Trace languages defined by regular string languages. (English) Zbl 0612.68071

A concurrent alphabet is a pair \({\mathcal C}=<\Sigma,C>\), where \(\Sigma\) is an alphabet and C is a relation over \(\Sigma\), called the concurrency relation. Two words over \(\Sigma\) are called C-equivalent, if they can be obtained from each other by successively interchanging adjacent symbols which are related to C. A trace (over \({\mathcal C})\) is now simply an equivalence class with respect to C-equivalence. We consider trace languages (i.e., sets of traces) as they are defined by regular string languages in the following ways: (i) existentially regular trace languages (the trace language defined existentially by a regular string language L consists of all traces which have a representative in L), (ii) universally regular trace languages (the trace language defined universally by a regular string language L consists of all traces which have all representatives in L), and (iii) consistently regular trace languages (a regular string language L defines a consistently regular trace language T if and only if L is the union of all the traces in T).
In particular, the main result characterizes those concurrent alphabets for which the family of existentially regular trace languages equals the family of universally regular trace languages. Furthermore, using this result, a number of decidability results and characterizations of closure properties for the three above mentioned families of trace languages are derived.


68Q45 Formal languages and automata
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