chunk_uid stringlengths 40 40 | chunk_type stringclasses 2
values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
values | discipline stringclasses 8
values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
e11ec2fc48e05cf5894e90f757b69771307e299f | subsection | 80 | 84 | Line-search: The Newton-secant solver | The details are left to the readerNewton-Secant solver
[1]
k = 0, k_{\max } \ge 1, \ell _{l} < \ell _{r},
g(\ell _{l}) > 0, g(\ell _{r}) < 0, \textup {tol} > 0
k \le k_{\max } and \ell _{r} - \ell _{l} > \textup {tol}
k := k + 1
\ell _l^{\textup {aux}} := \ell _{l}
g(\ell _{l}) > g(\ell _{r})
s_l := \frac{g(\ell _{r... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04438937082886696,
-0.003353990614414215,
-0.029836978763341904,
-0.024986181408166885,
0.035267431288957596,
-0.017206601798534393,
0.03664029762148857,
0.03630470857024193,
-0.004797407891601324,
0.029775962233543396,
-0.017221854999661446,
0.0099990488961339,
-0.02573363110423088,
0.... |
601b7447a6a706324e8c94a37af355e0e01d53dc | subsection | 81 | 84 | Line-search: The Newton-secant solver | For
instance, if the initial guess \ell ^0 \in [0,1] for Newton's
method is such that \ell ^0 > \ell ^* (g(\ell ^0) < 0), then
Newton's method produces a sequence \lbrace \ell ^k\rbrace _{k \in }
satisfying \ell ^* < \ell ^k for all k\in . This implies that
g(\ell ^k) < 0 for all k\in , which is incompatible with the
c... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05571071803569794,
0.008298639208078384,
-0.006033293902873993,
-0.014743814244866371,
0.012348802760243416,
0.0021032628137618303,
-0.009313086979091167,
0.000015016495126474183,
-0.004332378040999174,
0.03999822214245796,
-0.02157036028802395,
0.010686024092137814,
-0.030936839058995247... |
f20902cd841eb2c99c04df658c644487dd579492 | subsection | 82 | 84 | Relaxing the bounds | In
general the quantity \Psi _i^{\min } defined in
(REF ) is accurate enough to make the limited
high-order solution second-order in the L^1-norm in space. But it is
too tight to make the method higher-order or even second-order in the
L^\infty -norm in the presence of smooth extrema. The situation is
even worse when u... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06335566937923431,
0.009987062774598598,
-0.04052809625864029,
-0.003936840686947107,
-0.0070535060949623585,
-0.0050354935228824615,
0.015602401457726955,
-0.019150136038661003,
0.03111324831843376,
0.04348835349082947,
-0.03820871561765671,
0.017349565401673317,
-0.022766536101698875,
... |
98c541174a800f66e4d57720b016ea7d83db12ee | subsection | 83 | 84 | Relaxing the bounds | The exponent 1.5 is somewhat ad hoc; in principle one could take
r_i = (\frac{m_i}{||})^{\frac{\delta }{d}} with \delta <2. | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04135126993060112,
0.0023746462538838387,
-0.01707456447184086,
0.03344722464680672,
-0.01988217793405056,
-0.007438650820404291,
-0.002738949377089739,
0.06088249757885933,
0.04708856716752052,
-0.0006594649748876691,
-0.04428095370531082,
-0.00854491163045168,
-0.027816738933324814,
0... |
87190a34c5fd82003c63dabf35b5c650b285a535 | abstract | 0 | 25 | Abstract | In this work, we address some optimal control problems related to the
evolution of two isothermal, incompressible, immisible fluids in a two
dimensional bounded domain. A distributed optimal control problem is formulated
as the minimization of a suitable cost functional subject to the controlled
nonlocal Cahn-Hilliard-... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
0.0014987668255344033,
-0.008283261209726334,
-0.010891801677644253,
0.02707695960998535,
-0.015895318239927292,
0.00018686914700083435,
-0.020776798948645592,
-0.018290294334292412,
0.0244073998183012,
0.06254395842552185,
-0.02963973581790924,
-0.03432290628552437,
-0.005941676441580057,
... | |
f0ec1e55fd749b28a3945be271f1cff2f9984bdf | subsection | 1 | 25 | Body | lemmatheorem
lemmapropositiontheorem
propositionassumptiontheorem
assumptioncorollarytheorem
corollarydefinitiontheorem
definitionexampletheorem
exampleremarktheorem
remarkhypothesistheorem
hypothesispropertytheorem
property[Pontryagin Maximun Principle and Second order optimality conditions]Pontryagin Maximun... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.022460253909230232,
-0.00843022484332323,
-0.024382803589105606,
0.020339347422122955,
-0.009803473949432373,
-0.00044177621020935476,
-0.028487293049693108,
0.02082761377096176,
-0.00005900680480408482,
0.02627483569085598,
-0.044096559286117554,
0.010787636041641235,
-0.0227043870836496... | |
71d6fa48c58d1b5c1171a41e9b640a6b52736287 | subsection | 2 | 25 | Body | We describe the first order necessary conditions of optimality via Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.Key words: optimal control, nonlocal Cahn-Hilliard-Navier-Stokes systems, Pontryagin maximum principle, necessary and sufficient optima... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.01175758522003889,
0.0009089048253372312,
-0.025056397542357445,
0.001581284566782415,
0.019593432545661926,
-0.007538283243775368,
-0.020570050925016403,
-0.005111993756145239,
0.027879439294338226,
0.05160824581980705,
-0.027513207867741585,
-0.01782330870628357,
-0.010124798864126205,
... | |
2cb07a6fb14679cd0ba669749af2d5be9d7d6409 | subsection | 3 | 25 | Body | In , author studies the local Cahn-Hilliard-Navier-Stokes system and establishes the existence of weak solutions in dimensions 2 and 3. The existence and uniqueness of strong solutions in 2 and 3 dimensions (global in 2D and local in time for 3D) is also established in . The authors in proved the existence of a weak so... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.019896341487765312,
-0.008483408950269222,
-0.014487405307590961,
0.014182246290147305,
0.00817062146961689,
0.026792926713824272,
0.02631993032991886,
0.021635744720697403,
-0.010344876907765865,
0.05285347253084183,
-0.019133444875478745,
0.0009240584331564605,
0.006114615593105555,
0... | |
a978db4e65cab6095f95216c7e3ced9f4946eb66 | subsection | 4 | 25 | Body | An optimal distributed control problem for the two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential is studied in . A distributed optimal control problem for the nonlocal Cahn-Hilliard-Navier-Stokes system with non-constant viscosity and regular potential is exami... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
0.007280521094799042,
-0.026788046583533287,
-0.01697898469865322,
0.010922688990831375,
-0.0031902336049824953,
0.03243245184421539,
-0.0008585790055803955,
-0.0003758964885491878,
0.024026865139603615,
0.056627124547958374,
-0.030830662697553635,
0.0003646934637799859,
-0.00490453140810132... | |
759f13ed9de3b82072023567822c499b73f16ba0 | subsection | 5 | 25 | Body | In the system (REF )-(), \mathrm {U} is the distributed control acting on the system.Let us introduce the following functional spaces required for getting the unique global solvability results of the system (REF )-().Let us denote \Vert \cdot \Vert and (\cdot , \cdot ) the norm and the scalar product, respectively, on ... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.06408021599054337,
-0.0016229839529842138,
-0.038539670407772064,
0.0006641647196374834,
0.008414342068135738,
0.006472864653915167,
-0.013289015740156174,
0.0208870992064476,
0.022290760651230812,
0.03634263575077057,
-0.028622495010495186,
0.023297734558582306,
0.009680689312517643,
0... | |
bf9c2e5cf8f4b00a4dbcbc7afa97f144a5e5f79b | subsection | 6 | 25 | Body | We know that \mathbf {u} can be expressed as \mathbf {u}=\sum \limits _{j=1}^{\infty }\langle \mathbf {u},\mathbf {e}_j\rangle \mathbf {e}_j, so that \mathrm {A}\mathbf {u}=\sum \limits _{j=1}^{\infty }\lambda _j\langle \mathbf {u},\mathbf {e}_j\rangle \mathbf {e}_j. Thus, it is immediate thatwhich is the Poincaré ineq... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.0748710110783577,
0.0003308404702693224,
-0.08323068916797638,
-0.023111160844564438,
0.03609307482838631,
0.048113930970430374,
0.009686856530606747,
0.031226763501763344,
0.010030091740190983,
0.03685582056641579,
-0.04225604981184006,
0.0008170901564881206,
0.0322488434612751,
-0.035... | |
6a5c839fc77a7ff4a44ba866ee683009f891da4d | subsection | 7 | 25 | Body | Then, if
0< \alpha < 1 and
\frac{n}{2} = \alpha s_1 + (1-\alpha )s_2, the following inequality holdsFor \mathbf {u}\in \mathbb {H}^2(\Omega )\cap \mathbb {H}_0^1(\Omega ), the Agmon's inequality in 2D states that there exists a constant
C>0 such thatFor every f \in \mathrm {V}^{\prime } we denote \overline{f} the avera... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.029976287856698036,
-0.013935541734099388,
-0.027596490457654,
0.020045211538672447,
0.013516025617718697,
0.0018191717099398375,
0.026452356949448586,
0.02169276401400566,
0.03499521687626839,
0.001042114570736885,
-0.04405675455927849,
-0.012158321216702461,
0.009603090584278107,
0.00... | |
2dca79807e2a32877188378a33f4e44327fdab09 | subsection | 8 | 25 | Body | Let us first make the following assumptions:Assumption Let
\mathrm {J} and \mathrm {F} satisfy:\mathrm {J}\in \mathrm {W}^{1,1}(\mathbb {R}^2;\mathbb {R}), \ \mathrm {J}(x)= \mathrm {J}(-x) \; \text{and} \ a(x) = \int _\Omega \mathrm {J}(x-y)\mathrm {d}y \ge 0, a.e., in \Omega .
\mathrm {F}\in \mathrm {C}^{2}(\mathbb ... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.01427211333066225,
0.04164924472570419,
-0.046683769673109055,
0.022609589621424675,
-0.007147498894482851,
-0.033594004809856415,
0.009344382211565971,
0.04186283051967621,
-0.000031108211260288954,
-0.0007513645687140524,
0.00918419286608696,
-0.04528020694851875,
-0.011259026825428009,... | |
5b26bcf6b84fa614b38ce74620dc4655350a79e1 | subsection | 9 | 25 | Body | Indeed, it can be replaced by \mathrm {J}\in \mathrm {W}^{1,1}(\mathrm {B}_\delta ;\mathbb {R}), where \mathrm {B}_\delta := \lbrace z \in \mathbb {R}^2 : |z| < \delta \rbrace with \delta := \emph {diam}(\Omega )=\sup \limits _{x,y\in \Omega }d(x,y), where d(\cdot ,\cdot ) is the Euclidean metric on \mathbb {R}^2, or a... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.05280432850122452,
0.021152254194021225,
-0.04242659732699394,
-0.019580332562327385,
-0.01893935538828373,
-0.04691343754529953,
0.024296095594763756,
0.06083180382847786,
-0.01190386526286602,
-0.00541778514161706,
-0.026051152497529984,
0.0033784848637878895,
-0.0132773881778121,
0.0... | |
21e59bb9a035b25fb10538bc1003d4c0ee8704c8 | subsection | 10 | 25 | Body | Then (\mathbf {u},\varphi ) is said to be a weak solution to the uncontrolled system (REF )-() on [0,T] corresponding to the initial conditions \mathbf {u}_0 and \varphi _0 if(i)
\mathbf {u},\varphi and \mu satisfy
\mathopen {}\mathclose {\left\lbrace
\begin{aligned}& \mathbf {u}\in \mathrm {L}^{\infty }(0,T;\mathbb ... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
0.004951984155923128,
0.029482999816536903,
-0.014115825295448303,
-0.02931513637304306,
-0.021181367337703705,
-0.024782810360193253,
0.05557820200920105,
0.0827721506357193,
0.0016700547421351075,
0.0591491237282753,
-0.005951537285000086,
-0.013635124079883099,
0.025408485904335976,
0.0... | |
afd6e9d286c9795a1befca08a2a31f26c758b034 | subsection | 11 | 25 | Body | (iii)
Moreover, the following initial conditions hold in the weak sense
\mathbf {u}(0)=\mathbf {u}_0,\ \varphi (0)=\varphi _{0},
i.e., for every \mathbf {v}\in \mathbb {V}_{\text{div}}, we have (\mathbf {u}(t),\mathbf {v}) \rightarrow (\mathbf {u}_0,\mathbf {v}) as t\rightarrow 0, and for every \chi \in \mathrm {V}, ... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.016293879598379135,
0.02062670886516571,
-0.03725622966885567,
-0.007910464890301228,
0.003909463062882423,
-0.02265581488609314,
0.06358885020017624,
0.08128631860017776,
0.014730094000697136,
0.03335057944059372,
0.0004128774453420192,
-0.00858937669545412,
-0.006617481354624033,
0.01... | |
21099e29435b1b448ce838ed968ba5237368b142 | subsection | 12 | 25 | Body | Furthermore, settingthe following energy estimate holds for almost any t>0:or the weak solution (\mathbf {u},\varphi ) satisfies the following energy identity,Furthermore, if in addition \mathbf {h} \in \mathrm {L}^2_{\text{tb}}([0,\infty );\mathbb {V}_{\text{div}}^{\prime }), then the following dissipative estimate is... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.046526942402124405,
0.02425502985715866,
-0.03792327269911766,
-0.016475114971399307,
-0.02152443118393421,
-0.05259832739830017,
0.055008575320243835,
0.025002511218190193,
0.011517325416207314,
0.016017472371459007,
0.009656247682869434,
-0.01575814187526703,
-0.013462304137647152,
0.... | |
3eb138d555310ed60677d0abd371bc89dc3ccf9c | subsection | 13 | 25 | Body | Then the following continuous dependence estimate holds:for all t \in [0,T], where \Lambda _0(t), \Lambda _1(t) and \Lambda _2(t) are continuous functions which depend on the norms of the two solutions. The functions \mathbb {Q} and \Lambda _i(t) also depend on \mathrm {F}, \mathrm {J} and \Omega .The following theorem... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.045850109308958054,
0.027595536783337593,
-0.04447643458843231,
0.0010846282821148634,
-0.0009844646556302905,
-0.04817008599638939,
0.0231692623347044,
0.0349828377366066,
0.0007388254744000733,
0.028725000098347664,
-0.0015873538795858622,
-0.010294905863702297,
-0.006471519824117422,
... | |
4b5b1bbaa6252815e0605841ddf1e93df44cb2eb | subsection | 14 | 25 | Body | Then, \mathbf {u}_1 = \mathbf {u}_2 and \varphi _1=\varphi _2, satisfying the following differential inequality:where \mathbf {u}=\mathbf {u}_1-\mathbf {u}_2, \varphi =\varphi _1-\varphi _2 and the function \Pi is given byTheorem 0.5 (Lemma 2.6, )
Let the Assumption REF be satisfied. Let \mathbf {u}_0\in \mathbb {G}_{... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.02054215595126152,
0.013994916342198849,
-0.04377036169171333,
-0.025074860081076622,
0.008203127421438694,
0.000338140525855124,
0.027104657143354416,
0.07704071700572968,
0.004517443012446165,
0.05530815199017525,
-0.029851751402020454,
0.005890989676117897,
-0.010759450495243073,
0.0... | |
f62fc7ca0bb07f30b9abd5181166db0b388a7bf2 | subsection | 15 | 25 | Body | Since the pressure is also an unknown quantity, in order to linearize, we substitute \mathbf {u}= \mathbf {w}+ \widehat{\mathbf {u}}, =\widetilde{}+\widehat{} and \varphi = \psi + \widehat{\varphi } in (REF ) and (REF ) to getwhere \widetilde{}_\mathbf {w}= \widetilde{}-(\mathrm {F}^{\prime }(\widehat{\varphi })+a\wide... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
0.010609597899019718,
0.0293346606194973,
-0.027885470539331436,
-0.009190917015075684,
0.00871039554476738,
-0.02277516759932041,
0.018519125878810883,
0.0551607571542263,
-0.012463035993278027,
0.05860830470919609,
-0.03352968394756317,
-0.017375027760863304,
-0.016307203099131584,
0.005... | |
48abf92338fcd48b1aa41043c1af7f0d322d7ef4 | subsection | 16 | 25 | Body | The associated cost functional is defined bywhere \mathbf {u}_d(\cdot )\in \mathrm {L}^2(0,T;\mathbb {V}_{\text{div}}) and \varphi _d(\cdot )\in \mathrm {L}^2(0,T;\mathrm {V}) are the desired states. Note that the cost functional is the sum of total energy and total effort by control.Let us assume thatand the initial d... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.019535934552550316,
-0.019154373556375504,
-0.04212435707449913,
-0.008249353617429733,
-0.008890376426279545,
0.016636069864034653,
0.011752085760235786,
0.010515827685594559,
0.01706341840326786,
0.05604371055960655,
-0.008424871601164341,
-0.006135504227131605,
0.02251211181282997,
0... | |
ce639b2aaf189cd57526159bf114f515b9936705 | subsection | 17 | 25 | Body | Then the system (REF )-() can be written asWe define the augmented cost functional \widetilde{\mathcal {J}} bywhere \mathbf {p} and \eta denote the adjoint variables corresponding to \mathbf {u} and \varphi respectively. Corresponding to in the system (REF ), we have q in the adjoint system.Before establishing the Pont... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.038200657814741135,
0.025965463370084763,
-0.03591227903962135,
0.02253289520740509,
-0.002303633838891983,
-0.011197797022759914,
0.027399513870477676,
0.02619430050253868,
0.000013587245121016167,
0.022471873089671135,
0.01862739771604538,
0.01899353787302971,
-0.03801758587360382,
0.... | |
a53b71f202cabfc7fa6005920679caa3f971b057 | subsection | 18 | 25 | Body | Then, there exists a unique weak solution of the system (REF ) satisfyingand for all \mathbf {v}\in \mathbb {V} and \zeta \in \mathrm {H} and for almost all t \in (0,T), we havewhere \mathbf {p}(T)=\mathbf {p}_T\in \mathbb {G}_{\text{div}}, \eta (T)=\eta _T\in \mathrm {V} are satisfied in the weak sense.Using Galerkin ... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04146832227706909,
0.013601548038423061,
-0.04534358158707619,
-0.013929571956396103,
-0.00028964318335056305,
-0.006091327406466007,
0.04931038245558739,
0.02178688906133175,
-0.007121169473975897,
0.03249724954366684,
-0.018094712868332863,
-0.0005015713977627456,
0.023556692525744438,
... | |
d6b95157a542b1c25deebbc0dd21202f7e886b30 | subsection | 19 | 25 | Body | From the definition of \mathcal {J}(\cdot ,\cdot ,\cdot ), this impliesSince \mathbf {u},\mathbf {u}_d \in \mathrm {L}^{2}(0,T;\mathbb {G}_{\text{div}}) and \varphi ,\varphi _d \in \mathrm {L}^{2}(0,T;\mathrm {H}), from the above relation, it is clear that, there exist a K>0, large enough such thatIn particular, there ... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.03439965099096298,
0.03885604068636894,
-0.03034006617963314,
-0.038581330329179764,
-0.010324474424123764,
-0.0023579176049679518,
0.006264887750148773,
0.019382234662771225,
-0.007432400714606047,
0.03259878233075142,
-0.02441856451332569,
0.0009309577872045338,
0.016207821667194366,
... | |
8aa638deb7896a85db32839137b71abf460e05c9 | subsection | 20 | 25 | Body | Since \mathbf {u}^*\in \mathrm {C}([0,T];\mathbb {V}_{\text{div}}) and \mathrm {J}\in \mathrm {W}^{2,1}(\mathbb {R}^2;\mathbb {R}), we know that \varphi ^*\in \mathrm {C}([0,T];\mathrm {H}^2) and hence we haveHence (\mathbf {u}^*,\varphi ^*) is a unique strong solution of (REF )-() with control \mathrm {U}^*\in \mathrm... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.05226123705506325,
0.012218200601637363,
-0.05226123705506325,
-0.023032186552882195,
0.015995845198631287,
0.008036080747842789,
-0.005674098618328571,
0.034800123423337936,
-0.018789013847708702,
0.06471601873636246,
-0.00528870290145278,
-0.005323044955730438,
0.035135913640260696,
0... | |
60fa87dfe627a25a7b437ec5debe50da43c6740b | subsection | 21 | 25 | Body | Equivalently the above minimum principle may be written in terms of the Hamiltonian formulation.
Let us first define the Lagrangian byThen, we can define the corresponding Hamiltonian bywhere {N}_1 and {N}_2 are defined by (REF ).
