma_t$, $\kappa_b$, and $\kappa_t$, satisfy $|\sigma_i|^2 + |\kappa_i|^2 = 1$ and $\sigma_i^* \kappa_i = \sigma_i \kappa_i^*$, where $i = b, t$. Combining the effects of phase accumulation with those of coupling [Eq. (1)], we determine an expression that relates $l(d)$ and $u(d)$ to $l(0)$ and $u(0)$.
+
+Searching for the Bloch solution, we write $l(d) = \exp(ikd)l(0)$ and $u(d) = \exp(ikd)u(0)$, where $k$ is the Bloch wave number. Tracing the fields through the system, we find that
+
+$$
+\left[ \begin{array}{cc} \exp(ivd)(\beta_b\beta_t - \alpha^2) - \beta_t \exp(ikd) & \alpha \\ -\alpha & -\beta_t \exp(ikd) \end{array} \right] \left[ \begin{array}{c} l(0) \\ u(0) \end{array} \right] = 0, \quad (2)
+$$
+
+We begin in the linear regime and denote the elec-
+tric field in the bottom channel **L**(**r**) = **S**(x,y)l(z)ŷ, in
+the top channel **U**(**r**) = **S**(x,y)u(z)ŷ, and in the mi-
+croresonator **Q**(y,R,θ) = **T**(y,R)q(θ)ŷ, where **S**(x,y)
+[**T**(y,R)] is the mode profile associated with the chan-
+nel waveguides (resonator waveguides), **R** is the ra-
+dial variable, and θ is the angle within the resonator,
+measured counterclockwise from the bottom coupling
+points [see Fig. 1(b)]. We consider only the largest
+Cartesian component of the electric field in the chan-
+nel and resonator, and to make the nonlinear term
+in the equation more tractable we take it to be the y
+component.
+
+Away from the coupling points, the effect of propaga-
+tion is the accumulation of phase by means of propaga-
+tion constant v = n<sub>eff</sub>ω/c, where ω is the frequency of
+the light and n<sub>eff</sub> is the effective index of refraction as-
+
+where $\alpha = i\gamma\kappa_b\kappa_t \exp(i\pi\nu\rho)$, $\beta_i = [\sigma_i + i\gamma\sigma_i\kappa_i] \times \exp(2i\pi\nu\rho)$], and $\gamma = i[1 - \sigma_b\sigma_t \exp(2i\pi\nu\rho)]^{-1}$. Equation (2) has nontrivial solutions only when the determinant of the matrix vanishes, from which we find an expression for the wave number, $k(\omega)$, that we can invert to determine the dispersion relation, $\omega(k)$.
+
+We define the Bragg frequency, $\omega_b/c = \pi/(n_{\text{eff}}d)$,
+and the resonator frequency, $\omega_r/c = 1/(n_{\text{eff}}\rho)$.
+In Fig. 2 we plot the dispersion relation in the
+reduced band picture for a symmetric, two-channel
+SCISSOR structure with $n_{\text{eff}} = 3.47$, $\sigma_b = \sigma_t = 0.98$,
+$2\pi\rho = 26$ µm, and $d = 16$ µm. There are two types of
+gap: the 72nd-order Bragg gap at $\omega/c \approx 4.075$ µm$^{-1}$
+and the 59th-order resonator gap at $\omega/c \approx 4.11$ µm$^{-1}$.
+The upper and lower edges of the photonic bandgap
+occur at $k=0$, whereas for the resonator gap they
+occur at $k=0$, $\pi/d$. In the vicinity of a Bragg gap the

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+Fig. 1. (a) Schematic of the two-channel SCISSOR.
+(b) One unit cell of the structure. Filled circles, coupling points at the top and the bottom of the microresonator.
+
+curvature of the dispersion relation is high, whereas
+near a resonator gap the bands are almost completely
+flat.
+
+We now derive the NLSE that is relevant to the
+two-channel SCISSOR structure. We require the
+Bloch functions, $\mathbf{E}_{mk}(\mathbf{r})$, of the electric field,⁵ where
+$m$ is the index of the band. We can determine these
+functions by using the eigenvectors of Eq. (2) to find
+the electric field everywhere within one unit cell and
+then normalizing the field according to
+
+$$
+\int \frac{\mathrm{d}\mathbf{r}}{\Omega_{\mathrm{cell}}} n^2(\mathbf{r}) \mathbf{E}_{mk}^*(\mathbf{r}) \cdot \mathbf{E}_{m'k'}(\mathbf{r}) = \delta_{mn'} \delta_{kk'}, \quad (3)
+$$
+
+where $\Omega_{\text{cell}}$ is a normalization volume associated with
+one unit cell of the periodic medium. We assume that
+light is propagating in either a bottom or a top mode
+but not in both. We label the carrier wave number
+of the light $\bar{k}$, which corresponds to a frequency $\bar{\omega} =$
+$\omega(\bar{k})$, and introduce a field, $g_{m\bar{k}}(z, t)$, that is related to
+the energy in the electromagnetic field to lowest order
+through
+
+$$
+\epsilon = \int |g_{m\bar{k}}(z,t)|^2 dz . \tag{4}
+$$
+
+The NLSE is
+
+$$
+\left(i \frac{\partial}{\partial t} + i \omega_m \bar{k}' \frac{\partial}{\partial z}\right) g_{m\bar{k}}(\bar{z}, t) = -\frac{1}{2} \omega_{m\bar{k}''} \frac{\partial^2 g_{m\bar{k}}(\bar{z}, t)}{\partial z^2} \\
+- \Gamma_{m\bar{k}} |g_{m\bar{k}}(\bar{z}, t)|^2 g_{m\bar{k}}(\bar{z}, t), \quad (5)
+$$
+
+where $\omega_{m\bar{k}'} = \partial\omega/\partial k|_{\bar{k}}$ is the group velocity at the carrier wave number and $\omega_{m\bar{k}''} = \partial^2\omega/\partial k^2|_{\bar{k}}$ is the group-velocity dispersion. The nonlinear coefficient is given by
+
+$$
+\Gamma_{m\bar{k}} = \frac{3\bar{\omega}}{4A_{\text{eff}}\epsilon_0} \int_{\text{cell}} \frac{\mathrm{d}\mathbf{r}}{\Omega_{\text{cell}}} \chi^{(3)}(\mathbf{r}) |\mathbf{E}_{m\bar{k}}(\mathbf{r})|^4, \quad (6)
+$$
+
+where $\chi^{(3)}(\mathbf{r})$ is the nonlinear susceptibility of the
+medium and is assumed to have the same periodicity
+as the device and $A_{\text{eff}}$ is an effective area associated
+with the cross section of the channel waveguides.
+
+The region of validity of Eq. (5) has been extensively
+discussed.⁵,⁶
+
+When the carrier frequency of the field is at an up-
+per band edge, the NLSE [Eq. (5)] supports gap soliton
+solutions.⁷ We set $m = u$ to represent the upper band.
+The solitons have the form⁷
+
+$$
+g_{u\bar{k}}(z,t) = A \exp(iB_2z) \exp[-i(\delta + \Delta)t] \operatorname{sech}(B_1z - Ct),
+$$
+
+with $A = (-2\delta/\Gamma_{uk\bar{k}})^{1/2}$, $B_1 = (-2\delta/\omega_{uk''})^{1/2}$,
+$B_2 = (+2\Delta/\omega_{uk''})^{1/2}$, and $C = \omega_{uk''}B_1B_2$, where
+$\omega_{uk''}$ is the group-velocity dispersion at the upper
+band edge and where the signs of the detunings $\delta$ and
+$\Delta$ are chosen such that these coefficients come out to be
+real; $\delta$ determines the height and the spatial width of
+the soliton, whereas $\Delta$ determines the velocity. The
+center frequency of the soliton is $\omega_c = \omega_{uk\bar{k}} + \delta + \Delta$,
+and the frequency width of the pulse is denoted $C$. For most of the frequencies of the pulse to be contained
+within the gap we require that $\omega_c + 2C \le \omega_{uk\bar{k}}$. It
+is easy to confirm that this condition can be met for
+an arbitrary value of $C$ if we set $\Delta = C/(2M)$ and
+$\delta = -C(M/2)$, where $M \ge 4$. However, the pulse
+width is limited by the fact that NLSE (5) is valid
+only for frequencies slightly inside the gap. If we fix
+$M=4$ then we should have $C(\delta\bar{\omega})/20$, where $\bar{\delta}\bar{\omega}$ is
+the width of the gap. We have verified that a pulse
+of this width and central frequency is well described
+by the NLSE.
+
+Using the form of $g_{u\bar{k}}$ [Eq. (7)] in the expression for the energy [Eq. (4)], we find that $\epsilon_{u\bar{k}}^{\text{soliton}} = (2\sqrt{2|\delta|}/\Gamma_{u\bar{k}})\sqrt{\omega_{u\bar{k}}''}$. Because the group-velocity dispersion near a resonator gap is so much smaller than near a Bragg gap, the energy required for exciting a gap soliton with the same pulse width (C), and the same depth within the gap ($\delta + \Delta$), is much lower in a resonator gap; furthermore, the resonator soliton will travel with a slower group velocity and will
+
+Fig. 2. Dispersion relation for the two-channel SCIS-
+SOR with material parameters given in the text. The
+gap at $\omega/c = 4.11~\mu\text{m}^{-1}$ is associated with the 59th-
+order resonance of the microresonator. The gap at
+$\omega/c = 4.075~\mu\text{m}^{-1}$ is associated with Bragg reflection.
+These gaps are at typical communications wavelengths, as
+indicated by the right-hand axis.

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+Fig. 3. $S_{\text{sol}}$ is the ratio of the energy required for forming a gap soliton in a resonator gap to the energy required for forming the same gap soliton in a Bragg gap with the same gap width relative to its center frequency.
+
+consequently have a smaller spatial width. We define a quantity $S_{\text{sol}}(\bar{\delta}\omega) = \epsilon_{mk}^{\text{solitonのres}}/\epsilon_{mk}^{\text{solitonのBragg}}$, where $\epsilon_{mk}^{\text{solitonのres(Bragg)}}$ is the energy required for exciting a gap soliton in a resonator (Bragg) gap; $S_{\text{sol}}$ is a measure of how much easier it is to form a gap soliton in a resonator gap than in a Bragg gap. To make this comparison we consider one system in which $\omega_{uk}^-$ corresponds to the upper band edge of a resonator gap and another in which the same frequency $\omega_{uk}$ corresponds to the upper band edge of a Bragg gap. The overlap integrals that we use to determine the nonlinear coefficient, $\Gamma_{mk}$, are roughly equal at the gaps, so $S_{\text{sol}} \approx (\omega_{\text{res}''}/\omega_{\text{Bragg}'})^{1/2}$. We use physical parameters defined above but vary the values of $\sigma$ and $d$ to achieve different gap widths and center frequencies.
+
+In Fig. 3 we plot the value of $S_{\text{sol}}$ as a function of the gap width ($\bar{\delta}\omega/\omega_{uk}$). For a small gap width, $(\bar{\delta}\omega/\omega_{uk}) = 10^{-6}$, $S_{\text{sol}} = 10^{-4}$; for gap width $(\bar{\delta}\omega/\omega_{uk}) = 10^{-4}$, which is more realistic, $S_{\text{sol}} = 10^{-2}$. Of course, material and mode dispersion, both neglected in our calculations, will set a lower bound on $S_{\text{sol}}$. The low energy requirements for gap solitons in a resonator gap are balanced by a much longer soliton
+
+formation length,⁸ but for switching applications this restriction is not so important. A pulse with a form similar to Eq. (7) but with a much lower amplitude will be unable to propagate, because all its frequencies lie within the gap. By contrast, if the pulse has the correct amplitude, it will form into a soliton while it propagates. If the initial pulse is close to a soliton, then reshaping should be minimal.
+
+In conclusion, we have investigated optical propagation in a two-channel SCISSOR structure with a weak Kerr nonlinearity. We have presented a NLSE that accurately describes propagation near the band edges of a resonator gap if the light is propagating in only one mode of the system. The energy required for forming a gap soliton is much smaller than in a Bragg gap of similar width. We note, too, that whereas the one-channel SCISSOR structure investigated by Heebner et al.¹ supports solitons that can travel with a small group velocity, that velocity can never vanish; furthermore, that structure possesses no gap, so a true gap soliton could not be launched. We intend to extend the analysis to coupled gap solitons and to discuss the issues involved in experimentally launching and observing gap solitons.
+
+This research was supported by the Natural Science and Engineering Research Council of Canada and by Photonics Research Ontario. S. N. Pereira's e-mail address is pereira@physics.utoronto.ca.
+
+## References
+
+1. J. E. Heebner, R. W. Boyd, and Q.-H. Park, "Slow light, induced dispersion, enhanced nonlinearity and optical solitons in a resonator-array waveguide," submitted to Phys. Rev. Lett.
+
+2. J. E. Heebner, R. W. Boyd, and Q.-H. Park, "SCISSOR solitons and other novel propagation effects in micro-resonator modified waveguides," J. Opt. Soc. Am. B (to be published).
+
+3. B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, Opt. Lett. **25**, 344 (2000).
+
+4. A. Melloni, Opt. Lett. **26**, 917 (2001).
+
+5. N. A. R. Bhat and J. E. Sipe, Phys. Rev. E **64**, 0566-04 (2001).
+
+6. S. Pereira and J. E. Sipe, Phys. Rev. E **62**, 5745 (2000).
+
+7. C. M. de Sterke and J. E. Sipe, Opt. Lett. **14**, 871 (1989).
+
+8. G. P. Agrawal, *Non-Linear Fiber Optics* (Academic, San Diego, Calif., 1989).

samples/texts/142615/page_1.md

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+# One-to-One Disjoint Path Covers in DCell
+
+Xi Wang, Jianxi Fan, Baolei Cheng, Wenjun Liu, Yan Wang
+
+► To cite this version:
+
+Xi Wang, Jianxi Fan, Baolei Cheng, Wenjun Liu, Yan Wang. One-to-One Disjoint Path Covers in DCell. 10th International Conference on Network and Parallel Computing (NPC), Sep 2013, Guiyang, China. pp.61-70, 10.1007/978-3-642-40820-5_6 . hal-01513760
+
+HAL Id: hal-01513760
+
+https://hal.inria.fr/hal-01513760
+
+Submitted on 25 Apr 2017
+
+**HAL** is a multi-disciplinary open access
+archive for the deposit and dissemination of sci-
+entific research documents, whether they are pub-
+lished or not. The documents may come from
+teaching and research institutions in France or
+abroad, or from public or private research centers.
+
+L'archive ouverte pluridisciplinaire **HAL**, est
+destinée au dépôt et à la diffusion de documents
+scientifiques de niveau recherche, publiés ou non,
+émanant des établissements d'enseignement et de
+recherche français ou étrangers, des laboratoires
+publics ou privés.

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+Case 2. $\alpha \neq \beta$ and $(x, y) \in E(DCell_{\tau+1})$. Let $P_1 = \langle x, y \rangle$. Select $x_0 \in V(DCell_{\tau}^{\alpha})$ (resp. $y_0 \in V(DCell_{\tau}^{\beta})$), such that $(x, x_0) \in E(DCell_{\tau}^{\alpha})$ (resp. $(y, y_0) \in E(DCell_{\tau}^{\beta})$). According to the induction hypothesis, there exist $(\tau + 1)$ vertex disjoint paths $\{P'_i|2 \le i \le \tau + 2\}$ (resp. $\{Q'_j|2 \le j \le \tau + 2\}$) between any two distinct vertices $x$ and $x_0$ (resp. $y_0$ and $y$) in $DCell_{\tau}^{\alpha}$ (resp. $DCell_{\tau}^{\beta}$). Let $P''_2 = \langle x, x_0 \rangle$ (resp. $Q''_2 = \langle y_0, y \rangle$), $P'_i = \langle x, \dots, x_i, x_0 \rangle$ (resp. $Q'_j = \langle y_0, y_j, \dots, y \rangle$), and $P''_i = P'_i - (x_i, x_0)$ (resp. $Q''_j = Q'_j - (y_0, y_j)$) with $3 \le i \le \tau+2$ (resp. $3 \le j \le \tau+2$). Furthermore, let $z_i \in V(DCell_{\tau}^{\gamma_i})$ (resp. $w_j \in V(DCell_{\tau}^{\delta_j})$) with $2 \le i \le \tau+2$ (resp. $2 \le j \le \tau+2$) and $(x_i, z_i) \in E(DCell_{\tau+1})$ (resp. $(y_i, w_j) \in E(DCell_{\tau+1})$). Let $W_0 = \bigcup_{\theta=2}^{\tau+2} DCell_{\tau}^{\theta}$, $W_1 = \bigcup_{\theta=2}^{\tau+2} DCell_{\tau}^{\delta_{\theta}}$ and $W = W_0 \cup W_1 \cup DCell_{\tau}^{\alpha} \cup DCell_{\tau}^{\beta}$. For $2 \le i \le \tau + 2$, we can claim the following two subcases with respect to $DCell_{\tau}^{\gamma_i}$.
+
+Case 2.1. $DCell_{\tau}^{\gamma_i} \subseteq W_1$. Select $w_j \in V(DCell_{\tau}^{\gamma_i})$ such that $2 \le j \le \tau + 2$. According to Theorem 1, there exists a path a $S$ from $z_i$ to $w_j$ in $DCell_{\tau}^{\gamma_i}$. Furthermore, let $W = W \cup DCell_{\tau}^{\gamma_i}$ and $P_i = P_i'' + (x_i, z_i) + S + (w_j, y_j) + Q_j''$.
+
+Case 2.2. $DCell_{\tau}^{\gamma_i} \not\subseteq W_1$. Select $DCell_{\tau+1}^{\delta_j} \not\subseteq W$ such that $2 \le j \le \tau + 2$. Then, choose $DCell_{\tau}^{p}$ and $DCell_{\tau}^{q}$, such that three graphs $DCell_{\tau}^{p}$, $DCell_{\tau}^{q}$, and $W$ are internally vertex-independent with $p, q \in \{0, 1, \dots, t_k\}$. Let $W'_i = DCell_{\tau}^{\gamma_i} \cup DCell_{\tau}^{\delta_j} \cup DCell_{\tau}^{p} \cup DCell_{\tau}^{q}$, according to Lemma 4, there exists a path $S$ from $z_i$ to $w_j$ in $DCell_{\tau}[W'_i]$. Furthermore, let $W = W \cup W'_i$ and $P_i = P_i'' + (x_i, z_i) + S + (w_j, y_j) + Q_j''$.
+
+Furthermore, select $P_i$, such that $z_i \notin V(W_1)$ and $w_j \in V(W'_i)$ where $2 \le i \le \tau + 2$ and $2 \le j \le \tau + 2$. According to Lemma 4, there exists path $S$ from $z_i$ to $w_j$ in $DCell_{\tau+1}[V(W'_i) \cup (V(DCell_{\tau+1}) - V(W))]$. Furthermore, let $P_i = P_i'' + (x_i, z_i) + S + (w_j, y_j) + Q_j''$.
+
+According to above discussions, there exist $(\tau + 2)$ vertex disjoint paths $\{P_i|1 \le i \le \tau + 2\}$ between any two distinct vertices $x$ and $y$ of $DCell_{\tau+1}$.
+
+Case 3. $\alpha \neq \beta$ and $(x, y) \notin E(DCell_{\tau+1})$. Select $u \in V(DCell_{\tau+1})$ (resp. $v \in V(DCell_{\tau+1})$), such that $(x, u) \in E(DCell_{\tau+1})$ (resp. $(y, v) \in E(DCell_{\tau+1})$), $u \in DCell_{\tau}^{\phi}$ (resp. $v \in DCell_{\tau}^{\psi}$), and $\phi, \psi \in \{0, 1, \dots, t_k\}$, where $DCell_{\tau}^{\alpha}$ and $DCell_{\tau}^{\beta}$ (resp. $DCell_{\tau}^{\phi}$ and $DCell_{\tau}^{\psi}$) are internally vertex-independent. We can claim the following three subcases with respect to $u$ and $v$.
+
+Case 3.1. $u \in V(DCell_{\tau}^{\beta})$. Select $x_0 \in V(DCell_{\tau}^{\alpha})$, such that $(x, x_0) \in E(DCell_{\tau}^{\alpha})$. Let $y_0 = u$. According to the induction hypothesis, there exist $(\tau + 1)$ vertex disjoint paths $\{P'_i|2 \le i \le \tau + 2\}$ (resp. $\{Q'_j|1 \le j \le \tau + 1\}$) between any two distinct vertices $x$ and $x_0$ (resp. $y_0$ and $y$) in $DCell_{\tau}^{\alpha}$ (resp. $DCell_{\tau}^{\beta}$). Let $P_1 = (x, y_0) + Q'_1$ and $Q''_{\tau+2} = \emptyset$. Then, let $P''_2 = \langle x, x_0 \rangle$, $P'_i = \langle x, \dots, x_i, x_0 \rangle$ (resp. $Q'_j = \langle y_0, y_j, \dots, y \rangle$), and $P''_i = P'_i - (x_i, x_0)$ (resp. $Q''_j = Q'_j - (y_0, y_j)$) with $3 \le i \le \tau + 2$ (resp. $2 \le j \le \tau + 1$). Furthermore, let $z_i \in V(DCell_{\tau}^{\gamma_i})$ (resp. $w_j \in V(DCell_{\tau}^{\delta_j})$), where $2 \le i \le \tau + 2$ (resp. $1 \le j \le \tau + 1$), $w_{\tau+2} = v \in V(DCell_{\tau}^{\delta_{\tau+2}})$, and $(x_i, z_i) \in E(DCell_{\tau+1})$ (resp. $(y_i, w_j) \in E(DCell_{\tau+1})$). The required $\{P_i|2 \le i \le \tau + 2\}$ paths can be derived by the similar approach as the Case 2, so we skip it.

