Thomas Wilke

Stutter-invariant temporal properties are expressible without the next-time operator

We show that every stutter-invariant propositional linear temporal property is expressible without the next-time operator.

Specifying Timed State Sequences in Powerful Decidable Logics and Timed Automata

A monadic second-order language, denoted by \(\mathcal{L}d\), is introduced for the specification of sets of timed state sequences. A fragment of \(\mathcal{L}d\), denoted by , is proved to be expressively complete for timed automata (Alur and Dill), i.e., every timed regular language is definable by a -formula and every -formula defines a timed regular language. As a consequence the satisfiability problem for is decidable.

Translating Regular Expressions into Small ε-Free Nondeterministic Finite Automata

We prove that every regular expression of size n can be converted into an equivalent nondeterministic ?-free finite automaton (NFA) with O(n(logn)2) transitions in time O(n2logn). The best previously known conversions result in NFAs of worst-case size ?(n2). We complement our result by proving an almost matching lower bound. We exhibit a sequence of regular expressions of size O(n) and show the number of transitions required in equivalent NFAs is ?(nlogn). This also proves there does not exist a linear-size conversion from regular expressions to NFAs.

Fair Simulation Relations, Parity Games, and State Space Reduction for Büchi Automata

We give efficient algorithms, improving optimal known bounds, for computing a variety of simulation relations on the state space of a Buchi automaton. Our algorithms are derived via a unified and simple parity-game framework. This framework incorporates previously studied notions like fair and direct simulation, but also a new natural notion of simulation called delayed simulation, which we introduce for the purpose of state space reduction. We show that delayed simulation---unlike fair simulation---preserves the automaton language upon quotienting and allows substantially better state space reduction than direct simulation. Using our parity-game approach, which relies on an algorithm by Jurdzinski, we give efficient algorithms for computing all of the above simulations. In particular, we obtain an O(mn3)-time and O(mn)-space algorithm for computing both the delayed and the fair simulation relations. The best prior algorithm for fair simulation requires time and space O(n6). Our framework also allows one to compute bisimulations: we compute the fair bisimulation relation in O(mn3) time and O(mn) space, whereas the best prior algorithm for fair bisimulation requires time and space O(n10).

Classifying discrete temporal properties

This paper surveys recent results on the classiffication of discrete temporal properties, gives an introduction to the methods that have been developed to obtain them, and explains the connections to the theory of finite automata, the theory of finite semigroups, and to first-order logic.

Over words, two variables are as powerful as one quantifier alternation

We show a property of strings is expressible in the two-variable fragment of first-order logic if and only if it is express ible by both a 2 and a 2 sentence. We thereby establish: UTL = FO2 = 2 \ 2 = UL ; where UTL stands for the string properties expressible in th e temporal logic with ‘eventually in the future’ and ‘eventua lly in the past’ as the only temporal operators and UL stands for the class of unambiguous languages. This enables us to show that the problem of determining whether or not a given temporal string property belongs to UTL is decidable (in exponential space), which settles a hitherto open problem. Our proof of 2 \ 2 = FO2 involves a new combinatorial characterization of these two classes and introduce s a new method of playing Ehrenfeucht-Fraı̈ssé games to verif y identities in semigroups.

Complementation, Disambiguation, and Determinization of Büchi Automata Unified

We present a uniform framework for (1) complementing Buchi automata, (2) turning Buchi automata into equivalent unambiguous Buchi automata, and (3) turning Buchi automata into equivalent deterministic automata. We present the first solution to (2) which does not make use of McNaughtons theorem (determinization) and an intuitive and conceptually simple solution to (3). Our results are based on Muller and Schupps procedure for turning alternating tree automata into non-deterministic ones.

An algorithmic approach for checking closure properties of temporal logic specifications and Ω-regular languages

Abstract In concurrency theory, there are several examples where the interleaved model of concurrency can distinguish between execution sequences which are not significantly different. One such example is sequences that differ from each other by stuttering, i.e., the number of times a state can adjacently repeat. Another example is executions that differ only by the ordering of independently executed events. Considering these sequences as different is semantically rather meaningless. Nevertheless, specification languages that are based on interleaving semantics, such as linear temporal logic (LTL), can distinguish between them. This situation has led to several attempts to define languages that cannot distinguish between such equivalent sequences. In this paper, we take a different approach to this problem: we develop algorithms for deciding if a property cannot distinguish between equivalent sequences, i.e., is closed under the equivalence relation. We focus on properties represented by regular languages, ω-regular languages, or prepositional LTL formulas and show that for such properties there is a wide class of equivalence relations for which determining closure is decidable, in fact is in PSPACE. Hence, checking the closure of a specification is no more difficult than checking satisfiability of a temporal formula. Among the closure properties we are able to handle, one finds trace closedness, stutter closedness and projective closedness, for all of which we are also able to prove a PSPACE lower bound. Being able to check that a property is closed under an equivalence relation has an immediate application in state-space exploration based verification. Indeed, the knowledge that the specification does not distinguish between equivalent execution sequences allows constructing a reduced state space where it is sufficient that at least one sequence per equivalence class is represented.

An Eilenberg theorem for ∞ -languages

We use a new algebraic structure to characterize regular sets of finite and infinite words through recognizing morphisms. A one-to-one correspondence between special classes of regular ∞-languages and pseudovarieties of right binoids according to Eilenbergs theorem for regular sets of finite words is established. We give the connections to semigroup theoretical characterizations and classifications of regular ω-languages, and treat concrete classes of ∞-languages in the new framework.

An algebraic characterization of frontier testable tree languages

The class of frontier testable (i.e., reverse definite) tree languages is characterized by a finite set of pseudoidentities for tree algebras, which are introduced here for this characterization. An efficient algorithm is presented that decides whether a given tree automaton recognizes a frontier testable tree language. The algorithm runs in time O(mn3 + m2n2), where m is the cardinality of the alphabet and n is the number of states of the automaton.