H. J. Sander Bruggink

Recognizable Graph Languages for Checking Invariants

We generalize the order-theoretic variant of the Myhill-Nerode theorem to graph languages, and characterize the recognizable graph languages as the class of languages for which the Myhill-Nerode quasi order is a well quasi order. In the second part of the paper we restrict our attention to graphs of bounded interface size, and use Myhill-Nerode quasi orders to verify that, for such bounded graphs, a recognizable graph property is an invariant of a graph transformation system. A recognizable graph property is a recognizable graph language, given as an automaton functor. Finally, we present an algorithm to approximate the Myhill-Nerode ordering.

Treewidth, pathwidth and cospan decompositions with applications to graph-accepting tree automata ☆ ☆☆

Abstract We will revisit the categorical notion of cospan decompositions of graphs and compare it to the well-known notions of path decomposition and tree decomposition from graph theory. More specifically, we will define several types of cospan decompositions with appropriate width measures and show that these width measures coincide with pathwidth and treewidth. Such graph decompositions of small width are used to efficiently decide graph properties, for instance via graph automata. Hence we will give an application by defining graph-accepting tree automata, thus integrating previous work by Courcelle into the setting of cospan decompositions. Furthermore we will show that regardless of whether we consider path or tree decompositions, we arrive at the same notion of recognizability.

A Logic on Subobjects and Recognizability

We introduce a simple logic that allows to quantify over the subobjects of a categorical object. We subsequently show that, for the category of graphs, this logic is equally expressive as second-order monadic graph logic (msogl). Furthermore we show that for the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle’s result that every msogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into so-called automaton functors which accept recognizable languages of cospans.

Efficient symbolic implementation of graph automata with applications to invariant checking

We introduce graph automata as a more automata-theoretic view on (bounded) automaton functors and we present how automaton-based techniques can be used for invariant checking in graph transformation systems. Since earlier related work on graph automata suffered from the explosion of the size of the automata and the need of approximations due to the non-determinism of the automata, we here employ symbolic bdd-based techniques and recent antichain algorithms for language inclusion to overcome these issues. We have implemented techniques for generating, manipulating and analyzing graph automata and perform an experimental evaluation.

Conditional Reactive Systems

We lift the notion of nested application conditions from graph transformation systems to the general categorical setting of reactive systems as defined by Leifer and Milner. This serves two purposes: first, we enrich the formalism of reactive systems by adding application conditions for rules; second, it turns out that some constructions for graph transformation systems (such as computing weakest preconditions and strongest postconditions and showing local confluence by means of critical pair analysis) can be done very elegantly in the more general setting.

Proving Termination of Graph Transformation Systems Using Weighted Type Graphs over Semirings

We introduce techniques for proving uniform termination of graph transformation systems, based on matrix interpretations for string rewriting. We generalize this technique by adapting it to graph rewriting instead of string rewriting and by generalizing to ordered semirings. In this way we obtain a framework which includes the tropical and arctic type graphs of [6] and a new variant of arithmetic type graphs. These type graphs can be used to assign weights to graphs and to show that these weights decrease in every rewriting step in order to prove termination. We present an example involving counters and discuss the implementation in the tool Grez.

Towards Process Mining with Graph Transformation Systems

This paper is about process mining with graph transformation systems (gtss). Given a set of observed transition sequences, the goal is to find a gts – that is a finite set of graph transformation rules – that models these transition sequences as well as possible. In this paper the focus is on real-word processes such as business processes or (human) problem solving strategies, with the aim of better understanding such processes. The observed behaviour is not assumed to be either complete or error-free and the given model is expected to generalize the observed behaviour and be robust to erroneous input. The paper presents some basic algorithms that obtain gtss from observed transition sequences and gives a method to compare the resulting gtss.

Robustness and closure properties of recognizable languages in adhesive categories

We consider recognizable languages of cospans in adhesive categories defined via automaton functors, of which recognizable graph languages are a special case.There are three contributions in this paper: we first show that the notion of recognizable languages is robust in the sense that also semi-functors, i.e., functors that do not necessarily preserve identities, characterize recognizable languages. This is done by converting semi-functors to functors with a procedure akin to epsilon elimination for non-deterministic finite automata.Second, relying on this result, we show that recognizable languages are closed under concatenation, i.e. under cospan composition, by providing a concrete construction that creates a concatenation automaton from two given automata. The construction is considerably more complex than the corresponding construction for finite automata. We conclude by showing negative closure properties for Kleene star and substitution. We show that recognizable languages are closed under concatenation by constructing an automaton accepting the concatenation.Using counter-examples, we show that recognizable languages are not closed under Kleene-star and substitution.We introduce the notion of semi-automata.We show that (for suitable categories) semi-automata accept the same class of languages as automata.

Towards Alternating Automata for Graph Languages

In this paper we introduce alternating automata for languages of arrows of an arbitrary category, and as an instantiation thereof alternating automata for graph languages. We study some of their closure properties and compare them, with respect to expressiveness, to other methods for describing graph languages. We show, by providing several examples, that many graph properties (of graphs of bounded path width) can be naturally expressed as alternating automata.