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+According to discussions in Case 3 and Case 3.1, there exist $(\tau + 2)$ vertex disjoint paths $\{P_i|1 \le i \le \tau + 2\}$ between any two distinct vertices $x$ and $y$ of $DCell_{\tau+1}$.
+
+Case 3.2. $v \in V(DCell_{\tau}^{\alpha})$. The required paths can be derived by the similar approach as the Case 3.1, so we skip it.
+
+Case 3.3. $u \notin V(DCell_{\tau}^{\beta})$ and $v \notin V(DCell_{\tau}^{\alpha})$. Let $P_1'' = Q_1'' = \emptyset$, $x_1 = x$, $z_1 = u$, $w_1 = v$ and $y_1 = y$. Select $x_0 \in V(DCell_{\tau}^{\alpha})$ (resp. $y_0 \in V(DCell_{\tau}^{\beta})$), such that $(x, x_0) \in E(DCell_{\tau}^{\alpha})$ (resp. $(y, y_0) \in E(DCell_{\tau}^{\beta})$). According to the induction hypothesis, there exist $(\tau+1)$ vertex disjoint paths $\{P_i'|2 \le i \le \tau+2\}$ (resp. $\{Q_j'|2 \le j \le \tau+2\}$) between any two distinct vertices $x$ and $x_0$ (resp. $y_0$ and $y$) in $DCell_{\tau}^{\alpha}$ (resp. $DCell_{\tau}^{\beta}$). Let $P_2'' = \langle x, x_0 \rangle$ (resp. $Q_2'' = \langle y_0, y \rangle$), $P_i' = \langle x, \dots, x_i, x_0 \rangle$ (resp. $Q_j' = \langle y_0, y_j, \dots, y \rangle$), and $P_i'' = P_i' - (x_i, x_0)$ (resp. $Q_j'' = Q_j' - (y_0, y_j)$) with $3 \le i \le \tau+2$ (resp. $3 \le j \le \tau+2$). Furthermore, let $z_i \in V(DCell_{\tau+1}^{\gamma_i})$ (resp. $w_j \in V(DCell_{\tau+1}^{\delta_j})$), where $2 \le i \le \tau+2$ (resp. $2 \le j \le \tau+2$) and $(x_i, z_i) \in E(DCell_{\tau+1})$ (resp. $(y_i, w_j) \in E(DCell_{\tau+1}))$. The required $\{P_i|1 \le i \le \tau+2\}$ paths can be derived by the similar approach as the Case 2, so we skip it.
+
+According to discussions in Case 3 and Case 3.3, there exist $(\tau + 2)$ vertex disjoint paths $\{P_i|1 \le i \le \tau + 2\}$ between any two distinct vertices $x$ and $y$ of $DCell_{\tau+1}$.
+
+In summary, for any two distinct vertices $x, y \in V(DCell_{\tau+1})$, there exist $(\tau+2)$ vertex disjoint paths $\{P_i|1 \le i \le \tau+2\}$ between any two distinct vertices $x$ and $y$ of $DCell_{\tau+1}$. $\square$
+
+**Lemma 6.** For any $t_0 \ge 3$ and $k \ge 0$, $DCell_k$ is $(k+t_0-1)$-DPC-able.
+
+*Proof.* We will prove this lemma by induction on the dimension $k$ of DCell. For any $t_0 \ge 3$, by Lemma 1, the lemma holds for $k=0$. For any $t_0 \ge 3$, supposing that the lemma holds for $k=\tau$, where $\tau \ge 0$, the proof that the lemma holds for $k=\tau+1$ is similar to that of lemma 5 and thus omitted. $\square$
+
+**Theorem 3.** $DCell_k$ is $(k+t_0-1)$-DPC-able with $k \ge 0$.
+
+*Proof.* By Lemma 1, the theorem holds for $k=0$ and $t_0 \ge 2$. By Lemma 2, the theorem holds for $k=1$ and $t_0=2$. By Lemma 5, the theorem holds for $k \ge 2$ and $t_0=2$. By Lemma 6, the theorem holds for $t_0 \ge 3$ and $k \ge 0$. $\square$
+
+# 4 Conclusions
+
+DCell has been proposed for one of the most important data center networks and can support millions of servers with outstanding network capacity and provide good fault tolerance by only using commodity switches. In this paper, we prove that there exist $r$ vertex disjoint paths $\{P_i|1 \le i \le r\}$ between any two distinct vertices $u$ and $v$ of $DCell_k$ ($k \ge 0$) where $n$ is the vertex number of $DCell_0$ and $r=n+k-1$. The result is optimal because of any vertex in $DCell_k$ has $r$ neighbors with $r=n+k-1$. According to our result, the method in [8,9] can be used to decrease deadlock and congestion in multi-cast routing in DCell.

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+Acknowledgments
+
+This work is supported by National Natural Science Foundation of China (61170021),
+Specialized Research Fund for the Doctoral Program of Higher Education (20103201110018),
+Application Foundation Research of Suzhou of China (SYG201240), and Grad-
+uate Training Excellence Program Project of Soochow University (58320235).
+
+References
+
+1. S.Ghemawat, H.Gobioff, S.Leung: The Google file system. ACM SIGOPS Operating Systems Review. 37(5), 29-43 (2003)
+
+2. J.Dean, S.Ghemawat: MapReduce: Simplified data processing on large clusters. Communications of the ACM. 51(1), 107-113 (2008)
+
+3. M.Isard, M.Budiu, Y.Yu, A.Birrell, D.Fetterly: Dryad: distributed data-parallel programs from sequential building blocks. ACM SIGOPS Operating Systems Review, 41(3), 59-72 (2007)
+
+4. C.Guo, H.Wu, K.Tan, L.Shi, Y.Zhang, S.Lu: Dcell: a scalable and fault-tolerant network structure for data centers. ACM SIGCOMM Computer Communication Review, 38(4), 75-86 (2008)
+
+5. C.Guo, G.Lu, D.Li, H.Wu, X.Zhang, Y.Shi, C.Tian, Y.Zhang, S.Lu: BCube: a high performance, server-centric network architecture for modular data centers. ACM SIGCOMM Computer Communication Review, 39(4), 63-74 (2009)
+
+6. D.Li, C.Guo, H.Wu, K.Tan, Y.Zhang, S.Lu: FiConn: Using backup port for server interconnection in data centers. IEEE INFOCOM, 2276-2285 (2009)
+
+7. D.Li, C.Guo, H.Wu, K.Tan, Y.Zhang, S.Lu, J.Wu: Scalable and cost-effective interconnection of data-center servers using dual server ports. IEEE/ACM Transactions on Networking, 19(1), 102-114 (2011)
+
+8. X.Lin, P.Philip, L.Ni: Deadlock-free multicast wormhole routing in 2-D mesh multi-computers. IEEE Transactions on Parallel and Distributed Systems, 5(8), 793-804 (1994)
+
+9. N.Wang, C.Yen, C.Chu: Multicast communication in wormhole-routed symmetric networks with hamiltonian cycle model. Journal of Systems Architecture, 51(3), 165-183 (2005)
+
+10. C.Lin, H.Huang, L.Hsu: On the spanning connectivity of graphs, Discrete Mathematics. Discrete Mathematics, 307(2):285-289, 2007.
+
+11. C.Lin, H.Huang, J.Tan, L.Hsu: On spanning connected graphs. Discrete Mathematics, 308(7):1330-1333, 2008.
+
+12. D.B.West and others: Introduction to graph theory. Prentice hall Englewood Cliffs, 2, (2001)
+
+13. J.Park, C.Kim, S.Lim: Many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements. IEEE Transactions on Parallel and Distributed Systems, 17(3), 227-240 (2006)
+
+14. R.Caha, V.Koubek: Spanning multi-paths in hypercubes. Discrete mathematics, 307(16), 2053-2066 (2007)
+
+15. X.Chen: Many-to-many disjoint paths in faulty hypercubes. Information Sciences, 179(18), 3110-3115 (2009)
+
+16. X.Chen: Paired many-to-many disjoint path covers of hypercubes with faulty edges.
+Information Processing Letters, 112(3), 61-66 (2012)

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+17. J.Park, H.Kim, H.Lim: Many-to-many disjoint path covers in the presence of faulty elements. IEEE Transactions on Computers, 58(4), 528-540 (2009)
+
+18. M.Ma: The spanning connectivity of folded hypercubes. Information Sciences, 180(17), 3373-3379 (2010)
+
+19. Y.Shih, S.Kao: One-to-one disjoint path covers on k-ary n-cubes. Theoretical Computer Science, 412(35), 4513-4530 (2011)
+
+20. H.Hsu, C.Lin, H.Hung, L.Hsu: The spanning connectivity of the (n,k)-star graphs. International Journal of Foundations of Computer Science, 17(2), 415-434 (2006)
+
+21. X.Chen: Unpaired many-to-many vertex-disjoint path covers of a class of bipartite graphs. Information processing letters, 110(6), 203-205 (2010)
+
+22. P.Huanga, L.Hsub: The spanning connectivity of line graphs. Applied Mathematics Letters, 24(9), 1614-1617 (2011)
+
+23. M.Kliegl, J.Lee, J.Li, X.Zhang, C.Guo, D.Rincon: Generalized DCell structure for load-balanced data center networks. IEEE INFOCOM Conference on Computer Communications Workshops, 1-5 (2010)

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+# One-to-One Disjoint Path Covers in DCell
+
+Xi Wang, Jianxi Fan*, Baolei Cheng, Wenjun Liu, and Yan Wang
+
+School of Computer Science and Technology, Soochow University, Suzhou 215006,
+China
+{20124027002, jxfan, chengbaolei, 20114027003, wangyanme}@suda.edu.cn
+
+**Abstract.** DCell has been proposed for one of the most important data center networks as a server centric data center network structure. DCell can support millions of servers with outstanding network capacity and provide good fault tolerance by only using commodity switches. In this paper, we prove that there exist $r$ vertex disjoint paths $\{P_i|1 \le i \le r\}$ between any two distinct vertices $u$ and $v$ of $DCell_k$ ($k \ge 0$) where $r = n+k-1$ and $n$ is the vertex number of $DCell_0$. The result is optimal because of any vertex in $DCell_k$ has $r$ neighbors with $r = n + k - 1$.
+
+**Keywords:** DCell, Data Center Network, Disjoint Path Covers, Hamiltonian
+
+## 1 Introduction
+
+Data centers become more and more important with the development of cloud computing. Specifically, in recent years, data centers are critical to the business of companies such as Amazon, Google, Facebook, and Microsoft, which have already owned tremendous data centers with more than hundreds of thousands of servers. Their operations are important to offer both many on-line applications such as web search, on-line gaming, email, cloud disk and infrastructure services such as GFS [1], Map-reduce [2], and Dryad [3].
+
+Researches showed that the traditional tree-based data center networks [4] have issues of bandwidth bottleneck, failure of single switch, etc.. In order to solve the defects of tree-based data center networks, there are many data center networks which have been proposed such as DCell [4], BCube [5], and FiConn [6, 7]. DCell has many good properties including exponential scalability, high network capacity, small diameter, and high fault tolerantly. In comparison with good capabilities of DCell, BCube is meant for container-based data center networks which only supports thousands of servers, and FiConn is not a regularly network which may raises the construction complexity.
+
+DCells use servers as routing and forwarding infrastructure, and the multi-cast routing frequency between servers are quite high in data center networks. Multi-cast routing algorithms in DCells can be based on the Hamiltonian model as methods on [8, 9]. One-to-one disjoint path covers (also named spanning connectivity [10, 11]) are the extension of the Hamiltonian-connectivity which could
+
+* Corresponding author.

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+as well as used on multi-cast routing algorithms in DCells to largely decrease
+deadlock and congestion, compared with tree-based multi-cast routing. How-
+ever, the problem of finding disjoint path covers is NP-complete [13]. Therefore,
+a large amount researches on problems of disjoint path covers focused on dif-
+ferent special networks, such as hypercubes [13–16], their variants [17–19], and
+others [20–22].
+
+So far there is no work reported about the one-to-one disjoint path cover
+properties of DCell. In this paper, we prove that there exist $r$ vertex disjoint
+paths $\{P_i|1 \le i \le r\}$ between any two distinct vertices $u$ and $v$ of $DCell_k$
+$(k \ge 0)$ where $n$ is the vertex number of $DCell_0$ and $r = n + k - 1$. The result
+is optimal because of any vertex in $DCell_k$ has $r$ neighbors with $r = n + k - 1$.
+
+This work is organized as follows. Section 2 provides the preliminary knowl-
+edge. Some basic one-to-one disjoint path covers properties are given in Section
+3. We make a conclusion in Section 4.
+
+## 2 Preliminaries
+
+A data center network can be represented by a simple graph $G = (V(G), E(G))$, where $V(G)$ represents the vertex set and $E(G)$ represents the edge set, and each vertex represents a server and each edge represents a link between servers (switches can be regarded as transparent network devices [4]). The edge from vertex $u$ to vertex $v$ is denoted by $(u, v)$. In this paper all graphs are simple and undirected.
+
+We use $G_1 \cup G_2$ to denote the subgraph induced by $V(G_1) \cup V(G_2)$ of $G$. For $U \subseteq V(G)$, we use $G[U]$ to denote the subgraph induced by $U$ in $G$, i.e., $G[U] = (U, E')$, where $E' = \{(u, v) \in E(G) | u, v \in U\}$. A path in a graph is a sequence of vertices, $P: <u_0, u_1, \dots, u_j, \dots, u_{n-1}, u_n>$, in which no vertices are repeated and $u_j, u_{j+1}$ are adjacent for $0 \le j < n$. Let $V(P)$ denote the set of all vertices appearing in $P$. We call $u_0$ and $u_n$ the terminal vertices of $P$. $P$ can be denoted by $P(u_0, u_n)$, which is a path beginning with $u_0$ and ending at $u_n$. Let $P_1$ denote $<u_1, u_2, \dots, u_{k-1}, u_k>$ and $P_2$ denote $<u_k, u_{k+1}, \dots, u_{k+n}>$, then $P_1 + P_2$ denotes the path $<u_1, u_2, \dots, u_k, u_{k+1}, \dots, u_{k+n}>$. If $e = (u_k, u_{k+1})$, then $P_1 + e$ denote the path $<u_1, u_2, \dots, u_k, u_{k+1}>$. Furthermore, if $e = (u_{k-1}, u_k)$, $P_1 - e$ denote the path $<u_1, u_2, \dots, u_{k-1}>$.
+
+A path in a graph $G$ containing every vertex of $G$ is called a Hamiltonian path ($HP$). $HP(u,v,G)$ can be denoted by a Hamiltonian path beginning with a vertex $u$ and ending with another vertex $v$ in graph $G$. Obviously, if $(v,u) \in E(G)$, then $HP(u,v,G) + (v,u)$ is a Hamiltonian cycle in $G$. A Hamiltonian graph is a graph containing a Hamiltonian cycle. $G$ is called a Hamiltonian-connected graph if there exists a Hamiltonian path between any two different vertices of $G$. Obviously, if $G$ is a Hamiltonian-connected graph, then $G$ must be the Hamiltonian graph. Suppose that $u$ and $v$ are two vertices of a graph $G$. We say a set of $r$ paths between $u$ and $v$ is an $r$-disjoint path cover in $G$ if the $r$ paths do not contain the same vertex besides $u$ and $v$ and their union covers all vertices of $G$. An $r$-disjoint path cover is abbreviated as an $r$-DPC for simplicity.

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+A graph $G$ is one-to-one $r$-disjoint path coverable ($r$-DPC-able for short) if there
+is an $r$-DPC between any two vertices of $G$. In this paper $G$ is $r$-DPC-able is
+not same as $G$ is $(r+1)$-DPC-able.
+
+For any other fundamental graph theoretical terminology, please refer to [12].
+
+DCell uses recursively-defined structure to interconnect servers. Each server
+connects to different levels of DCell through multiple links. We build high-level
+DCell recursively form many low-level ones. Due to this structure, DCell uses
+only mini-switches to scale out instead of using high-end switches to scale up,
+and it scales doubly exponentially with server vertex degree.
+
+We use $\textit{DCell}_k$ to denote a $k$-dimension DCell ($k \ge 0$), $\textit{DCell}_0$ is a complete graph on $n$ vertices ($n \ge 2$). Let $t_0$ denote the number of vertices in a $\textit{DCell}_0$, where $t_0 = n$. Let $t_k$ denote the number of vertices in a $\textit{DCell}_k$ ($k \ge 1$), where $t_k = t_{k-1} \times (t_{k-1} + 1)$. The vertex of $\textit{DCell}_k$ can be labeled by $[\alpha_k, \alpha_{k-1}, \dots, \alpha_i, \dots, \alpha_0]$, where $\alpha_i \in \{0, 1, \dots, t_{i-1}\}, i \in \{1, 2, \dots, k\}$, and $\alpha_0 \in \{0, 1, \dots, t_0 - 1\}$. According to the definition of $\textit{DCell}_k$ [4, 23], we provide the recursive definition as Definition 1.
+
+**Definition 1.** The *k*-dimensional DCell, *DCell*<sub>*k*</sub>, is defined recursively as
+follows.
+
+(1) $\mathrm{DCell}_0$ is a complete graph consisting of $n$ vertices labeled with $[0],[1],\dots,[n-1]$.
+
+(2) For any $k \ge 1$, $\mathrm{DCell}_k$ is built from $t_{k-1} + 1$ disjoint copies $\mathrm{DCell}_{k-1}$, according to the following steps.
+
+(2.1) Let $\mathcal{DCell}_{k-1}^0$ denote the graph obtained by prefixing the label of each vertex of one copy of $\mathcal{DCell}_{k-1}$ with 0. Let $\mathcal{DCell}_{k-1}^1$ denote the graph obtained by prefixing the label of each vertex of one copy of $\mathcal{DCell}_{k-1}$ with 1. ... . Let $\mathcal{DCell}_{k-1}^{t_{k-1}}$ denote the graph obtained by prefixing the label of each vertex of one copy of $\mathcal{DCell}_{k-1}$ with $t_{k-1}$. Clearly, $\mathcal{DCell}_{k-1}^0 \cong \mathcal{DCell}_{k-1}^1 \cong \cdots \cong \mathcal{DCell}_{k-1}^{t_{k-1}}$.
+
+(2.2) For any $\alpha_k, \beta_k \in \{0, 1, \dots, t_{k-1}\}$ and $\alpha_k \ge \beta_k (\text{resp. } \alpha_k < \beta_k)$, connecting the vertex $[\alpha_k, \alpha_{k-1}, \dots, \alpha_i, \dots, \alpha_1, \alpha_0]$ of $\mathcal{DCell}_{k-1}^{\alpha_k}$ with the vertex $[\beta_k, \beta_{k-1}, \dots, \beta_i, \dots, \beta_1, \beta_0]$ of $\mathcal{DCell}_{k-1}^{\beta_k}$ as follow:
+
+$$
+\left\{
+\begin{aligned}
+\alpha_k &= \beta_0 + \sum_{j=1}^{k-1} (\beta_j \times t_{j-1}) + 1 \\
+\beta_k &= \alpha_0 + \sum_{j=1}^{k-1} (\alpha_j \times t_{j-1})
+\end{aligned}
+\right.
+\qquad (1)
+$$
+
+```
+```
+
+```
+```
+
+```
+\[
+\text{(resp.}
+\]
+\begin{equation}
+\left\{
+\begin{aligned}
+\alpha_k &= \beta_0 + \sum_{j=1}^{k-1} (\beta_j \times t_{j-1}) \\
+\beta_k &= \alpha_0 + \sum_{j=1}^{k-1} (\alpha_j \times t_{j-1}) + 1
+\end{aligned}
+\right.
+\tag{2}
+\end{equation}
+```
+
+), where $\alpha_i, \beta_i \in \{0, 1, \dots, t_{i-1}\}, i \in \{1, 2, \dots, k\}$, and $\alpha_0, \beta_0 \in \{0, 1, \dots, t_0 - 1\}$.
+
+$$
+\text{(resp.}
+$$

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+By Definition 1, $DCell_{k-1}^{\alpha_k}$ is a subgraph of $DCell_k$, where $\alpha_k \in \{0, 1, \dots, t_{k-1}\}$.
+Figure 1(1), 1(2), and 1(3) demonstrate $DCell_0$, $DCell_1$, and $DCell_2$ with
+$t_0 = 2$ respectively. 1(4) and 1(5) demonstrate $DCell_0$ and $DCell_1$ with $t_0 = 3$
+respectively.
+
+**3 Main results**
+
+In this section, we will study one-to-one disjoint path cover properties of DCell.
+
+**Theorem 1.** $DCell_k$ ($k \ge 0$) is Hamiltonian-connected with $t_0 \ge 2$ except for $DCell_1$ with $t_0 = 2$. In other word, $DCell_k$ ($k \ge 0$) is 1-DPC-able with $t_0 \ge 2$ except for $DCell_1$ with $t_0 = 2$.
+
+*Proof.* We omit the proof due to the page limitation. $\square$
+
+**Theorem 2.** $DCell_k$ is a Hamiltonian graph for any $k \ge 0$. In other word, $DCell_k$ is 2-DPC-able for any $k \ge 0$.
+
+*Proof.* We omit the proof due to the page limitation. $\square$
+
+**Lemma 1.** $DCell_0$ is $(t_0 - 1)$-DPC-able with $t_0 \ge 2$.
+
+*Proof.* The lemma holds for $DCell_0$ which is a complete graph [12]. $\square$
+
+**Lemma 2.** $DCell_1$ is 2-DPC-able with $t_0 = 2$.
+
+*Proof.* $DCell_1$ is a cycle with 6 vertices. Therefore, $DCell_1$ is 2-DPC with $t_0 = 2$ [12]. $\square$
+
+**Lemma 3.** $DCell_2$ is 3-DPC-able with $t_0 = 2$.
+
+*Proof.* For $t_0 = 2$, we use construction method to proof this lemma. We can construct an 3-DPC between *u* and *v* in $DCell_2$ for any pair of vertices $\{u, v\} \in V(DCell_2)$.
+
+For example, the 3-DPC $\{P_1, P_2, P_3\}$ (resp. $\{R_1, R_2, R_3\}$, $\{T_1, T_2, T_3\}$, $\{S_1, S_2, S_3\}$, $\{U_1, U_2, U_3\}$) from $[0, 0, 0]$ to $[0, 0, 1]$ (resp. $[0, 1, 0]$, $[0, 1, 1]$, $[0, 2, 0]$, $[0, 2, 1]$) whose union covers $V(DCell_2)$ with $t_0 = 2$ are listed below (Similarly for the other cases).
+
+$$P_1 =< [0, 0, 0], [0, 0, 1] >,$$
+
+$$P_2 =< [0, 0, 0], [0, 1, 0], [0, 1, 1], [0, 2, 1], [0, 2, 0], [0, 0, 1] >,$$
+
+$$P_3 =< [0, 0, 0], [1, 0, 0], [1, 0, 1], [2, 0, 1], [2, 2, 0], [2, 2, 1], [2, 1, 1], [4, 1, 0], \\ 
+[4, 0, 0], [4, 0, 1], [4, 2, 0], [5, 2, 0], [5, 2, 1], [6, 2, 1], [6, 1, 1], [6, 1, 0], [6, 0, 0], [6, 0, 1], \\ 
+[6, 2, 0], [4, 2, 1], [4, 1, 1], [3, 1, 1], [3, 2, 1], [3, 2, 0], [5, 1, 1], [5, 1, 0], [5, 0, 0], [5, 0, 1], \\ 
+[1, 2, 0], [1, 2, 1], [1, 1, 1], [1, 1, 0], [3, 0, 1], [3, 0, 0], [3, 1, 0], [2, 1, 0], [2, 0, 0], [0, 0, 1] >.$$
+
+$$R_1 =< [0, 0, 0], [0, 1, 0] >,$$
+
+$$R_2 =< [0, 0, 0], [0, 0, 1], [0, 2, 0], [0, 2, 1], [0, 1, 1], [0, 1, 0] >,$$
+
+$$R_3 =< [0, 0, 0], [1, 0, 0], [1, 0, 1], [1, 2, 0], [1, 2, 1], [1, 1, 1], [1, 1, 0], [3, 0, 1], \\ 
+[3, 2, 0], [3, 2, 1], [3, 1, 1], [4, 1, 1], [4, 2, 1], [6, 2, 0], [6, 0, 1], [6, 0, 0], [6, 1, 0], [6, 1, 1], \\ 
+[6, 2, 1], [5, 2, 1], [5, 1, 1], [5, 1, 0], [5, 0, 0], [5, 0, 1], [5, 2, 0], [4, 2, 0], [4, 0, 1], [4, 0, 0], \\ 
+[4, 1, 0], [2, 1, 1], [2, 2, 1], [2, 2, 0], [2, 0, 1], [2, 0, 0], [2, 1, 0], [3, 1, 0], [3, 0, 0], [0, 1, 0] >.$$
+
+$$T_1 =< [0, 0, 0], [0, 1, 0], [0, 1, 1] >,$$
+
+$$T_2 =< [0, 0, 0], [0, 0, 1], [0, 2, 0], [0, 2, 1], [0, 1, 1] >,$$
+
+$$T_3 =< [0, 0, 0], [1, 0, 0], [1, 0, 1], [1, 2, 0], [1, 2, 1], [1, 1, 1], [1, 1, 0], [3, 0, 1], \\ 
+[3{.}{.} {o} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], \\ 
+{[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], \\ 
+{[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], {[} {o} {r} {l} {a} {n} {g} {o}], \\ 
+{[} {\bar{o}} {{\bar{o}}} {{\bar{o}}} {{\bar{o}}} {{\bar{o}}} ]$$

samples/texts/142615/page_8.md

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+**Fig. 1.** (1), (2), and (3) demonstrate $DCell_0$, $DCell_1$, and $DCell_2$ with $t_0 = 2$ respectively. (4) and (5) demonstrate $DCell_0$ and $DCell_1$ with $t_0 = 3$ respectively.

