Rational Trace Languages Research Articles

Globally deterministic CD-systems of stateless R-automata with window size 1

It is known that cooperating distributed systems (CD-systems) of stateless deterministic restarting automata with window size 1 accept a class of semi-linear languages that properly includes all rational trace languages. Although the component automata of such a CD-system are all deterministic, the CD-system itself is not. Here, we study CD-systems of stateless deterministic restarting automata with window size 1 that are themselves completely deterministic. In fact, we consider two such types of CD-systems: the strictly deterministic systems and the globally deterministic systems.

Asynchronous trace-matching automata

This paper presents algorithms for solving several matching problems on Mazurkiewicz traces. The creation of the algorithms is reduced to the construction of automata that recognize corresponding rational trace languages. Such languages and their properties were studied by many authors. This paper considers trace languages used in solving concrete problems that have analogs in stringology.

A Note on the Logical Definability of Rational Trace Languages

The regular languages in the free monoid generated by a finite alphabet A are exactly the languages that are the models of some sentence of the second-order monadic logic of one successor and a unary predicate for each letter. For trace monoids the natural extension obtained by adapting the successor to the partial order underlying the traces is insufficient to capture the family of their rational subsets. We show that these subsets can be expressed by formulas of the form ∃Γp where p is a first-order formula over the structure of traces and Γ is an n-ary predicate semantically restricted, where n is the cardinality of the alphabet.

On CD-systems of stateless deterministic R-automata with window size one

Here we study cooperating distributed systems (CD-systems) of restarting automata that are very restricted: they are deterministic, they cannot rewrite, but only delete symbols, they restart immediately after performing a delete operation, they are stateless, and they have a read/write window of size 1 only, that is, these are stateless deterministic R(1)-automata. We study the expressive power of these systems by relating the class of languages that they accept by mode =1 computations to other well-studied language classes, showing in particular that this class only contains semi-linear languages. Our model can be viewed as a nondeterministic finite-state acceptor with translucent letters, that is, it processes its input in a different way than the usual left-to-right order. In this way all commutative semi-linear languages, and in fact all rational trace languages, can be accepted. In addition, we investigate the closure and non-closure properties of the class of languages accepted by our model and some of its algorithmic properties.

Efficient recognition of trace languages defined by repeat-until loops

A sequence of operations may be validly reordered, provided that only pairs of independent operations are commuted. Focusing on a program scheme, idealized as a local finite automaton, we consider the problem of checking whether a given string is a valid permutation of a word recognized by the automaton. Within the framework of trace theory, this is the membership problem for rational trace languages. Existing general algorithms, although time-polynomial, have unbounded degree related to some properties of the dependence graph. Here we present two original linear-time solutions. A straightforward algorithm is suitable for any finite automaton such that all the transitions starting from the same state are labelled by dependent symbols. The second approach is currently restricted to automata representing programs of nested repeat-until loops. Using integer compositions to represent loop iterations and under suitable conditions, the algorithm constructs the syntax tree of a possible word equivalent to the input string. The same procedures show that, under our hypotheses, the uniform version of the membership problem (which is NP-complete in the general case) is solvable in polynomial time.

On Bounded Rational Trace Languages

In this paper, for a finitely generated monoid M, we tackle the following three questions: • Is it possible to give a characterization of rational subsets of M which have polynomial growth? • What is the structure of the counting function of rational sets which have polynomial growth? • Is it true that every rational subset of M has either exponential growth or it has polynomial growth? Can one decide for a given rational set which of the two options holds? We give a positive answer to all the previous questions in the case that M is a direct product of free monoids. Some of the proved results also extend to trace monoid.

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.

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.