Hence, we get the minimum principle asfor all \mathrm {W} \in \mathbb {G}_{\text{div}} an... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.046847276389598846,
-0.0028459338936954737,
-0.01384054683148861,
-0.04144534096121788,
0.00995695311576128,
-0.04300183057785034,
0.029130764305591583,
-0.009262637235224247,
0.01866261102259159,
0.032320041209459305,
-0.03296094760298729,
0.007149168755859137,
0.0062564765103161335,
0... | |
abb45de18c1323f5b56d6d4fbaf6ce88c123162f | subsection | 22 | 25 | Body | Let \mathrm {U}^*+\lambda \mathrm {U}\in \mathbb {G}_{\text{div}} such that (\mathbf {u}_{\mathbf {u}^*+\lambda \mathrm {U}},\varphi _{\mathbf {u}^*+\lambda \mathrm {U}},\mathrm {U}_{\mathbf {u}^*+\lambda \mathrm {U}})\in {A}_{\text{ad}}, for all 0\le \lambda \le 1. Then, for \lambda \in [0,1], we can deduceSince (\mat... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.039073724299669266,
0.020941073074936867,
-0.008577903732657433,
-0.026328973472118378,
0.0038234249223023653,
-0.015003698877990246,
-0.0033197400625795126,
0.04459899291396141,
0.02918318659067154,
0.030068451538681984,
-0.014293961226940155,
0.016255280002951622,
-0.025001076981425285,... | |
454ee8842a382271e719900fd659af1fd5799f31 | subsection | 23 | 25 | Body | Let (\mathbf {w},\psi ) satisfy the linearized system (REF )-() with control \mathrm {U}, and initial data and forcing term to be equal to zero, that is, \mathbf {w}(0)=\widetilde{\mathbf {h}}=\mathbf {0} and \psi (0)=0. From Lemma REF (see below), we haveandsince \mathbf {u}_{\mathrm {U}^*}-\mathbf {u}_d\in \mathrm {L... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.03821032494306564,
0.015366531908512115,
0.004101048223674297,
-0.013130984269082546,
-0.0004730511282104999,
-0.0021611570846289396,
0.00798846036195755,
0.03640967607498169,
-0.005859730299562216,
0.04699992015957832,
-0.00629081716760993,
0.007618412375450134,
0.021424638107419014,
-... | |
0803a39d4ad58163ab9b463b1fbb978c59dff2eb | subsection | 24 | 25 | Body | Thus it is immediate thatSince the above equality is true for all \mathrm {U}\in \mathbb {G}_{\text{div}}, we getLemma
Let (\mathbf {u}_0, \varphi _0) satisfies (REF ) and \mathrm {F}(\cdot ) satisfies REF , the mapping \mathrm {U}\mapsto (\mathbf {u}_{\mathrm {U}},\varphi _{\mathrm {U}}) from {U}_{\text{ad}} into \ma... | {
"cite_spans": []
} | 1802.08413 | Pontryagin's maximum principle and second order optimality condition for
optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems
in two dimensions | [
"Tania Biswas",
"Sheetal Dharmatti",
"Manil T Mohan"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.015987740829586983,
0.035667307674884796,
-0.014035039581358433,
-0.032646723091602325,
-0.0060526104643940926,
-0.02634621039032936,
-0.010411863215267658,
0.03728438541293144,
0.009405001997947693,
0.02598007768392563,
-0.0030472816433757544,
-0.0032913691829890013,
0.001724822446703910... | |
02b9c7614332f10d30f9bb2b0611a6d762de3c9c | abstract | 0 | 59 | Abstract | Completion is one of the most studied techniques in term rewriting and
fundamental to automated reasoning with equalities. In this paper we present
new correctness proofs of abstract completion, both for finite and infinite
runs. For the special case of ground completion we present a new proof based on
random descent. ... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.028061101213097572,
0.030670372769236565,
0.0009226861293427646,
0.022308500483632088,
-0.00024891068460419774,
-0.05654945224523544,
0.01760876178741455,
0.004688294604420662,
0.030761925503611565,
0.007190752774477005,
-0.055389776825904846,
-0.018432741984725,
0.033539045602083206,
0... |
acd03cddde230b7c2c43727b92bdfc7ad5b79806 | subsection | 1 | 59 | Introduction | Reasoning with equalities is pervasive in computer science and
mathematics, and has consequently been one of the main research areas of
automated deduction. Indeed completion as introduced by Knuth and
Bendix has evolved into a fundamental technique whose ideas
appear throughout automated reasoning whenever equalities... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.010411947965621948,
0.029595866799354553,
-0.01621670462191105,
0.019679725170135498,
-0.02212062105536461,
-0.03258596360683441,
0.031487561762332916,
-0.006033590063452721,
0.03655242174863815,
0.01739138551056385,
-0.03524043783545494,
-0.010923010297119617,
0.016765905544161797,
0.0... |
5267acfa910b8914c6f2ed02876154873090172d | subsection | 2 | 59 | Introduction | In this paper we revisit and extend his work.A special case of KB_\mathsf {f} that is known to be decidable is the
completion of ground systems . We present new
correctness and completeness proofs for the
corresponding inference system KB_\mathsf {g}, based on the recent notion of
random descent .On a given set of inpu... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.02664361149072647,
0.046969667077064514,
-0.025087112560868263,
0.001968514174222946,
-0.008881203830242157,
-0.045870959758758545,
0.05252423509955406,
-0.023988407105207443,
0.022004632279276848,
0.052402153611183167,
-0.018891632556915283,
-0.01169663667678833,
0.005562197417020798,
... |
b3016a3bd68dfd9f56c000f387e515f6c0bb41d2 | subsection | 3 | 59 | Introduction | It moreover incorporated canonicity results (Section ).
In addition to these results we present
new and
formalized proofs of correctness and completeness of
ground completion (Section ), as well as
completeness of ordered completion for two different cases
(Section ).
At the end of each section, we remark on the novelt... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
0.014446365647017956,
0.03432346507906914,
-0.010907234624028206,
0.03346919268369675,
-0.011097921058535576,
-0.056565072387456894,
0.014499757438898087,
0.005457431077957153,
0.016109146177768707,
0.03578793257474899,
-0.032981038093566895,
0.018931297585368156,
0.0060142339207232,
0.024... |
a6311a319c36caa007bd235a9514fc41f2eda972 | subsection | 4 | 59 | Preliminaries | We assume familiarity with the basic notions of
abstract rewrite systems, term rewrite systems, and
completion , , but nevertheless shortly recapitulate
terminology and notation that we use in the remainder. | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.021959353238344193,
0.027605608105659485,
-0.002212721388787031,
0.06641216576099396,
-0.02417207509279251,
-0.08399185538291931,
0.0299556702375412,
0.019731370732188225,
0.0021230680868029594,
0.00004631693445844576,
0.0003254703478887677,
-0.02826179377734661,
-0.037356842309236526,
... |
76b7f33eb9d9496ba993483fb408004917066043 | subsection | 5 | 59 | Rewrite Systems | For an arbitrary binary relation \mathchoice{\xrightarrow[\alpha ]{}}{\rightarrow {\alpha }{}{_{\alpha }}{}{}{^{}}}{\rightarrow {\alpha }{}{_{\alpha }}{}{}{^{}}}{\rightarrow {\alpha }{}{_{\alpha }}{}{}{^{}}}, we write
\mathchoice{\xleftarrow[\alpha ]{}}{\textstyle \vphantom{\leftarrow }{\alpha }{}{_{\vphantom{\alpha }}... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.049182258546352386,
-0.011876051314175129,
-0.002431272529065609,
0.02730652689933777,
-0.0030452879145741463,
-0.01855013146996498,
0.029610037803649902,
0.0473516546189785,
0.01775687001645565,
-0.010960748419165611,
0.0012242172379046679,
0.004919751547276974,
0.002467503072693944,
-... |
259c0d24ea41dd868a4fb4ec02d2163b8a345e9f | subsection | 6 | 59 | Rewrite Systems | We further use as abbreviation for the
\emph {joinability relation} []******* []*\textstyle \vphantom{\leftarrow }{\alpha }{}{_{\vphantom{\alpha }}}{*}{}{^{*}}\textstyle \vphantom{\leftarrow }{\alpha }{}{_{\alpha }}{*}{}{^{\vphantom{*}}}\scriptstyle \vphantom{\leftarrow }{\alpha }{}{_{\vphantom{\alpha }}}{*}{}{^{*}}\sc... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.029517093673348427,
-0.005277991760522127,
-0.03407813236117363,
0.019678061828017235,
-0.004118664190173149,
-0.06171894818544388,
0.028174715116620064,
0.02086789906024933,
0.04975956678390503,
0.014758546836674213,
-0.050064653158187866,
0.010914459824562073,
0.009175468236207962,
0.... |
68af364ae1d12c46050facc2efc6cc72be03e3b6 | subsection | 7 | 59 | Rewrite Systems | The subset consisting of the positions addressing
function symbols in t is denoted by \mathcal {P}\mathsf {os}_\mathcal {F}(t) whereas
\mathcal {P}\mathsf {os}_\mathcal {V}(t) = \mathcal {P}\mathsf {os}(t) - \mathcal {P}\mathsf {os}_\mathcal {F}(t) is the set of variable positions in t.
We write
p \leqslant q if p is a... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03621398285031319,
0.008969596587121487,
-0.035603806376457214,
0.05476335063576698,
0.014781529083848,
-0.053939614444971085,
0.022942641749978065,
0.04924125224351883,
0.02736642211675644,
0.02846474014222622,
0.008420437574386597,
0.013980671763420105,
0.014339150860905647,
0.0346275... |
5be121cc7dc334615d852ca4dabf0bb1dcc89092 | subsection | 8 | 59 | Rewrite Systems | Two TRSs \mathcal {R}_1 and \mathcal {R}_2
are called literally similar, denoted by \mathcal {R}_1 \doteq \mathcal {R}_2, if
every rewrite rule in \mathcal {R}_1 has a variant in \mathcal {R}_2 and vice versa.
The following result is folklore; we formalized the non-trivial proof.Two terms s and t are variants of each o... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.018177548423409462,
0.027212904766201973,
-0.03305841237306595,
0.031776368618011475,
-0.009577171877026558,
-0.04191061854362488,
0.008981937542557716,
-0.003790804883465171,
0.04703879356384277,
0.042948465794324875,
-0.03971283137798309,
0.013835388235747814,
0.004254401195794344,
0.... |
59da412df7c76f6c9f69318ebdd4bb56e6b63247 | subsection | 9 | 59 | Rewrite Systems | \hfill {\normalfont \href {http://cl-informatik.uibk.ac.at/isafor/v2.37/LMCS2019/Abstract_Completion.html\#lem:SN_encomp_Un_less_relto_encompeq}{{linkblue}{{check-square-o}}}}First we show the inclusion {\mathrel {\mbox{[}0pt]{\mbox{[}9pt][r]{\cdot }}}{\unrhd }}\cdot \mathrel {R}
R \cdot .
Suppose s \cdot t R u. So s =... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.04237886890769005,
0.032340966165065765,
-0.03298168256878853,
0.012966896407306194,
-0.01990799978375435,
-0.02552190236747265,
0.01516364049166441,
-0.011677834205329418,
0.034354645758867264,
0.03362239897251129,
-0.027001654729247093,
0.008733585476875305,
0.010976096615195274,
0.03... |
764199e13b8e169759dcc0c5f9fd748c357ae46b | subsection | 10 | 59 | Abstract Confluence Criteria | We use the following simple confluence criterion for ARSs to replace
Newman's Lemma in the correctness proof of abstract completion.
