The Complexity of Enriched μ-Calculi

Electron. Proc. Theor. Comput. Sci.

et al. 2016

Self Cite

Strategy Logic (SL) is a logical formalism for strategic reasoning in multi-agent systems. Its main feature is that it has variables for strategies that are associated to specific agents with a binding operator. We introduce Graded Strategy Logic (GRADEDSL), an extension of SL by graded quantifiers over tuples of strategy variables, i.e., "there exist at least g different tuples (x 1 , ..., x n ) of strategies" where g is a cardinal from the set N ∪ {ℵ 0 , ℵ 1 , 2 ℵ 0 }. We prove that the model-checking problem of GRADEDSL is decidable. We then turn to the complexity of fragments of GRADEDSL. When the g's are restricted to finite cardinals, written GRADED N SL, the complexity of model-checking is no harder than for SL, i.e., it is non-elementary in the quantifier rank. We illustrate our formalism by showing how to count the number of different strategy profiles that are Nash equilibria (NE), or subgame-perfect equilibria (SPE). By analyzing the structure of the specific formulas involved, we conclude that the important problems of checking for the existence of a unique NE or SPE can both be solved in 2EXPTIME, which is not harder than merely checking for the existence of such equilibria.

Section: Introductionmentioning

confidence: 99%

Extended Graded Modalities in Strategy Logic

Aminof

Malvone

Electron. Proc. Theor. Comput. Sci.

et al. 2016

Self Cite

“…In fact, although CTL * is a very powerful logic, there are several important but complex properties that require a more powerful framework. To overcome this limitation, several attempts have been carried out in literature in order to extend these logics by introducing appropriate semantics or operators usually guided by embedded contexts [AHK02,BLMV06,BMM09].…”

Section: Introductionmentioning

confidence: 99%

Branching-Time Temporal Logics with Minimal Model Quantifiers

Mogavero

Developments in Language Theory

2009

Self Cite

Abstract. Temporal logics are a well investigated formalism for the specification and verification of reactive systems. Using formal verification techniques, we can ensure the correctness of a system with respect to its desired behavior (specification), by verifying whether a model of the system satisfies a temporal logic formula modeling the specification. From a practical point of view, a very challenging issue in using temporal logic in formal verification is to come out with techniques that automatically allow to select small critical parts of the system to be successively verified. Another challenging issue is to extend the expressiveness of classical temporal logics, in order to model more complex specifications. In this paper, we address both issues by extending the classical branching-time temporal logic CTL * with minimal model quantifiers (MCTL * ). These quantifiers allow to extract, from a model, minimal submodels on which we check the specification (also given by an MCTL * formula). We show that MCTL * is strictly more expressive than CTL * . Nevertheless, we prove that the model checking problem for MCTL * remains decidable and in particular in PSPACE. Moreover, differently from CTL * , we show that MCTL * does not have the tree model property, is not bisimulation-invariant and is sensible to unwinding. As far as the satisfiability concerns, we prove that MCTL * is highly undecidable. We further investigate the model checking and satisfiability problems for MCTL * sublogics, such as MPML, MCTL, and MCTL + , for which we obtain interesting results. Among the others, we show that MPML retains the finite model property and the decidability of the satisfiability problem.

“…In [BP04], it has been shown that satisfiability is undecidable in the fully enriched µ-calculus. On the other hand, it has been shown in [SV01,BLMV06] that satisfiability for enriched µ-calculus is decidable and Exptime-complete. The upper bound result is based on an automata-theoretic approach via two-way graded alternating parity tree automata ( 2GAPT).…”

Section: Introductionmentioning

confidence: 99%

“…Intuitively, these automata generalize alternating automata on infinite trees in a similar way as the fully enriched µ-calculus extends the standard µ-calculus: 2GAPT can move up to a node's predecessor (by analogy with inverse programs), move down to at least n or all but n successors (by analogy with graded modalities), and jump directly to the roots of the input forest (which are the analogues of nominals). Using these automata, along with the fact that the en-riched µ-calculus enjoys the quasi-forest model property 1 , it has been shown in [SV01,BLMV06] that it is possible to build a 2GAPT accepting all trees encoding of all quasi-forest models of any enriched µ-calculus formula. Then, the exponential-upper bound follows from the fact that 2GAPT can be exponentially translated in nondeterministic graded parity tree automata ( GNPT), and the emptiness problem for GNPT is solvable in Ptime [KPV02].…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the set exec(M) is a set of quasi-forests, and the automaton A M we build will accept all trees encodings of all quasi-forests of exec (M). From the formula side, accordingly to [BLMV06], we can build in a polynomial time a 2GAPT A ¬ϕ accepting all models of ¬ϕ, with the intent to check that no models of ¬ϕ are in exec(M). Thus, we check that M |= r ϕ by checking whether L(A M ) ∩ L(A ¬ϕ ) is empty.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Enriched μ–Calculus Pushdown Module Checking

Ferrante

Logic for Programming, Artificial Intelligence, and Reasoning

Parente

Self Cite

Abstract. The model checking problem for open systems (called module checking) has been intensively studied in the literature, both for finite-state and infinite-state systems. In this paper, we focus on pushdown module checking with respect to decidable fragments of the fully enriched µ-calculus. We recall that finite-state module checking with respect to fully enriched µ-calculus is undecidable and hence the extension of this problem to pushdown systems remains undecidable as well. On the contrary, for the fragments of the fully enriched µ-calculus we consider here, we show that pushdown module checking is decidable and solvable in double-exponential time in the size of the formula and in exponential time in the size of the system. This result is obtained by exploiting a classical automata-theoretic approach via pushdown nondeterministic parity tree automata. In particular, we reduce in exponential time our problem to the emptiness problem for these automata, which is known to be decidable in Exptime. As a key step of our algorithm, we show an exponential improvement of the construction of a nondeterministic parity tree automaton accepting all models of a formula of the considered logic. This result, does not only allow our algorithm to match the known lower bound, but also to investigate decision problems related to the fragments of the enriched µ-calculus in a greatly simplified manner.