samples/texts/142615/page_9.md

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+$[6, 1, 1], [6, 1, 0], [6, 0, 0], [6, 0, 1], [6, 2, 0], [4, 2, 1], [4, 2, 0], [4, 0, 1], [4, 0, 0], [0, 1, 1] >$
+
+$S_1 = <[0, 0, 0], [0, 0, 1], [0, 2, 0]>$,
+
+$S_2 = <[0, 0, 0], [0, 1, 0], [0, 1, 1], [0, 2, 1], [0, 2, 0]>,$
+
+$S_3 = <[0, 0, 0], [1, 0, 0], [1, 0, 1], [1, 2, 0], [1, 2, 1], [1, 1, 1], [1, 1, 0], [3, 0, 1], [3, 0, 0], [3, 1, 0], [3, 1, 1], [4, 1, 1], [4, 1, 0], [4, 0, 0], [4, 0, 1], [4, 2, 0], [4, 2, 1], [6, 2, 0], [6, 0, 1], [6, 0, 0], [6, 1, 0], [2, 2, 1], [2, 1, 1], [2, 1, 0], [2, 0, 0], [2, 0, 1], [2, 2, 0], [5, 1, 0], [5, 1, 1], [3, 2, 0], [3, 2, 1], [6, 1, 1], [6, 2, 1], [5, 2, 1], [5, 2, 0], [5, 0, 1], [5, 0, 0], [0, 2, 0]>.
+
+$U_1 = <[0, 0, 0], [0, 0, 1], [0, 2, 0], [0, 2, 1]>,$
+
+$U_2 = <[0, 0, 0], [0, 1, 0], [0, 1, 1], [0, 2, 1]>,$
+
+$U_3 = <[0, 0, 0], [1, 0, 0], [1, 0, 1], [1, 2, 0], [1, 2, 1], [1, 1, 1], [1, 1, 0], [3, 0, 1], [3, 0, 0], [3, 1, 0], [2, 1, 0], [2, 0, 0], [2, 0, 1], [2, 2, 0], [5, 1, 0], [5, 0, 0], [5, 0, 1], [5, 2, 0], [4, 2, 0], [4, 0, 1], [4, 0, 0], [4, 1, 0], [2, 1, 1], [2, 2, 1], [6, 1, 0], [6, 1, 1], [6, 2, 1], [5, 2, 1], [5, 1, 1], [3, 2, 0], [3, 2, 1], [3, 1, 1], [4, 1, 1], [4, 2, 1], [6, 2, 0], [6, 0, 1], [6, 0, 0], [0, 2, 1]>$.
+
+□
+
+**Lemma** **4.** For any $\alpha,\beta \in \{0,1,\cdots,t_k\}$ , $m \in \{1,2,\cdots,t_k-3\}$ , and $\alpha \neq \beta$ ,
+let $x \in V(DCell_k^\alpha)$ be an arbitrary white vertex , $y \in V(DCell_k^\beta)$ be an arbitrary
+black vertex , and $G_0 = DCell_k^\alpha \cup DCell_k^\beta \cup (\bigcup_{\theta=0}^m DCell_k^{\omega_\theta})$, where $DCell_k^\alpha$,
+$DCell_k^\beta$, $DCell_k^{\omega_0}$ , ..., $DCell_k^{\omega_i}$ , ..., $DCell_k^{\omega_m}$ are internally vertex-independent
+with $i \in \{0,1,\cdots,m\}$ and $\omega_i \in \{0,1,\cdots,t_k\}$. Then there exists a path between
+$x$ and $y$ that containing every vertex in $DCell_k[V(G_0)]$ where $k \geq 1$ and $t_0 = 2$.
+
+*Proof.* Let $G_1 = DCell_k^\alpha \cup DCell_k^\beta$. Select $z \in V(DCell_k^\alpha)$ and $u \in V(DCell_k^\gamma)$,
+such that $z \neq x$, $(u,z) \in E(DCell_k)$, and $DCell_k^\gamma \subseteq G_0$, where two graphs
+$G_1$ and $DCell_k^\gamma$ are internally vertex-independent. Select $\omega \in V(DCell_k^\beta)$ and
+$v \in V(DCell_k^\delta)$, such that $\omega \neq y$, $(\omega,v) \in E(DCell_k)$, and $DCell_k^\delta \subseteq G_0$ where
+three graphs $G_1$, $DCell_k^\gamma$, and $DCell_k^\delta$ are internally vertex-independent. Ac-
+cording to Theorem **1**, there exists a path $P$ from $x$ to $z$ that containing every
+vertex in $DCell_k^\alpha$ and a path $Q$ from $\omega$ to $y$ that containing every vertex in
+$DCell_k^\beta$. Let $G_2 = G_0[V(\bigcup_{\theta=0}^m DCell_k^{\omega_\theta})]$. We can construct a path $S$ from $u$ to
+$v$ that containing every vertex in $G_2$ which is similar to Theorem **1**. Then there
+exists a path $P + (z,u) + S + (v,\omega) + Q$ between $x$ and $y$ that containing every
+vertex in $DCell_k[V(G_0)]$ where $k \geq 1$ and $t_0 = 2$. $\square$
+
+**Lemma** **5.** *DCell*<sub>*k*</sub> is (*k* + 1)-DPC-able with *k* ≥ 2 and *t*₀ = *2*.
+
+*Proof.* We will prove this lemma by induction on the dimension *k* of DCell.
+By lemma **3**, the lemma holds for *t*₀ = *2* and *k* = *2*. For *t*₀ = *2*, supposing
+that the lemma holds for *k* = *τ* (*τ* ≥ *2*), we will prove that the lemma holds for
+*k* = *τ* + *1*.
+
+For any vertex $x,y \in V(DCell_{\tau+1})$ with $x \neq y$. Let $x \in V(DCell_{\tau}^{\alpha})$ and $y \in V(DCell_{\tau}^{\beta})$ with $\alpha,\beta \in \{0,1,\dots,t_{\tau}\}$. We can identify $\alpha$ and $\beta$ as follows.
+
+Case **1.** $\alpha = \beta$. There exist $(\tau + 1)$ vertex disjoint paths $\{P_i|1 \le i \le \tau + 1\}$ between any two distinct vertices $x$ and $y$ of $DCell_{\tau}^{\alpha}$. Select $u \in V(DCell_{\tau}^{\gamma})$ and $v \in V(DCell_{\tau}^{\delta})$, such that $(x,u), (y,v) \in E(DCell_{\tau+1})$, where three graphs $DCell_{\tau}^{\alpha}$,$DCell_{\tau}^{\gamma}$,$$DCell_{\tau}^{\delta}$ are internally vertex-independent. According to Lemma **4**, there exists a path $P_{\tau+2}$ from $u$ to $v$ that visits every vertex in $DCell_{\tau+1}[V(DCell_{\tau+1}-DCell_{\tau}^{\alpha})]$. Then there exist $(\tau+2)$ vertex disjoint paths $\{P_i|1 \le i \le \tau + 2\}$ between any two distinct vertices $x$ and $y$ of $DCell_{\tau+1}$.

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+Fachbereich Informatik der Universität Hamburg
+
+Vogt-Kölln-Str. 30 ◊ D-22527 Hamburg / Germany
+
+University of Hamburg - Computer Science Department
+
+Mitteilung Nr. 297/00 • Memo No. 297/00
+
+# Obstacles on the Way to Spatial Reasoning with Description Logics: Undecidability of $\mathcal{ALC}_{RA}\ominus$
+
+(Slightly Revised Version)
+
+Michael Wessel
+
+Arbeitsbereich KOGS
+
+FBI-HH-M-297/00
+
+Oktober 2000

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+**Definition 11 (Concatenation Grammar)** A context-free grammar $G = (\mathcal{V}, \Sigma, \mathcal{P}, S)$ is called a *concatenation grammar* iff $\mathcal{P} \subseteq \mathcal{V} \times ((\mathcal{V} \cup \Sigma) \times (\mathcal{V} \cup \Sigma))$. $\square$
+
+We say that a language is a concatenation language iff it has a generating
+concatenation grammar. For example, the language $\{a, b\}$ is not a concatenation
+language. The language $\{a^n b^n \mid n \ge 1\}$ is a concatenation language, since it is
+generated by the grammar $(\{S, X\}, \{a, b\}, \{S \to a b, S \to a X, X \to S b\}, S)$.
+
+**Lemma 1** The intersection problem for concatenation languages is undecidable.
+$\square$
+
+**Proof 1** (Thanks to Harald Ganzinger who has suggested this proof) Let $G_1 = (\mathcal{V}_1, \Sigma_1, \mathcal{P}_1, S_1)$ and $G_2 = (\mathcal{V}_2, \Sigma_2, \mathcal{P}_2, S_2)$ be two arbitrary context-free grammars in Chomsky Normal Form.⁴ Let $\#\notin \mathcal{V}_1 \cup \mathcal{V}_2 \cup \Sigma_1 \cup \Sigma_2$ be a new terminal symbol, for $i \in \{1, 2\}: \Sigma'_i =_{def} \Sigma_i \cup \{\#\}, \mathcal{P}'_i =_{def} \{A \to B C \mid A \to B C \in \mathcal{P}_i\} \cup \{A \to a\# \mid A \to a \in \mathcal{P}_i\}$, and $G'_i = (\mathcal{V}_i, \Sigma'_i, \mathcal{P}'_i, S_i)$.
+
+Then, $G'_1 \cap G'_2 = \emptyset$ iff $G_1 \cap G_2 = \emptyset$. Since the latter is an undecidable problem for context-free grammars (e.g. see [21]), the former is undecidable as well. $\square$
+
+Given an arbitrary concatenation grammar, the key-observation is now that one
+can simply reverse the productions $\mathcal{P}$ of the grammar and get a role box $\mathfrak{R}$. If
+a word can be derived “top down” by the grammar using a derivation tree, then
+it is possible to “parse” this word in a bottom-up style using the role axioms.
+The following Lemma fixes the relationship between words that are derivable by
+a concatenation grammar and the models of the role box corresponding to this
+grammar:
+
+**Lemma 2** Let $\mathcal{G} = (\mathcal{V}, \Sigma, \mathcal{P}, S)$ be an arbitrary concatenation grammar. Let $w = w_1...w_n$ be a word, $w \in \Sigma^+$ with $|w| \ge 2$, and $\mathcal{I}$ be a model of $(\exists w_1...\exists w_n.\top,\mathfrak{R})$ with $\mathfrak{R}=_{def} \{B \circ C \sqsubseteq A \mid A \to B C \in \mathcal{P}\}$. Let $\langle x_0, x_1 \rangle \in w_1^\mathcal{I}, ... \langle x_{n-1}, x_n \rangle \in w_n^\mathcal{I}$ be an arbitrary path in the model $\mathcal{I}$ corresponding to $w$.
+
+Let $V \in V$ be an arbitrary non-terminal of $\mathcal{G}$. Then, $\langle x_0, x_n \rangle \in V^\mathcal{I}$ holds in all models $\mathcal{I}$ of $(\exists w_1...\exists w_n.\top,\mathfrak{R})$ iff there is a derivation of $w$ having $V$ as the root node: we write $V \xrightarrow{\cdot} w$. As a consequence, $\langle x_0, x_n \rangle \in S^\mathcal{I}$ in all models $\mathcal{I}$ of $(\exists w_1...\exists w_n.\top,\mathfrak{R})$ iff $w \in L(\mathcal{G})$. $\square$
+
+**Proof 2** "⇔" This can be shown using induction over the length of $w$.
+
+⁴A context-free grammar $G = (\mathcal{V}, \Sigma, \mathcal{P}, S)$ is in Chomsky Normal Form, iff $\mathcal{P} \subseteq \mathcal{V} \times ((\mathcal{V} \times \mathcal{V}) \cup \Sigma))$ (see [21]).

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+• If $|w| = 2$, $w = w_1w_2$, and $V \xrightarrow{+} w$, then there must be a production of the form $V \to w_1w_2 \in \mathcal{P}$. Note that there cannot be productions of the form $V \to w_1B$, $V \to Aw_2$, $V \to AB$, since $\mathcal{G}$ is a concatenation grammar – we would additionally need productions of the form $A \to w_1\ldots$, or even productions of the form $A \to \epsilon$. If $\mathcal{I}$ is a model of $\mathfrak{R}$ and $\langle x_0, x_1 \rangle \in w_1^\mathcal{I}$, $\langle x_1, x_2 \rangle \in w_2^\mathcal{I}$, then, due to $w_1 \circ w_2 \subseteq V \in \mathfrak{R}$ we have $\langle x_0, x_2 \rangle \in V^\mathcal{I}$ in every model $\mathcal{I}$.
+
+• Let $w = w_1 \dots w_n$, $n \ge 3$. Let $V \xrightarrow{+} w$. Since $\mathcal{G}$ is a concatenation grammar, there must be a production of the form $V \to XY \in \mathcal{P}$, and the following cases can occur:
+
+1. $X \in \mathcal{V}$, $Y \in \mathcal{\Sigma}$: then, there is a derivation $X \xrightarrow{+} w_1 \dots w_{n-1}$, and $Y = w_n$. Due to the induction hypothesis we have $\langle x_0, x_{n-1} \rangle \in X^{\mathcal{I}}$ in every model $\mathcal{I}$. Since we consider a model of $(\exists w_1...\exists w_{n-1}.\exists w_n.\top, \mathfrak{R})$, with $\langle x_{n-1}, x_n \rangle \in w_n^{\mathcal{I}}$, we have $\langle x_0, x_n \rangle \in V^{\mathcal{I}}$, because $\mathcal{I}$ is a model of $\mathfrak{R}$ with $X \circ w_n \subseteq V \in \mathfrak{R}$.
+
+2. $X \in \mathcal{\Sigma}$, $Y \in \mathcal{V}$: same argumentation.
+
+3. $X \in \mathcal{V}$, $Y \in \mathcal{V}$: let $w = uv$ be the partition of $w$ corresponding to the derivations $X \xrightarrow{+} u$, $Y \xrightarrow{+} v$. Let $u = w_1\dots w_i$, $v = w_{i+1}\dots w_n$. Due to the induction hypothesis we have $\langle x_0, x_i \rangle \in X^{\mathcal{I}}$ and $\langle x_{i+1}, x_n \rangle \in Y^{\mathcal{I}}$, since both $u$ and $v$ have a length smaller than $n$. We have $X \circ Y \subseteq V \in \mathfrak{R}$. This shows that $\langle x_0, x_n \rangle \in V^{\mathcal{I}}$.
+
+Summing up we have shown that $\langle x_0, x_n \rangle \in V^{\mathcal{I}}$ holds in all models $\mathcal{I}$ of $(\exists w_1...\exists w_n.\top, \mathfrak{R})$, if $V \xrightarrow{+} w$.
+
+“⇒” If $\langle x_0, x_n \rangle \in V^\mathcal{I}$ holds in all models $\mathcal{I}$ of $(\exists w_1...\exists w_n.\top, \mathfrak{R})$, then the presence of $\langle x_0, x_n \rangle \in V^\mathcal{I}$ is a logical consequence of $(\exists w_1...\exists w_n.\top, \mathfrak{R})$. Therefore, $\langle x_0, x_n \rangle \in V^\mathcal{I}$ is enforced by the role axioms in $\mathfrak{R}$. One can easily construct a derivation tree for $w$, showing that $V \xrightarrow{+} w$, by inspecting one of these models. More formally this could be shown using induction as well, and the proof would be very similar to the previous one.
+
+□
+
+Since we are trying to reduce the intersection problem of concatenation grammars to the satisfiability problem of $\mathcal{ALC}_{\mathbb{R}\mathcal{A}}^\ominus$, we have to deal with two grammars. Please note that concatenation grammars are not closed under intersection (i.e. for two grammars $\mathcal{G}_1$ and $\mathcal{G}_2$ there is in general no concatenation grammar $\mathcal{G}_{1,2}$ such that $\mathcal{L}(\mathcal{G}_{1,2}) = \mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2)$). In order to deal with this problem we have two put two concatenation grammars into one role box:
+
+**Lemma 3** Let $\mathcal{G}_1 = (\nu_1, \Sigma_1, P_1, S_1)$ and $\mathcal{G}_2 = (\nu_2, \Sigma_2, P_2, S_2)$ be two arbitrary concatenation grammars. Without loss of generality can assume $\nu_1 \cap \nu_2 = \emptyset$,

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+since we can always consistently rename the variables in one of the grammars
+and get $\mathcal{V}_1 \cap \mathcal{V}_2 = \emptyset$.
+
+For $i \in \{1, 2\}$, we define $\mathfrak{R}_i =_{def} \{ B \circ C \subseteq A \mid A \to B C \in \mathcal{P}_i \}$.
+Let $\Sigma =_{def} \Sigma_1 \cup \Sigma_2$ and $\mathfrak{R} =_{def} \mathfrak{R}_1 \cup \mathfrak{R}_2$.
+
+Then, for $i \in \{1, 2\}$, $w \in \mathcal{L}(\mathcal{G}_i)$ iff $\langle x_0, x_n \rangle \in S_i^\mathcal{I}$ in all models $\mathcal{I}$ of $(\exists w_1...\exists w_n.\top, \mathfrak{R})$. Obviously, $w \in \mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2)$ iff $\langle x_0, x_n \rangle \in S_1^\mathcal{I} \cap S_2^\mathcal{I}$ in all models $\mathcal{I}$ of $(\exists w_1...\exists w_n.\top, \mathfrak{R})$. $\square$
+
+**Proof 3** An easy consequence of the previous Lemma and of the requirement that $\mathcal{V}_1 \cap \mathcal{V}_2 = \emptyset$ (the derivation trees do not become "mixed", i.e. each grammar solely uses its own productions). $\square$
+
+As an application of this Lemma, let us consider the two grammars
+
+* $\mathcal{G}_1 = (\{\mathcal{S}_1\}, \{\boldsymbol{a}, \boldsymbol{b}\}, \mathcal{P}_1, S_1)$, where
+  $\mathcal{P}_1 = \{\boldsymbol{S}_1 \to \boldsymbol{a}\boldsymbol{b} \mid a\boldsymbol{S}_1\boldsymbol{b}\}$,
+
+* $\mathcal{G}_2 = (\{\mathcal{S}_2\}, \{\boldsymbol{a}, \boldsymbol{b}\}, \mathcal{P}_2, S_2)$, where
+  $\mathcal{P}_2 = \{\boldsymbol{S}_2 \to \boldsymbol{a}\boldsymbol{a}\boldsymbol{b} \mid aa\boldsymbol{S}_2\boldsymbol{b}\boldsymbol{b}\}$.
+
+Obviously, $\mathcal{L}(\mathcal{G}_1) = \{\boldsymbol{a}^n\boldsymbol{b}^n \mid n \ge 1\}$ and $\mathcal{L}(\mathcal{G}_2) = \{\boldsymbol{a}^{2n}\boldsymbol{b}^{2n} \mid n \ge 1\}$. Transformed into concatenation grammars we get
+
+* $\mathcal{G}'_1 = (\{\mathcal{S}_1, A\}, \{\boldsymbol{a}, \boldsymbol{b}\}, \mathcal{P}'_1, S_1)$, where
+  $\mathcal{P}'_1 = \{\boldsymbol{S}_1 \to \boldsymbol{a}\boldsymbol{b} \mid aA, A \to S_1\boldsymbol{b}\}$, and
+
+* $\mathcal{G}'_2 = (\{\mathcal{S}_2, B, C, D, E, F\}, \{\boldsymbol{a}, \boldsymbol{b}\}, \mathcal{P}'_2, S_2)$, where
+  $\mathcal{P}'_2 = \{\begin{aligned}[t]
+    &\boldsymbol{S}_2 \to aB, B \to aC, C \to bb, \\
+    &S_2 \to aD, D \to aE, E \to S_2F, F \to bb
+  \end{aligned}}$
+
+The corresponding role box is
+
+$$
+\begin{align*}
+\mathfrak{R} &= \left\{
+  \begin{array}{@{}l@{}}
+    a \circ b \sqsubseteq S_1, a \circ A \sqsubseteq S_1, S_1 \circ b \sqsubseteq A \\
+    a \circ B \sqsubseteq S_2, a \circ C \sqsubseteq B, b \circ b \sqsubseteq C, \\
+    a \circ D \sqsubseteq S_2, a \circ E \sqsubseteq D, S_2 \circ F \sqsubseteq E, b \circ b \sqsubseteq F
+  \end{array}
+\right\} \\
+&\cup \\
+&\left\{
+  \begin{array}{@{}l@{}}
+    a^0 b^0 c^0 d^0 e^0 f^0 \\
+    a^1 b^1 c^1 d^1 e^1 f^1 \\
+    a^2 b^2 c^2 d^2 e^2 f^2 \\
+    a^3 b^3 c^3 d^3 e^3 f^3
+  \end{array}
+\right\}.
+\end{align*}
+$$
+
+The “first part” of this role box corresponds to $\mathcal{P}'_1$, and the “sec-
+ond part” to $\mathcal{P}'_2$. The symbols of the grammars correspond to roles
+now. Please consider $(\forall S_1.C \sqcap \forall S_2.D \sqcap \exists a.\exists a.\exists b.\exists b.\neg(C \sqcap D), \mathfrak{R})$. Any
+model of $(\forall S_1.C \sqcap \forall S_2.D \sqcap \exists a.\exists a.\exists b.\exists b.\neg(C \sqcap D), \mathfrak{R})$ would also be a model of
+$(\exists a.\exists a.\exists b.\exists b.\top, \mathfrak{R})$, and must therefore contain $\langle x_0, x_4\rangle \in S_1^\mathcal{I} \cap S_2^\mathcal{I}$, because
+$w = aabb \in \mathcal{L}(\mathcal{G}'_1) \cap \mathcal{L}(\mathcal{G}'_2)$, due to Lemma 3. Since $x_0 \in (\forall S_1.C \sqcap \forall S_2.D)^\mathcal{I}$ also

