Trace Languages Research Articles

Iteration Lemmata for Certain Classes of Word, Trace and Graph Languages

Using general iteration lemmata for languages defined by rational, linear, and algebraic systems of equations on ω-complete semirings, it can be shown that such iteration lemmata hold for certain classes of word and trace languages, and for process algebra classes related to them. The basic operation has to be associative with a neutral element, and is extended to sets of elements, such that the structure defined by it together with union and ∅ is an ω-complete semiring. With this property rational, linear and algebraic systems of equations can be defined, having a unique minimal solution as least fixed points. To guarantee iteration lemmata for the corresponding rational, linear and algebraic classes, a norm on the structure has to exist, fulfilling some monotonicity on union and extended operation on sets. This article is an extended version of the paper presented at the workshop CS&P'98 in Berlin, September 28-30, 1998. (This paper has been presented on the workshop CS&P'98 at Berlin.)

Determinants and Möbius functions in trace monoids

We show that the independence relation defining a trace monoid M admits a transitive orientation if and only if the characteristic series ξ of a lexicographic cross section of M is the inverse of the determinant of (Id- X), where X is a matrix representing the minimum finite automaton recognizing ξ and Id is the identity matrix. This implies that, if the independence relation of a trace monoid M admits a transitive orientation, then any unambiguous lifting of the Möbius function of M is the determinant of a matrix defined by the smallest acceptor of the corresponding cross section.

Formalizing and Executing Message Sequence Charts via Timed Rewriting

Message Sequence Charts (MSC) is a graphical trace language for describing and specifying the communication behaviour of distributed systems by means of message interchange. (Timed) Maude is a formal object-oriented specification language which combines algebraic specification techniques for describing complex data structures with (timed) term rewriting to deal with dynamic behaviour. In this paper we show first how to formalize MSC in Timed Maude. Then we give a translation of timed rewriting to untimed rewrite systems and use this translation to execute Message Sequence Charts with the Elan system, a powerful tool which combines Rewriting Logic with a language of rewriting strategies. We illustrate our approach with the bench mark example of a railroad crossing.

On the inclusion problem for finitely ambiguous rational trace languages

In this work we prove that the Inclusion is decidable for a particular class of trace languages that is the class R FIN (Σ, C) of finitely rational trace languages over an alphabet Σ = A ∪ B with the commutation relation C = (A x Σ ∪ Σ x A)\I.

Event Structures for Local Traces

Local traces and local event structures have been introduced in order to lift the semantical theory of 1-safe Petri nets to the level of more general Petri nets. In this paper, we cut out Petri nets and establish a direct link between these two classes. Whereas it is relatively easy to associate a local independence relation to a local event structure, the opposite connection is much less clear. The problem here is the identification of the events in a local trace language. We consider various extensions of the classical technique which relates Mazurkiewicz' traces to prime event structures, one of which leads to a precise connection in terms of a coreflection between main subclasses of each model.

Geometric Transformations of Language Families: The Power of Symmetry

We consider certain isomorphisms between language families. When viewed as a group, the isomorphisms reduce to well-known geometric transformations. We obtain new characterizations for linear, simple matrix, simple contextual and trace languages. Our results make it possible to transfer results concerning languages in one family into results concerning languages in another family.

Any Lifting of a Trace Coding Is a Word Coding

A trace coding is an injective morphism between two trace monoids. This definition naturally extends the classical notion of word coding between two free monoids. We prove that any lifting of a trace coding is a word coding. More precisely, given a trace coding F , let us choose for any letter a , a representative word f ( a ) of the trace F ( a ). Then, the word morphism f is also a coding. This result positively solves a problem proposed by Ochmański.

Regular Trace Event Structures

We propose trace event structures as a starting point for constructing effective branching time temporal logics in a non-interleaved setting. As a first step towards achieving this goal, we define the notion of a regular trace event structure. We then provide some simple characterizations of this notion of regularity both in terms of recognizable trace languages and in terms of finite 1-safe Petri nets.

