## 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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