In the sequel, we will refer to a conversion of the form
\mathrel {{\vphantom{\rightarrow }_{\mathcal {A}}}{\mathrel {\leftarrow }}} \cdot \rightarrow _\mathcal {A} as a peak.[Peak Decrea... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.00804161187261343,
0.03735458478331566,
-0.06866651773452759,
0.011627529747784138,
-0.022614171728491783,
-0.06585881859064102,
0.03118985705077648,
-0.009521757252514362,
0.0424511656165123,
0.045350417494773865,
-0.03894154354929924,
-0.00007552124588983133,
-0.012306565418839455,
0.... |
97c96d1cfe5c09380347604a7d9f1362a840063d | subsection | 11 | 59 | Abstract Confluence Criteria | We denote by
\mathcal {M}(J) the set of all multisets over a set J.Every peak decreasing ARS is confluent.
linkbluecheck-square-o
Let > be a well-founded order on I which shows that the ARS
\mathcal {A}= \langle A, {\lbrace {\mathchoice{\xrightarrow[\alpha ]{}}{\rightarrow {\alpha }{}{_{\alpha }}{}{}{^{}}}{\rightarro... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.001250954926945269,
0.00003212007868569344,
-0.08427164703607559,
0.02953474223613739,
-0.021922223269939423,
-0.054340261965990067,
-0.016353948041796684,
-0.0013577437493950129,
0.02064075693488121,
0.029794085770845413,
-0.03405038267374039,
0.017376068979501724,
0.008565990254282951,
... |
4ba577ec0b4efb2cbc5756ae633dca746f94e07b | subsection | 12 | 59 | Abstract Confluence Criteria | Here an ARS \mathcal {A} is called
Church-Rosser modulo an ARS \mathcal {B} if the inclusion{[\mathcal {A}\hspace{0.85358pt}\cup \hspace{0.85358pt}\mathcal {B}\,]{*}} \subseteq {\xrightarrow[\mathcal {A}]{*} \cdot [\mathcal {B}]{*} \cdot \xleftarrow[\mathcal {A}]{*}}holds.[Peak Decreasingness Modulo
linkbluecheck-squar... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.006319943815469742,
-0.011274443939328194,
-0.07976020872592926,
0.03716447204351425,
-0.01850595511496067,
-0.026576565578579903,
0.013547640293836594,
0.0060872845351696014,
0.037530627101659775,
0.053458258509635925,
-0.028529377654194832,
0.014508790336549282,
0.010175987146794796,
... |
54a23317ab46c61a3f6140dac0cc3290f219b6f6 | subsection | 13 | 59 | Abstract Confluence Criteria | If the given conversion is not of the desired shape, there is an
index 1 i < n such that
xi xi+1 xi+2 or
xi xi+1 xi+2
for some I and I J.
As the reasoning is similar, we only consider the former case. By peak
decreasingness there are labels
1,...,m with
xi []1 []m xi+2 such that
j for all 1 j m.
Writing N for the multi... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.051866885274648666,
0.012440425343811512,
-0.07432219386100769,
0.034659285098314285,
-0.011273420415818691,
-0.007902072742581367,
0.008390231989324093,
0.007574090734124184,
0.027291135862469673,
0.003655471373349428,
-0.036886509507894516,
-0.0043476652354002,
0.004374361597001553,
0... |
5f1e06011c1a2dea7534095fe0667449ade46769 | subsection | 14 | 59 | Critical Peaks | Completion is based on critical pair analysis. In this subsection we
present a version of the critical pair lemma that incorporates
primality (cf. Definition REF below).[Overlaps
linkbluecheck-square-o]
An overlap of a TRS \mathcal {R} is a triple
\langle \ell _1 \rightarrow r_1, p, \ell _2 \rightarrow r_2 \rangle ,
co... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.036222781985998154,
-0.00444392766803503,
-0.010848524048924446,
0.030699340626597404,
0.004878784529864788,
0.008834452368319035,
0.035704005509614944,
0.0013102912344038486,
-0.014571505598723888,
0.006843267474323511,
-0.06353481858968735,
0.010322119109332561,
0.001213020645081997,
... |
929fed5934acd0c03a68bdd175526a0ee91cbd0e | subsection | 15 | 59 | Critical Peaks | The term \ell _2\sigma {[\ell _1\sigma ]}_p = \ell _2\sigma
can be reduced in two different ways:[minimum height=6mm]
s)\ell _2\sigma [\ell _1\sigma ]_p = \ell _2\sigma ;
(swest) at (s.west);
(seast) at (s.east);
t)[below left of=swest, anchor=north east]\ell _2\sigma [r_1\sigma ]_p;
u)[below right of=seast, anchor=no... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.04128022864460945,
0.02744387835264206,
-0.02764219418168068,
-0.004549825564026833,
0.001191802672110498,
-0.03255432844161987,
0.011105693876743317,
0.03083050437271595,
-0.01601782813668251,
0.021814756095409393,
-0.028404947370290756,
0.0467720553278923,
-0.01760435476899147,
0.0076... |
24517643658fef0cf6a4f1590ec8cebc62ad2d60 | subsection | 16 | 59 | Critical Peaks | If t \mathrel {{\vphantom{\xleftarrow{}}_{\mathcal {R}}}{\xleftarrow{}}} s \xrightarrow{}_\mathcal {R}u then
one of the following holds:
linkbluecheck-square-ot \mathrel {\downarrow }_\mathcal {R}u,
p_2 \leqslant p_1 and
{t|_{p_2} \xleftarrow{} s|_{p_2} \xrightarrow{} u|_{p_2}}
is an instance of a critical peak, or
... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.050134941935539246,
0.03487780690193176,
-0.02636432833969593,
0.013586477376520634,
-0.009741679765284061,
-0.0017059382516890764,
0.06322555989027023,
0.04082809016108513,
0.012243850156664848,
0.019696960225701332,
-0.017133761197328568,
0.00805576704442501,
-0.029553068801760674,
0.... |
6ba886ca365dee8a998366bd69ba666c9fddf67f | subsection | 17 | 59 | Critical Peaks | Let
\sigma ^{\prime }(x) = \sigma _1(x) for x \in \mathcal {V}\mathsf {ar}(\ell _1 \rightarrow r_1) and
\sigma ^{\prime }(x) = \sigma _2(x), otherwise. The substitution \sigma ^{\prime } is a
unifier of {\ell _2}|_p and \ell _1: ({\ell _2}|_p)\sigma ^{\prime } =
(\ell _2\sigma _2)|_p = \ell _1\sigma _1 = \ell _1\sigma ... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03204420581459999,
0.014084191992878914,
-0.014839519746601582,
-0.031494878232479095,
-0.028137866407632828,
-0.01505314838141203,
0.010223628021776676,
0.06494292616844177,
0.03176954388618469,
-0.0016479878686368465,
0.014755594544112682,
-0.0007939524948596954,
-0.024002637714147568,
... |
c2d13745334c830c686865b0b89518d77e62360f | subsection | 18 | 59 | Critical Peaks | We also have
\ell _2\sigma _2{[r_1\sigma _1]}_p = \ell _2\sigma _2{[\sigma _2^{\prime }(x)]}_{q_1}
\rightarrow ^* \ell _2\sigma _2^{\prime } \rightarrow r_2\sigma _2^{\prime }
Consequently, t \rightarrow ^* s{[r_2\sigma _2^{\prime }]}_{p_2} \mathrel {{\vphantom{\rightarrow }^{*}}{\mathrel {\leftarrow }}} u. Hence, (R... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.04033772647380829,
0.027110492810606956,
-0.023754101246595383,
-0.018704256042838097,
-0.0013644876889884472,
0.003364020027220249,
0.02132834494113922,
0.023479485884308815,
0.008543542586266994,
0.0404902920126915,
-0.020443476736545563,
0.03319776803255081,
-0.010671800002455711,
-0... |
e90ff8b84f6ba840d9af3cd7ed64b9290f5780a9 | subsection | 19 | 59 | Critical Peaks | Now, if
\smash{v \xleftarrow{} s \xrightarrow{} u} is an instance of a critical peak then
v \rightarrow _{\mathsf {PCP}(\mathcal {R})} u. Otherwise, v \mathrel {\downarrow }_\mathcal {R}u by Lemma REF ,
since q \lnot \leqslant \epsilon . In both cases we obtain
v \mathrel {\triangledown _{\hspace{-1.42262pt}s}} u. Fina... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.024784788489341736,
0.01176666934043169,
-0.03845915198326111,
-0.01687929593026638,
0.014086428098380566,
0.0020164349116384983,
0.03687195107340813,
0.005707826931029558,
0.030752060934901237,
0.03699404373764992,
-0.01846649870276451,
0.024418510496616364,
-0.004895148333162069,
0.01... |
fb83cd91bb18a567d82bbf114bac847974a7a877 | subsection | 20 | 59 | Critical Peaks | Lemma REF yields a
term v
such that t \mathrel {\triangledown _{\hspace{-1.42262pt}s}} v \mathrel {\triangledown _{\hspace{-1.42262pt}s}} u. From the assumption
\mathsf {PCP}(\mathcal {R}) \subseteq {\mathrel {\downarrow }_\mathcal {R}} we obtain t \mathrel {\downarrow }_\mathcal {R}v \mathrel {\downarrow }_\mathcal {R... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.060378555208444595,
-0.001196200493723154,
-0.04444447532296181,
0.010119054466485977,
0.03044874034821987,
0.038431040942668915,
0.014407824724912643,
-0.010989018715918064,
0.02651100791990757,
0.012026870623230934,
-0.05894387513399124,
0.023290615528821945,
-0.006643014494329691,
-0... |
9cc6bf885e56edb2a96a15b44d13aa1846196b5b | subsection | 21 | 59 | Correctness for Finite Runs | The original completion procedure by Knuth and Bendix was
presented as a concrete algorithm. Later on, Bachmair, Dershowitz, and
Hsiang presented an inference system for completion and
showed that all fair implementations thereof (in particular the
original procedure) are correct. Abstracting from a concrete strategy... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.026695141568779945,
0.03711983188986778,
-0.018758336082100868,
0.012477574869990349,
-0.0013364742044359446,
-0.04606400057673454,
0.07344598323106766,
0.009371536783874035,
0.03507457673549652,
0.03226616978645325,
-0.02811460942029953,
0.0025489358231425285,
-0.000944403582252562,
0.... |
8b524fd647eb6cb6e377808772663d303331651c | subsection | 22 | 59 | Correctness for Finite Runs | Then, the following two inclusions
hold:If s \xrightarrow[\mathcal {E}\hspace{0.85358pt}\cup \hspace{0.85358pt}\mathcal {R}]{} t then
s \xrightarrow[\mathcal {R}^{\prime }]{=} \cdot \xrightarrow[\mathcal {E}^{\prime } \hspace{0.85358pt}\cup \hspace{0.85358pt}\mathcal {R}^{\prime }]{=}
\cdot \xleftarrow[\mathcal {R}^{\p... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.04199666902422905,
0.05258740112185478,
-0.011575027368962765,
0.023012710735201836,
0.011300339363515377,
-0.018419325351715088,
-0.017152709886431694,
0.027407711371779442,
0.048436567187309265,
0.04214927554130554,
-0.006191914435476065,
0.004322513472288847,
-0.01077385526150465,
0.... |
ea6583a11372ac78f17a8ca01a72c954149f9aad | subsection | 23 | 59 | Correctness for Finite Runs | If s \approx t \in \mathcal {E} then
s \approx t \in \mathcal {E}^{\prime } because \mathcal {E}\subseteq \mathcal {E}^{\prime }. If
s \approx t \in \mathcal {R} then either s \approx t \in \mathcal {R}^{\prime } or
s \rightarrow _\mathcal {R}u with u \approx t \in \mathcal {E}^{\prime } and thus
s \rightarrow _{\mathc... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.04981257766485214,
0.0221592728048563,
-0.0027164947241544724,
0.014185291714966297,
-0.00879045482724905,
-0.021152032539248466,
0.017229903489351273,
0.02859950251877308,
0.04138839244842529,
0.0443490669131279,
-0.01384191494435072,
0.019610650837421417,
-0.015390926972031593,
0.0551... |
cf25159fe91d91a568b192aa4ef57c2c701bf068 | subsection | 24 | 59 | Correctness for Finite Runs | It is the final result
in this section whose proof refers to the inference rules.If (\mathcal {E},\mathcal {R}) \vdash _\mathsf {f}^* (\mathcal {E}^{\prime },\mathcal {R}^{\prime }) and \mathcal {R}\subseteq {>} then
\mathcal {R}^{\prime } \subseteq {>}.