Probabilistic asynchronous automata

Asynchronous automata were introduced by W. Zielonka as an algebraic model of distributed systems, showing that the class of trace languages recognizable by automata over free partially commutative monoids coincides with the class of trace languages recognizable by deterministic asynchronous automata. In this paper we extend the notion of asynchronous automata to the probabilistic case. Our main result is a nontrivial generalization to Zielonka's theorem: we prove that the sets of behaviors of probabilistic automata and of probabilistic asynchronous automata coincide in the case of concurrent alphabets with acyclic dependency graphs.

Logical definability on infinite traces

The main results of the present paper are the equivalence of definability by monadic second-order logic and recognizability for real trace languages, and that first-order definable, star-free, and aperiodic real trace languages form the same class of languages. This generalizes results on infinite words and on finite traces to infinite traces. It closes an important gap in the different characterizations of recognizable languages of infinite traces.

Determinizing Asynchronous Automata on Infinite Inputs

Asynchronous automata are a natural distributed machine model<br />for recognizing trace languages - languages defined over an alphabet<br />equipped with an independence relation.<br />To handle infinite traces, Gastin and Petit introduced Buchi asynchronous<br />automata, which accept precisely the class of omega-regular trace<br />languages. Like their sequential counterparts, these automata need to<br />be non-deterministic in order to capture all omega-regular languages. Thus<br />complementation of these automata is non-trivial. Complementation<br />is an important operation because it is fundamental for treating the<br />logical connective "not" in decision procedures for monadic second-order<br />logics. Subsequently, Diekert and Muscholl solved the complementation<br />problem by showing that with a Muller acceptance condition, deterministic<br />automata suffice for recognizing omega-regular trace languages.<br />However, a direct determinization procedure, extending the classical<br />subset construction, has proved elusive.<br />In this paper, we present a direct determinization procedure for<br />Buchi asynchronous automata, which generalizes Safra's construction<br />for sequential Buchi automata. As in the sequential case, the blow-up<br />in the state space is essentially that of the underlying subset construction.

On some decision problems for trace codings

The problem we mainly deal with is the existence of a coding between two trace monoids. We introduce a new notion of coding: the strong coding (independent letters are mapped into independent traces). We prove that the existence of a strong coding between two trace monoids is decidable when the first monoid belongs to one of two large families of trace monoids that we specify. Our decision conditions are based on graph-theoretical properties of the dependence relations. A related question of Ochmański (1988) is completely solved for strong codings.

Logical definability of some rational trace languages

Trace monoids are obtained from free monoids by defining a subsetI of pairs of letters that are allowed to commute. Most of the work of this theory is an attempt to relate the properties of these monoids to the properties ofI. Following the work initiated by Buchi we show that when the reflexive closure ofI is transitive (the trace monoid is then a free product of free commutative monoids) it is possible to define a second-order logic whose models are the traces viewed as dependence graphs and which characterizes exactly the sets of traces that are rational. This logic essentially utilizes a predicate based on the partial ordering defined by the dependence graph and a predicate related to a restricted use of the comparison of cardinality.

New Results on the Star Problem in Trace Monoids

We prove that if it is decidable whether X* is recognizable for a recognizable subset X of a free partially commutative monoid, then it is decidable whether a recognizable subset of a free partially commutative monoid possesses the finite power property. We prove that if any trace of a set X is connected, we can decide whether X possesses the finite power property. Finally, we also show that if X is a finite set containing at most four traces or at most two connected traces, it is decidable whether X* is recognizable.

A Trace Semantics for Petri Nets

A generalization of the notion of trace is proposed. This enables us to associate with each Petri net a single behavioural object, namely a poset of (generalized) traces. A characterization is given of the trace languages defined by Petri nets. We show that the general event structures of Winskel and the stable event structures can also be characterized in terms of our trace languages. One consequence is that in this framework, stable event structures, general event structures, and Petri nets constitute a strictly ascending chain in terms of expressive power.

