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1.

De Oliveira Oliveira, Mateus

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

Canonizable partial order generators2012In: Language and Automata Theory and Applications, Springer Berlin/Heidelberg, 2012, Vol. 7183 LNCS, p. 445-457Conference paper (Refereed)

Abstract [en]

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. This result allow us to 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 which can be represented by weakly saturated slice graphs is closed under union, intersection and a suitable notion of complementation (bounded cut-width 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.

Slices are digraphs that can be composed together to form larger digraphs.In this thesis we introduce the foundations of a theory whose aim is to provide ways of defining and manipulating infinite families of combinatorial objects such as graphs, partial orders, logical equations etc. We give special attentionto objects that can be represented as sequences of slices. We have successfully applied our theory to obtain novel results in three fields: concurrency theory,combinatorics and logic. Some notable results are:

Concurrency Theory:

We prove that inclusion and emptiness of intersection of the causalbehavior of bounded Petri nets are decidable. These problems had been open for almost two decades.

We introduce an algorithm to transitively reduce infinite familiesof DAGs. This algorithm allows us to operate with partial order languages defined via distinct formalisms, such as, Mazurkiewicztrace languages and message sequence chart languages.

Combinatorics:

For each constant z ∈ N, we define the notion of z-topological or-der for digraphs, and use it as a point of connection between the monadic second order logic of graphs and directed width measures, such as directed path-width and cycle-rank. Through this connection we establish the polynomial time solvability of a large numberof natural counting problems on digraphs admitting z-topological orderings.

Logic:

We introduce an ordered version of equational logic. We show thatthe validity problem for this logic is fixed parameter tractable withrespect to the depth of the proof DAG, and solvable in polynomial time with respect to several notions of width of the equations being proved. In this way we establish the polynomial time provability of equations that can be out of reach of techniques based on completion and heuristic search.

We introduce the notion of z-topological orderings for digraphs. We prove that given a digraph G on n vertices admitting a z-topological ordering, together with such an ordering, one may count the number of subgraphs of G that at the same time satisfy a monadic second order formula φ and are the union of k directed paths, in time f(φ,k,z)·nO(k·z). Our result implies the polynomial time solvability of many natural counting problems on digraphs admitting z-topological orderings for constant values of z and k. Concerning the relationship between z-topological orderability and other digraph width measures, we observe that any digraph of directed path-width d has a z-topological ordering for z ≤ 2d + 1. On the other hand, there are digraphs on n vertices admitting a z-topological order for z = 2, but whose directed path-width is Θ(log n). Since graphs of bounded directed path-width can have both arbitrarily large undirected tree-width and arbitrarily large clique width, our result provides for the first time a suitable way of partially transposing metatheorems developed in the context of the monadic second order logic of graphs of constant undirected tree-width and constant clique width to the realm of digraph width measures that are closed under taking subgraphs and whose constant levels incorporate families of graphs of arbitrarily large undirected tree-width and arbitrarily large clique width.