linkbluecheck-square-o
We consider a single step (\mathcal {E},... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.029347211122512817,
0.027149604633450508,
0.015093287453055382,
0.014398904517292976,
-0.016008958220481873,
-0.005196425132453442,
0.006642420310527086,
0.027805835008621216,
0.029331950470805168,
0.05481809005141258,
-0.023593753576278687,
0.0187254436314106,
0.014040268026292324,
0.0... |
f6a7bc0b74872623ff4be1c54e3b9e506a6e0ab3 | subsection | 25 | 59 | Correctness for Finite Runs | The run is fair if \mathcal {E}_n= \varnothing and\mathsf {PCP}({\mathcal {R}_n}) ~\subseteq ~ {\mathrel {\downarrow }_{{\mathcal {R}_n}}} \cup \bigcup _{i=0}^n {[\mathcal {E}_i]{}}The reason for writing []{}_{\mathcal {E}_i} instead of \mathcal {E}_i in the
definition of fairness is that critical pairs are ordered, so... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.042668987065553665,
0.014131431467831135,
-0.018175523728132248,
0.0305672325193882,
-0.006123365834355354,
0.0325053445994854,
-0.0477660670876503,
0.029453199356794357,
0.05197802931070328,
0.02882750891149044,
-0.042668987065553665,
-0.006756685674190521,
0.007576949894428253,
0.0299... |
71c73fd0e3140dedfb5b495bb6502b58e48db83f | subsection | 26 | 59 | Correctness for Finite Runs | If
\smash{t \mathrel {\smash{[\mathcal {E}\hspace{0.85358pt}\cup \hspace{0.85358pt}\mathcal {R}]{M}}^{*}\vphantom{[\mathcal {E}\hspace{0.85358pt}\cup \hspace{0.85358pt}\mathcal {R}]{M}}} u} and \mathcal {R}^{\prime } \subseteq {>} then
\smash{t \mathrel {\smash{[\mathcal {E}^{\prime } \hspace{0.85358pt}\cup \hspace{0.8... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.024056684225797653,
0.02956363558769226,
0.004050126299262047,
-0.016993751749396324,
0.017832759767770767,
0.03886901214718819,
-0.019800618290901184,
0.00329692498780787,
0.02501773089170456,
0.026421165093779564,
0.025063494220376015,
0.009313003160059452,
0.008100252598524094,
0.004... |
daa92c4b42646d0fe73434042c83a7b6f385fe5a | subsection | 27 | 59 | Correctness for Finite Runs | Hencet \mathrel {\smash{\xrightarrow[\mathcal {R}^{\prime }]{M}}^{=}\vphantom{\xrightarrow[\mathcal {R}^{\prime }]{M}}}
v \mathrel {\smash{\xrightarrow[\mathcal {E}^{\prime } \hspace{0.85358pt}\cup \hspace{0.85358pt}\mathcal {R}^{\prime }]{M}}^{=}\vphantom{\xrightarrow[\mathcal {E}^{\prime } \hspace{0.85358pt}\cup \hsp... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.021215317770838737,
0.04737578704953194,
0.008379287086427212,
-0.0477115698158741,
0.004040831234306097,
0.02037586271762848,
-0.023611579090356827,
-0.0050634401850402355,
0.010149774141609669,
0.02072690799832344,
-0.02678624540567398,
0.032143495976924896,
0.012011838145554066,
0.00... |
850f839371bec92d2542c0c456be3267846c6497 | subsection | 28 | 59 | Correctness for Finite Runs | Because s > v and
s > w we have M_1 >_{\mathsf {mul}}\lbrace v, w \rbrace and M_2 >_{\mathsf {mul}}\lbrace v, w \rbrace . Hence by
repeated applications of Lemma we obtain a conversion
in \mathcal {R}_n between v and w in which each step is labeled with a
multiset that is smaller than both M_1 and M_2. It follows that... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.0238033477216959,
0.037230879068374634,
-0.0160672590136528,
0.00939164124429226,
-0.023421883583068848,
0.028136778622865677,
0.00396722462028265,
-0.008262508548796177,
0.04437188059091568,
0.02249111235141754,
-0.04138120636343956,
0.0313410758972168,
0.004356317687779665,
0.02673299... |
f6b3d293e8cce4cb07912d89427d3a2a71f8ace2 | subsection | 29 | 59 | Correctness for Finite Runs | One possible run is(\mathcal {E},\varnothing ) ~
& \mathrel {{\vdash _\mathsf {f}^\textsf {\small \tiny orient}}^+} &&
(\lbrace \mathsf {a} \approx \mathsf {c}, \mathsf {f}(\mathsf {a}) \approx \mathsf {d} \rbrace ,
\lbrace \mathsf {a} \rightarrow \mathsf {b},\mathsf {f}(\mathsf {b}) \rightarrow \mathsf {b} \rbrace ) \... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.019064050167798996,
0.05424630269408226,
-0.02378045581281185,
0.03602174296975136,
-0.036662809550762177,
0.014744493179023266,
-0.013172358274459839,
0.05620002746582031,
0.019888276234269142,
0.031595341861248016,
-0.010134931653738022,
0.028924239799380302,
-0.003375130472704768,
-0... |
65e3f8452703b0741d90f9f060ce5aace814a579 | subsection | 30 | 59 | Correctness for Finite Runs | However, the run(\mathcal {E},\varnothing ) ~
& \vdash _\mathsf {f}^\textsf {\small \tiny orient} &&
(\lbrace \mathsf {a} \approx \mathsf {b}, \mathsf {f}(\mathsf {b}) \approx \mathsf {b},
\mathsf {f}(\mathsf {a}) \approx \mathsf {d} \rbrace ,
\lbrace \mathsf {a} \rightarrow \mathsf {c} \rbrace ) \\
& \mathrel {{\vdash... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03246375918388367,
0.06730738282203674,
-0.013630812987685204,
0.005781844723969698,
-0.03093820810317993,
-0.012929058633744717,
0.002635392127558589,
0.0037395108956843615,
0.011006861925125122,
0.0249275304377079,
-0.019649118185043335,
0.021342482417821884,
0.004672004841268063,
0.0... |
5de226b5e697c8c381f4707c9cff19ba275f2c65 | subsection | 31 | 59 | Correctness for Finite Runs | When applying compose steps in a naive way by
simplifying the rules in descending order, exponentially many steps are
required to obtain a canonical system . However, when
processing the rules in reverse order only a polynomial number of steps
is necessary.This section resumes our results on finite runs .