Rational and Recognizable Complex Trace Languages

Mazurkiewicz defined traces as an algebraic model of finite concurrent processes. In order to model non-terminating processes a good notion of infinite traces was needed, which finally led to the notion of complex traces. For complex traces an associative concatenation and an ω-iteration are defined. This paper defines and investigates rational and recognizable complex trace languages. We prove various closure results such as the closure under boolean operations (for recognizable languages), concatenation, and left and right quotients by recognizable sets. Then we study sufficient conditions ensuring the recognizability of the finite and infinite iterations of complex trace languages. We introduce a generalization of the notion of concurrent iteration which leads to the main result of the paper: the generalization of Kleene′s and Ochmański′s theorems to complex trace languages.

Object-oriented graphics programming in C++

Display modes - VGA, VESA super VGA, and undocumented selecting colours with an Octree displaying TGA files displaying four pictures on a single screen saving and restoring PCX files vector and matrix classes ray tracing language description using object-oriented programming to handle primitives ray tracing spheres planes and polygons rings and circles quadric curves constructive solid geometry textures and surfaces animation halftoning, shading and area fills using pictures from Kodak CD-ROMs.

Asynchronous automata versus asynchronous cellular automata

In this paper we compare and study some properties of two mathematical models of concurrent systems, asynchronous automata (Zielonka, 1987) and asynchronous cellular automata (Zielonka, 1989). First, we show that these models are “polynomially” related, exhibiting polynomial-time reductions between them. Subsequently, we prove that, in spite of that, the classes of asynchronous automata and of asynchronous cellular automata recognizing a given trace language are, in general, deeply different. In fact, we exhibit a recognizable trace language T with the following properties: there exists a unique minimum asynchronous automaton accepting T, does not exist a unique minimum asynchronous cellular automaton, but there are infinitely many minimal (i.e., unreducible) nonisomorphic asynchronous cellular automata accepting T. We characterize the class of concurrent alphabets for which every recognizable trace language admits a minimum finite state asynchronous cellular automaton as the class of alphabets with full concurrency relation. Finally, extending a result of (Bruschi et al., 1988), we show that for every concurrent alphabet with nontransitive dependency relation, there exists a trace language accepted by infinitely many minimal nonisomorphic asynchronous automata.

On Confluent Semi-commutations: Decidability and Complexity Results

The subject of this paper is the confluence of finite semi-commutation systems. Confluence of such systems is proved to be decidable property. Existence of a finite complete presentation of a trace monoid using rules only from another given trace monoid is proved to be reducible to the existence of a confluent semi-commutation system. Complexity results related to the preceding problems are proved: deciding the existence of finite complete presentations is Σ P 2 -complete, whereas deciding confluence of semi-commutation systems is Co-NP-complete. Additionally, an open problem about trace synchronizations is solved: The local checking property is Co-NP-complete.

An extension of Kleene's and Ochmański's theorems to infinite traces

As was noted by Mazurkiewicz, traces constitute a convenient tool for describing finite behaviour of concurrent systems. Extending in a natural way Mazurkiewicz's original definition, infinite traces have recently been introduced enabling one to deal with infinite behaviour of nonterminating concurrent systems. In this paper we examine the basic families of recognizable sets and of rational sets of infinite traces. The seminal Kleene characterization of recognizable subsets of the free monoid and its subsequent extensions to infinite words due to Büchi and to finite traces due to Ochmański are the cornerstones of the corresponding theories. The main result of our paper is an extension of these characterizations to the domain of infinite traces. Using recognizing and weakly recognizing morphisms, as well as a generalization of the Schützenberger product of monoids, we prove various closure properties of recognizable trace languages. Moreover, we establish normal-form representations for recognizable and rational sets of infinite traces.