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+Figure 6: “Bottom up parsing” of $aabb \in \mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2)$
+
+$x_4 \in (C \sqcap D)^I$ must hold, which obviously contradicts $x_4 \in (\neg(C \sqcap D))^I$. The example is therefore unsatisfiable. Considering Figure 6, it can be seen that the role box performs a “bottom up parsing” of the word $aabb$ – the two derivation trees shown in the figure can be immediately discovered as role compositions in the graph.
+
+We can now prove the main result of this section by showing how to reduce the intersection problem of concatenation grammars to the satisfiability problem of $\mathcal{ALC}_{\mathcal{R}A\ominus}$:
+
+**Theorem 1** The satisfiability problem of $\mathcal{ALC}_{\mathcal{R}A\ominus}$ is undecidable. $\square$
+
+**Proof 4** We give an example for a pair $(E, \mathfrak{R})$ for which no algorithm exists that is capable of checking its satisfiability.
+
+Let $\mathcal{G}_1 = (\mathcal{V}_1, \Sigma_1, \mathcal{P}_1, S_1)$ and $\mathcal{G}_2 = (\mathcal{V}_2, \Sigma_2, \mathcal{P}_2, S_2)$ be two arbitrary concatenation grammars. Without loss of generality we assume $\mathcal{V}_1 \cap \mathcal{V}_2 = \emptyset$.
+
+For $i \in \{1, 2\}$, we define $\mathfrak{R}_i =_{def} \{B \circ C \subseteq A \mid A \to B C \in \mathcal{P}_i\}$.
+Let $\Sigma =_{def} \Sigma_1 \cup \Sigma_2$ and $\mathfrak{R} =_{def} \mathfrak{R}_1 \cup \mathfrak{R}_2$. Let $R? \notin \text{roles}(\mathfrak{R})$, and let
+
+$$ 
+\begin{array}{l}
+\mathfrak{R}' =_{def} \mathfrak{R} \cup \{ R \circ S \sqsubseteq R? \mid R, S \in (\{R?\} \cup \text{roles}(\mathfrak{R})), \\
+\quad \neg\exists ra \in \mathfrak{R} : \text{pre}(ra) = (R, S) \}
+\end{array}
+$$
+
+be the completion of $\mathfrak{R}$.
+
+Then, $(E, \mathfrak{R}')$ is satisfiable iff $\mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2) = \emptyset$, where
+
+$$ 
+\begin{align*}
+E &=_{def} X \sqcap \neg(C \sqcap D) \sqcap Y \sqcap \forall S_1.C \sqcap \forall S_2.D, &\text{with} \\
+X &=_{def} \sqcap_{a \in \Sigma} \exists a.\top &\text{and} \\
+Y &=_{def} \sqcap_{R \in \text{roles}(\mathfrak{R}')}\forall R.(X \sqcap \neg(C \sqcap D)).
+\end{align*}
+$$

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+Since $\mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2) = \emptyset$ is undecidable, the satisfiability of $(E, \mathfrak{R}')$ is undecid-
+able as well.
+
+We have to show that $(E, \mathfrak{R}')$ is satisfiable iff $\mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2) = \emptyset$:
+
+⇒ We prove the contra-positive: if $\mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2) \neq \emptyset$, then $(E, \mathfrak{R}')$ is unsatisfiable. Assume to the contrary that $\mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2) \neq \emptyset$, but $(E, \mathfrak{R}')$ is satisfiable. Let $\mathcal{I}$ be a model of $(E, \mathfrak{R}')$. Because $\mathcal{I}$ satisfies $\mathfrak{R}'$, it holds that $\langle x_0, x_n \rangle \in (\bigcup_{R \in \text{roles}(\mathfrak{R}')} R^\mathcal{I})^+$ implies $\langle x_0, x_n \rangle \in *^\mathcal{I}$, where $*^\mathcal{I} =_{def} \bigcup_{R \in \text{roles}(\mathfrak{R}')} R^\mathcal{I}$ is the so-called universal relation. This is ensured by the fact that the composition of two arbitrary roles from $\text{roles}(\mathfrak{R}')$ is always defined in $\mathfrak{R}'$, due to the completion process. Since $\mathcal{I}$ is a model of $E$, there is some $x_0 \in E^\mathcal{I}$. Due to $x_0 \in (X \sqcap Y)^\mathcal{I}$ it holds that $x_0 \in ((\coprod_{a \in \Sigma} \exists a.\top) \sqcap (\coprod_{R \in \text{roles}(\mathfrak{R}')}\forall R.(\coprod_{a \in \Sigma} \exists a.\top)))^\mathcal{I}$. The model $\mathcal{I}$ therefore represents all possible words $w \in \Sigma^+$. Let $w \in \mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2)$, with $w = w_1...w_{n-1}w_n$. Obviously, $\mathcal{I}$ is also a model of $\exists w_1...\exists w_n.\top$, with $x_0 \in (\exists w_1...\exists w_n.\top)^\mathcal{I}$. Let $\langle x_0, x_1 \rangle = w_1^\mathcal{I}, ..., x_{n-1}, x_n \rangle = w_n^\mathcal{I}$ be a path in the model corresponding to $w$; $\langle x_0, x_n \rangle \in *^\mathcal{I}$ holds. If $\mathcal{I}$ is a model of $\mathfrak{R}'$, then it is also a model of $\mathfrak{R}$ due to $\mathfrak{R} \subseteq \mathfrak{R}'$, and therefore Lemma 3 is applicable. Due to Lemma 3 we then have $\langle x_0, x_n \rangle \in S_1^\mathcal{I} \cap S_2^\mathcal{I}$ in every model, and since $x_0 \in (\forall S_1.C \sqcap \forall S_2.D)^\mathcal{I}$, $x_n \in (C \sqcap D)^\mathcal{I}$ must also hold in every model. However, this obviously contradicts $x_n \in (\neg(C \sqcap D)^\mathcal{I})$ which must hold because $\langle x_0, x_n \rangle \in *^\mathcal{I}$ and $x_0 \in (\coprod_{R \in \text{roles}(\mathfrak{R}')}\forall R.(\neg(C \sqcap D)^\mathcal{I}))^\mathcal{I}$. This shows that there are no models. $(E, \mathfrak{R}')$ is therefore unsatisfiable.
+
+⇔ If $\mathcal{L}(\mathcal{G}_1) \cap \mathcal{L}(\mathcal{G}_2) = \emptyset$, then we show that $(E, \mathfrak{R}')$ is satisfiable by constructing an infinite model. The model $\mathcal{I}$ is constructed incrementally, e.g. $\mathcal{I}_0 \subset \mathcal{I}_1 \subset \mathcal{I}_2 \subset \dots \subset \mathcal{I}_\omega$, $\mathcal{I} = \mathcal{I}_\omega$. We refer to the set $\bigcup_{a \in \Sigma} a^\mathcal{I}$ as the *skeleton* of the model $\mathcal{I}$. The skeleton has the form of an infinite tree. An illustration of $\mathcal{I}$ is given in Figure 7; the thick lines correspond to the skeleton. Each node in the model $\mathcal{I}$ has $|\Sigma|$ different *direct successors* in the skeleton; the skeleton of $\mathcal{I}$ is a tree with branching factor $|\Sigma|$.
+
+For each $n \in \mathbb{N} \cup \{\emptyset\}$, the skeleton of the interpretation $\mathcal{I}_n$ is a tree of depth $n$, encoding all words $w$ with $|w| \le n$, i.e. $w \in \bigcup_{i \in \{0, ..., n\}} \Sigma^i$. Each word $w$ of length $i = |w|$, $i \le n$, corresponds to a path from the root node $x_{0,0}$ to some node $x_{i,m}$ at depth $i$, in all skeletons of the models $\mathcal{I}_n$. Therefore, the skeleton of $\mathcal{I}$ represents all words from $\Sigma^+$.
+
+Intuitively, the terminal symbols of the words to be parsed by the role box are represented as *direct edges* in the skeleton of the model, whereas the *indirect edges* in this model are inserted to mimic the “bottom-up parsing process” of these words, which is performed by the role box. The construction of $\mathcal{I}$ therefore works as follows:

samples/texts/2395852/page_16.md

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+Figure 7: Illustration of the constructed model for (E, 𝓶')

samples/texts/2395852/page_17.md

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+4. $C^{\mathcal{I}_{n+1}} := C^{\mathcal{I}_{n+1}} \cup \{x_{n+1,j} | <x_{0,0}, x_{n+1,j}> \in S_1^{\mathcal{I}_{n+1}}\}$
+
+5. $D^{\mathcal{I}_{n+1}} := D^{\mathcal{I}_{n+1}} \cup \{x_{n+1,j} | <x_{0,0}, x_{n+1,j}> \in S_2^{\mathcal{I}_{n+1}}\}$
+
+$\mathcal{I}$ is a model: due to step 3 in the construction we have $\mathcal{I}_n \models \mathfrak{R}'$ for all $n \in \mathbb{N} \cup \{0\}$ (since we have a finite number of role axioms and $\mathcal{I}_n$ is finite as well, the while-loop terminates in finite time), and therefore obviously $\mathcal{I} \models \mathfrak{R}'$. We prove that $x_{0,0} \in E^\mathcal{I}$. Due to the construction it is obviously the case that $x_0 \in ((\coprod_{a \in \Sigma} \exists a.\top) \cap (\coprod_{R \in \text{roles}(\mathfrak{R}')}\forall R.(\coprod_{a \in \Sigma} \exists a.\top)) \cap \forall S_1.C \sqcap \forall S_2.D)^\mathcal{I}$: for each node $x_{i,j} \in \Delta^\mathcal{I}$, we have $<x_{0,0}, x_{i,j}> \in *^\mathcal{I}$ (recall that $*^\mathcal{I} =_{def} \bigcup_{R \in \text{roles}(\mathfrak{R}')}\mathbb{R}_R^\mathcal{I}$), and each node has the required $k = |\Sigma|$ successors, $a_1, \dots, a_k$. This holds for $x_{0,0}$ as well as for $x_{i,j}$. This shows that $x_{0,0} \in X^\mathcal{I}$, $x_{i,j} \in X^\mathcal{I}$, and therefore $x_{0,0} \in (\forall R.(\coprod_{a \in \Sigma} \exists a.\top))^\mathcal{I}$. It also holds that $x_{0,0} \in (\coprod_{R \in \text{roles}(\mathfrak{R}')}\forall R.\neg(C \sqcap D))^\mathcal{I}$. Assume the contrary: then there must be some successor node $x_{n,i_n} \in \Delta^\mathcal{I}$ with $x_{n,i_n} \in C^\mathcal{I}$, $x_{n,i_n} \in D^\mathcal{I}$. Since this node lies at depth $n$, it holds that $x_{n,i_n} \in \Delta_{i_n}^\mathcal{I}$ with $x_{n,i_n} \in C^\mathcal{I}_n$, $x_{n,i_n} \in D^\mathcal{I}_n$. Due to the construction, $x_{n,i_n} \in C^\mathcal{I}_n$ iff $\langle x_{0,0}, x_{n,i_n} \rangle \in S_1^{\mathcal{I}_n}$, and $x_{n,i_n} \in D^{\mathcal{I}_n}$ iff $\langle x_{0,0}, x_{n,i_n} \rangle \in S_2^{\mathcal{I}_n}$. Let $w$ be the corresponding path of length $n$ in the skeleton with $w = w_1 \dots w_n$, $\langle x_{0,0}, x_{1,i_1} \rangle = w_1^\mathcal{I}$, $\dots, \langle x_{n-1,i_{n-1}}, x_{n,i_n} \rangle = w_n^\mathcal{I}$, with $w_i \in \{a_1, a_2, i_1, \dots, i_k\}$, leading from $x_{0,0}$ to $x_{n,i_n}$. But then, due to Lemma 3, $w \in L(G_1) \cap L(G_2)$, due to $\langle x_{0,0}, x_{n,i_n} \rangle \in S_1^{\mathcal{I}_n} \cap S_2^{\mathcal{I}_n}$. Contradiction. Summing up we have shown that $\mathcal{I} \models (E, \mathfrak{R}')$. $\square$
+
+# 6 Discussion & Conclusion
+
+We have proven that the satisfiability problem of $\mathcal{ALC}_{\mathcal{RA}\ominus}$ is undecidable. As already noted, this is a severe result, due to the high relevance of axioms having the form $S \circ T \sqsubseteq R_1 \sqcup \dots \sqcup R_n$ in the field of qualitative (spatial or temporal) reasoning.
+
+Considering the proof, it can be seen that not the whole expressiveness of $\mathcal{ALC}$ was needed in order to show the undecidability. In fact, the existential restrictions used in the proof have only the form $\exists R.\top$, no qualified existential restrictions were needed ($\exists R \equiv \exists R.\top$). Additionally, we did not make use of disjunctions on the right hand side of the role axioms in the proof – all role axioms were of the form $R \circ S \sqsubseteq T$. The negation operator was only used within $E$ in the form $\neg(C \sqcap D)$, which can be rewritten as $\neg C \sqcup \neg D$. Therefore, no full negation operator is needed; it is sufficient if the DL provides negation for concept names. Summing up, the language $\mathcal{ALU}$ with deterministic role boxes, called $\mathcal{ALU}_{\mathcal{RA}\ominus}$, is already undecidable.⁵
+
+⁵$\mathcal{ALU}$ provides negation for concept names, disjunction $\sqcup$, universal qualification $\forall R.C$

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+Figure 8: Illustration of an $\mathcal{ALC}_{RA}\ominus$ model of $\mathfrak{R}$ and $\exists R.((\exists S.\exists T.\top) \sqcap \forall Y.\bot) \sqcap \forall A.\bot$. The same concept is unsatisfiable in $\mathcal{ALC}_{RA}$ w.r.t. $\mathfrak{R}$.
+
+It is obvious that special kinds of role boxes lead to decidability. For example, if we restrict the set of admissible role boxes to role boxes of the form $\{R_1 \circ R_1 \sqsubseteq R_1, \dots, R_n \circ R_n \sqsubseteq R_n\}$, we get a syntactic variation of the logic $\mathcal{ALC}_{R+}$, with $R_1 \dots R_n$ declared as transitively closed roles. However, before considering special role boxes that might yield decidability and therefore special $\mathcal{ALC}_{RA}\ominus$ fragments, it is very important to understand the principal limitations which was the motivation for this investigation. The question remains whether admissible role boxes can be found which are still useful for spatial reasoning tasks. Obviously, the syntactic restrictions should not be stronger than necessary. For example, considering a syntactic variation of $\mathcal{ALC}_{R+}$ makes no sense.
+
+In the following we will only briefly sketch why the undecidability result given here does not immediately apply to $\mathcal{ALC}_{RA}$. Let us examine the difference between $\mathcal{ALC}_{RA}$ and $\mathcal{ALC}_{RA}\ominus$. Considering the two different but very similar looking languages, the question arises whether $\mathcal{ALC}_{RA}$ is in fact subsumed by $\mathcal{ALC}_{RA}\ominus$.
+
+The language $\mathcal{ALC}_{RA}$ requires that all roles have to be interpreted as disjoint: for any two different roles $R, S, R \neq S$ and any interpretation $\iota$, $R^\iota \cap S^\iota = \emptyset$ must hold. The calculus given in [22] requires that the role boxes are unique: for any pair of roles $R, S$, there is at most one role axiom $ra \in \mathfrak{R}$ with $\text{pre}(ra) = (R, S)$. The disjointness for roles really makes a difference: for example, the concept $\exists R.((\exists S.\exists T.\top) \sqcap \forall Y.\bot) \sqcap \forall A.\bot$ w.r.t. $\mathfrak{R} = \{R \circ S \sqsubseteq A \cup B, S \circ T \sqsubseteq X \cup Y, A \circ T \sqsubseteq U, B \circ T \sqsubseteq V, R \circ X \sqsubseteq U, R \circ Y \sqsubseteq V\}$ is satisfiable in $\mathcal{ALC}_{RA}\ominus$, but unsatisfiable in $\mathcal{ALC}_{RA}$ (see Figure 8). This is due to the fact that a non-empty role intersection between $U$ and $V$ is enforced, violating the disjointness requirement, yielding unsatisfiability only in the case of $\mathcal{ALC}_{RA}$.
+
+$\mathcal{ALC}_{RA}$ cannot be easily reduced to $\mathcal{ALC}_{RA}\ominus$, even though $\mathcal{ALC}_{RA}\ominus$ seems to
+
+and unqualified existential quantification $\exists R$.

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+be the more general description logic, since there are less restrictions. If (C, 𝓶) is satisfiable in 𝒜LC<sub>RA</sub>, then it is also satisfiable in 𝒜LC<sub>RA∘</sub> as well, but not vice versa. One idea to enforce role disjointness within 𝒜LC<sub>RA∘</sub> might be the following: for each role pair R, S with R ≠ S to be declared as disjoint, create a new atomic “marker concept”, e.g. [RS], and add two universal value restrictions like ∀R.[RS]∩∀S.¬[RS] conjunctively to the original concept C. (C, 𝓶) would be transformed into (C ∩ ∀R.[RS] ∩ ∀S.¬[RS] ∩ ... ∩ 𝓶), for each pair of disjoint roles R, S. Let 𝒯 be a model of the latter, and let x₀ ∈ (C ∩ ∀R.[RS] ∩ ∀S.¬[RS] ∩ ...)ᵀ. Unfortunately, this only ensures that ({⟨x₀, xᵢ⟩ | xᵢ ∈ Δᵀ} ∩ Rᵀ ∩ Sᵀ) = ∅, which is obviously a much too weak requirement. As a “solution” one might think that the universal role * might be used in order to propagate ∀R.[RS] ∩ ∀S.¬[RS] ∩ ... to every individual in the model. (C, 𝓶) would be transformed into (C ∩ ∀R.[RS] ∩ ∀S.¬[RS] ∩ ... ∩ ∀*. (∀R.[RS] ∩ ∀S.¬[RS] ∩ ...)’, 𝓶’). Unfortunately, this is a much stronger requirement than disjointness for roles, since the additional conjunct now enforces {xⱼ | ⟨xᵢ, xⱼ⟩ ∈ Rᵀ} ∩ {xⱼ | ⟨xᵢ, xⱼ⟩ ∈ Sᵀ} = ∅, which obviously implies Rᵀ ∩ Sᵀ = ∅.
+
+Instead, one would need some kind of “counting construct” that would enable
+the distinction of different individuals in order to simulate the role disjointness
+of $\mathcal{ALC}_{RA}$ within $\mathcal{ALC}_{RA\ominus}$. We therefore believe that disjoint roles are really
+something very special that cannot be simulated by means of any $\mathcal{ALC}_{RA\ominus}$ con-
+struction. Therefore, we conjecture that $\mathcal{ALC}_{RA}$ is not subsumed by $\mathcal{ALC}_{RA\ominus}$.
+
+In the undecidability proof we enforced the existence of the appropriate successors $a_1, \dots, a_k$ with $\Sigma = \{a_1, \dots, a_k\}$ for every node in the model. The existence of every word $w$ in the model was therefore granted. For this purpose the role box was completed, using the auxiliary role $R_?$. In the case of $\mathcal{ALC}_{RA}$, we can make use of the same construction. However, due to the disjointness requirement, things get more complicated.
+
+Considering the reduction, the unsatisfiability is due to the fact that $\forall S_1.C \sqcap \forall S_2.D$ is used to assert $C$ and $D$ to one and the same individual from the root node in order to produce an inconsistency via $\neg(C \sqcap D)$ which holds for all individuals. Obviously, with a disjointness requirement on $S_1$ and $S_2$, the presence of $S_1$ and $S_2$ connecting the nodes $x_{0,0}$ and $x_{i,j}$ such that $\langle x_{0,0}, x_{i,j} \rangle \in S_1^T \sqcap S_2^T$ is an inconsistency by itself. Therefore, $\forall S_1.C \sqcap \forall S_2.D$ is useless in order to yield an inconsistency, since the inconsistency is present in the first place due to the disjointness requirement. If the same proof technique could be applied for showing the undecidability of $\mathcal{ALC}_{RA}$, then neither disjunctions nor atomic negation would be needed for this undecidability proof: if $X =_{def} \exists a_1 \sqcap \dots \sqcap \exists a_k$, then $(X \sqcap (\sqcap_{R \in \text{roles}(\mathfrak{R}')}) \forall R.X), \mathfrak{R}')$ would suffice to prove the undecidability! This concept is already expressible in the language $\mathcal{FL}^{-}$, showing even the undecidability of $\mathcal{FL}_{RA}^{-}$. This would indicate that composition-based role axioms are really highly problematic language constructs, since adding them to one of the

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+An interpretation $\mathcal{I}$ satisfies / is a model of $(C, \mathfrak{R}, \mathfrak{T})$, written $\mathcal{I} \models (C, \mathfrak{R}, \mathfrak{T})$, iff $\mathcal{I} \models C$, $\mathcal{I} \models \mathfrak{R}$ and $\mathcal{I} \models \mathfrak{T}$.
+
+**Definition 7 (Satisfiability)** A syntactic entity (concept, role box, concept with role box, etc.) is called *satisfiable* iff there is an interpretation which satisfies this entity; i.e. the entity has a model. ■
+
+Then, the satisfiability problem is to decide whether a syntactic entity is satisfi-
+able or not.
+
+An important relationship between concepts is the subsumption relationship,
+which is a partial ordering on concepts w.r.t. their specificity:
+
+**Definition 8 (Subsumption Relationship)** A concept *D* subsumes a concept *C*, *C* $\subseteq$ *D*, iff $C^I \subseteq D^I$ holds for all interpretations *I*. $\square$
+
+Since $\mathcal{ALC}_{RA\ominus}$ provides a full negation operator, the subsumption problem can
+be reduced to the concept satisfiability problem: $C \sqsubseteq D$ iff $C \sqcap \neg D$ is unsatis-
+fiable.
+
+It should be noticed that a satisfiability tester for $\mathcal{ALC}_{RA\ominus}$ would also be able
+to determine satisfiability resp. subsumption w.r.t. free TBoxes. Each concept
+inclusion axiom can be dealt with by a technique called *internalization* (see
+[13, 14, 1]). Internalization for $\mathcal{ALC}_{RA\ominus}$ works as follows. Let $(C, \mathfrak{R}, \mathfrak{T})$ be the
+concept, role box and free TBox to be tested for satisfiability. Let $R? \in \mathcal{N}_R$ be
+some role such that $R? \notin \text{roles}(C) \cup \text{roles}(\mathfrak{R})$. Referring to $R?$ and $(C, \mathfrak{R}, \mathfrak{T})$,
+the role box $\mathfrak{R}$ is completed: $\mathfrak{R}' = \mathfrak{R} \cup \{ R \circ S \sqsubseteq R? \mid R, S \in (\{R?\} \cup \text{roles}(\mathfrak{R})),$
+$\neg\exists ra \in \mathfrak{R}: \text{pre}(ra) = (R, S) \}$.
+
+Now, $(C, \mathfrak{R}, \mathfrak{T})$ is satisfiable iff $((C \sqcap M_{\mathfrak{T}} \sqcap \forall *. M_{\mathfrak{T}}), \mathfrak{R}')$ is satisfiable, where
+$\forall *. M_{\mathfrak{T}}$ is an abbreviation for $\forall (\sqcup_{R \in \text{roles}(\mathfrak{R}')} R). M_{\mathfrak{T}}$. $M_{\mathfrak{T}}$ is the so-called *meta-constraint* corresponding to the TBox $\mathfrak{T}: M_{\mathfrak{T}} =_{def} \sqcap_{C \sqsubseteq D \in \mathfrak{T}} (\neg C \sqcup D)$.
+
+# 3 Relationships to Other Logics
+
+In order to judge the expressive power of $\mathcal{ALC}_{RA\ominus}$ we consider other logics and examine whether they are subsumed¹ by $\mathcal{ALC}_{RA\ominus}$. We briefly sketch the relationships to the most important base description logics offering some form of transitivity. Additionally, (un)decidability results regarding transitivity extensions of the so-called (loosely) guarded fragment of FOPL are briefly discussed
+
+¹We say that a language *A* is “subsumed” by a language *B* (resp. provides the same expressive power) iff the satisfiability problem of *A* can be reduced to the satisfiability problem of *B*.

