Abstract
Extensions to finite-state automata on strings, such as multi-head automata or multi-counter automata, have been successfully used to encode many infinite-state non-regular verification problems. In this paper, we consider a generalization of automata-theoretic infinite-state verification from strings to labelled series–parallel graphs. We define a model of non-deterministic, 2-way, concurrent automata working on series–parallel graphs and communicating through shared registers on the nodes of the graph. We consider the following verification problem: given a family of series–parallel graphs described by a context-free graph transformation system (GTS), and a concurrent automaton over series–parallel graphs, is some graph generated by the GTS accepted by the automaton? The general problem is undecidable already for (one-way) multi-head automata over strings. We show that a bounded version, where the automata make a fixed number of reversals along the graph and use a fixed number of shared registers is decidable, even though there is no bound on the sizes of series–parallel graphs generated by the GTS. Our decidability result is based on establishing that the number of context switches can be bounded and on an encoding of the computation of bounded concurrent automata that allows us to reduce the reachability problem to the emptiness problem for pushdown automata.
Highlights
Automata theory studies abstract models of computation and the computational and decision problems associated with them
We consider the following verification problem: given a family of series–parallel graphs described by a context-free graph transformation system (GTS), and a concurrent automaton over series– parallel graphs, is some graph generated by the GTS accepted by the automaton? The general problem is undecidable already for multi-head automata over strings
We show that a bounded version, where the automata make a fixed number of reversals along the graph and use a fixed number of shared registers is decidable, even though there is no bound on the sizes of series–parallel graphs generated by the GTS
Summary
Automata theory studies abstract models of computation and the computational and decision problems associated with them. We study the emptiness problem: given a context-free GTS defining a language of series– parallel graphs, and a concurrent finite-state automaton, check if there is a graph in the language of the GTS accepted by the automaton This problem is, not surprisingly, undecidable: for example, we can encode linear bounded automata over strings. The work [21] studies the emptiness problem for concurrent automata with auxiliary storage and provides a generalization of the decidability results for a number of classes of such automata for which the emptiness problem can be reduced to emptiness of finite-state graph automata defined MSO definable graphs with bounded tree width The conference version of this paper appeared as [8]
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