Automata-based Decision Procedure Research Articles

Logic programming approach to automata-based decision procedures

We propose a novel technique that reduces the decision problem of WSnS (weak monadic second-order logic with n successors) to the problem of evaluation of Complex-value Datalog queries. We then show how the use of advanced implementation techniques for Logic Programs, in particular the use of tabling in the XSB system, yields a considerable improvement in performance over more traditional approaches. We also explore various optimizations of the proposed technique based on variants of tabling and goal reordering. Although our primary focus is on WS1S, the logic of single successor, we show that it is straightforward to adapt our approach for other logics with existing automata-theoretic decision procedures, for example WS2S.

A Faster Tableau for CTL*

There have been several recent suggestions for tableau systems for deciding satisfiability in the practically important branching time temporal logic known as CTL*. In this paper we present a streamlined and more traditional tableau approach built upon the author's earlier theoretical work. Soundness and completeness results are proved. A prototype implementation demonstrates the significantly improved performance of the new approach on a range of test formulas. We also see that it compares favourably to state of the art, game and automata based decision procedures.

Ehrenfeucht–Fraïssé goes automatic for real addition

The decision problem of various logical theories can be decided by automata-theoretic methods. Notable examples are Presburger arithmetic FO ( Z , + , < ) and the linear arithmetic over the reals FO ( R , + , < ) . Despite the practical use of automata to solve the decision problem of such logical theories, many research questions are still only partly answered in this area. One of these questions is the complexity of the automata-based decision procedures and the related question about the minimal size of the automata of the languages that can be described by formulas in the respective logic. In this article, we establish a double exponential upper bound on the automata size for FO ( R , + , < ) and an exponential upper bound for the first-order theory of the discrete order over the integers FO ( Z , < ) . The proofs of these upper bounds are based on Ehrenfeucht–Fraïssé games. The application of this mathematical tool has a similar flavor as in computational complexity theory, where it can often be used to establish tight upper bounds of the decision problem for logical theories.

Complete axiomatization and decidability of Alternating-time temporal logic

Alternating-time Temporal Logic (ATL), introduced by Alur, Henzinger and Kupferman, is a logical formalism for the specification and verification of open systems involving multiple autonomous players (agents, components). In particular, this logic allows for the explicit expression of coalition abilities in such systems, modelled as infinite transition games between the coalition and its complement. Formally, ATL is a non-normal multi-modal extension of CTL (regarded as a one-player fragment of ATL) with temporal operators indexed by coalitions of players, and thus expressing selective quantification over those paths which can be effected as outcomes of infinite transition games between the coalition and its complement. We present a sound and complete axiomatization of the logic ATL, based on Pauly's axiomatization of his Coalition Logic, augmented with axioms and rules for fixed point formulae characterizing the temporal operators. The completeness proof is by construction of a bounded branching tree model for each ATL-consistent formula. These models can be folded into finite models, thus rendering the finite model property for ATL. We also describe an automata-based decision procedure for ATL by translating the satisfiability problem to the nonemptiness problem for alternating automata on infinite trees. When considering formulae over a fixed finite set of players the decidability problem is shown to be EXPTIME-complete.

Abstract abstract reduction

We propose novel algebraic proof techniques for rewrite systems. Church–Rosser theorems and further fundamental statements that do not mention termination are proved in Kleene algebra. Certain reduction and transformation theorems for termination that depend on abstract commutation, cooperation or simulation properties are proved in an extension with infinite iteration. Benefits of the algebraic approach are simple concise calculational proofs by equational reasoning, connection with automata-based decision procedures and a natural formal semantics for rewriting diagrams. It is therefore especially suited for mechanization and automation.

PITL2MONA: Implementing a Decision Procedure for Propositional Interval Temporal Logic

Interval Temporal Logic (ITL) is a finite-time linear temporal logic with applications in hardware verification, temporal logic programming and specification of multimedia documents. Due to the logic's non-elementary complexity, efficient ITL-based verification tools have been difficult to develop, even for propositional subsets. MONA is an efficient implementation of an automata-based decision procedure for the logic WS1S. Despite the non-elementary complexity of WS1S, MONA has been successfully applied in problems such as hardware synthesis, protocol verification and theorem proving. Here we consider a rich propositional subset of ITL, PITL, whose expressive power is equivalent to that of WS1S, and in turn to that of regular languages. PITL not only includes operators such as chop (;), star (*) and projection (proj), but also past operators such as previous (-), chop in the past (˜) and since (S). We provide an interpretation of PITL formulas in WS1S, which led us to a direct translation from PITL formulas to MONA specifications. We present the tool PITL2MONA as an implementation of such translation. With PITL2MONA acting as a front-end, MONA is used as a decision procedure for PITL. To our knowledge, this is one of the few implementations of a decision procedure for PITL, the first one based on automata, and the only one which handles both projection and past operators. We have tested our implementation on a number of examples; we show in this paper the application of PITL and its MONA-based decision procedure in solutions to the dining-philosophers and a multimedia synchronisation problem, together with some experimental results on these and some other examples.