On the complexity of parity games

数理解析研究所講究録, 2002

We present the notion of robust models for amixed strategy Nash equilibrium of a strategic form game and to introduce a group structure on the class of these models. Prom the algebraic point of view we give $\mathrm{a}$ characterization of the class of robust models by the class of epistemic models with common-knowledge of conjectures about the other players' actions.

AiML-2004: Advances in Modal Logic, 2004

We study and give a summary of the complexity of 15 basic normal monomodal logics under the restriction to the Horn fragment and/or bounded modal depth. As new results, we show that: a) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by k ≥ 2 in the modal logics K 4 and KD4 is PSPACE-complete, in K is NP-complete; b) the satisfiability problem of modal formulas with modal depth bounded by 1 in K 4, KD4, and S 4 is NP-complete; c) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by 1 in K , K 4, KD4, and S 4 is PTIME-complete. In this work, we also study the complexity of the multimodal logics Ln under the mentioned restrictions, where L is one of the 15 basic monomodal logics. We show that, for n ≥ 2 : a) the satisfiability problem of sets of Horn modal clauses in K5n, KD5n, K45n, and KD45n is PSPACE-complete; b) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by k ≥ 2 in Kn

Journal of Logic and Computation, 2019

We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke-models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler-Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler-Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic FO and (basic) modal logic ML. We also present a version of the game for the modal µ-calculus L µ and show that FO is also non-elementarily more succinct than L µ .

This paper gives a novel approach to analyze SAT problem more deeply. First, I define new elements of Boolean formula such as dominant variable, decision chain, and chain coupler. Through the analysis of the SAT problem using the elements, I prove that we can construct a k-SAT (k>2) instance where the coefficients of cutting planes take exponentially large values in the input size. This exponential property is caused by the number system formed from the calculation of coefficients. In addition, I show that 2-SAT does not form the number system and Horn-SAT partially forms the number system according to the feasible value of the dominant variable. Whether or not the coefficients of cutting planes in cutting plane proof are polynomially bounded was open problem. Many researchers believed that cutting plane proofs with large coefficients are highly non-intuitive 20. However, we can construct a k-SAT (k>2) instance in which cutting planes take exponentially large coefficients by the number system. In addition, this exponential property is so strong that it gives definite answers for several questions: why Horn-SAT has the intermediate property between 2-SAT and 3-SAT; why random-SAT is so easy; and why k-SAT (k>2) cannot be solved with the linear programming technique. As we know, 2-SAT is NL-complete, Horn-SAT is P-complete, and k-SAT (k>2) is NP-complete. In terms of computational complexity, this paper gives a clear mathematical property by which SAT problems in three different classes are distinguished. Two questions, NL =? P and P =? NP, have been open problems for several decades. This study presents a definite supporting evidence for the conjecture that NL⊊ P ⊊ NP and a new solving direction for the P versus NP problem.

Journal of Logic and Computation, 2007

There has been a great deal of work on characterizing the complexity of the satisfiability and validity problem for modal logics. In particular, Ladner showed that the consistency (i.e., nonprovability) problem for all logics between K and S4 is PSPACE-hard, while for S5 it is NP-complete. We show that it is negative introspection, the axiom ¬Kp ⇒ K¬Kp, that causes the gap: if we add this axiom to any modal logic between K and S4, then the consistency problem becomes NPcomplete. Indeed, the consistency problem is NPcomplete for any modal logic that includes the negative introspection axiom.

Logical Methods in Computer Science, 2012

Ong has shown that the modal mu-calculus model checking problem (equivalently, the alternating parity tree automaton (APT) acceptance problem) of possibly-infinite ranked trees generated by order-n recursion schemes is n-EXPTIME complete. We consider two subclasses of APT and investigate the complexity of the respective acceptance problems. The main results are that, for APT with a single priority, the problem is still n-EXPTIME complete; whereas, for APT with a disjunctive transition function, the problem is (n-1)-EXPTIME complete. This study was motivated by Kobayashi's recent work showing that the resource usage verification of functional programs can be reduced to the model checking of recursion schemes. As an application, we show that the resource usage verification problem is (n-1)-EXPTIME complete.

2009

The complexity class of Π p k -Polynomial Local Search (PLS) problems with Π p -goal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by translating Bounded Arithmetic proofs using the Paris-Wilkie-translation. For ≤ k, the Σ b +1 -definable search problems of T k+1 2 are exactly characterised by Π p k -PLS problems with Π pgoals. These Π p k -PLS problems can be defined in a weak base theory such as S 1 2 , and proved to be total in T k+1

Trying to overcome Dugundji’s (1940) result on uncharacterisability of modal logics by finite logical matrices, Kearns (1981) and Ivlev (1988) proposed, independently, a characterisation of some modal systems by means of four-valued multivalued truth-functions (by restricting the valuations using level valuations, in Kearns’s approach), as an alternative to Kripke semantics. This constitutes an antecedent of the non-deterministic matrices introduced by Avron and Lev (2001). In this paper we propose a reconstruction of Kearns’s and Ivlev’s results (which did not have the dissemination or impact they deserved) in a uniform way, obtaining an extension to another modal systems. The first part of the paper is devoted to four-valued Nmatrices, including Kearns’s and Ivlev’s. Besides proving with full details Kearns’s results for T, S4 and S5, we also obtain a characterisation of the system B by four-valued Nmatrices with level valuations. Concerning Ivlev’s results, two new modal systems are introduced and characterised by Nmatrices. In the second part of this paper, six-valued Nmatrices are introduced which characterise a variant of the eight systems studied in the first part, by replacing axiom (T) with axiom (D). As a by-product, novel decision procedures for T, S4, S5, D, KDB, KD4 and KD45 are obtained, which open up interesting possibilities in the study of the complexity of modal logics and, in particular, of intuitionistic propositional logic (IPC), taking into account the Gödel-McKinsey-Tarski translation between IPC and S4.

Electronic Proceedings in Theoretical Computer Science, 2020

We introduce a new game-theoretic semantics (GTS) for the modal mu-calculus. Our so-called bounded GTS replaces parity games with alternative evaluation games where only finite paths arise; infinite paths are not needed even when the considered transition system is infinite. The novel games offer alternative approaches to various constructions in the framework of the mu-calculus. For example, they have already been successfully used as a basis for an approach leading to a natural formula size game for the logic. While our main focus is introducing the new GTS, we also consider some applications to demonstrate its uses. For example, we consider a natural model transformation procedure that reduces model checking games to checking a single, fixed formula in the constructed models, and we also use the GTS to identify new alternative variants of the mu-calculus with PTime model checking.