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Notes on finite asynchronous automata. (English) Zbl 0623.68055

Let \(\Sigma\) be a finite alphabet and \(I\subset \Sigma \times \Sigma\) an irreflexive relation. Intuitively (a,b)\(\in I\) denotes that the actions a, b can be executed simultaneously. Let \(\sim\) be the minimal congruence relation over \(\Sigma^*\) created by \(\{\) ab\(\equiv ba|\) (a,b)\(\in I\}\). The free partially commutative monoid E(\(\Sigma\),I) is a quotient of \(\Sigma^*\) by \(\sim\). As it was shown by A. Mazurkiewicz [Concurrent program schemes and their interpretations, DAIMI-PB-78, Aarhus University (1977)] elements of E(\(\Sigma\),I), called traces, model in a natural way executions in concurrent systems. The trace language \(T\subset E(\Sigma,I)\) is called regular if the language UT is regular. In the paper the notion of a finite asynchronous automaton is introduced. Such an automaton is a tuple \({\mathbb{A}}=({\mathbb{P}}_ 1,...,{\mathbb{P}}_ u,\Delta,{\mathbb{F}})\), where \({\mathbb{P}}_ i=(\Sigma_ i,S_ i,s^ 0_ i)\) is an ith process, \(\Sigma_ i\) is an alphabet of \({\mathbb{P}}_ i\), \(S_ i\) is a finite and nonempty set of states of \({\mathbb{P}}_ i\), \(s^ 0_ i\in S_ i\) is an initial state. \({\mathbb{F}}\subset S=\times^{n}_{i=1}S_ i\) the family of final states.
Let \(Dom(a)=\{i\in \Pr oc:\) \(a\in \Sigma_ i\}\) for \(a\in \Sigma =\cup^{n}_{i=1}\Sigma_ i\). Then, \(\Delta =\{\delta_ a:\) \(a\in \Sigma \}\) is the family of the next-state functions \[ \forall a\in \Sigma,\quad \delta_ a:\times_{i\in Dom(a)}S_ i\quad \to \quad {\mathcal P}(\times_{i\in Dom(a)}S_ i). \] Let \((s_ 1',...,s_ n')\), \((s_ 1'',...,s_ n'')\in \times^{n}_{i=1}S_ i\), \(a\in \Sigma\). Then \((s_ 1',...,s_ n')\Rightarrow^{a}(s_ 1'',...,s_ n'')\) iff \((s_ i''=s_ i')\) for \(i\not\in Dom(a)\) and \((s''_{i_ 1},...,s''_{i_ k})\in \delta_ a(s'_{i_ 1},...,s'_{i_ k})\), where \(Dom(a)=\{i_ 1,...,i_ k\}.\)
This relation is extended in a natural way to traces from \(E(\Sigma,I_{{\mathbb{A}}})\), where \(I_{{\mathbb{A}}}=\{(a,b)\in \Sigma^ 2:\) \(Dom(a)\cap Dom(b)=\emptyset \}\). The main theorem of the paper is as follows: \(T\subset E(\Sigma,I)\) is regular iff there exists a finite asynchronous automaton \({\mathbb{A}}\) such that \(I=I_{{\mathbb{A}}}\), and \(T=T({\mathbb{A}})=\{t\in E(\Sigma,I):\exists s\in {\mathbb{F}}(s^ 0_ 1,...,s^ 0_ n)\Rightarrow^{t}S\}\). As a corollary a new characterization of regular subsets of f.p.c. monoids is given.

MSC:

68Q45 Formal languages and automata
20M35 Semigroups in automata theory, linguistics, etc.
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References:

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