A resolution calculus for the branching-time temporal logic CTL

Zhang, Lan; Hustadt, Ullrich; Dixon, Clare

doi:10.1145/2529993

Cited by 9 publications

(9 citation statements)

References 38 publications

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“…building graphs or tableaux by recursion over the formula, or bottomup; the two groups perform very differently [18]. We compare our implementation with the top-down solvers TreeTab [14], GMUL [18], MLSolver [11] and the bottom-up solvers CTL-RP [36] and BDDCTL [18]. Out of the top-down solvers, only TreeTab is singlepass like COOL; however, TreeTab has suboptimal (doubly exponential) worst-case runtime.…”

Section: Implementation and Benchmarkingmentioning

confidence: 99%

Global Caching for the Flat Coalgebraic µ-Calculus

Hausmann

Schröder

2015

2015 22nd International Symposium on Temporal Representation and Reasoning (TIME)

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We present a sound, complete, and optimal single-pass tableau algorithm for the alternation-free µ-calculus. The algorithm supports global caching with intermediate propagation and runs in time 2 O(n) . In game-theoretic terms, our algorithm integrates the steps for constructing and solving the Büchi game arising from the input tableau into a single procedure; this is done onthe-fly, i.e. may terminate before the game has been fully constructed. This suggests a slogan to the effect that global caching = game solving on-the-fly. A prototypical implementation shows promising initial results.Parikh's game logic [31], or probabilistic fixpoint logic. To aid readability, we phrase our results in terms of the relational µ-calculus, and discuss the coalgebraic generalization only at the end of Section 4. The model construction in the completeness proof yields models of size 2 O(n) .We have implemented of our algorithm as an extension of the Coalgebraic Ontology Logic Reasoner COOL, a generic reasoner for coalgebraic modal logics [21]; given the current state of the implementation of instance logics in COOL, this means that we effectively support alternation-free fragments of relational, monotone, and alternating-time [1] µ-calculi, thus in particular covering CTL and ATL. We have evaluated the tool in comparison with existing reasoners on benchmark formulas for CTL [18] (which appears to be the only candidate logic for which well-developed benchmarks are currently available) and on random formulas for ATL and the alternation-free relational µ-calculus, with promising results; details are discussed in Section 5. Related WorkThe theoretical upper bound ExpTime has been established for the full coalgebraic µ-calculus [5] (and earlier for instances such as the alternating-time µ-calculus AMC [35]), using a multi-pass algorithm that combines games and automata in a similar way as for the standard relational case, in particular involving the Safra construction. Global caching has been employed successfully for a variety of description logics [17, 20], and lifted to the level of generality of coalgebraic logics with global assumptions [15] and nominals [16].A tableaux-based non-optimal (NExpTime) decision procedure for the full µ-calculus has been proposed in [23]. Friedmann and Lange [12] describe an optimal tableau method for the full µ-calculus that, unlike most other methods including the one we present here, makes do without requiring guardedness. Like earlier algorithms for the full µ-calculus, the algorithm constructs and solves a parity game, and in principle allows for an on-thefly implementation. The models constructed in the completeness proof are asymptotically larger than ours, but presumably the proof can be adapted for the alternation-free case by using determinization of co-Büchi automata [28] instead of Safra's determinization of Büchi automata [34] to yield models of size 2 O(n) , like ours. For non-relational instances of the coalgebraic µ-calculus, including the alternation-free fragment of the alternatin...

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Section: Implementation and Benchmarkingmentioning

confidence: 99%

Global Caching for the Flat Coalgebraic µ-Calculus

Hausmann

Schröder

2015

2015 22nd International Symposium on Temporal Representation and Reasoning (TIME)

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“…We also introduced an branching-time variant of MTL and provided a translation using 'gaps' into the branching-time temporal logic CTL. This opens up the opportunity to use CTL solvers such as [58] in a similar way as we have done for LTL solvers.…”

Section: Discussionmentioning

confidence: 98%

Theorem Proving for Pointwise Metric Temporal Logic Over the Naturals via Translations

2020

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We study translations from metric temporal logic (MTL) over the natural numbers to linear temporal logic (LTL). In particular, we present two approaches for translating from MTL to LTL which preserve the complexity of the satisfiability problem for MTL. In each of these approaches we consider the case where the mapping between states and time points is given by (i) a strict monotonic function and by (ii) a non-strict monotonic function (which allows multiple states to be mapped to the same time point). We use this logic to model examples from robotics, traffic management, and scheduling, discussing the effects of different modelling choices. Our translations allow us to utilise LTL solvers to solve satisfiability and we empirically compare the translations, showing in which cases one performs better than the other. We also define a branching-time version of the logic and provide translations into computation tree logic.

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“…This section reviews the basic semantics and syntax of CTL (with indexes) and the resolution procedure for CTL formulas in a normal form [41].…”

Section: Preliminariesmentioning

confidence: 99%

“…In this subsection, we recall the clausal forms of CTL with index, the rules of transforming CTL formulas into the clausal forms, and the sound and complete resolution refutation procedure for the clausal theories [41].…”

Section: Separated Normal Form and Resolutionmentioning

confidence: 99%