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+simplest of all description logics $\mathcal{FL}^{-}$ would already yield undecidability. However, it is an open question whether the exploited proof technique can indeed be applied.
+
+Considering the tableaux calculus for $\mathcal{ALC}_{RA}$ given in [22], we only conjectured that $\mathcal{ALC}_{RA}$ might be decidable. We did not prove it. The given tableaux calculus was incomplete since it suffered from the definition of a so-called *blocking condition*. The tableaux calculus was presented in the expectation that an appropriate blocking condition could be found in the future. However, we still have not found a correct blocking condition for $\mathcal{ALC}_{RA}$. On the other hand, the reader should be informed that we have also carefully tried to reduce various other known undecidable problems to $\mathcal{ALC}_{RA}$, but without success (e.g. the Domino Problem).
+
+As the investigation has shown, the exact position of the boundary line between
+decidable and undecidable description logics with composition-based role axioms
+of the form $S \circ T \sqsubseteq R_1 \sqcup \dots \sqcup R_n$ must be investigated much more thoroughly
+in the future.
+
+7 Acknowledgments
+
+I would like to thank Harald Ganzinger, Volker Haarslev, Ullrich Hustadt, Amar Isli, Carsten Lutz, Thomas Mantay, Bernd Neumann, Ralf Möller and Anni-Yasmin Turhan for valuable discussions on the topics covered in this paper. I am especially grateful to Harald Ganzinger and Carsten Lutz. Both read a draft of this paper and suggested modifications in the proof to improve its comprehensibility.
+
+References
+
+[1] F. Baader. Augmenting concept languages by transitive closure of roles: An alternative to terminological cycles. In *Twelfth International Conference on Artificial Intelligence, Darling Harbour, Sydney, Australia, Aug. 24-30, 1991*, pages 446–451, August 1991.
+
+[2] R.J. Brachman and J.G. Schmolze. An overview of the KL-ONE knowledge representation system. *Cognitive Science*, pages 171–216, August 1985.
+
+[3] M. Buchheit, F.M. Donini, and A. Schaerf. Decidable reasoning in terminological knowledge representation systems. *Journal of Artificial Intelligence Research*, 1:109–138, 1993.

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+[4] F.M. Donini, M. Lenzerini, D. Nardi, and W. Nutt. The complexity of concept languages. Technical Report RR-95-07, German Center for AI (DFKI), 1995.
+
+[5] F.M. Donini, M. Lenzerini, D. Nardi, and A. Schaerf. Reasoning in description logics. In G. Brewka, editor, *Principles of Knowledge Representation*. CSLI Publications, 1996.
+
+[6] M.J. Egenhofer. Reasoning about binary topological relations. In O. Günther and H.-J. Schek, editors, *Advances in Spatial Databases, Second Symposium, SSD’91, Zurich, Aug. 28-30, 1991*, volume 525 of *Lecture Notes in Computer Science*, pages 143–160. Springer Verlag, Berlin, August 1991.
+
+[7] H. Ganzinger, Chr. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In *Proc. 14th IEEE Symposium on Logic in Computer Science*, pages 24–34. IEEE Computer Society Press, 1999. To appear in LICS’99.
+
+[8] E. Grädel. Guarded fragments of first-order logic: a perspective for new description logics? Extended abstract, Proceedings of 1998 International Workshop on Description Logics DL ‘98, Trento 1998, CEUR Electronic Workshop Proceedings, http://sunsite.informatik.rwth-aachen.de/Publications/CEUR-WS/Vol-11.
+
+[9] E. Grädel. On the restraining power of guards. (24 pages), to appear in *Journal of Symbolic Logic*.
+
+[10] E. Grädel, M. Otto, and E. Rosen. Undecidability Results for Two-Variable Logics. *Archive for Mathematical Logic*, 38:313–354, 1999. See also: Proceedings of 14th Symposium on Theoretical Aspects of Computer Science STACS’97, Lecture Notes in Computer Science No. 1200, Springer 1997, 249–260.
+
+[11] V. Haarslev, C. Lutz, and R. Möller. A description logic with concrete domains and a role-forming predicate operator. *Journal of Logic and Computation*, 9(3):351–384, June 1999.
+
+[12] V. Haarslev, R. Möller, A.-Y. Turhan, and M. Wessel. On terminological default reasoning about spatial information: Extended abstract. In P. Lambrix et al., editor, *Proceedings of the International Workshop on Description Logics (DL’99), July 30 - August 1, 1999, Linköping, Sweden*, pages 155–159, June 1999.

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+[13] I. Horrocks and U. Sattler. A description logic with transitive and inverse roles and role hierarchies. *Journal of Logic and Computation*, 9(3):385–410, 1999.
+
+[14] I. Horrocks, U. Sattler, and S. Tobies. Practical reasoning for expressive description logics. In *Proceedings of the 6th International Conference on Logic for Programming and Automated Reasoning (LPAR '99)*, 1999.
+
+[15] C. Lutz and R. Möller. Defined topological relations in description logics. In M.-C. Rousset et al., editor, *Proceedings of the International Workshop on Description Logics, DL'97, Sep. 27-29, 1997, Gif sur Yvette, France*, pages 15–19. Universite Paris-Sud, Paris, September 1997.
+
+[16] D. A. Randell, Z. Cui, and A. G. Cohn. A Spatial Logic based on Regions and Connections. In B. Nebel, C. Rich, and W. Swartout, editors, *Principles of Knowledge Representation and Reasoning*, pages 165–176, 1992.
+
+[17] U. Sattler. A concept language extended with different kinds of transitive roles. In G. Görz and S. Hölldobler, editors, *20. Deutsche Jahrestagung für Künstliche Intelligenz*, number 1137 in Lecture Notes in Artificial Intelligence, pages 333–345. Springer Verlag, Berlin, 1996.
+
+[18] K. Schild. A correspondence theory for terminological logics: Preliminary report. In *Twelfth International Conference on Artificial Intelligence, Darling Harbour, Sydney, Australia, Aug. 24-30, 1991*, pages 466–471, August 1991.
+
+[19] M. Schmidt-Schauß. Subsumption in KL-ONE is Undecidable. In *Principle of Knowledge Representation and Reasoning – Proceedings of the First International Conference KR '89*, 1989.
+
+[20] M. Schmidt-Schauß and G. Smolka. Attributive concept descriptions with complements. *Artificial Intelligence*, 48:1–26, 1991.
+
+[21] U. Schöning. *Theoretische Informatik kurz gefasst*. BI-Wissenschaftsverlag, 1. edition, 1992.
+
+[22] M. Wessel, V. Haarslev, and R. Möller. *ALC<sub>RA</sub> – ALC* with Role Axioms. In F. Baader and U. Sattler, editors, *Proceedings of the International Workshop in Description Logics 2000 (DL2000)*, number 33 in CEUR-WS, pages 21–30, Aachen, Germany, August 2000. RWTH Aachen. Proceedings online available from http://SunSITE.Informatik.RWTH-Aachen.DE/Publications/CEUR-WS/Vol-33/.
+
+[23] W.A. Woods and J.G. Schmolze. The KL-ONE family. In F. Lehmann, editor, *Semantic Networks in Artificial Intelligence*, pages 133–177. Pergamon Press, Oxford, 1992.

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+Obstacles on the Way to
+Spatial Reasoning with Description Logics:
+Undecidability of $\mathcal{ALC}_{RA}\ominus$
+
+Michael Wessel
+
+University of Hamburg, Computer Science Department,
+Vogt-Kölln-Str. 30, 22527 Hamburg, Germany
+
+Abstract
+
+This paper presents the new description logic $\mathcal{ALC}_{RA}\ominus$. $\mathcal{ALC}_{RA}\ominus$ combines the well-known standard description logic $\mathcal{ALC}$ with composition-based role axioms of the form $S \circ T \sqsubseteq R_1 \sqcup \cdots \sqcup R_n$. We argue that these axioms are nearly indispensable components in a description logic framework suitable for qualitative spatial reasoning tasks. An $\mathcal{ALC}_{RA}\ominus$ spatial reasoning example is presented, and the relationships to other descriptions logics are discussed (namely $\mathcal{ALC}_{R_A}$, $\mathcal{ALC}_{R+}$, $\mathcal{ALC}_\oplus$, $\mathcal{ALCH}_{R+}$). Unfortunately, the satisfiability problem of this new logic is undecidable. Due to the high relevance of role axioms of the proposed form for all kinds of qualitative reasoning tasks, the undecidability of $\mathcal{ALC}_{RA}\ominus$ is an important result.
+
+# 1 Introduction and Motivation
+
+Since the introduction of KL-ONE (see [2]), knowledge representation systems based on description logics (DLs) have been proven valuable tools in the field of formal knowledge representation. Description logic systems offer formally defined syntax and semantics, which enables the unambiguous specification of the services offered to users of these systems. In fact, many early knowledge representation systems and frameworks suffered from unclear semantics (e.g. see [23] for an overview and discussion). In many cases the underlying base description logic of a DL-based system can be seen as a subset of first order predicate logic (FOPL). In contrast to FOPL, decidability of diverse inference problems is usually guaranteed for description logics, for example, for the *satisfiability problem*

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+of formulas. Please recall that the satisfiability problem is only semi-decidable for FOPL. Moreover, for (less expressive) description logics even tractable (de-terministic polynomial-time) inference algorithms have been found (see [4, 5]). The merits of description logics are widely recognized, and a remarkable amount of research covering theory and practice has been carried out during the last 20 years. However, mediation between expressiveness and tractability remained a problem.
+
+Description logics focus on the structural description of unary and binary predi-cates. Unary predicates are called *concepts*, and binary predicates correspond to so-called *roles*. Sometimes DLs are even called *concept description languages* – indicating that the focus has traditionally been more on the side of the concept descriptions than on the side of the role descriptions. In our opinion the ability to interrelate roles via some kind of constraints has not been investigated as thoroughly as the concept description side of DLs. For example, see [3] for a description logic providing *role conjunction*. In contrast to role conjunction, *role disjunction* is not interesting in most description logics. *Role negation* has only been considered very recently. Role inclusion axioms have also been considered. However, the space of possibilities for role axioms resp. formulas relating roles to one another has not been exhaustively examined. To the best of our knowledge, the concept satisfiability problem w.r.t. to a set of role axioms of the proposed form has not been considered before.
+
+Like for formulas in FOPL, the syntax of the concepts is determined by a set of concept forming operators and a set of atomic components, so-called role names and concept names. The semantics of the syntactic elements is then specified by giving a Tarski-style *interpretation*. An interpretation maps concepts and roles to unary resp. binary relations on the non-empty interpretation domain: concepts are therefore mapped to subsets of the interpretation domain, and roles to sets of tuples of domain objects. The denoted (unary or binary) relation is also called the *extension* of the concept or role.
+
+If the semantics of the operators is preserved by the mapping and the extension of a concept is non-empty, then the interpretation is said to be a *model* of that concept. Given an arbitrary concept *C* of the language, the most important inference problem is to decide whether *C* has a model. In this case, *C* is called *satisfiable*.
+
+Before we discuss the modeling of *spatial* concepts, let us consider some non-spatial concepts. For example, the unary FOPL predicate `father_withSon(x)` could be defined by means of the FOPL formula `human(x) ∧ male(x) ∧ ∃y : has_child(x, y) ∧ human(y) ∧ male(y)`. Here, `human` and `male` are unary predicate names, whereas `has_child` is a binary predicate name. Translated into the variable-free description logic syntax we would get `human ⊓ male ⊓`

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+Figure 1: Simple Example
+
+∃has\_child.(human ⊓ male). The whole expression is a concept, human and male are concept names or atomic concepts, has\_child is a role (name), and ⊓ and ∨ are concept-forming operators.
+
+Obviously, if roles are not related to one another by some kind of con-
+straints, we cannot claim to have represented inherent properties that would
+be natural for some relationships. For example, in order to appropriately
+capture the meaning of the relationship “niece”, one would have to en-
+sure that a brother’s or a sister’s daughter is indeed a niece of this very
+same person. In FOPL this requirement could be expressed by means of
+two (conjunctively combined) universally quantified statements of the form
+∀x,y,z : (has\_brother(x,y) ∧ has\_daughter(y,z) ⇒ has\_niece(x,z)) and
+∀x,y,z : (has\_sister(x,y) ∧ has\_daughter(y,z) ⇒ has\_niece(x,z)). The in-
+terpretation of the role has\_niece is then no longer independent from the inter-
+pretations of the roles has\_brother (resp. has\_sister) and has\_daughter. The
+role axioms of $\mathcal{ALC}_{RA}\ominus$ allow to express global universally quantified impli-
+cation statements exactly like these: in fact, these formulas are equivalent
+to the $\mathcal{ALC}_{RA}\ominus$ role axioms has\_brother $\circ$ has\_daughter $\sqsubseteq$ has\_niece and
+has\_sister $\circ$ has\_daughter $\sqsubseteq$ has\_niece.
+
+In the following we assume that the reader is familiar with description logics, at least with the basic logic $\mathcal{ALC}$ (see [20] and [23] for an introduction). Basically $\mathcal{ALC}_{RA}\ominus$ augments the standard description logic $\mathcal{ALC}$ with composition-based role axioms of the form $S \circ T \sqsubseteq R_1 \sqcup \dots \sqcup R_n$, $n \ge 1$, enforcing $S^\mathcal{T} \circ T^\mathcal{T} \sqsubseteq R_1^\mathcal{T} \cup \dots \cup R_n^\mathcal{T}$ on the models $\mathcal{I}$. This corresponds to a universally quantified FOPL formula of the form $\forall x,y,z : (S(x,y) \land T(y,z) \Rightarrow R_1(x,z) \lor \dots \lor R_n(x,z))$. A finite set of these role axioms is called a *role box* and is denoted by $\mathfrak{R}$.
+
+Please consider the $\mathcal{ALC}$ concept $(\exists R.\exists S.C) \sqcap \forall T.\neg C$; in FOPL:
+
+((∃[x>((∃[y](R(x,y) ∧ ∃[x](S(y,x) ∧ C(x)))) ∧ (∀[y](T(x,y) ⇒ ¬C(y))))).
+
+Obviously, this concept is satisfiable, since the FOPL formula is satisfiable.
+However, the same concept is unsatisfiable in $\mathcal{ALC}_{RA}\ominus$ w.r.t. the role box

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+Figure 2: Complex Example
+
+{R ⋄ S ⊑ T}. Again, translated into FOPL we have
+
+$$
+\begin{array}{l}
+(\forall[x,y,z](R(x,y) \land S(y,z) \Rightarrow T(x,z))) \land \\
+(\exists[x]((\exists[y](R(x,y) \land \exists[x](S(y,x) \land C(x)))) \land (\forall[y](T(x,y) \Rightarrow \neg C(y))))).
+\end{array}
+$$
+
+The only difference to the former formula is the additional con-
+junct in the first line, expressing the role axiom. In fact,
+($\forall[x,y,z](R(x,y) \land S(y,z) \Rightarrow T(x,z))$) enforces the presence of $T(x,z)$
+because of $[\exists[x]((\exists[y](R(x,y) \land \exists[x](S(y,x) \land C(x))))...)$, and then the
+qualification $\forall[y](T(x,y) \Rightarrow \neg C(y))$ is applicable, yielding an inconsistency
+since also $C$ holds for this individual.
+
+As another example taken from the realm of genealogy, let us consider the concept expression
+
+$(\exists has\_brother . \exists has\_sister . \exists has\_sister . \exists has\_daughter . \exists has\_sister .$
+computer\_science\_student) \sqcap (\forall has\_niece . \neg computer\_science\_student)
+
+w.r.t. the role box
+
+{ has_brother ○ has_sister ⊑ has_sister,
+has_sister ○ has_daughter ⊑ has_niece,
+has_daughter ○ has_sister ⊑ has_daughter,
+has_sister ○ has_sister ⊑ has_sister }.
+
+A careful inspection reveals that this concept is inconsistent w.r.t. this role box,
+since the computer science student plays also the role of a niece and is therefore
+a filler of the *has_niece* role, see Figure 2.
+
+Note that composition of roles is not allowed to appear on the right hand side of role axioms. One can therefore not write axioms like *has_niece* ⊑ (*has_brother* ◦ *has_daughter*) ∪ (*has_sister* ◦ *has_daughter*). The rationale for this restriction is that it is known since 1989 that allowing composition also on the right hand side of role axioms would yield a form of undecidability that is

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+also present in the so-called *role value maps* (see [19]). For the same reason, $\mathcal{ALC}_{RA\ominus}$ does not include *inverse roles*. As we show in this paper, it suffices to allow composition on the left-hand side to make the resulting logic undecid-able. This is a new and unexpected result. The proof techniques applied in [19] to show the undecidability of role value maps cannot be exploited to show the undecidability of $\mathcal{ALC}_{RA\ominus}$, because the proof given in [19] strongly depends on the presence of role compositions on the right hand side of implication axioms.
+
+As discussed below in the spatial reasoning example, axioms of the form $S \circ T \sqsubseteq R_1 \sqcup \dots \sqcup R_n$ seem to be indispensable components in a description logic framework suitable for qualitative spatial reasoning tasks. The discov-ered undecidability result is therefore a big obstacle on the way to a full-fledged spatial-reasoning description-logic framework which would even need more ex-pressiveness than provided by $\mathcal{ALC}_{RA\ominus}$. For example, in order to truly capture the semantics of qualitative spatial relationships like the ones discussed below, *inverse roles* and additional role disjointness declarations would be needed.
+
+In [22], we presented the logic $\mathcal{ALC}_{RA}$. The only difference between $\mathcal{ALC}_{RA}$ and $\mathcal{ALC}_{RA\ominus}$ is that the former requires that all roles are interpreted as disjoint, i.e. for any two roles $R, S$ with $R \neq S$ and any interpretation $\mathcal{I}, R^\mathcal{I} \cap S^\mathcal{I} = \emptyset$ must hold. Even though this seems to be a minor variation of $\mathcal{ALC}_{RA\ominus}$, in fact it is not, because the disjointness requirement for roles has a number of non-obvious and far-reaching consequences (see [22]). The undecidability proof given here does not apply to $\mathcal{ALC}_{RA}$, so the question whether $\mathcal{ALC}_{RA}$ is decidable or not is still open.
+
+The structure of this paper is as follows: first we will formally define the syntax and semantics of $\mathcal{ALC}_{RA\ominus}$. Then, the relationships to other known description logics providing some kind of transitive roles are sketched. The usefulness of $\mathcal{ALC}_{RA\ominus}$ in a spatial reasoning scenario is exemplified in the next section. The main contribution of this paper is the undecidability proof in Section 5. Finally, we conclude by discussing whether $\mathcal{ALC}_{RA}$ might be undecidable as well, and future work is outlined. In the search of a decidable description logic with composition-based role axioms of the proposed form, a promising idea is to impose certain syntactic restrictions on the allowed role boxes. These syntactic restrictions have to be worked out in the future.
+
+## 2 Syntax and Semantics of $\mathcal{ALC}_{RA\ominus}$
+
+In the following the set of well-formed concepts of $\mathcal{ALC}_{RA\ominus}$ is specified:
+
+**Definition 1 (Concept Expressions)** Let $\mathcal{N}_C$ be a set of concept names, and let $\mathcal{N}_R$ be a set of role names (roles for short), such that $\mathcal{N}_C \cap \mathcal{N}_R = \emptyset$. The set

