Canonizable Partial Order Generators and Regular Slice Languages

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. In the present work we show that any slice graph can be transitive reduced into a Hasse diagram generator representing the same set of partial orders. By employing this result we establish unknown connections between the true concurrent behavior of bounded p/t-nets and traditional approaches for representing infinite families of partial orders, such as Mazurkiewicz trace languages and Message Sequence Chart (MSC) languages. Going further, we identify the family of weakly saturated slice graphs. The class of partial order languages that can be represented by weakly saturated slice graphs is closed under union, intersection and even under a suitable notion of complementation (globally bounded complementation). The partial order languages in this class also admit canonical representatives in terms of Hasse diagram generators, and have decidable inclusion and emptiness of intersection. Our transitive reduction algorithm plays a fundamental role in these decidability results.