The presented... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
0.0010469711851328611,
0.012388205155730247,
-0.038263075053691864,
0.050376664847135544,
0.0005663942429237068,
-0.021175896748900414,
0.02578333392739296,
0.014859743416309357,
-0.020321538671851158,
0.014813974499702454,
-0.038293588906526566,
-0.03338102623820305,
0.003169519128277898,
... |
d5f05e464a02f0c808e10ebf8aaeca1bdfac8562 | subsection | 32 | 59 | Canonicity and Normalization Equivalence | A natural question arising in the context of completion concerns
uniqueness of resulting systems: Is there a single
complete presentation of a given equational theory conforming to a
certain reduction order? Métivier showed that for reduced
and hence canonical systems this is indeed the case, up to renaming
variables.... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.01052834838628769,
0.015975624322891235,
-0.06115597113966942,
0.06909038126468658,
-0.00819380208849907,
-0.04864402115345001,
0.0007285922183655202,
0.007942036725580692,
0.0178524162620306,
0.04864402115345001,
-0.06732039898633957,
-0.03771895170211792,
0.008476083166897297,
0.03964... |
d72a20e7206aac66ea9ac431b88b7ae4606e37ac | subsection | 33 | 59 | Canonicity and Normalization Equivalence | However, the present version suffices to prove the
following lemma that we employ in our proof of Métivier's transformation
result (Theorem below).Let \mathcal {A} and \mathcal {B} be ARSs such that \mathsf {NF}(\mathcal {B}) \subseteq \mathsf {NF}(\mathcal {A}) and
either{\rightarrow _\mathcal {B}} \subseteq {\right... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
0.007959475740790367,
0.020711425691843033,
-0.03791243955492973,
0.012011711485683918,
0.005960067734122276,
-0.006742279045283794,
0.003140291664749384,
0.011653038673102856,
0.06746094673871994,
0.05955488979816437,
-0.021734023466706276,
-0.012538272887468338,
0.01633104309439659,
0.02... |
fafaa83ab5a8f0cfddf698f82c86d5192eea4581 | subsection | 34 | 59 | Canonicity and Normalization Equivalence | \cdot \mathrel {\vphantom{\mathrel {\leftarrow }}^{!}_{\mathcal {A}}}\scriptstyle !}\hspace{0.0pt}\copy 1\scriptstyle \mathcal {A}
B! \scriptstyle !\scriptstyle \mathcal {B}where we obtain the inclusions fromB A,
confluence of A, termination of A, and normalization equivalence
of A and B, respectively.In the above ... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.049656543880701065,
0.02490457147359848,
-0.01448188629001379,
0.005611921660602093,
0.009384995326399803,
-0.02760561928153038,
0.0432167574763298,
0.008713548071682453,
0.04166022315621376,
0.019853461533784866,
-0.04568890109658241,
0.021547337993979454,
0.0014106105081737041,
0.0299... |
ee9e12493fbf1a57488b1645da281a4047c57786 | subsection | 35 | 59 | Canonicity and Normalization Equivalence | (This is the only place in the paper where
variant-freeness of TRSs is important.)The following example shows why the result of \dot{\mathcal {R}} has to be
variant-free.Consider the TRS \mathcal {R} consisting of the four rules\mathsf {f}(x) &\rightarrow \mathsf {a} &
\mathsf {f}(y) &\rightarrow \mathsf {b} &
\mathsf ... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.025600528344511986,
0.02581411972641945,
-0.014379318803548813,
0.02932313084602356,
0.03213034197688103,
-0.018445195630192757,
0.014943812042474747,
0.0299944207072258,
0.02132868766784668,
0.03396112844347954,
-0.039941705763339996,
-0.009077660739421844,
0.01117543876171112,
-0.0006... |
1818536f22f53255c41bf6a3fc789c4b9990ea09 | subsection | 36 | 59 | Canonicity and Normalization Equivalence | \begin{}
\item If \ell \mathrel {{\hss \rhd \hss \cr \hspace{1.69997pt}\scalebox {1.0}{\cdot }}}\end{}\ell ^{\prime } then we obtain
\ell ^{\prime } \notin \mathsf {NF}(\ddot{\mathcal {R}}) from the induction hypothesis and hence
\ell \notin \mathsf {NF}(\ddot{\mathcal {R}}) as desired.If \ell \doteq \ell ^{\prime } th... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.062757708132267,
0.028509581461548805,
0.011660235933959484,
-0.004410743713378906,
0.007936285808682442,
-0.027624379843473434,
-0.009630377404391766,
0.04212336614727974,
0.01904708705842495,
0.041634976863861084,
-0.04807557910680771,
0.014995002187788486,
-0.011347362771630287,
0.03... |
2e28c91d72e352949c71023ded0bccd1704d88e5 | subsection | 37 | 59 | Canonicity and Normalization Equivalence | \hfill {\normalfont \href {http://cl-informatik.uibk.ac.at/isafor/v2.37/LMCS2019/Normalization_Equivalence.html\#lem:right_reduced_min_step_rule}{{linkblue}{{check-square-o}}}}
Let \ell \rightarrow r be the rewrite rule that is used in the first step from
s to t. So s \mathrel {\mbox{[}0pt]{\mbox{[}9pt][r]{\cdot }}}{\... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.050742991268634796,
0.03957526013255119,
-0.017819546163082123,
0.0444878414273262,
0.009176760911941528,
0.012502670288085938,
0.0150733832269907,
0.012983248569071293,
0.03621883690357208,
0.041833218187093735,
-0.028178684413433075,
-0.009275928139686584,
-0.014615689404308796,
0.018... |
02c213e7efee225673f5a62dcca82bb2bf16e602 | subsection | 38 | 59 | Canonicity and Normalization Equivalence | Let \mathcal {R} and \mathcal {S} be equivalent canonical TRSs. If \mathcal {R} and \mathcal {S}
are compatible with the same reduction order then \mathcal {R}\doteq \mathcal {S}.
linkbluecheck-square-o
Suppose \mathcal {R} and \mathcal {S} are compatible with the reduction order >.
We show that {\rightarrow _\mathcal ... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.027677731588482857,
0.05007624626159668,
-0.02702164277434349,
-0.002826515818014741,
0.03289591521024704,
-0.052975237369537354,
0.019011275842785835,
-0.006602832581847906,
0.04818427562713623,
0.0783642902970314,
-0.01920962706208229,
0.0012063233880326152,
0.004581167828291655,
0.04... |
1c2f8fa62193ec16365f90bc48c4335c00215206 | subsection | 39 | 59 | Canonicity and Normalization Equivalence | Consider the ES \mathcal {E} consisting of the ground equations
\mathsf {f}(\mathsf {f}(\mathsf {f}(\mathsf {a}))) &\approx \mathsf {f}(\mathsf {b}) &
\mathsf {f}(\mathsf {f}(\mathsf {b})) &\approx \mathsf {c} &
\mathsf {f}(\mathsf {c}) &\approx \mathsf {a} &
\mathsf {f}(\mathsf {a}) &\approx \mathsf {f}(\mathsf {f}(\... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
0.009495565667748451,
0.022743595764040947,
-0.012165002524852753,
-0.007695603184401989,
-0.013484465889632702,
-0.039873749017715454,
0.02838754653930664,
0.019402988255023956,
0.02556557208299637,
0.04927016422152519,
-0.050124384462833405,
0.013553109019994736,
0.006143516860902309,
0.... |
b6c7db439004cea099c0ab7940db625c6c1ef429 | subsection | 40 | 59 | Canonicity and Normalization Equivalence | We start by applying \textsf {\small orient}
to the last two equations:}
\mathsf {f}(\mathsf {f}(\mathsf {f}(\mathsf {a}))) &\approx \mathsf {f}(\mathsf {b}) &
\mathsf {f}(\mathsf {f}(\mathsf {b})) &\approx \mathsf {c} &
\mathsf {f}(\mathsf {c}) &\mathrel {\leftarrow }\mathsf {a} &
\mathsf {f}(\mathsf {a}) &\rightarrow... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.026432327926158905,
0.061349861323833466,
-0.024448378011584282,
-0.0003145229711662978,
-0.008256287313997746,
0.0007754576508887112,
0.002853836864233017,
0.023319052532315254,
0.038915958255529404,
0.04444049671292305,
-0.029988178983330727,
0.004929354414343834,
0.027943182736635208,
... |
fc22187392826076533cb2b0ff099943d9dbe2f0 | subsection | 41 | 59 | Canonicity and Normalization Equivalence | The absence of deduce from KB_\mathsf {g} does not hurt for ground
systems. If s \mathrel {\leftarrow }\cdot \rightarrow t and the two contracted redexes are at
parallel positions then trivially s \rightarrow \cdot \mathrel {\leftarrow }t. If the steps are
identical then s = t. In the remaining case one of the contract... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.061639539897441864,
0.023343687877058983,
-0.0325591042637825,
0.00384674989618361,
-0.022672366350889206,
-0.013083145022392273,
0.03893665969371796,
0.011313296854496002,
0.026395149528980255,
0.03631240129470825,
-0.026074746623635292,
0.0033508872147649527,
-0.0037666489370167255,
0... |
038b4a9f29fca5aa0741fe8283916b7535e9e669 | subsection | 42 | 59 | Canonicity and Normalization Equivalence | Furthermore let M(\mathcal {E},\mathcal {R}) denote the (finite)
multiset of left-hand sides and right-hand sides occurring in \mathcal {E} and
\mathcal {R}
M(\mathcal {E},\mathcal {R}) =
\bigcup \,\lbrace \lbrace s, t \rbrace \mid (s,t) \in \mathcal {E}\rbrace
~\cup ~
\bigcup \,\lbrace \lbrace s, t \rbrace \mid (s,t... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.02487003244459629,
0.026273740455508232,
-0.01942303776741028,
0.0100471880286932,
-0.009749663062393665,
0.0052524590864777565,
-0.006240394432097673,
0.008681625127792358,
0.021696433424949646,
0.03860195353627205,
-0.03259042277932167,
0.016997065395116806,
0.02319168671965599,
0.033... |
e03582671cbf237b9d67e812dddcfaf807014103 | subsection | 43 | 59 | Canonicity and Normalization Equivalence | In particular, it holds for any LPO or KBO
based on a total precedence.
Next we consider completeness of ground completion. Our proof makes
use of the following concept.
[Random Descent
linkbluecheck-square-o]
An ARS \mathcal {A} has random descent if for every conversion a b
with normal form b we have a \rightarrow ^n... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.01898578554391861,
0.03965038061141968,
-0.021580306813120842,
0.01009573694318533,
0.003588450839743018,
0.009897332638502121,
0.04712870344519615,
-0.013621232472360134,
0.01897052302956581,
0.04715922474861145,
-0.05192093551158905,
0.01367464940994978,
0.017642740160226822,
0.051188... |
25547b1ed848e5a00420d1ecba227118750f815c | subsection | 44 | 59 | Canonicity and Normalization Equivalence | Left-reduced TRSs enjoy the WCR1 property.
linkbluecheck-square-o
This follows from a straightforward case analysis on the relative positions
of the two redexes that are part of a peak together with the fact that for
left-reduced TRSs the left-hand side alone uniquely determines the employed
rewrite rule.
Left-reduce... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.05631302669644356,
0.05136846750974655,
-0.008111212402582169,
0.019671406596899033,
0.005535921547561884,
-0.010545339435338974,
0.018938880413770676,
0.01456660870462656,
0.04730904847383499,
0.042883362621068954,
-0.06128810718655586,
0.005326082929968834,
0.01906096749007702,
0.0097... |
fef02cfb5c5f2f3ab38bcba7d815f5dfcdf4acfb | subsection | 45 | 59 | Canonicity and Normalization Equivalence | Let \sqsupset be a total precedence
and define s > t if and only if s _\mathcal {E}t and either
d_\mathcal {R}(s) > d_\mathcal {R}(t) or both d_\mathcal {R}(s) = d_\mathcal {R}(t) and
s \sqsupset _{\mathsf {lpo}} t.In the formalization we actually use
\sqsupset _{\mathsf {kbo}} with all weights set to 1, since in contr... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.013167215511202812,
0.033780086785554886,
-0.03448193147778511,
0.026609066873788834,
-0.008132243528962135,
-0.03866248577833176,
0.045741960406303406,
0.0015352851478382945,
0.027448227629065514,
0.05157032236456871,
-0.024900227785110474,
0.012663718312978745,
0.02654803730547428,
0.... |
1a294ee2d5e65831b9ba37967aea4f412df84232 | subsection | 46 | 59 | Canonicity and Normalization Equivalence | Correctness for Infinite Runs
Completion as presented in the preceding sections does not always
succeed in producing a finite complete presentation. It may fail because
an unorientable equation is encountered or it may run forever.
