Satisfiability Games for Branching-Time Logics

Friedmann, Oliver; Lange, Martin; Latte, Markus

doi:10.2168/lmcs-9(4:5)2013

Cited by 8 publications

(7 citation statements)

References 46 publications

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“…For alternation-free fixpoint logics, the game-based approach (e.g. [13]) is to (1.) define a nondeterministic co-Büchi automaton of size O(n) that recognizes unsuccessful branches of the tableau.…”

Section: Algorithm (Global Caching) Decide Satisfiability Of a Closed...mentioning

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: Algorithm (Global Caching) Decide Satisfiability Of a Closed...mentioning

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 omit a formal proof of these easily checkable facts. The estimation on the running time can be deduced from the observation that the value of the array count grows lexicographically in each iteration of the loop in lines [5][6][7][8][9][10][11][12][13][14][15]…”

Section: Solving Parity Games By Fixpoint Iterationmentioning

confidence: 99%

“…They are used as the algorithmic backbone in satisfiability checking for temporal logics [10], in controller synthesis [1] and, most commonly known, in model checking [19].…”

Section: Introductionmentioning

confidence: 99%

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The Fixpoint-Iteration Algorithm for Parity Games

Bruse

Falk

Lange

2014

Electron. Proc. Theor. Comput. Sci.

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It is known that the model checking problem for the modal µ-calculus reduces to the problem of solving a parity game and vice-versa. The latter is realised by the Walukiewicz formulas which are satisfied by a node in a parity game iff player 0 wins the game from this node. Thus, they define her winning region, and any model checking algorithm for the modal µ-calculus, suitably specialised to the Walukiewicz formulas, yields an algorithm for solving parity games. In this paper we study the effect of employing the most straight-forward µ-calculus model checking algorithm: fixpoint iteration. This is also one of the few algorithms, if not the only one, that were not originally devised for parity game solving already. While an empirical study quickly shows that this does not yield an algorithm that works well in practice, it is interesting from a theoretical point for two reasons: first, it is exponential on virtually all families of games that were designed as lower bounds for very particular algorithms suggesting that fixpoint iteration is connected to all those. Second, fixpoint iteration does not compute positional winning strategies. Note that the Walukiewicz formulas only define winning regions; some additional work is needed in order to make this algorithm compute winning strategies. We show that these are particular exponential-space strategies which we call eventually-positional, and we show how positional ones can be extracted from them.

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“…The main novelty of our work is to use techniques from game theory to go beyond reach-avoid and safety specifications for self-triggered control. Game theory, and in particular parity games [7], is a well-known technique to deal with expressive logic like the µ-calculus [7] and CTL * [8], as the parity winning conditions provide complex scenarios and strategies while keeping computability. In particular, parity games can be used for control synthesis of reactive systems under Linear Temporal Logic (LTL) specifications [9].…”

Section: Introductionmentioning

confidence: 99%

Symbolic Self-triggered Control of Continuous-time Non-deterministic Systems without Stability Assumptions for 2-LTL Specifications

Pruekprasert¹,

Eberhart²,

Dubut³

2020

Preprint

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We propose a symbolic self-triggered controller synthesis procedure for non-deterministic continuous-time nonlinear systems without stability assumptions. The goal is to compute a controller that satisfies two objectives. The first objective is represented as a specification in a fragment of LTL, which we call 2-LTL. The second one is an energy objective, in the sense that control inputs are issued only when necessary, which saves energy. To this end, we first quantise the state and input spaces, and then translate the controller synthesis problem to the computation of a winning strategy in a meanpayoff parity game. We illustrate the feasibility of our method on the example of a navigating nonholonomic robot.

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Satisfiability Games for Branching-Time Logics

Cited by 8 publications

References 46 publications

Global Caching for the Flat Coalgebraic µ-Calculus

Global Caching for the Flat Coalgebraic µ-Calculus

The Fixpoint-Iteration Algorithm for Parity Games

Symbolic Self-triggered Control of Continuous-time Non-deterministic Systems without Stability Assumptions for 2-LTL Specifications

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