Volker Diekert

The Book of Traces

The theory of traces belongs to both formal language theory and the theory of concurrent systems. In both these disciplines it is a well-recognized and dynamic research area. Within formal language theory it yields the theory of partially commutative monoids, and provides an important connection between languages and graphs. Within the theory of concurrent systems it provides an important formal framework for the analysis and synthesis of concurrent systems. This monograph covers all important research lines of the theory of traces - each chapter of the book is devoted to one research line and is written by leading experts. It is organized in such a way that each chapter can be read independently - and hence is suitable for advanced courses/seminars on formal language theory and the theory of concurrent systems.

Combinatorics on traces

Free partialy commutative monoids.- Recognizable and rational trace languages.- Petri nets and synchronization.- Complete semi-thue systems and mobius functions.- Trace replacement systems.

Characterization of the expressive power of silent transitions in timed automata

Timed automata are among the most widely studied models for real-time systems. Silent transitions, i.e., e-transitions, have already been proposed in the original paper on timed automata by Alur and Dill [3]. We show that the class TL e of timed languages recognized by automata with e-transitions, is more robust and more expressive than the corresponding class TL without e-transitions. We then focus on e-transitions without reset, i.e. e-transitions which do not reset clocks. We propose an algorithm to construct, given a timed automaton, an equivalent one without such transitions. This algorithm is in two steps, it first suppresses the cycles of e-transitions without reset and then the remaining ones. Then, we prove that a timed automaton such that no e-transition which resets clocks lies on any directed cycle, can be effectively transformed into a timed automaton without etransitions. Interestingly, this main result holds under the assumption of non-Zenoness and it is false otherwise. To complete the picture, we exhibit a simple timed automaton with an e-transition, which resets some clock, on a cycle and which is not equivalent to any e-free timed automaton. To show this, we develop a promising new technique based on the notion of precise action. This paper presents a synthesis of the two conference communications [9] and [13].

Partial commutation and traces

Parallelism and concurrency are fundamental concepts in computer science. Specification and verification of concurrent programs are of first importance. It concerns our daily life whether software written for distributed systems behaves correctly.

A SURVEY ON SMALL FRAGMENTS OF FIRST-ORDER LOGIC OVER FINITE WORDS

We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simons theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose.

The existential theory of equations with rational constraints in free groups is PSPACE-complete

It is well-known that the existential theory of equations in free groups is decidable. This is a celebrated result of Makanin which was published 1982. Makanin did not discuss complexity issues, but later it was shown that the scheme of his algorithm is not primitive recursive. In this paper we present an algorithm that works in polynomial space. This improvement is based upon an extension of Plandowskis techniques for solving word equations. We present a pspace-algorithm in a more general setting where each variable has a rational constraint, that is, the solution has to respect a specification given by a regular word language. We obtain our main result about the existential theory in free groups as a corollary of the corresponding statement in free monoids with involution.

Solvability of Equations in Free Partially Commutative Groups Is Decidable

Trace monoids are well-studied objects in computer science where they serve as a basic algebraic tool for analyzing concurrent systems. The question whether the existential theory of trace equations is decidable has been solved positively in 1996. Free partially commutative groups (graph groups) generalize trace monoids in the sense that every element has an inverse. In this paper we show that the existential theory of equations over graph groups is decidable, too. Our decision procedure is non-elementary, but if a certain graph theoretical parameter is viewed as a constant, then we obtain a PSPACE-completeness result. Restricting ourselves to trace monoids we still obtain a better complexity result, as it was known previously.

Pure future local temporal logics are expressively complete for Mazurkiewicz traces

The paper settles a long standing problem for Mazurkiewicz traces: the pure future local temporal logic defined with the basic modalities exists-next and until is expressively complete. This means every first-order definable language of Mazurkiewicz traces can be defined in a pure future local temporal logic. The analogous result with a global interpretation has been known, but the treatment of a local interpretation turned out to be much more involved. Local logics are interesting because both the satisfiability problem and the model checking problem are solvable in PSPACE for these logics whereas they are non-elementary for global logics. Both, the (previously known) global and the (new) local results generalize Kamps Theorem for words, because for sequences local and global viewpoints coincide.

Deterministic asynchronous automata for infinite traces

This paper shows the equivalence between the family of recognizable languages over infinite traces and the family of languages which are recognized by deterministic asynchronous cellular Muller automata. We thus give a proper generalization of McNaughtons Theorem from infinite words to infinite traces. Thereby we solve one of the main open problems in this field. As a special case we obtain that every closed (w.r.t. the independence relation) word language is accepted by someI-diamond deterministic Muller automaton.

On the concatenation of infinite traces

There is a straightforward generalization of traces to infinite traces as dependence graphs where every vertex has finitely many predecessors, or what is the same, as a backward closed and directed set of traces with respect to prefix ordering. However, this direct approach has a drawback since it allows no good notion of a concatenation. We solve this problem by adding to an infinite trace a second component. This second component is a finite alphabetic information which is called the alphabet at infinity. We obtain a compact and complete ultra-metric space where the concatenation is uniformly continuous and where the set of finite traces is an open, discrete, and dense subset. Our objects arise in a natural way from the consideration of dependence graphs where the induced partial order is well-founded. Such a graph splits into a so-called real part and a transfinite (or imaginary) part. From the transfinite part only the alphabet is of importance.