In the latter case it is possible that in the limit a possibly
infinite complete present... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.04088892042636871,
0.02753899246454239,
-0.004100334830582142,
0.05486438423395157,
-0.03222290799021721,
-0.004054563585668802,
-0.01144279446452856,
0.014898518100380898,
0.01775921694934368,
0.024716436862945557,
-0.026165856048464775,
-0.0006827533943578601,
0.002284744754433632,
0.... |
a5a6730fd1bd5aa0a4f4ee0ed7ee22da8df29752 | subsection | 47 | 59 | Canonicity and Normalization Equivalence | After two
orient steps, we apply deduce to generate the two critical
pairs:
\mathsf {a}\mathsf {b}\mathsf {a} &\rightarrow \mathsf {a}\mathsf {b} &
\mathsf {b}\mathsf {b} &\rightarrow \mathsf {b} &
\mathsf {a}\mathsf {b}\mathsf {a}\mathsf {b} &\approx \mathsf {a}\mathsf {b}\mathsf {b}\mathsf {a} &
\mathsf {b}\mathsf {... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.044394418597221375,
0.02508055791258812,
-0.016705116257071495,
0.026636650785803795,
-0.006365488283336163,
0.030511626973748207,
0.017101766541600227,
0.01106046512722969,
0.03045060485601425,
0.013562418520450592,
-0.028879255056381226,
0.013775999657809734,
0.017696743831038475,
0.0... |
4017d6da40f0f671e6b9cd134b6b6ce35843b6b1 | subsection | 48 | 59 | Canonicity and Normalization Equivalence | Since none of the rules
\mathsf {a}\mathsf {b}\mathsf {a} \rightarrow \mathsf {a}\mathsf {b}^n survives, in the
limit we obtain the TRS consisting of the single rule
\mathsf {b}\mathsf {b} \rightarrow \mathsf {b}. This TRS is complete but
not equivalent to \mathcal {E} as witnessed by non-joinability of
\mathsf {a}\mat... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.01496393047273159,
0.03748231753706932,
-0.006226703990250826,
-0.005032489541918039,
-0.0015127991791814566,
-0.04682237282395363,
0.020877772942185402,
0.020969342440366745,
0.020313097164034843,
0.047280218452215195,
-0.035834070295095444,
0.01341488491743803,
0.01307913102209568,
0.... |
dbd8af90a4b9f86a27f59d3ca773f0d27b0fb627 | subsection | 49 | 59 | Ground Completion | In this section we focus on the special case of ground equations, that is,
equations where both sides are ground terms.[Ground Completion
linkbluecheck-square-o]
The inference system KB_\mathsf {g} consists of the inference
rules of KB_\mathsf {f} except for deduce.Snyder proved that sets of ground equations can alway... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.029732907190918922,
0.03682669997215271,
-0.011441601440310478,
0.0060640485025942326,
0.034751955419778824,
-0.0946449264883995,
0.0500379353761673,
-0.01162466686218977,
0.013882475905120373,
0.04137283191084862,
-0.017116636037826538,
0.014782548882067204,
0.011029703542590141,
0.051... |
7b681406e99044119facbb91f4dd1822bea2708b | subsection | 50 | 59 | Ground Completion | We start by applying \textsf {\small orient}
to the last two equations:}
\mathsf {f}(\mathsf {f}(\mathsf {f}(\mathsf {a}))) &\approx \mathsf {f}(\mathsf {b}) &
\mathsf {f}(\mathsf {f}(\mathsf {b})) &\approx \mathsf {c} &
\mathsf {f}(\mathsf {c}) &\mathrel {\leftarrow }\mathsf {a} &
\mathsf {f}(\mathsf {a}) &\rightarrow... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.030484797433018684,
0.05965743586421013,
-0.03835774585604668,
-0.001669759745709598,
-0.017027543857693672,
-0.006770584732294083,
0.01864485628902912,
0.01830918714404106,
0.027265431359410286,
0.043819986283779144,
-0.03225468471646309,
0.005492756143212318,
0.039303719997406006,
0.0... |
f383d56d5c07489c4d7cf7df7a86bf2dca91ac09 | subsection | 51 | 59 | Ground Completion | If s \mathrel {\leftarrow }\cdot \rightarrow t and the two contracted redexes are at
parallel positions then trivially s \rightarrow \cdot \mathrel {\leftarrow }t. If the steps are
identical then s = t. In the remaining case one of the contracted
redexes is a subterm of the other contracted redex, and the effect of
ded... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.0579441636800766,
0.0402161180973053,
-0.01788061112165451,
0.0021702363155782223,
-0.028209257870912552,
0.012060258537530899,
0.026973480358719826,
0.01022185105830431,
0.026775145903229713,
0.03554762899875641,
-0.018597666174173355,
0.011945834383368492,
0.007105711847543716,
0.0503... |
e704df69e96265a96f779ea4f41f6bfbdfd3ea89 | subsection | 52 | 59 | Ground Completion | Now it is straightforward to verify that any
infinite \vdash _\mathsf {g}-sequence would give rise to an infinite
sequence P(\mathcal {E}_0,\varnothing ) \succ P(\mathcal {E}_1,\mathcal {R}_1) \succ \cdots ,
contradicting the well-foundedness of \succ .If > is total on \mathcal {E}-equivalent ground terms then every ma... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.0440068393945694,
0.05590882524847984,
-0.007137378212064505,
0.007347188889980316,
0.002582578919827938,
-0.007743921596556902,
0.026535330340266228,
0.00487142289057374,
0.016784854233264923,
0.05194149538874626,
-0.03701823577284813,
0.008155914023518562,
0.03067050874233246,
0.02856... |
0c4fe962ae1dd4206503467d0102d065ea00f8a1 | subsection | 53 | 59 | Ground Completion | We formalized a new, short and direct proof of the following result due
to van Oostrom .
Here an element a is said to be complete if it is both terminating
(there are no infinite rewrite sequences starting at a) and
confluent (if b \mathrel {{\vphantom{\rightarrow }^{*}}{\mathrel {\leftarrow }}} a \rightarrow ^* c then... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.01786678656935692,
0.03967006504535675,
-0.01687503606081009,
0.000027714213501894847,
0.010306588374078274,
0.0190263744443655,
0.029508426785469055,
-0.01614266447722912,
0.029569456353783607,
0.041867177933454514,
-0.027448633685708046,
0.021650701761245728,
0.04238593950867653,
0.03... |
d7d10ce803506486ba4fecbd324bf9b2ad2caa30 | subsection | 54 | 59 | Ground Completion | The remainder of the proof proceeds by induction on k
together with Lemma .Right-reduced ground TRSs are terminating.
linkbluecheck-square-o
Let \mathcal {R} be a right-reduced ground TRS. For the sake of a contradiction,
assume that \mathcal {R} is non-terminating. Then there is a minimal
non-terminating term t (tha... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.02522602491080761,
0.03488608077168465,
-0.010529919527471066,
0.01516155805438757,
-0.007779168896377087,
-0.015123406425118446,
0.033024270087480545,
0.009805033914744854,
0.022906390950083733,
0.057136259973049164,
-0.044530875980854034,
0.026950489729642868,
0.02054097317159176,
0.0... |
95c789e539df87be8795e1832d2e7a8cc8e9b1c9 | subsection | 55 | 59 | Ground Completion | Both are
basically handled by the following observation:
d_\mathcal {R}(C[t\sigma ]) = d_\mathcal {R}(C[t{\downarrow }\sigma ]) + d_\mathcal {R}(t) for any
term t (which holds due to random descent together with
termination). This allows us to lift d_\mathcal {R}(s) = d_\mathcal {R}(t) and
d_\mathcal {R}(s) > d_\mathca... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.021959489211440086,
0.02009773999452591,
-0.0066000549122691154,
-0.008568625897169113,
0.00991915725171566,
-0.035495325922966,
0.0405922457575798,
-0.0016709965420886874,
0.031069854274392128,
0.05621873214840889,
-0.02519465982913971,
0.03105459362268448,
0.009446090087294579,
0.0352... |
cd93136f1141b27bacb9a3e4c21f231c30cb964e | subsection | 56 | 59 | Correctness for Infinite Runs | Completion as presented in the preceding sections does not always
succeed in producing a finite complete presentation. It may fail because
an unorientable equation is encountered or it may run forever.