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+of concept expressions (or concepts for short) is the smallest inductively defined set such that
+
+1. Every concept name $C \in \mathcal{N}_C$ is a concept.
+
+2. If $C$ and $D$ are concepts, and $R \in \mathcal{N}_R$ is a role, then the following expressions are concepts as well: $(\neg C)$, $(C \sqcap D)$, $(C \sqcup D)$, $(\exists R.C)$, and $(\forall R.C)$.
+
+3. Nothing else is a concept. ■
+
+The set of concepts is the same as for the language $\mathcal{ALC}$. If a concept starts with “(”, we call it a compound concept, otherwise a concept name or atomic concept. Brackets may be omitted for the sake of readability if the concept is still uniquely parsable.
+
+We use the following abbreviations: if $R_1, \dots, R_n$ are roles, and $C$ is a concept, then we define $(\forall R_1 \sqcup \dots \sqcup R_n.C) =_{def} (\forall R_1.C) \sqcap \dots \sqcap (\forall R_n.C)$ and $\exists R_1 \sqcup \dots \sqcup R_n.C =_{def} (\exists R_1.C) \sqcup \dots \sqcup (\exists R_n.C)$. Additionally, for some $CN \in \mathcal{N}_C$ we define $\top =_{def} CN \sqcup \neg CN$ and $\bot =_{def} CN \sqcap \neg CN$ (therefore, $\top^\mathcal{T} = \Delta^\mathcal{T}, \bot^\mathcal{T} = \emptyset$).
+
+The set of *roles* being used within a concept term $C$ is defined:
+
+**Definition 2 (Used Roles, roles($C$))**
+
+$$ \text{roles}(C) =_{def} \left\{ \begin{array}{ll} \emptyset & \text{if } C \in \mathcal{N}_C \\ \text{roles}(D) & \text{if } C = (\neg D) \\ \text{roles}(D) \sqcup \text{roles}(E) & \text{if } C = (D \sqcap E) \\ \text{or } C = (D \sqcup E) \\ \{R\} \sqcup \text{roles}(D) & \text{if } C = (\exists R.D) \\ \text{or } C = (\forall R.D) & \hfill \blacksquare \end{array} \right. $$
+
+For example, $\text{roles}(\forall R.\exists S.SC \sqcap \exists T.D) = \{R, S, T\}$.
+
+As already noted, $\mathcal{ALC}_{RA\odot}$ provides role axioms of the form $S \circ T \subseteq R_1 \sqcup \dots \sqcup R_n$. More formally, the syntax of these role axioms is as follows:
+
+**Definition 3 (Role Axioms, Role Box)** If $S, T, R_1, \dots, R_n \in \mathcal{N}_R$, then the expression $S \circ T \subseteq R_1 \sqcup \dots \sqcup R_n$, $n \ge 1$, is called a *role axiom*. If $ra = S \circ T \subseteq R_1 \sqcup \dots \sqcup R_n$, then $\text{pre}(ra) =_{def} (S, T)$ and $\text{con}(ra) =_{def} \{R_1, \dots, R_n\}$. A finite set $\mathfrak{R}$ of role axioms is called a *role box*. Let $\text{roles}(ra) =_{def} \{S, T, R_1, \dots, R_n\}$, and $\text{roles}(\mathfrak{R}) =_{def} \bigcup_{ra \in \mathfrak{R}} \text{roles}(ra)$. $\square$
+
+Additionally, a set of global *concept inclusion axioms (GCIs)* can be specified. A set of these GCIs is called a free TBox:

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+**Definition 4 (Generalized Concept Inclusion Axiom, TBox)** If *C* and *D* are $\mathcal{ALC}_{RA}\ominus$ concepts, then the expression $C \dot{\sqsubseteq} D$ is called a generalized concept inclusion axiom, or GCI for short. A finite set of such GCIs is called a free TBox, $\mathfrak{T}$. We use $C \doteq D \in \mathfrak{T}$ as a shorthand for $\{C \dot{\sqsubseteq} D, D \dot{\sqsubseteq} C\} \subseteq \mathfrak{T}$. $\square$
+
+The semantics of an $\mathcal{ALC}_{RA}\ominus$ concept is specified by giving a Tarski-style interpretation $\mathcal{I}$ that has to satisfy the following conditions:
+
+**Definition 5 (Interpretation)** An interpretation $\mathcal{I} =_{def} (\Delta^{\mathcal{I}}, \cdot^{\mathcal{I}})$ consists of a non-empty set $\Delta^{\mathcal{I}}$, called the domain of $\mathcal{I}$, and an interpretation function $\cdot^{\mathcal{I}}$ that maps every concept name to a subset of $\Delta^{\mathcal{I}}$, and every role name to a subset of $\Delta^{\mathcal{I}} \times \Delta^{\mathcal{I}}$.
+
+The interpretation function $\cdot^{\mathcal{I}}$ can then be extended to arbitrary concepts $C$ by using the following definitions (we write $X^{\mathcal{I}}$ instead of $\cdot^{\mathcal{I}}(X)$):
+
+$$ 
+\begin{align*}
+(\neg C)^{\mathcal{I}} &=_{def} \Delta^{\mathcal{I}} \setminus C^{\mathcal{I}} \\
+(C \sqcap D)^{\mathcal{I}} &=_{def} C^{\mathcal{I}} \cap D^{\mathcal{I}} \\
+(C \sqcup D)^{\mathcal{I}} &=_{def} C^{\mathcal{I}} \cup D^{\mathcal{I}} \\
+(\exists R.C)^{\mathcal{I}} &=_{def} \{i \in \Delta^{\mathcal{I}} | \exists j \in C^{\mathcal{I}} : <i,j> \in R^{\mathcal{I}}\} \\
+(\forall R.C)^{\mathcal{I}} &=_{def} \{i \in \Delta^{\mathcal{I}} | \forall j : <i,j> \in R^{\mathcal{I}} \Rightarrow j \in C^{\mathcal{I}}\} \quad \square
+\end{align*}
+$$
+
+It is therefore sufficient to provide the interpretations for the concept names and the roles, since the interpretation of every concept is uniquely determined then by using the definitions.
+
+In the following we specify under which conditions a given interpretation is a model of a syntactic entity (we also say an interpretation satisfies a syntactic entity):
+
+**Definition 6 (Model Relationship)** An interpretation $\mathcal{I}$ satisfies / is a model of a concept $C$, written $\mathcal{I} \models C$, iff $C^{\mathcal{I}} \neq \emptyset$.
+
+An interpretation $\mathcal{I}$ satisfies / is a model of a role axiom $S \circ T \subseteq R_1 \sqcup \cdots \sqcup R_n$, written $\mathcal{I} \models S \circ T \subseteq R_1 \sqcup \cdots \sqcup R_n$, iff $S^{\mathcal{I}} \circ T^{\mathcal{I}} \subseteq R_1^{\mathcal{I}} \cup \cdots \cup R_n^{\mathcal{I}}$.
+
+An interpretation $\mathcal{I}$ satisfies / is a model of a role box $\mathfrak{R}$, written $\mathcal{I} \models \mathfrak{R}$, iff for all role axioms $ra \in \mathfrak{R}: \mathcal{I} \models ra$.
+
+An interpretation $\mathcal{I}$ satisfies / is a model of a GCI $C \dot{\sqsubseteq} D$, written $\mathcal{I} \models C \dot{\sqsubseteq} D$, iff $C^{\mathcal{I}} \subseteq D^{\mathcal{I}}$.
+
+An interpretation $\mathcal{I}$ satisfies / is a model of a TBox $\mathfrak{T}$, written $\mathcal{I} \models \mathfrak{T}$, iff for all GCIs $g \in \mathfrak{T}: \mathcal{I} \models g$.
+
+An interpretation $\mathcal{I}$ satisfies / is a model of $(C, \mathfrak{R})$, written $\mathcal{I} \models (C, \mathfrak{R})$, iff $\mathcal{I} \models C$ and $\mathcal{I} \models \mathfrak{R}$.

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+($C', \mathfrak{R}$) is satisfiable in $\mathcal{ALC}_{\mathcal{R}\mathfrak{A}\ominus}$ iff the original concept $C$ is.$^2$
+
+$C''$ is constructed from $C$ as follows: The role $\oplus(R)$ in $C$ is replaced by the role $R_{\oplus}$. Then, for every role $R_{\oplus}$, we add the role axioms $\{R \circ R \subseteq R_{\oplus}, R_{\oplus} \circ R \subseteq R_{\oplus}\}$ to $\mathfrak{R}$. Please note that this only ensures $(\oplus(R))^\mathcal{I} = R^\mathcal{I} \cup R_{\oplus}^\mathcal{I}$, and not $(\oplus(R))^\mathcal{I} = R_{\oplus}^\mathcal{I}$, since $R^\mathcal{I} \not\subseteq R_{\oplus}^\mathcal{I}$. Therefore, in order to get an equi-satisfiable concept $C'$, we have to rewrite the original concept $C$ in the following way:
+
+$$ \exists \oplus (R).D \rightarrow \exists R_{\oplus}.D $$
+
+$$ \exists R.D \rightarrow \exists R_{\oplus}.D \sqcap \exists R.D $$
+
+$$ \forall \oplus (R).D \rightarrow \forall R_{\oplus}.D \sqcap \forall R.D $$
+
+Now, $C'$ is satisfiable w.r.t. the role box $\mathfrak{R}$ iff $C$ is satisfiable.
+
+**$\mathcal{ALCH}_{R+}$**: The description logic $\mathcal{ALCH}_{R+}$ (see [13, 14]) extends $\mathcal{ALC}_{R+}$ by an additional set of role inclusion axioms of the form $R \subseteq S$, enforcing $R^\mathcal{I} \subseteq S^\mathcal{I}$ on the models $\mathcal{I}$. Adding the identity role $Id$ with the fixed semantics of the identity relationship $Id^\mathcal{I} =_{def} \{\langle x, x \rangle \mid x \in \Delta^\mathcal{I} \}$ to $\mathcal{ALC}_{RA\ominus}$ would obviously enable the simulation of these role inclusion axioms: for each role inclusion axiom $R \subseteq S$, add the role axiom $R \circ Id \subseteq S$ to a role box $\mathfrak{R}$ and consider the concept satisfiability w.r.t. $\mathfrak{R}$. Currently, neither $\mathcal{ALC}_{RA\ominus}$ nor $\mathcal{ALC}_{RA}$ provide the identity role.
+
+**Other Fragments of FOPL:** In the following we will briefly discuss whether decidability or undecidability of $\mathcal{ALC}_{RA\ominus}$ follows from already known results in logic, namely from results in bounded number of variables FOPL, or results from research carried out in the so-called (loosely) guarded fragment of FOPL. To the best of our knowledge, no previously known decidability resp. undecidability result is exploitable in the case of $\mathcal{ALC}_{RA\ominus}$.
+
+It is well-known that certain fragments of FOPL are decidable, for example, the class of all closed FOPL formulas containing at most two variables, denoted by $FO^2$. $FO^2$ has the finite model property – each satisfiable formula has a finite model. We already noted that one would need at least three variables if one translates $\mathcal{ALC}_{RA\ominus}$ role boxes and concepts into FOPL. In fact, there is no way even to express the transitivity axiom $\forall x, y, z : R(x, y) \land R(y, z) \Rightarrow R(x, z)$ in $FO^2$ (see [10]). If $FO^2$ is augmented by transitivity on an extra-logical level (since transitivity cannot be expressed within the language itself), $FO^2$ becomes undecidable, as Grädel et al. have shown (see [10]). However, the class $FO^2$ is much too large to capture the concept-side of $\mathcal{ALC}_{RA\ominus}$, since $\mathcal{ALC}$ concepts are expressible in a proper subset of $FO^2$, namely $GF_\beta^2$, see below. Recall that $\mathcal{ALC}_{R+}$ is decidable.
+
+$^2$It follows that if $\mathcal{ALC}_{RA\ominus}$ was decidable it would be EXPTIME-hard, since $\mathcal{ALC}_\oplus$ is EXPTIME-complete.

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+Figure 4: Illustration of $\forall a, b, c : EC(a, b) \land EC(b, c) \Rightarrow (DC(a, c) \lor EC(a, c) \lor PO(a, c) \lor TPP(a, c) \lor TPPI(a, c))$
+
+means of a so called *composition table* that lists, given the “column” relationship $R(a, b)$ and the “row” relationship $S(b, c)$, all possible relationships $T_1(a, c)$, $T_2(a, c), ..., T_n(a, c)$ that may hold between $a$ and $c$. For example, in the case of RCC8, the composition table contains the entry $\{DC, EC, PO, TPP, TPPI\}$, given the relationship *EC* for the row as well as for the column – please consider Figure 4. This corresponds to the FOPL axiom $\forall a, b, c : EC(a, b) \land EC(b, c) \Rightarrow (DC(a, c) \lor EC(a, c) \lor PO(a, c) \lor TPP(a, c) \lor TPPI(a, c))$, which is equivalent to the role axiom $EC \circ EC \sqsubseteq DC \sqcup EC \sqcup PO \sqcup TPP \sqcup TPPI$.
+
+Usually, also the *disjointness* of the base relations must be captured. As already noted, $\mathcal{ALC}_{RA\ominus}$ lacks this expressiveness (and it cannot be simulated by means of other constructs easily, see below), but $\mathcal{ALC}_{RA}$ does not. For an adequate modeling of spatial relationships, also *inverse roles* must be taken into account. For example, the RCC8 relationship *TPPI* is the inverse of *TPP*, and *NTPPI* is the inverse of *NTPP*. Of course, $TPP^I = (TPPI^I)^{-1}$ and $NTPP^I = (NTPPI^I)^{-1}$ should be ensured. However, both $\mathcal{ALC}_{RA}$ and $\mathcal{ALC}_{RA\ominus}$ lack inverse roles, since undecidability would follow immediately then by previously known undecidability results. Since we use $\mathcal{ALC}_{RA\ominus}$ in the following example, we can neither rely on $TPP^I = (TPPI^I)^{-1}$ nor on the disjointness of roles.
+
+The possibility to approximate composition tables, which are very widely used in the field of relation algebra-based knowledge representation and reasoning, is the distinguishing feature of $\mathcal{ALC}_{RA\ominus}$ and $\mathcal{ALC}_{RA}$. Usually, the $\mathcal{ALC}_{RA}$ approximation will be better, since the disjointness of the base relations is also enforced. Nevertheless, as the example demonstrates, we can still solve some interesting spatial reasoning task using $\mathcal{ALC}_{RA\ominus}$. Consider the following TBox:
+
+$$
+\begin{array}{lcl}
+\textit{circle} & \dot{\sqsubseteq} & \textit{figure} \\
+\textit{figure touching } a\textit{-figure} & \doteq & \textit{figure} \sqcap \exists \textit{EC}. \textit{figure} \\
+\textit{special figure} & \doteq & \textit{figure} \sqcap \\
+& & \forall \textit{PO}. \neg \textit{figure} \sqcap \\
+& & \forall \textit{NTPPI}. \neg \textit{figure} \sqcap \\
+& & \forall \textit{TPPI}. \neg \textit{circle} \sqcap \\
+& & \exists \textit{TPPI}. (\textit{figure} \sqcap \exists \textit{EC}. \textit{circle})
+\end{array}
+$$

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+stricted $\mathcal{ALCRP}(D)$. The strong syntactic requirements make modeling with $\mathcal{ALCRP}(D)$ much more complicated, and many interesting spatial-reasoning tasks cannot be addressed within the decidable fragment. One the other hand, the special $\mathcal{ALCRP}(D)$ instantiation $\mathcal{ALCRP}(S_2)$ captures the semantics of the RCC8 spatial relationships much more appropriately than it would be possible with $\mathcal{ALC}_{RA}\ominus$ inverse roles are present and disjointness is ensured as well. For example, see [12] for an $\mathcal{ALCRP}(S_2)$ spatial reasoning application. However, $\mathcal{ALCRP}(S_2)$ suffers from the same strong syntax restrictions which nearly make it impossible to address more complex spatial reasoning tasks. One of the motivations for our work on $\mathcal{ALC}_{RA}$ and $\mathcal{ALC}_{RA}\ominus$ was to create a logic that might be used more freely than $\mathcal{ALCRP}(S_2)$ for spatial modeling and reasoning.
+
+# 5 Proving Undecidability of $\mathcal{ALC}_{RA}\ominus$
+
+The structure of the proof is as follows: first we show that the intersection problem for a special class of context-free grammars – so called concatenation grammars – is undecidable. Then we show that the intersection problem of concatenation grammars could be solved iff the satisfiability problem of $\mathcal{ALC}_{RA}\ominus$ was decidable. This obviously shows that the latter must be undecidable as well, since the former is. It should be noted that the underlying idea of the proof given below is nearly identical to the idea exploited in the proof given by Ganzinger et al. in [7] for showing the undecidability of $LGF_-$ with one transitive relation. However, the proof has been found independently, and for different classes of languages ($\mathcal{ALC}_{RA}\ominus$ is not in $LGF$). We start with some basic definitions needed for the proofs:
+
+**Definition 9 (Context-Free Grammar, Language)** A context-free grammar $G$ is a quadruple $(V, \Sigma, P, S)$, where $V$ is a finite set of variables or non-terminal symbols, $\Sigma$ is finite alphabet of terminal symbols with $V \cap \Sigma = \emptyset$, and $P \subseteq V \times (V \cup \Sigma)^+$ is a set of productions or grammar rules. $S \in V$ is the start variable. The language generated by a context-free grammar $G$ is defined as $\mathcal{L}(G) = \{w \mid w \in \Sigma^*, S \xrightarrow{*} w\}$ (see [21]). In the following, we will only consider languages with $\epsilon \notin \mathcal{L}(G)$ (therefore we write $\mathcal{L}(G) = \{w \mid w \in \Sigma^+, S \xrightarrow{+} w\}$). $\square$
+
+**Definition 10 (Intersection-Problem for Languages)** Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be formal languages (e.g. context-free languages). The intersection problem is to decide whether $\mathcal{L}_1 \cap \mathcal{L}_2$ is empty or not. $\square$
+
+For lack of a better name we will consider special context-free grammars that we call *concatenation grammars* (for reasons that will become clear later):

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+Intermittent Fault Detection for Nonlinear
+Stochastic Systems
+
+Yichun Niu*, Li Sheng*, Ming Gao*, Donghua Zhou**
+
+* College of Control Science and Engineering, China University of Petroleum (East China), Qingdao, 266580, China. Corresponding author: Li Sheng. (email: shengli@upc.edu.cn).
+
+** College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China.
+
+**Abstract:** In this paper, the problem of intermittent fault detection is investigated for nonlinear stochastic systems. The moving horizon estimation with dynamic weight matrices is proposed, where the weight matrices are adjusted by an unreliability index of prior estimate to avoid the smearing effects of intermittent faults. Based on the particle swarm optimization algorithm, the nonlinear optimization problem is solved and the approximate estimate is derived. Finally, the feasibility and effectiveness of the proposed algorithm are validated by a numerical example.
+
+**Keywords:** Intermittent fault detection, Nonlinear stochastic systems, Moving horizon estimation, Dynamic weight matrices, Particle swarm optimization.
+
+# 1. INTRODUCTION
+
+For the sake of strengthening the reliability and safety of industrial processes, during the past several decades, tremendous effort has been devoted to the study of fault diagnosis techniques and a large number of research results have been effectively applied in various fields, such as chemical processes, aerospace systems, power systems and so on, see Fazai et al. (2019); Mandal et al. (2019); Shen et al. (2019). Nevertheless, it should be pointed out that most existing literature has concentrated on permanent faults, while little attention has been paid on another kind of common faults, intermittent faults (IFs). Different from permanent faults, a IF usually recurs by the same reason and lasts within a limited period of time. Since the appearing and disappearing times of IFs are nonde-terministic, the system can recover without fault-tolerant operations (Rashid et al. (2015)). Nonetheless, if IFs are not treated properly and promptly, the destructiveness of IFs may become larger over time and finally lead to major accidents (Correcher et al. (2012)). In fact, in power systems, mechanical equipment, electrical industries and many other engineering applications with electronics, the occurrence frequency of IFs is much larger than permanent faults. Therefore, it is an urgent need to develop the fault diagnosis methods for IFs.
+
+Generally speaking, the objective of fault diagnosis consists of fault detection, isolation and estimation, which respectively study the time, location and size of faults. It should be noted that the IF detection is more difficult than the permanent fault, since its aim is to detect all appearing
+
+and disappearing times of IFs. Especially for the detection of disappearing times, the residual is affected by previous IFs and then remains above the threshold for an uncertain period of time, which is the so-called smearing effects of IFs. Up to now, there have been some research results on the IF detection based on qualitative or quantitative analysis methods, see Constantinescu (2008); Correcher et al. (2012); Kim (2009); Yan et al. (2018, 2016). For example, in Yan et al. (2018) and Yan et al. (2016), the intermittent actuator and sensor fault detection problems for linear stochastic systems have been investigated, respectively.
+
+On the other hand, it is well known that nonlinearity pervasively exists in almost all dynamic systems. In order to solve the fault detection for nonlinear systems, fruitful methods have been proposed by a variety of communities. These methods include, but are not limited to, the extended Kalman filter (EKF) method (Wang et al. (2019)), particle filter (PF) method (Daroogheh et al. (2018); Yin and Zhu (2015)), strong tracking filter (STF) method (Qin et al. (2016)). However, after a thorough literature search, it has been revealed that, for IFs in nonlinear systems, the corresponding research results on the fault detection are still in the blank.
+
+In order to fill the research gap of existing literature, this paper studies the IF detection for nonlinear systems with stochastic noises. The main contributions are listed as follows.
+1. *This paper represents the first of few attempts to investigate the IF detection problem for nonlinear systems.*
+2. *By means of the moving horizon estimation with dynamic weight matrices (MHEDWM), the smearing effects of IFs are properly suppressed.*
+
+The rest of this paper is organized as follows. Section 2 gives the problem description about the IF detection for nonlinear systems and analyzes the deficiencies of existing methods for detecting IFs. Section 3 proposes the MHED-
+
+* This work is supported by National Natural Science Foundation of China (Nos. 61773400, 61751307), Key Research and Development Program of Shandong Province (No. 2019GGX101046), Fundamental Research Funds for the Central Universities of China (No. 19CX02044A), and Research Fund for the Taishan Scholar Project of Shandong Province of China.

