Petri Net DAG Languages and Regular Tree Languages with Synchronization
نویسنده
چکیده
A true-concurrency semantics is often given via pomsets. A pomset is an abstract directed acyclic graph that is in addition transitive: any grandson of a node is also a son of that node. We will drop this rather artificial transitive property in this paper and research abstract directed acyclic graphs (dags) as a true-concurrency semantics for Petri nets and their relation to regular languages of trees and forests. Therefor we regard regular graph grammars operating on unranked, unordered trees or on forests of them. Regular grammars over words with or without ǫ-rules generate the same languages (modulo the empty word). This is not true for regular graph grammars: regular graph grammars without ǫ-rules generate a class F ǫ−free 3 of tree languages where each language is of bounded out-degree, whereas general regular graph grammars generate F3, a class of regular forest languages including languages of unbounded out-degree. It turns out that nevertheless the synchronization closures of F3 and of F ǫ−free 3 are equal and coincide with the class of terminal dag languages of Petri nets. This implies that any dag of a true concurrency Petri net behavior language is the result of a uniform synchronization applied to a single tree of a regular tree language. As there are many concepts of graphs and graph grammars in the literature we will spend some effort in this Technical Report to give precise definitions of the concepts needed. The report can be read without any a priori knowledge about graph theory.
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