In the latter case it is possible that in the limit a possibly
infinite complete presentation is obtained. In this cas... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03229685500264168,
0.02561158873140812,
-0.004384343978017569,
0.0371505431830883,
-0.02313895709812641,
0.0055557917803525925,
-0.011424478143453598,
0.0244973786175251,
0.021521061658859253,
0.02713790535926819,
-0.02281842939555645,
0.00720421364530921,
-0.0009300990495830774,
0.0420... |
c0e1e0f539ed841e575f9cf8d6e1daa966dca9c7 | subsection | 57 | 59 | Correctness for Infinite Runs | After two
orient steps, we apply deduce to generate the two critical
pairs:\mathsf {a}\mathsf {b}\mathsf {a} &\rightarrow \mathsf {a}\mathsf {b} &
\mathsf {b}\mathsf {b} &\rightarrow \mathsf {b} &
\mathsf {a}\mathsf {b}\mathsf {a}\mathsf {b} &\approx \mathsf {a}\mathsf {b}\mathsf {b}\mathsf {a} &
\mathsf {b}\mathsf {b}... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.04450254514813423,
0.025028865784406662,
-0.016421377658843994,
0.02313643880188465,
-0.010293884202837944,
0.03116399049758911,
0.01517756562680006,
0.012995170429348946,
0.030065162107348442,
0.013102000579237938,
-0.029988855123519897,
0.011713203974068165,
0.019962046295404434,
0.01... |
546fd691a333782b55b785768c9c6e6f6fd40eab | subsection | 58 | 59 | Correctness for Infinite Runs | Since none of the rules
\mathsf {a}\mathsf {b}\mathsf {a} \rightarrow \mathsf {a}\mathsf {b}^n survives, in the
limit we obtain the TRS consisting of the single rule
\mathsf {b}\mathsf {b} \rightarrow \mathsf {b}. This TRS is complete but
not equivalent to \mathcal {E} as witnessed by non-joinability of
\mathsf {a}\mat... | {
"cite_spans": []
} | 10.23638/LMCS-15(3:19)2019 | 1802.08437 | Abstract Completion, Formalized | [
"Nao Hirokawa",
"Aart Middeldorp",
"Christian Sternagel",
"Sarah Winkler"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.0155452536419034,
0.03758930787444115,
-0.007635327987372875,
-0.004397369455546141,
-0.00137775344774127,
-0.04527803137898445,
0.020884644240140915,
0.021067708730697632,
0.018901441246271133,
0.046712037175893784,
-0.034294143319129944,
0.015499487519264221,
0.012631472200155258,
0.0... |
36eebc33303501ffbe508b72476b2dd1fed91333 | abstract | 0 | 18 | Abstract | Due to the fact that it is prohibitively expensive to completely annotate
visual relationships, i.e., the (obj1, rel, obj2) triplets, relationship models
are inevitably biased to object classes of limited pairwise patterns, leading
to poor generalization to rare or unseen object combinations. Therefore, we are
interest... | {
"cite_spans": []
} | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.09718739986419678,
-0.013979845680296421,
-0.03247719630599022,
0.021870216354727745,
0.016498049721121788,
-0.014506379142403603,
0.04169534891843796,
-0.005574387032538652,
-0.00594067107886076,
0.012812314555048943,
-0.03650632128119469,
-0.03488856554031372,
0.0014670443488284945,
0... | |
9413a26fab0344a8adf2fab7820594b6f4b1f14a | subsection | 1 | 18 | Introduction | Thanks to the maturity of mid-level vision solutions such as object classification and detection , , , we are more ambitious to pursue higher-level vision-language tasks such as image captioning , , , , visual Q&A , , , and visual chatbot . Unfortunately, we gradually realize that many of the state-of-the-art systems m... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/cvpr.2016.90",
"end": 240,
"openalex_id": "https://openalex.org/W2194775991",
"raw": "He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: CVPR (2016)",
"source_ref_id": "77facf87265072cd14953... | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.08379649370908737,
-0.01568514108657837,
-0.042508564889431,
0.032560400664806366,
0.01299974787980318,
0.019270751625299454,
0.036832619458436966,
-0.024488961324095726,
0.019224978983402252,
0.0024698758497834206,
-0.02512979321181774,
-0.05398251861333847,
0.004012832883745432,
0.030... | |
c6485ddba532c8355f12e16dfe3c64a873a515ed | subsection | 2 | 18 | Introduction | REF (b), “shuffle” is to discard the original one-to-one paired object alignments, and thus no explicit obj1-obj2 class information is used; “assemble” is to pose the relationship modeling into an unsupervised pair recover problem by transferring Region-of-Interest (ROI) features between the two unpaired domains. Our i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/iccv.2017.244",
"end": 1236,
"openalex_id": "https://openalex.org/W2962793481",
"raw": "Zhu, J.Y., Park, T., Isola, P., Efros, A.A.: Unpaired image-to-image translation using cycle-consistent adversarial networks. In: ICCV (2017)",
... | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.06822612881660461,
-0.004607399459928274,
-0.045463744550943375,
0.02009253390133381,
-0.005477007944136858,
0.016888713464140892,
0.053152915090322495,
0.0016476793680340052,
0.02527967281639576,
0.008779995143413544,
-0.016080129891633987,
-0.002570684999227524,
0.002133973641321063,
... | |
ece0caa0be36b698ef846c1a30eb4280dfc2600d | subsection | 3 | 18 | Visual Relationships. | Modeling the object interactions such as verbs , , actions , , , and visual phrases , , , has been extensively studied in literature. In particular, our relationship model used in this paper follows the recent progress on modeling generic visual relationships, i.e., the (obj1, rel, obj2) triplets detected in images , .... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-540-88682-2_3",
"end": 134,
"openalex_id": "https://openalex.org/W1584193343",
"raw": "Gupta, A., Davis, L.S.: Beyond nouns: Exploiting prepositions and comparative adjectives for learning visual classifiers. In: ECCV (2008)",... | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.09744665026664734,
-0.0032511453609913588,
-0.0686458870768547,
0.02878550812602043,
0.004717497620731592,
0.04536730423569679,
0.04170618951320648,
0.018992027267813683,
-0.014941920526325703,
0.005331497173756361,
-0.04024174436926842,
-0.03716030716896057,
0.01313424576073885,
0.0346... | |
262d4eb55173767df61a7f4c74871a4aa189cf4c | subsection | 4 | 18 | Method | Fig. REF illustrates the overview of using Shuffle-Then-Assemble to enhance the relationship model. The goal of the feature learning process is to pre-train the Object-Agnostic (OA) conv-layers, which result in the desired OA feature map for better relationship modeling. We will first introduce the widely-used relation... | {
"cite_spans": []
} | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.06993185728788376,
-0.010145001113414764,
-0.02851279266178608,
-0.0008929317118600011,
0.0025229016318917274,
-0.020244235172867775,
0.015377686358988285,
0.004508041776716709,
0.0001860473130363971,
0.001973698614165187,
-0.010717087425291538,
-0.0005377613706514239,
-0.0163082797080278... | |
2db4cdae9124956ccc5c8bfd0612be08c5822e4f | subsection | 5 | 18 | Visual Relationship Model | The input of the visual relationship model is an image with a pair of object bounding boxes, and the output is an “obj1-rel-obj2” triplet, where “obj1” and “obj2” are the object classes of the two bounding boxes, and “rel” is the relationship class. In this paper, we adopt the common practice as in , that we do not dir... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-319-46448-0_51",
"end": 564,
"openalex_id": "https://openalex.org/W2479423890",
"raw": "Lu, C., Krishna, R., Bernstein, M., Fei-Fei, L.: Visual relationship detection with language priors. In: ECCV (2016)",
"source_ref_i... | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.09003668278455734,
-0.03128392994403839,
-0.061163902282714844,
0.04062332957983017,
0.006764196325093508,
0.03900572285056114,
0.015130740590393543,
0.03610623627901077,
-0.0038895083125680685,
-0.008202494122087955,
-0.04639178141951561,
-0.029574761167168617,
0.04156947880983353,
0.0... | |
02430d8383982a6bae5aa871a61614d140badc1d | subsection | 6 | 18 | Visual Relationship Model | For example, if most of the triplets of “stand on” is “person stand on street”, then the “stand on” classifier will mistakenly consider the joint pattern “person” and “street” into “stand on”, and fails in cases of “person stand on chair” or “dog stand on street”.
[Figure: (a) The overview of unsupervised domain transf... | {
"cite_spans": []
} | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.07896015048027039,
-0.010114099830389023,
-0.033378053456544876,
0.024179257452487946,
0.01537709217518568,
0.02233339659869671,
-0.0054880050010979176,
0.017924686893820763,
-0.020990950986742973,
0.03173050656914711,
-0.02440808340907097,
-0.0027706988621503115,
0.003872876288369298,
... | |
c08e9c88834203f30fd3f0dffe7a48bb57d891ff | subsection | 7 | 18 | Shuffle-Then-Assemble Feature Learning | To alleviate the bias, we detail our proposed Shuffle-then-Assemble strategy to pre-train the Object-Agnostic (OA) conv-layers for obtaining the OA feature map. As discussed above, the bias is mainly due to the dominant object pairs in training data, therefore, our key idea is to discard the original one-to-one pairwis... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/iccv.2017.244",
"end": 950,
"openalex_id": "https://openalex.org/W2962793481",
"raw": "Zhu, J.Y., Park, T., Isola, P., Efros, A.A.: Unpaired image-to-image translation using cycle-consistent adversarial networks. In: ICCV (2017)",
... | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.07806472480297089,
0.005889949854463339,
-0.07757643610239029,
0.015838777646422386,
-0.006557528395205736,
0.0010433298302814364,
0.027908597141504288,
0.010894881561398506,
0.018066581338644028,
0.023956531658768654,
-0.018112357705831528,
-0.010360818356275558,
0.009597871452569962,
... | |
1805d286f82c3bf82cb662afd14f0415477a8b15 | subsection | 8 | 18 | Shuffle-Then-Assemble Feature Learning | Recall that there is one-to-one supervision between the two domains, we adopt the adversarial objective \mathcal {L}_{adv} such that the mapped features \lbrace F(\mathbf {a})\rbrace and \lbrace G(\mathbf {b})\rbrace are indistinguishable from B and A, respectively; in particular, the indistinguishability is measured b... | {
"cite_spans": []
} | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.0520440973341465,
0.010920102708041668,
-0.03308844566345215,
-0.01062249019742012,
-0.004998370073735714,
0.0211534071713686,
0.03101278841495514,
-0.005128098651766777,
0.027029354125261307,
0.03232533484697342,
-0.04358883947134018,
-0.014025960117578506,
0.00022034799621906132,
0.02... | |
93e97583bf84516055bbd36831f75899ce75a318 | subsection | 9 | 18 | Shuffle-Then-Assemble Feature Learning | (REF ) together, the full objective for pre-training the OA conv-layers is:\phi ^* = \arg \min \limits _\phi \min \limits _{F, G}\max \limits _{D_A, D_B}\mathcal {L}_{adv}(A, B; \phi , F, G, D_A, D_B)+\lambda \mathcal {L}_{cycle}(A, B; \phi , F, G),where \lambda >0 is a trade-off hyper-parameter. Then, we can use \phi ... | {
"cite_spans": []
} | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.049934834241867065,
-0.019671760499477386,
-0.054848961532115936,
-0.026585111394524574,
0.010827862657606602,
0.023731255903840065,
0.014826311729848385,
0.009172015823423862,
-0.006352498661726713,
0.012407402507960796,
-0.053811196237802505,
-0.0017827056581154466,
0.008721808902919292... | |
cffe61f45c1cafe851d5d14dcffb97ca7a132419 | subsection | 10 | 18 | Implementation Details | Network Architecture.
For base CNN, we adopt Faster RCNN (VGG16) , which takes short width to be 600 and outputs the original 1/16\times 1/16 \times 512 feature map. As shown in Fig REF , our OA conv-layer has 1 filter of the size 1\times 1, stride 1, followed by a Leaky Relu . The transformation network is detailed in... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/tpami.2016.2577031",
"end": 165,
"openalex_id": "https://openalex.org/W639708223",
"raw": "Ren, S., He, K., Girshick, R., Sun, J.: Faster r-cnn: Towards real-time object detection with region proposal networks. In: NIPS (2015)",
... | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.06798789650201797,
-0.032285094261169434,
-0.04464375600218773,
0.0066904607228934765,
0.01187804713845253,
0.035214558243751526,
0.0024774540215730667,
0.02439691312611103,
0.010016619227826595,
0.03381085768342018,
-0.05697190389037132,
-0.010893931612372398,
-0.005687273107469082,
0.... | |
119312de61b1e113345132d8c50d16ea2a9df53d | subsection | 11 | 18 | Experiments | We evaluated our Shuffle-Then-Assemble method by performing visual relationship prediction on two benchmark datasets. We conducted experiments under extensive settings: supervised, weakly-supervised, and zero-shot, each of which has various ablative baselines and state-of-the-art methods. We also visualized qualitative... | {
"cite_spans": []
} | 1808.00171 | Shuffle-Then-Assemble: Learning Object-Agnostic Visual Relationship
Features | [
"Xu Yang",
"Hanwang Zhang",
"Jianfei Cai"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.055528003722429276,
0.006803630385547876,
-0.029015595093369484,
0.039989929646253586,
-0.0007469493430107832,
-0.015965444967150688,
0.052109017968177795,
0.002493646927177906,
-0.0021025240421295166,
0.031686678528785706,
-0.0018468631897121668,
-0.0032835244201123714,
0.002304763300344... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.