Abstract. The automata-theoretic approach to linear temporal logic uses the theory of automata as a unifying paradigm for program specification, verification, and synthesis. Both programs and specifications are in essence descriptions of computations. These computations can be viewed as words over some alphabet. Thus,programs and specificationscan be viewed as descriptions of languagesover some alphabet. The automata-theoretic perspective considers the relationships between programs and their specifications as relationships between languages.By translating programs and specifications to automata, questions about programs and their specifications can be reduced to questions about automata. More specifically, questions such as satisfiability of specifications and correctness of programs with respect to their specifications can be reduced to questions such as nonemptiness and containment of automata. Unlike classical automata theory, which focused on automata on finite words, the applications to program specification, verification, and synthesis, use automata on infinite words, since the computations in which we are interested are typically infinite. This paper provides an introduction to the theory of automata on infinite words and demonstrates its applications to program specification, verification, and synthesis. 1

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

Abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations. When the application domain is described by a logical theory, we speak about logic-based abduction. Candidates for abductive explanations are usually subjected to minimality criteria such as subsetminimality, minimal cardinality, minimal weight, or minimality under prioritization of individual hypotheses. This paper presents a comprehensive complexity analysis of relevant decision and search problems related to abduction on propositional theories. Our results indicate that abduction is harder than deduction. In particular, we show that with the most basic forms of abduction the relevant decision problems are complete for complexity classes at the second level of the polynomial hierarchy, while the use of prioritization raises the complexity to the third level in certain cases.

Of special interest in formal verification are safety properties, which assert that the system always stays within some allowed region. Proof rules for the verification of safety properties have been developed in the proof-based approach to verification, making verification of safety properties simpler than verification of general properties. In this paper we consider model checking of safety properties. A computation that violates a general linear property reaches a bad cycle, which witnesses the violation of the property. Accordingly, current methods and tools for model checking of linear properties are based on a search for bad cycles. A symbolic implementation of such a search involves the calculation of a nested fixed-point expression over the system's state space, and is often impossible. Every computation that violates a safety property has a finite prefix along which the property is violated. We use this fact in order to base model checking of safety properties on a search for ...

We present a theorem-prover for quantified Boolean formulae and evaluate it on random quantified formulae and formulae that represent problems from automated planning. Even though the notion of quantified Boolean formula is theoretically important, automated reasoning with QBF has not been thoroughly investigated. Universal quantifiers are needed in representing many computational problems that cannot be easily translated to the propositional logic and solved by satisfiability algorithms. Therefore efficient reasoning with QBF is important. The Davis-Putnam procedure can be extended to evaluate quantified Boolean formulae. A straightforward algorithm of this kind is not very efficient. We identify universal quantifiers as the main area where improvements to the basic algorithm can be made. We present a number of techniques for reducing the amount of search that is needed, and evaluate their effectiveness by running the algorithm on a collection of formulae obtained from planning and generated randomly. For the structured problems we consider, the techniques lead to a dramatic speed-up. 1

In this paper, we describe an automated reasoning system, called DeReS. DeReS implements default logic of Reiter by supporting several basic reasoning tasks such as testing whether extensions exist, finding one or all extensions (if at least one exists) and querying if a formula belongs to one or all extensions. If an input theory is a logic program, DeReS computes stable models of this program and supports queries on membership of an atom in some or all stable models. The paper contains an account of our preliminary experiments with DeReS and a discussion of the results. We show that a choice of a propositional prover is critical for the efficiency of DeReS. We also present a general technique that eliminates the need for some global consistency checks and results in substantial speedups. We experimentally demonstrate the potential of the concept of relaxed stratification for making automated reasoning systems practical. 1 INTRODUCTION The area of nonmonotonic l...

Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomial-time Turing machines. Well known examples of counting classes are NP, co-NP, \PhiP, and PP. Every counting class is a subset of P #P[1] , the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine. Using closure properties of #P, we systematically develop a complexity theory for counting classes defined in terms of thresholds and moduli. An unexpected result is that MOD k iP = MOD k P for prime k. Finally, we improve a result of Cai and Hemachandra by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in [28]). 1. Introduction Valiant [27] defined the class #P of functions whose values equal the number of accepting paths of polynomial-time bo...

In Floyd-Hoare logic, programs are dynamic while assertions are static (hold at states). In action logic the two notions become one, with programs viewed as on-the-fly assertions whose truth is evaluated along intervals instead of at states. Action logic is an equational theory ACT conservatively extending the equational theory REG of regular expressions with operations preimplication a!b (had a then b) and postimplication b/a (b if-ever a). Unlike REG, ACT is finitely based, makes a reflexive transitive closure, and has an equivalent Hilbert system. The crucial axiom is that of pure induction, (a!a) = a!a. This work was supported by the National Science Foundation under grant number CCR-8814921. 1 Introduction Many logics of action have been proposed, most of them in the past two decades. Here we define action logic, ACT, a new yet simple juxtaposition of old ideas, and show off some of its attractive aspects. The language of action logic is that of equational regular expressio...

Abstract. We show that approximating the shortest vector problem (in any ℓp norm) to within any constant factor less than p √ 2 is hardfor NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (random polynomial time), unless NP equals RP. We also prove a proper NP-hardness result (i.e., hardness under deterministic many-one reductions) under a reasonable number theoretic conjecture on the distribution of square-free smooth numbers. As part of our proof, we give an alternative construction of Ajtai’s constructive variant of Sauer’s lemma that greatly simplifies Ajtai’s original proof. Key words. NP-hardness, shortest vector problem, point lattices, geometry of numbers, sphere packing

this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies