Dietrich Kuske

Skew and infinitary formal power series

We investigate finite-state systems with weights. Departing from the classical theory, in this paper the weight of an action does not only depend on the state of the system, but also on the time when it is executed; this reflects the usual human evaluation practices in which later events are considered less urgent and carry less weight than close events. We first characterize the terminating behaviors of such systems in terms of rational formal power series. This generalizes a classical result of Schutzenberger. n nSecondly, we deal with nonterminating behaviors and their weights. This includes an extension of the Buchi-acceptance condition from finite automata to weighted automata and provides a characterization of these nonterminating behaviors in terms of ω-rational formal power series. This generalizes a classical theorem of Buchi.

Satisfiability and Model Checking for MSO-Definable Temporal Logics Are in PSPACE

Temporal logics over Mazurkiewicz traces have been extensively studied over the past fifteen years. In order to be usable for the verification of concurrent systems they need to have reasonable complexity for the satisfiability and the model checking problems. Whenever a new temporal logic was introduced, a new proof (usually non trivial) was needed to establish the complexity of these problems. In this paper, we introduce a unified framework to define local temporal logics over traces. We prove that the satisfiability problem and the model checking problem for asynchronous Kripke structures for local temporal logics over traces are decidable in PSPACE. This subsumes and sometimes improves all complexity results previously obtained on local temporal logics for traces.

Asynchronous cellular automata for promsets

Abstract This paper extends to pomsets without auto-concurrency the fundamental notion of asynchronous cellular automata (ACA) which was originally introduced for traces by Zielonka. We generalize to pomsets the notion of asynchronous mapping introduced by Cori, Metivier and Zielonka and we show how to construct a deterministic ACA~from an asynchronous mapping. Then we investigate the relation between the expressiveness of monadic second-order logic, nondeterministic ACAs and deterministic ACAs. We can generalize Buchis theorem for finite words to a class of pomsets without auto-concurrency which satisfy a natural axiom. This axiom ensures that an asynchronous cellular automaton works on the pomset as a concurrent read and exclusive owner write machine. More precisely, in this class nondeterministic ACAs, deterministic ACAs and monadic second-order logic have the same expressive power. Then we consider a class where deterministic ACAs are strictly weaker than nondeterministic ones. But in this class nondeterministic ACAs still capture monadic second-order logic. Finally, it is shown that even this equivalence does not hold in the class of all pomsets since there the class of recognizable pomset languages is not closed under complementation.

Propositional dynamic logic for message-passing systems

We examine a bidirectional Propositional Dynamic Logic (PDL) for message sequence charts (MSCs) extending LTL and TLC-. Every formula is translated into an equivalent communicating finite-state machine (CFM) of exponential size. This synthesis problem is solved in full generality, i.e., also for MSCs with unbounded channels. The model checking problems for CFMs and for HMSCs against PDL formulas are shown to be in PSPACE for existentially bounded MSCs. It is shown that CFMs are to weak to capture the semantics of PDL with intersection.

Theories of orders on the set of words

It is shown that small fragments of the first-order theory of the subword order, the (partial) lexicographic path ordering on words, the homomorphism preorder, and the infix order are undecidable. This is in contrast to the decidability of the monadic second-order theory of the prefix order [M.O. Rabin, Trans. Amer. Math. Soc., 1969] and of the theory of the total lexicographic path ordering [P. Narendran and M. Rusinowitch, Lect. Notes Artificial Intelligence, 2000] and, in case of the subword and the lexicographic path order, improves upon a result by Comon & Treinen [H. Comon and R. Treinen, Lect. Notes Comp. Sci., 1994]. Our proofs rely on the undecidability of the positive Σ 1 -theory of (N, +,.) [Y. Matiyasevich, Hilberts Tenth Problem, 1993] and on Treinens technique [R. Treinen, J. Symbolic Comput., 1992] that allows to reduce Posts correspondence problem to logical theories.

LOGICAL ASPECTS OF CAYLEY-GRAPHS: THE MONOID CASE

Cayley-graphs of monoids are investigated under a logical point of view. It is shown that the class of monoids, for which the Cayley-graph has a decidable monadic second-order theory, is closed under free products. This result is derived from a result of Walukiewicz, stating that the decidability of monadic second-order theories is preserved under tree-like unfoldings. Concerning first-order logic, it is shown that the class of monoids, for which the Cayley-graph has a decidable first-order theory, is closed under arbitrary graph products, which generalize both, free and direct products. For the proof of this result, tree-like unfoldings are generalized to so-called factorized unfoldings. It is shown that the decidability of the first-order theory of a structure is preserved by factorized unfoldings. Several additional results concerning factorized unfoldings are shown.

Uniform satisfiability problem for local temporal logics over Mazurkiewicz traces

We continue our study of the complexity of MSO-definable local temporal logics over concurrent systems that can be described by Mazurkiewicz traces. In previous papers, we showed that the satisfiability problem for any such logic is in PSPACE (provided the dependence alphabet is fixed, Gastin and Kuske (2003) [10]) and remains in PSPACE for all classical local temporal logics even if the dependence alphabet is part of the input, Gastin and Kuske (2007) [8]. In this paper, we consider the uniform satisfiability problem for arbitrary MSO-definable local temporal logics. For this problem, we prove multi-exponential lower and upper bounds that depend on the number of alternations of set quantifiers present in the chosen MSO-modalities.

The Isomorphism Problem on Classes of Automatic Structures

Several new undecidability results on isomorphism problems for automatic structures are shown: (i) The isomorphism problem for automatic equivalence relations is pi^0_1-complete. (ii) The isomorphism problem for automatic trees of height

The isomorphism problem on classes of automatic structures with transitive relations

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Construction of tree automata from regular expressions

??? 2 is pi^0_2