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+Then it can be found that there exist two following main
+difficulties for the IF detection of nonlinear systems.
+
+(1) In traditional fault detection methods for nonlinear systems, such as EKF, PF, STF and so on, it can be only ensured that the designed residual is larger than the detection threshold after the fault appears. However, when the fault disappears, owing to the smearing effects, it is hard to guarantee that the residual is smaller than the threshold, see Fig. 1.
+
+(2) The model linearization method cannot be applied to the IF detection of nonlinear systems, due to the fact that the omitted high order terms of Taylor expansion maybe larger than the reserved lower order terms after the fault occurs. Thus, the existing IF detection methods for linear systems are unsuitable to be extended to nonlinear systems by the model linearization. Similarly, EKF, STF and other similar methods containing Taylor expansion approximation will also meet such a problem. By employing the imprecise approximation model, the estimation error may tend to diverge, see Fig. 2.
+
+Based on the above analysis, it can be seen that the IF detection for nonlinear systems is a quite challenging problem, which cannot be properly solved by the existing methods. Hence, the main objective of this paper is to design a new algorithm to deal with such a problem.
+
+3. IF DETECTION ALGORITHM
+
+In this section, the following algorithm of MHEDWM is designed, where for each time $k \ge N$ ($N < \min\{\bar{d}_1 - 1, \bar{d}_2 - 1\}$), the system state $x(k - N)$ is estimated depending on the past measurement outputs $\{y(k-i)\}_{0\le i \le N}$. For facilitating the understanding, we respectively define $\hat{x}(k-N)$ and $\bar{x}(k-N|k) = g(\hat{x}(k-1-N))$ as the posteriori estimate and prior estimate of $x(k-N)$. Construct the following quadratic cost function (QCF)
+
+$$
+\begin{equation}
+\begin{aligned}
+\mathcal{J}(k, \hat{x}(k-N|k)) = & \| \hat{x}(k-N|k) - \bar{x}(k-N|k) \|_{P(k)}^2 \\
+& + \sum_{i=0}^{N} \| y(k-i) - \hat{y}(k-i|k) \|_{Q(k)}^2,
+\end{aligned}
+\tag{3}
+\end{equation}
+$$
+
+where $P(k) \ge 0$ and $Q(k) \ge 0$ is a set of weight matrices
+to be designed, $\hat{y}(k-i|k) = h(\hat{x}(k-i|k))$ ($0 \le i \le N$) and
+$\hat{x}(k-i+1|k) = g(\hat{x}(k-i|k))$ ($1 \le i \le N$). Therefore, the
+desired estimate $\hat{x}(k-N)$ can be derived by solving the
+following optimization problem (OP)
+
+$$
+\hat{x}(k-N) = \arg \min_{\hat{x}(k-N|k)} \mathcal{J}(k, \hat{x}(k-N|k)). \quad (4)
+$$
+
+In this paper, an unreliability index of prior estimate $\bar{x}(k-N|k)$ is designed as follows
+
+$$
+\rho(k) = \| \sigma(k) \|^{2}, \qquad (5)
+$$
+
+where
+
+$$
+\begin{align*}
+\sigma(k) &= Y(k) - \bar{Y}(k|k), \\
+Y(k) &= [y^T(k-N), \dots, y^T(k)]^T, \\
+\bar{Y}(k|k) &= [\bar{y}^T(k-N|k), \dots, \bar{y}^T(k|k)]^T, \\
+\bar{y}(k-i|k) &= h(\bar{x}(k-i|k)), \quad 0 \le i \le N, \\
+\bar{x}(k-i+1|k) &= g(\bar{x}(k-i|k)), \quad 1 \le i \le N.
+\end{align*}
+$$
+
+In order to avoid the smearing effects of IFs, the prior esti-
+mate $\bar{x}(k - N|k)$ should be properly discarded during the
+estimation process, which can be achieved by regulating
+the weight matrices $P(k)$ and $Q(k)$. Then the following
+rules are established
+
+1) If $\rho(k) \le \underline{\rho}$, let $P(k) = I$ and $Q(k) = 0$;
+
+2) If $\rho(k) \ge \bar{\rho}$, let $P(k) = 0$ and $Q(k) = I$;
+
+3) If $\underline{\rho} < \rho(k) < \bar{\rho}$, let $P(k) = \beta(k)I$ and $Q(k) = (1 - \beta(k))I$, where $\beta(k) = (\bar{\rho} - \rho(k))/(1 - \bar{\rho})$,
+
+where $\bar{\rho} \ge \underline{\rho} \ge 0$ are given scalars related to the stochastic noises.
+
+Defining $g^{(i)}(x) = g(g^{(i-1)}(x))$ $(i \in \mathbb{N}^{+})$ and $g^{(0)}(x) = x$, one has
+
+$$
+\hat{x}(k - N + i|k) = g^{(i)}(\hat{x}(k - N|k)). \quad (6)
+$$
+
+Then the QCF (3) can be rewritten as the following form
+
+$$
+\begin{equation}
+\begin{split}
+\mathcal{J}(k, \hat{x}(k-N|k)) = {}& \| \hat{x}(k-N|k) - \bar{x}(k-N|k) \|_{P(k)}^2 \\
+& + \| Y(k) - \hat{Y}(k|k) \|_{\tilde{Q}(k)}^2,
+\end{split}
+\tag{7}
+\end{equation}
+$$
+
+where
+
+$$
+\tilde{Q}(k) = \operatorname{diag}\left\{Q(k), \cdots, Q(k)\right\}_{N+1},
+$$
+
+$$
+\begin{align*}
+\hat{Y}(k|k) &= [\hat{y}^T(k-N|k), \dots, \hat{y}^T(k|k)]^T \\
+\hat{y}(k-N+i|k) &= h(g^{(i)}(\hat{x}(k-N|k))), \quad i=0, \dots, N.
+\end{align*}
+$$
+
+It can be found that for time instant $k > N$, the QCF $\mathcal{J}(k, \hat{x}(k-N|k))$ is a nonlinear function of $\hat{x}(k-N|k)$, which is related to functions $g(\cdot)$ and $h(\cdot)$. For general nonlinear functions $g(\cdot)$ and $h(\cdot)$, it is hardly possible to give a precise analytical solution of the nonlinear OP (4). Hence, in this paper, the particle swarm optimization (PSO) algorithm is introduced to search for an approximate solution $\hat{x}^\circ(k-N)$ of OP (4). Defining the residual
+
+$$
+r(k) = \hat{x}^{\circ}(k - N) - g(\hat{x}^{\circ}(k - 1 - N)), \quad (8)
+$$
+
+the evaluation function $J(k)$ and threshold $J_{\text{th}}$ can be
+given as follows
+
+$$
+J(k) = \sum_{l=0}^{L-1} \|r(k-l)\|^2, \qquad (9)
+$$
+
+$$
+J_{\text{th}} = \sup_{f(k)=0} J(k), \qquad (10)
+$$
+
+where *L* is a given positive integer satisfying *N* + *L* < max{*d*<sub>1</sub> − 1, *d*<sub>2</sub> − 1}. The IF *f*(*k*) can be detected by the following test
+
+$$
+\begin{cases}
+J(k) \geq J_{\text{th}} \\
+J(k) < J_{\text{th}}
+\end{cases}
+\Rightarrow
+\begin{array}{l}
+\text{faults occur} \\
+\text{no faults.}
+\end{array}
+$$
+
+**Remark 2.** As is known to all, the prior estimate derived by the previous estimates plays an important role in traditional estimation methods, such as Kalman filter, PF, Luenberger observer and so on. When the fault disappears, the posteriori estimate is still affected by IFs existing in

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+Fig. 6. Alarm times of IF in the case of $f_a = 1$
+
+Fig. 7. IF and residuals in the case of $f_a = 2$
+
+Fig. 8. Alarm times of IF in the case of $f_a = 2$
+
+## 5. CONCLUSIONS AND PERSPECTIVES
+
+In this paper, the IF detection problem for nonlinear stochastic systems has been investigated based on the moving horizon estimation (MHE) algorithm. By introducing the unreliability index of prior estimate, the weight matrices in MHE has been dynamically adjusted, which
+
+can avoid the smearing effects of IFs. The simulation has shown the proposed MHEDWM can guarantee the accuracy of estimator, in the meantime detect all appearing and disappearing times of IFs.
+
+Further research topics include 1) the convergence analysis for the estimation error of MHEDWM; 2) the reduction of the calculation load for nonlinear OP; 3) the simplification of QCF.
+
+## REFERENCES
+
+- Constantinescu, C. (2008). Intermittent faults and effects on reliability of integrated circuits. In *2008 Annual Reliability and Maintainability Symposium*, 370–374.
+- Correcher, A., Garcia, E., Morant, F., Quiles, E., and Rodriguez, L. (2012). Intermittent failure dynamics characterization. *IEEE Transactions on Reliability*, 61(3), 649–658.
+- Daroogheh, N., Meskin, N., and Khorasani, K. (2018). A dual particle filter-based fault diagnosis scheme for nonlinear systems. *IEEE Transactions on Control Systems Technology*, 26(4), 1317–1334.
+- Fazai, R., Mansouri, M., Abodayeh, K., Nounou, H., and Nounou, M. (2019). Online reduced kernel pls combined with glrt for fault detection in chemical systems. *Process Safety and Environmental Protection*, 128, 228–243.
+- Kim, C.J. (2009). Electromagnetic radiation behavior of low-voltage arcing fault. *IEEE Transactions on Power Delivery*, 24(1), 416–423.
+- Mandal, S., Santhi, B., Sridhar, S., Vinolia, K., and Swaminathan, P. (2019). Sensor fault detection in nuclear power plants using symbolic dynamic filter. *Annals of Nuclear Energy*, 134, 390–400.
+- Qin, X., Fang, H., and Liu, X. (2016). Strong tracking filter-based fault diagnosis of networked control system with multiple packet dropouts and parameter perturbations. *Circuits Systems and Signal Processing*, 35(7), 2331–2350.
+- Rashid, L., Pattabiraman, K., and Gopalakrishnan, S. (2015). Characterizing the impact of intermittent hardware faults on programs. *IEEE Transactions on Reliability*, 64(1), 297–310.
+- Shen, Q., Yue, C., Goh, C.H., and Wang, D. (2019). Active fault-tolerant control system design for spacecraft attitude maneuvers with actuator saturation and faults. *IEEE Transactions on Industrial Electronics*, 66(5), 3763–3772.
+- Wang, T., Liu, L., Zhang, J., Schaeffer, E., and Wang, Y. (2019). A M-EKF fault detection strategy of insulation system for marine current turbine. *Mechanical Systems and Signal Processing*, 115, 269–280.
+- Yan, R., He, X., Wang, Z., and Zhou, D.H. (2018). Detection, isolation and diagnosability analysis of intermittent faults in stochastic systems. *International Journal of Control*, 91(2), 480–494.
+- Yan, R., He, X., and Zhou, D. (2016). Detecting intermittent sensor faults for linear stochastic systems subject to unknown disturbance. *Journal of the Franklin Institute*, 353(17), 4734–4753.
+- Yin, S. and Zhu, X. (2015). Intelligent particle filter and its application to fault detection of nonlinear system. *IEEE Transactions on Industrial Electronics*, 62(6), 3852–3861.

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+# Half-Gain Tuning for Active Disturbance Rejection Control
+
+Gernot Herbst* Arne-Jens Hempel** Thomas Göhrt**
+Stefan Streif**
+
+* Siemens AG, Clemens-Winkler-Str. 3, 09116 Chemnitz, Germany
+** Technische Universität Chemnitz, 09107 Chemnitz, Germany
+
+**Abstract:** A new tuning rule is introduced for linear active disturbance rejection control (ADRC), which results in similar closed-loop dynamics as the commonly employed bandwidth parameterization design, but with lower feedback gains. In this manner the noise sensitivity of the controller is reduced, paving the way for using ADRC in more noise-affected applications. It is proved that the proposed tuning gains, while rooted in the analytical solution of an algebraic Riccati equation, can always be obtained from a bandwidth parameterization design by simply halving the gains. This establishes a link between optimal control and pole placement design.
+
+**Keywords:** Disturbance rejection (linear case); Lyapunov methods; Observers for linear systems; Time-invariant systems.
+
+## 1. INTRODUCTION
+
+ADRC was developed as a nonlinear general-purpose controller by Han (2009). A linear variant was proposed by Gao (2003), facilitating stability analysis and significantly reducing the number of tuning parameters with the introduction of the “bandwidth parameterization” approach.
+
+The seemingly unorthodox use of elements from modern control theory for creating an almost model-free controller from the user’s point of view is key to its appraisal as a “paradigm change”, cf. Gao et al. (2001); Gao (2006), and a differentiator to other model-free approaches, e. g. as introduced by Fliess and Join (2013). And indeed, the ease of tuning, its performance compared to traditional (PID-type) control, and its extendability with features desirable for industrial applications as in Herbst (2016) and Madoński et al. (2019), make it an attractive alternative for real-world control problems, cf. Zheng and Gao (2018).
+
+The core of ADRC is formed by an observer, which is denoted as “extended state observer” (ESO) and puts emphasis on rejecting disturbances in a broader sense. There are further possible extensions to the observer, such as tracking disturbance derivatives using Generalized Proportional Integral (GPI) observers, cf. Sira-Ramírez et al. (2017). However, we will focus on the (arguably) most common variant of the ESO, which incorporates a single lumped (“total”) disturbance state modeling both unknown dynamics and piecewise constant input disturbance signals of the plant.
+
+In this paper, we will explore the use of the so-called $\alpha$-controller design for tuning the observer and control loop within linear ADRC. It was put forward by Buhl and Lohmann (2009) and is based on the solution of an algebraic Riccati equation to obtain feedback gains leading to an exponentially decaying Lyapunov function for the controlled system. A similar approach has been proposed
+
+by Zhou et al. (2008), denoted as “low gain feedback”. As a matter of fact, applying $\alpha$-controller design to ADRC will lead to reduced controller/observer gains compared to the established bandwidth parameterization approach, which will in turn reduce noise sensitivity of the resulting design.
+
+The main contribution of this work is the introduction of a new ADRC tuning rule, which we will denote as “half-gain tuning”. We will show that $\alpha$-controller design aiming at closed-loop dynamics similar to bandwidth parameterization will always lead to exactly halved gains for the controller and/or observer. Therefore, while grounded in an algebraic Riccati equation, an $\alpha$-controller design for ADRC can be trivially obtained from bandwidth parameterization, superseding the need for solving the former. For an example, detailed insights are given into the frequency-and time-domain behavior when using ADRC with half-gain tuning for the controller and/or observer.
+
+## 2. ACTIVE DISTURBANCE REJECTION CONTROL
+
+This section provides a very brief overview of continuous-time linear ADRC and the prevalent tuning method. For a more detailed introduction we refer to Herbst (2013).
+
+### 2.1 Idea and Structure of the Controller
+
+The essence of linear ADRC can be described as follows:
+
+(1) assume an $n$-th order integrator chain behavior $P(s) = b_0/s^n$ for a single-input single-output (SISO) plant of order $n$, regardless of its actual structure;
+
+(2) apply the inverted gain $1/b_0$ at the controller output to compensate for the plant gain $b_0$;
+
+(3) set up a full-order observer for the integrator chain model (estimated states $\hat{x}_1,...,n$), extended by a constant input disturbance (estimate $\hat{x}_{n+1}$) that captures both actual disturbances and model uncertainties (“extended state observer”, ESO);

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+(4) compensate the disturbance using the estimate $\hat{x}_{n+1}$;
+
+(5) design a full-order state-feedback controller for the remaining “pure” integrator chain $1/s^n$ to achieve the desired closed-loop dynamics.
+
+Control law and observer equations are illustrated in Figure 1, and given in (1) and (2) for the n-th order case.
+
+$$ 
+\begin{gathered}
+u(t) = \frac{1}{b_0} \cdot (k_1 \cdot r(t) - (\mathbf{k}^\mathrm{T} \cdot 1) \cdot \hat{\mathbf{x}}(t)) \quad (1) \\
+\text{with } \mathbf{k}^\mathrm{T} = (k_1 \cdots k_n), \quad \hat{\mathbf{x}} = (\hat{x}_1 \cdots \hat{x}_{n+1})^\mathrm{T}
+\end{gathered}
+$$
+
+$$ 
+\begin{gathered}
+\dot{\mathbf{x}}(t) = \mathbf{A}_{\text{ESO}} \cdot \hat{\mathbf{x}}(t) + \mathbf{b}_{\text{ESO}} \cdot u(t) + \mathbf{l} \cdot (y(t) - \mathbf{c}_{\text{ESO}}^{\mathrm{T}} \cdot \hat{\mathbf{x}}(t)) \quad (2) \\
+\text{with } \mathbf{A}_{\text{ESO}} = \begin{pmatrix} \mathbf{0}^{n \times 1} & \mathbf{I}^{n \times n} \\ 0 & \mathbf{0}^{1 \times n} \end{pmatrix}, \quad \mathbf{b}_{\text{ESO}} = \begin{pmatrix} \mathbf{0}^{(n-1) \times 1} \\ b_0 \\ 0 \end{pmatrix}, \\
+\mathbf{l} = (l_1 \cdots l_{n+1})^\mathrm{T}, \quad \mathbf{c}_{\text{ESO}}^\mathrm{T} = (\mathbf{1} \ \mathbf{0}^{1 \times n})
+\end{gathered}
+$$
+
+Fig. 1. Control loop with ADRC, consisting of extended state observer (ESO) and state-feedback controller including disturbance compensation. Signals: controlled variable *y*, control action *u*, reference *r*; and possible disturbances at plant input (*d*) and output (*n*).
+
+## 2.2 Tuning by Bandwidth Parameterization
+
+Assuming perfect disturbance rejection and compensation of the plant gain $b_0$ by the control law (1), the state-feedback controller $\mathbf{k}^\mathrm{T}$ has to be designed for a “virtual” plant in form of a pure n-th order integrator chain:
+
+$$ 
+\begin{gathered}
+\dot{\mathbf{x}}(t) = \mathbf{A} \cdot \mathbf{x}(t) + \mathbf{b} \cdot u(t), \quad y(t) = x_1(t) \quad (3) \\
+\text{with } \mathbf{A} = \begin{pmatrix} \mathbf{0}^{(n-1) \times 1} & \mathbf{I}^{(n-1) \times (n-1)} \\ 0 & \mathbf{0}^{1 \times (n-1)} \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} \mathbf{0}^{(n-1) \times 1} \\ 1 \end{pmatrix}.
+\end{gathered}
+$$
+
+The predominant controller design approach in linear ADRC is called “bandwidth parameterization”, cf. Gao (2003), and is using pole placement with all poles at $\lambda = -\omega_{\mathrm{CL}}$, the desired closed-loop bandwidth:
+
+$$ 
+\begin{align}
+(\lambda + \omega_{\mathrm{CL}})^n &\stackrel{!}{=} \det (\lambda \mathbf{I} - (\mathbf{A} - \mathbf{b}\mathbf{k}^{\mathrm{T}})) && (4) \\
+&= \lambda^n + k_n \lambda^{n-1} + \dots + k_2 \lambda + k_1. && 
+\end{align}
+$$
+
+Binominal expansion of (4) leads to the controller gains:
+
+$$ k_i = \frac{n!}{(n-i+1)! (i-1)!} \omega_{\mathrm{CL}}^{n-i+1} \quad \forall i=1, \dots, n. \quad (5) $$
+
+For tuning the extended state observer (ESO) with the bandwidth approach, we will follow the notation of Herbst
+
+(2013) in placing the closed-loop observer poles at $\lambda = -k_{\mathrm{ESO}} \cdot \omega_{\mathrm{CL}}$, with $k_{\mathrm{ESO}}$ being the relative factor between observer and control loop bandwidth:
+
+$$ 
+\begin{align}
+(\lambda + k_{\mathrm{ESO}} \cdot \omega_{\mathrm{CL}})^{n+1} &\stackrel{!}{=} \det (\lambda \mathbf{I} - (\mathbf{A}_{\mathrm{ESO}} - l\mathbf{c}_{\mathrm{ESO}}^{\mathrm{T}})) && (6) \\
+&= \lambda^{n+1} + l_1\lambda^n + \dots + l_n\lambda + l_{n+1}. 
+\end{align}
+$$
+
+Binominal expansion of (6) yields the observer gains:
+
+$$ l_i = \frac{(n+1)!}{(n+1-i)! i!} (k_{\mathrm{ESO}} \cdot \omega_{\mathrm{CL}})^i, \quad \forall i=1, \dots, n+1. \quad (7) $$
+
+Comparing these two tuning tasks for linear ADRC, we can conclude that—in both cases—only integrator chains are to be handled: of order *n* (for the closed-loop dynamics) and *n* + 1 (for the extended state observer).
+
+## 3. α-CONTROLLER APPROACH
+
+### 3.1 Brief Overview of the Tuning Method
+
+Buhl and Lohmann (2009) introduced the so-called α-controller approach, a design method leading to an exponentially decreasing Lyapunov function for the closed-loop system. The rate of decay α is the only design parameter of this method:
+
+$$ 
+\dot{V} = -\alpha V, \quad \text{with } \alpha > 0, \text{ and } V = x^{\mathrm{T}} P x. 
+$$
+
+With a plant as in (3):
+
+$$ 
+\begin{align}
+\dot{V} &= \left(\frac{\partial V}{\partial x}\right)^{\mathrm{T}} \dot{x} = 2x^{\mathrm{T}} P (Ax + bu) && (9) \\
+&= x^{\mathrm{T}} (A^{\mathrm{T}} P + PA) x + 2x^{\mathrm{T}} Pbu \\
+&= -\alpha x^{\mathrm{T}} P x.
+\end{align}
+$$
+
+A suitable control law for achieving a negative $\dot{V}$ in (9) is:
+
+$$ u = -b^{\mathrm{T}}Px. 
+$$
+
+Combining these two equations we obtain an algebraic Riccati equation:
+
+$$ (\mathbf{A} + \frac{\alpha}{2}\mathbf{I})^{\mathrm{T}}\mathbf{P} + \mathbf{P}(\mathbf{A} + \frac{\alpha}{2}\mathbf{I}) - 2\mathbf{P}\mathbf{b}\mathbf{b}^{\mathrm{T}}\mathbf{P} = 0. 
+$$
+
+In summary, the state-feedback controller gains for the α-controller approach are $\mathbf{k}^\mathrm{T} = \mathbf{b}^\mathrm{T}\mathbf{P}$, with $\mathbf{P}$ being the solution of (11).
+
+### 3.2 Comparison to Bandwidth Parameterization
+
+When applying the α-controller tuning approach to a loop with the plant being an integrator chain, the closed-loop poles may be complex-valued, but will all have the same real part of $-\frac{\alpha}{2}$, cf. the proof in Buhl and Lohmann (2009). On the other hand, using the “bandwidth parameterization” pole placement design as given in (4), all closed-loop poles will be at $-\omega_{CL}$ and real-valued only.
+
+Being the respective counterparts of PI and PID controllers, first- and second-order ADRC are the most relevant cases in practical applications, resulting in tuning tasks for integrator chains of order up to three. When comparing the α-controller tuning results with pole placement (bandwidth parameterization for $\omega_{CL}$), two observations can be made:

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+(1) Selecting the tuning parameters of both methods as $\alpha = \omega_{CL}$ results in similar closed-loop dynamics for integrator chains of order two and above, with a slightly underdamped response for the $\alpha$-controller due to the complex-valued poles. For the second- and third-order case, the closed-loop step response achieved with these two methods is being compared in Figure 2. A first-order $\alpha$-controller design would be necessarily slower, since only one real-valued pole (at $-\frac{\alpha}{2}$, therefore at half the bandwidth of $\omega_{CL}$) can be placed.
+
+(2) When designing with $\alpha = \omega_{CL}$, the resulting controller gains of the $\alpha$-controller approach are exactly half of the controller gains obtained using pole placement with bandwidth parameterization. We will prove this relation in Section 4.
+
+Fig. 2. Comparison of the normalized closed-loop step responses using bandwidth parameterization (pole placement) and $\alpha$-controller design for full-order state-feedback control of integrator chain systems of order $n=2$ and $n=3$.
+
+The closed-loop pole configurations of $\alpha$-controller designs are presented in Figure 3, with the most important cases being:
+
+* $\lambda_{1/2} = \left(-\frac{1}{2} \pm \frac{1}{2}i\right) \cdot \alpha$ for the second-order integrator chain, and
+
+* $\lambda_1 = -\frac{1}{2} \cdot \alpha$, $\lambda_{2/3} = \left(-\frac{1}{2} \pm \frac{\sqrt{3}}{2}i\right) \cdot \alpha$ for the third-order integrator chain.
+
+Concluding this comparison: The $\alpha$-controller design leads to similar closed-loop dynamics for systems of order two and above, but with only half the controller gain compared
+
+Fig. 3. Closed-loop poles resulting from $\alpha$-controller design for integrator chain plants of orders $n=2$ to $n=5$. Bandwidth parameterization, by contrast, will always place all poles at $-\alpha$.
+
+to a pole placement design with bandwidth parameterization. Therefore the impact of measurement noise on the control action will be reduced, making the $\alpha$-controller design an interesting alternative for noise-affected systems, if the underdamped behavior is tolerable.
+
+## 4. THE HALF-GAIN TUNING METHOD FOR ADRC
+
+As pointed out in Section 3.2, there are up to three options of replacing bandwidth parameterization in linear ADRC with the $\alpha$-controller approach: (1) only for the controller (using $\alpha = \omega_{CL}$, for ADRC of order $n \ge 2$); (2) only for the observer (using $\alpha = k_{ESO} \cdot \omega_{CL}$, possible for any order $n \ge 1$); or (3) for both controller and observer (for $n \ge 2$).
+
+Applying $\alpha$-controller tuning to ADRC results in halved controller and/or observer gains, while maintaining similar (albeit slightly underdamped) dynamics for the control loop and/or the extended state observer. We will therefore denote this design approach for ADRC as the “half-gain tuning” method.
+
+This is the main result of this article, which will be proved in the following: To obtain the $\alpha$-controller gains, the Riccati equation (11) does not have to be solved. The gains can simply be obtained from the straightforward bandwidth tuning rules, i. e. equations (5) (controller) or (7) (observer), by halving these gains for a bandwidth $\omega_{CL} = \alpha$ (controller) or $k_{ESO} \cdot \omega_{CL} = \alpha$ (observer).
+
+**Theorem 1.** For plants as given in (3), the controller gains $k_{BW}^T$ obtained via bandwidth parameterization in (4) are related to the $\alpha$-controller gains $k_\alpha^T$ from (10), (11) by

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+an exact factor of two, if $k_{\text{BW}}^{\text{T}}$ has been designed for a bandwidth $\omega_{\text{CL}} = \alpha$:
+
+$$ k_{\alpha}^{\text{T}} = \frac{1}{2} \cdot k_{\text{BW}}^{\text{T}} = \frac{1}{2} \cdot (k_{\text{BW},1} \cdots k_{\text{BW},n}). \quad (12) $$
+
+**Proof.** We start by rewriting (11) as follows,
+
+$$ \alpha \mathbf{P} = -(\mathbf{A}^{\mathrm{T}} \mathbf{P} + \mathbf{P} \mathbf{A}) + \mathbf{S}, \quad (13) $$
+
+where, using (10) and (12),
+
+$$ \mathbf{S} = \frac{1}{2}\mathbf{k}_{\text{BW}}\mathbf{k}_{\text{BW}}^{\text{T}} = \frac{1}{2} \cdot (k_{\text{BW},1}\mathbf{k}_{\text{BW}} \cdots k_{\text{BW},n}\mathbf{k}_{\text{BW}}). \quad (14) $$
+
+Since **A** is an upper shift matrix, **PA** will result in **P**'s columns **p**ᵢ being shifted:
+
+$$ \mathbf{P}\mathbf{A} = (\mathbf{0} \ \mathbf{p}_1 \ \cdots \ \mathbf{p}_{n-1}). \quad (15) $$
+
+From the first column of (13) we obtain **p**₁, and, as an abbreviation, introduce **Φ**:
+
+$$ \begin{aligned} \alpha \mathbf{p}_1 &= -\mathbf{A}^{\mathrm{T}} \mathbf{p}_1 + \frac{k_{\mathrm{BW},1}}{2} \mathbf{k}_{\mathrm{BW}} \\ \mathbf{p}_1 &= (\alpha \mathbf{I} + \mathbf{A}^{\mathrm{T}})^{-1} \cdot \frac{k_{\mathrm{BW},1}}{2} \mathbf{k}_{\mathrm{BW}} = \mathbf{\Phi}^{-1} \cdot \frac{k_{\mathrm{BW},1}}{2} \mathbf{k}_{\mathrm{BW}}. \end{aligned} \quad (16) $$
+
+For all other columns ($i=2, \dots, n$):
+
+$$ \begin{aligned} \alpha p_i &= -A^T p_i - p_{i-1} + \frac{k_{BW,i}}{2} k_{BW} \\ p_i &= -\Phi^{-1} \cdot p_{i-1} + \Phi^{-1} \cdot \frac{k_{BW,i}}{2} k_{BW}. \end{aligned} \quad (17) $$
+
+We now recursively expand (17) for the final ($n$-th) column:
+
+$$ p_n = \sum_{i=1}^{n} (-1)^{(n-i)} \cdot \Phi^{-(n-i+1)} \cdot \frac{k_{\text{BW},i}}{2} k_{\text{BW}}. \quad (18) $$
+
+$\mathbf{p}_n^\mathrm{T}$ is the gain vector of the $\alpha$-controller, since, recalling (10) with (3), $\mathbf{k}_\alpha^\mathrm{T} = \mathbf{b}^\mathrm{T}\mathbf{P} = \mathbf{b}^\mathrm{T}\mathbf{P}^\mathrm{T} = \mathbf{p}_n^\mathrm{T}$. Multiplying (18) with $\mathbf{\Phi}^n$ one obtains:
+
+$$ \mathbf{\Phi}^n \cdot \mathbf{p}_n = \left( \sum_{i=1}^{n} (-1)^{(n-i)} \cdot \mathbf{\Phi}^{(i-1)} \cdot k_{\text{BW},i} \right) \cdot \frac{1}{2} k_{\text{BW}}. \quad (19) $$
+
+The characteristic polynomial of $\mathbf{\Phi}$ is:
+
+$$ \det (\lambda \mathbf{I} - \mathbf{\Phi}) = \det (\lambda \mathbf{I} - (\alpha \mathbf{I} + \mathbf{A}^{\mathrm{T}})) = (\lambda - \alpha)^n. \quad (20) $$
+
+Comparing (20) with (4) and (5) when $\alpha = \omega_{\text{CL}}$ we find the characteristic polynomial to be:
+
+$$ \det (\lambda\mathbf{I} - \mathbf{\Phi}) = \lambda^n - \sum_{i=1}^{n} (-1)^{(n-i)} \cdot k_{\text{BW},i} \cdot \lambda^{i-1}. \quad (21) $$
+
+This allows us to apply the Cayley-Hamilton theorem to (19), with $\mathbf{\Phi}^n = \sum_{i=1}^n (-1)^{(n-i)} \cdot (\mathbf{\Phi}^{(i-1)}) \cdot k_{\text{BW},i}$ we finally obtain:
+
+$$ p_n = k_\alpha = \frac{1}{2} k_{\text{BW}}. \quad (22) $$
+
+This concludes the proof. As the analytical solution of the algebraic Riccati equation (11), it provides a link between optimal control and pole placement for linear ADRC. □
+
+*Remark 2.* Due to the duality of the design problem, a proof of the half-gain relation for the extended state observer design (with $k_{\text{ESO}} \cdot \omega_{\text{CL}} = \alpha$) can be constructed in the same manner.
+
+## 5. EXAMPLES
+
+Aim of this section is to provide visual insights into an ADRC-based control loop when using half-gain tuning for the controller, the extended state observer, or both. For this purpose we can restrict ourselves to a second-order plant with normalized gain and eigenfrequency:
+
+$$ P(s) = \frac{1}{s^2 + 2s + 1}. \quad (23) $$
+
+Since ADRC is almost insensitive to the damping ratio, especially of underdamped systems, cf. Herbst (2013), the informative value of our example will not be compromised by the particular choice of critical damping in $P(s)$.
+
+Bandwidth parameterization is applied to a second-order ADRC ($n=2$) using $\omega_{\text{CL}} = 1$ rad/s, $k_{\text{ESO}} = 10$, and $b_0 = 1$. Four cases are being compared: (1) unmodified bandwidth tuning, (2) applying half-gain tuning only to the outer control loop (“K/2 controller”), (3) applying half-gain tuning only to the ESO (“L/2 observer”), and (4) half-gain tuning for both controller and observer.
+
+### 5.1 Impact on Open-Loop Characteristics
+
+For stability and dynamics, the feedback controller part of an ADRC control loop is essential. In Figure 4 the transfer functions from controlled variable $y$ to control signal $u$ are compared for the four possible cases. Additionally, the loop gain transfer functions are being compared in Figure 5.
+
+Fig. 4. Comparison of the feedback controller transfer functions with or without half-gain tuning.
+
+The most interesting result might be that half-gain observer tuning (“L/2” case) provides significantly improved high-frequency damping while having almost no impact on the lower frequencies up to and including the crossover frequency. On the other hand one has to expect some low-frequency performance penalty when (additionally or solely) applying half-gain controller tuning (“K/2” cases).

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+Fig. 5. Comparison of the open-loop gain transfer function with or without half-gain tuning.
+
+## 5.2 Impact on Closed-Loop Characteristics
+
+With the control loop signals denoted as in Figure 1, the “gang of six” transfer functions are defined as
+
+$$ \begin{pmatrix} y(s) \\ u(s) \end{pmatrix} = \begin{pmatrix} G_{yr}(s) & G_{yd}(s) & G_{yn}(s) \\ G_{ur}(s) & G_{ud}(s) & G_{un}(s) \end{pmatrix} \cdot \begin{pmatrix} r(s) \\ d(s) \\ n(s) \end{pmatrix}, \quad (24) $$
+
+providing frequency-domain insights for a two-degrees-of-freedom control loop as is the case with ADRC, cf. Åström and Murray (2008). For the four cases in our example, they are presented and discussed in Figure 6 and Figure 7.
+
+While not shown here for brevity, a discrete-time implementation of “K/2” and “L/2” design was successfully tested as well, exhibiting the desired noise reduction in the control signal $u$.
+
+# 6. CONCLUSION
+
+A new “half-gain tuning” rule for linear active disturbance rejection control (ADRC) based on the so-called $\alpha$-controller design was introduced. Compared to the common “bandwidth parameterization” approach, similar closed-loop dynamics can be achieved with lower (halved) feedback gains, therefore reducing the noise sensitivity of ADRC.
+
+In view of the examples presented in Section 5, a recommendation emerges to start with half-gain tuning for the observer. This has the least impact on the closed-loop dynamics compared to bandwidth parameterization, while already providing a significant reduction of control signal sensitivity to measurement noise.
+
+While being the analytical solution of an algebraic Riccati equation, the proposed feedback gains can simply be obtained from a bandwidth parameterization design by
+
+halving the gains, as proved in this paper, establishing a link between pole placement and optimal control.
+
+# ACKNOWLEDGEMENTS
+
+Gernot Herbst would like to thank Michael Buhl for drawing his attention to the $\alpha$-controller approach.
+
+# REFERENCES
+
+Åström, K.J. and Murray, R.M. (2008). *Feedback Systems: An Introduction for Scientists and Engineers*. Princeton University Press.
+
+Buhl, M. and Lohmann, B. (2009). Control with exponentially decaying Lyapunov functions and its use for systems with input saturation. In *2009 European Control Conference (ECC)*, 3148–3153. doi: 10.23919/ECC.2009.7074889.
+
+Fliess, M. and Join, C. (2013). Model-free control. *International Journal of Control*, 86(12), 2228–2252. doi: 10.1080/00207179.2013.810345.
+
+Gao, Z. (2003). Scaling and bandwidth-parameterization based controller tuning. In *Proceedings of the 2003 American Control Conference*, 4989–4996. doi: 10.1109/ACC.2003.1242516.
+
+Gao, Z. (2006). Active disturbance rejection control: A paradigm shift in feedback control system design. In *Proceedings of the 2006 American Control Conference*, 2399–2405. doi:10.1109/ACC.2006.1656579.
+
+Gao, Z., Huang, Y., and Han, J. (2001). An alternative paradigm for control system design. In *Proceedings of the 40th IEEE Conference on Decision and Control*. doi: 10.1109/CDC.2001.980926.
+
+Han, J. (2009). From PID to active disturbance rejection control. *IEEE Transactions on Industrial Electronics*, 56(3), 900–906. doi:10.1109/TIE.2008.2011621.
+
+Herbst, G. (2013). A simulative study on active disturbance rejection control (ADRC) as a control tool for practitioners. *Electronics*, 2(3), 246–279. doi: 10.3390/electronics2030246.
+
+Herbst, G. (2016). Practical active disturbance rejection control: Bumpless transfer, rate limitation, and incremental algorithm. *IEEE Transactions on Industrial Electronics*, 63(3), 1754–1762. doi: 10.1109/TIE.2015.2499168.
+
+Madoński, R., Shao, S., Zhang, H., Gao, Z., Yang, J., and Li, S. (2019). General error-based active disturbance rejection control for swift industrial implementations. *Control Engineering Practice*, 84, 218–229. doi: 10.1016/j.conengprac.2018.11.021.
+
+Sira-Ramírez, H., Luviano-Juárez, A., Ramírez-Neria, M., and Zurita-Bustamante, E.W. (2017). *Active Disturbance Rejection Control of Dynamic Systems: A Flatness Based Approach*. Butterworth-Heinemann.
+
+Zheng, Q. and Gao, Z. (2018). Active disturbance rejection control: Some recent experimental and industrial case studies. *Control Theory and Technology*, 16(4), 301–313. doi:10.1007/s11768-018-8142-x.
+
+Zhou, B., Duan, G., and Lin, Z. (2008). A parametric Lyapunov equation approach to the design of low gain feedback. *IEEE Transactions on Automatic Control*, 53(6), 1548–1554. doi:10.1109/TAC.2008.921036.

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+Fig. 6. Gang-of-six comparison (frequency domain) with or without half-gain tuning for controller and/or observer within ADRC. As predicted in Section 5.1, the half-gain observer (“L/2”) case provides enhanced high-frequency damping in $G_{un}(j\omega)$ almost without any side-effects on other performance criteria. The “K/2” cases, on the other hand, will—while still yielding some additional high-frequency damping in $G_{un}$—involve slower reaction to reference signal changes and more overshoot induced by disturbances at the plant input.
+
+Fig. 7. To ease and support the interpretability of Figure 6, a time-domain perspective is given in this figure with the step responses of the gang-of-six transfer functions with or without half-gain tuning.

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+**Theorem 7.** For $h \in \text{Def}(\mathbb{R}^n)$,
+
+$$ (21) \quad \int_{s=0}^{\infty} h \lfloor d\mu_k \rfloor = \int_{\sum_{n,k}}^{\infty} \mu_k\{h \ge s\} - \mu_k\{h < -s\} ds \quad \text{excursion sets} $$
+
+$$ (22) \quad = \int_{\mathcal{P}_{n,n-k}} \int_P h \lfloor d\chi \rfloor d\lambda(P) \quad \text{slices} $$
+
+$$ (23) \quad = \int_{G_{n,k}} \int_L \int_{\pi_L^{-1}(x)} h \lfloor d\chi \rfloor dx d\gamma(L) \quad \text{projections} $$
+
+$$ (24) \quad = \int_{s=0}^{\infty} (\mathbf{C}^{\{h \ge s\}}(\omega_k) - \mathbf{C}^{\{h < -s\}}(\omega_k)) ds \quad \text{conormal cycle} $$
+
+$$ (25) \quad = -\int h \lceil d\mu_k \rceil \quad \text{duality} $$
+
+*Proof.* Note that for $T > 0$ sufficiently large and $N = mT$,
+
+$$ 
+\begin{align*}
+\int h \lfloor d\mu_k \rfloor &= \lim_{m \to \infty} \frac{1}{m} \int \lfloor mh \rfloor d\mu_k = \lim_{m \to \infty} \frac{1}{m} \sum_{i=1}^{\infty} \mu_k\{mh \ge i\} - \mu_k\{mh < -i\} \\
+&= \lim_{N \to \infty} \frac{T}{N} \sum_{i=1}^{N} \mu_k\left\{h \ge \frac{iT}{N}\right\} - \mu_k\left\{h < -\frac{iT}{N}\right\} \\
+&= \int_0^T \mu_k\{h \ge s\} - \mu_k\{h < -s\} ds.
+\end{align*}
+$$
+
+Thus, (21); the same proof using $\lceil d\mu_k \rceil$ implies that
+
+$$ (26) \quad \int h \lceil d\mu_k \rceil = \int_{s=0}^{\infty} \mu_k\{h > s\} - \mu_k\{h \le -s\} ds, $$
+
+which, with (21), yields (25). For (22),
+
+$$ \int_0^\infty \mu_k\{h \ge s\} - \mu_k\{h < -s\} ds = \int_0^\infty \int_{P_{n,n-k}} \chi(\{h \ge s\} \cap P) - \chi(\{h < -s\} \cap P) d\lambda(P) ds. $$
+
+This integral is well-defined, since the excursion sets $\{h \ge s\}$ and $\{h < -s\}$ are definable, and $h$ is bounded and of compact support. The Fubini theorem yields (22) via
+
+$$ \int_{P_{n,n-k}} \int_0^\infty \chi(\{h \ge s\} \cap P) - \chi(\{h < -s\} \cap P) ds d\lambda(P) = \int_{P_{n,n-k}} \int_P h \lfloor d\chi \rfloor d\lambda(P). $$
+
+For (23), fix an $L \in G_{n,k}$ and let $\pi_L$ be the orthogonal projection map on to $L$. Then the affine subspaces perpendicular to $L$ are the fibers of $\pi_L$ and
+
+$$ \{P \in P_{n,n-k} : P^\perp L\} = \{\pi_L^{-1}(x) : x \in L\}. $$
+
+Instead of integrating over $P_{n,n-k}$, integrate over the fibers of orthogonal projections onto all linear subspaces of $G_{n,k}$:
+
+$$ \int h \lfloor d\mu_k \rfloor = \int_{P_{n,n-k}} \int_P h \lfloor d\chi \rfloor d\lambda(P) = \int_{G_{n,k}} \int_L \int_{\pi_L^{-1}(x)} h \lfloor d\chi \rfloor dx d\gamma(L). $$
+
+Finally, for (24), rewrite (21) by expressing the intrinsic volumes in terms of the conormal cycles, as in (12). □

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+FIGURE 3. An upper step function of h, depicted at left, composed with a decreasing function c, becomes a lower step function of c(h), depicted at right. As the step size approaches zero, we obtain Proposition 13.
+
+We may then rewrite the above sum as:
+
+$$ \int_{\mathbb{R}^n} \frac{1}{m} [\mathrm{mc}(h)] \, d\mu_k = \sum_{s \in S} c(s) \cdot \mu_k\{c(s) \le c(h) < c(s-\epsilon)\} = \sum_{s \in S} c(s) \cdot \mu_k\{s - \epsilon < h \le s\}, $$
+
+where $\epsilon \to 0$ as $m \to \infty$ by continuity of c. In the limit, both sides are equal:
+
+$$ \lim_{\epsilon \to 0} \sum_{s \in S} c(s) \cdot \mu_k\{s - \epsilon < h \le s\} = \lim_{m \to \infty} \sum_{i \in Z} c\left(\frac{i}{m}\right) \cdot \mu_k\left\{\frac{i-1}{m} < h \le \frac{i}{m}\right\}. \quad \square $$
+
+Proposition 13 implies that if $c : \mathbb{R} \to \mathbb{R}$ is increasing on some interval and decreasing on another, then the maps $v, u : \mathrm{Def}(\mathbb{R}^n) \to \mathbb{R}$ defined
+
+$$ v(h) = \int_{\mathbb{R}^n} c(h) [\, d\mu_k] \quad \text{and} \quad u(h) = \int_{\mathbb{R}^n} c(h) [\, d\mu_k] $$
+
+are neither lower- nor upper-continuous.
+
+Lemma 12 and Proposition 13 provide a generalization of Hadwiger's Theorem:
+
+**Theorem 14.** Any $E_n$-invariant definably lower-continuous valuation $v : \mathrm{Def}(\mathbb{R}^n) \to \mathbb{R}$ is of the form:
+
+$$ (36) \qquad v(h) = \sum_{k=0}^{n} \left( \int_{\mathbb{R}^n} c_k \circ h [\, d\mu_k] \right), $$
+
+for some $c_k \in C(\mathbb{R})$ continuous and monotone, satisfying $c_k(0) = 0$. Likewise, an upper-continuous valuation can be similarly written in terms of upper Hadwiger integrals.
+
+*Proof.* Let $v : \mathrm{Def}(\mathbb{R}^n) \to \mathbb{R}$ be a lower valuation, and $h \in \mathrm{Def}(\mathbb{R}^n)$. First approximate $h$ by lower step functions. That is, for $m > 0$, let $h_m = \frac{1}{m} [\mathrm{mh}]$. In the lower flat topology, $\lim_{m\to\infty} h_m = h$. On each of these step functions, Lemma 12 implies that $v$ is a linear combination of Hadwiger integrals:
+
+$$ (37) \qquad v(h_m) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h_m) \, d\mu_k. $$
+
+for some $c_k : \mathbb{R} \to \mathbb{R}$ with $c_k(0) = 0$, depending only on $v$ and not on $m$. By Proposition 13, the $c_k$ must be increasing functions since we are approximating $h$ with lower step functions in the lower flat topology.