Canonizable Partial Order Generators and Regular Slice Languages

TL;DR

This paper introduces methods to effectively transform slice graphs into Hasse diagram generators, enabling canonical representations and closure properties for partial order languages, with applications to Petri nets and other formal models.

Contribution

It presents an effective transitive reduction method for slice graphs, defines saturated slice graphs, and proves their closure properties and canonical forms for partial order languages.

Findings

Effective transformation of slice graphs to Hasse diagram generators.

Closure of saturated slice graph languages under union, intersection, and complementation.

Existence of canonical representatives for partial order languages.

Abstract

In a previous work we introduced slice graphs as a way to specify both infinite languages of directed acyclic graphs (DAGs) and infinite languages of partial orders. Therein we focused on the study of Hasse diagram generators, i.e., slice graphs that generate only transitive reduced DAGs, and showed that they could be used to solve several problems related to the partial order behavior of p/t-nets. In the present work we show that both slice graphs and Hasse diagram generators are worth studying on their own. First, we prove that any slice graph SG can be effectively transformed into a Hasse diagram generator HG representing the same set of partial orders. Thus from an algorithmic standpoint we introduce a method of transitive reducing infinite families of DAGs specified by slice graphs. Second, we identify the class of saturated slice graphs. By using our transitive reduction algorithm, we prove that the class of partial order languages representable by saturated slice graphs is closed under union, intersection and even under a suitable notion of complementation (cut-width complementation). Furthermore partial order languages belonging to this class can be tested for inclusion and admit canonical representatives in terms of Hasse diagram generators. As an application of our results, we give stronger forms of some results in our previous work, and establish some unknown connections between the partial order behavior of $p/t$-nets and other well known formalisms for the specification of infinite families of partial orders, such as Mazurkiewicz trace languages and message sequence chart (MSC) languages.