## Deterministic Asynchronous Automata for Infinite Traces (1993)

Venue: | Acta Informatica |

Citations: | 13 - 3 self |

### BibTeX

@INPROCEEDINGS{Diekert93deterministicasynchronous,

author = {Volker Diekert and Anca Muscholl},

title = {Deterministic Asynchronous Automata for Infinite Traces},

booktitle = {Acta Informatica},

year = {1993},

pages = {617--628},

publisher = {Springer}

}

### OpenURL

### Abstract

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 McNaughton's 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 some I-diamond deterministic Muller automaton. 1 Introduction A. Mazurkiewicz introduced the concept of traces as a suitable semantics for concurrent systems [Maz77]. A concurrent system is given by a set of atomic actions \Sigma = fa; b; c; : : :g together with an independence relation I ` \Sigma \Theta \Sigma, which specifies pairs of actions which can be performed concurrently. This leads to an equivalence relation on \Sigma generated by the independence relation I. More precisely, if a and b denote independent...

### Citations

134 |
Probl`emes combinatoires de commutations et r'earrangements
- Cartier, Foata
- 1969
(Show Context)
Citation Context ...ence, it defines a quotient monoid of \Sigma , which is called trace monoid and is denoted by IM(\Sigma; D). In fact, these monoids have been introduced and studied independently by Cartier and Foata =-=[CF69]-=- in combinatorics and are also called free partially commutative monoids. This research has been supported by the ESPRIT Basic Research Action No. 6317 ASMICS 2. 1 Hence, a trace is a congruence class... |

27 |
Asynchronous mappings and asynchronous cellular automata
- Cori, Metivier, et al.
- 1993
(Show Context)
Citation Context ...ng the main theorem of this section, let us recall some important concepts of the construction of Zielonka for asynchronous cellular automata which we will need in our construction (see also [Die90], =-=[CMZ89]-=-). The a-prefix (A-prefix, respectively) @ a (t) (@ A (t), respectively) of a finite trace t has been defined as the minimal prefix of t containing all a (all letters a 2 A, respectively), which occur... |

16 |
Combinatorics on Traces. Number 454
- Diekert
- 1990
(Show Context)
Citation Context ...ore stating the main theorem of this section, let us recall some important concepts of the construction of Zielonka for asynchronous cellular automata which we will need in our construction (see also =-=[Die90]-=-, [CMZ89]). The a-prefix (A-prefix, respectively) @ a (t) (@ A (t), respectively) of a finite trace t has been defined as the minimal prefix of t containing all a (all letters a 2 A, respectively), wh... |

11 |
Asynchronous cellular automata for infinite traces
- Gastin, Petit
(Show Context)
Citation Context ...zing morphisms [Gas91] and by c-rational expressions [GPZ91]. Concerning characterizations by finite automata, P. Gastin and A. Petit investigated asynchronous (cellular) automata for infinite traces =-=[GP92]-=-. This type of automaton has been defined by W. Zielonka [Zie87, Zie89], who showed that for finite traces, the recognizable languages are exactly the languages recognized by deterministic asynchronou... |

9 |
A syntactic congruence for rational !-languages
- Arnold
- 1985
(Show Context)
Citation Context ...IM the following saturation condition holds: t 0 t 1 t 2 : : : 2 L =) j \Gamma1 j(t 0 ) j \Gamma1 j(t 1 ) j \Gamma1 j(t 2 ) : : : ` L An equivalent definition uses the syntactic congruence of Arnold (=-=[Arn85]-=-). For L ` IR, two finite traces u; v 2 IM are syntactically congruent if and only if : 8x; y 2 IM : x(uy) ! 2 L , x(vy) ! 2 L 8x; y; z 2 IM : xuyz ! 2 L , xvyz ! 2 L We denote the syntactic congruenc... |

9 |
Automates et commutations partielles. R.A.I.R.O.- Informatique Th'eorique et Applications
- Cori, Perrin
- 1985
(Show Context)
Citation Context ...; se = s; e 2 = e g. The following technical propositions represent important tools for all our results. We begin by recalling the well-known decomposition lemma (Levi's Lemma) for traces: Lemma 3.5 (=-=[CP85]-=-) Let t 1 ; t 2 ; x 1 ; x 2 2 IM satisfying t 1 x 1 = t 2 x 2 . Then t 1 = pf; t 2 = pg; x 1 = gy; and x 2 = fy for p = t 1 u t 2 and some f; g; y 2 IM such that alph(f) \Theta alph(g) ` I. Lemma 3.6 ... |

8 |
On the concatenation of infinite traces
- Diekert
- 1991
(Show Context)
Citation Context ...the following these objects are called here real traces. Unfortunately, there can be no convenient associative concatenation on real traces. This led to the definition of the monoid of complex traces =-=[Die91]-=-. It is a quotient monoid of the set of infinite dependence graphs under the largest congruence which respects real parts. Recognizable subsets form a well-studied family in the context of both free p... |

5 |
Recognizable and rational trace languages of finite and infinite traces
- Gastin
- 1991
(Show Context)
Citation Context ...bility is equivalent to definability in certain monadic second order theories. As regards recognizable real trace languages, there exist equivalent characterizations by means of recognizing morphisms =-=[Gas91]-=- and by c-rational expressions [GPZ91]. Concerning characterizations by finite automata, P. Gastin and A. Petit investigated asynchronous (cellular) automata for infinite traces [GP92]. This type of a... |

4 | A Kleene theorem for infinite trace languages - Gastin, Petit, et al. - 1991 |