The Reachability Problem for Two-Dimensional Vector Addition Systems with States

We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This, in turn, enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.

Acceleration For Presburger Petri Nets

The reachability problem for Petri nets is a central problem of net theory. The problem is known to be decidable by inductive invariants definable in the Presburger arithmetic. When the reachability set is definable in the Presburger arithmetic, the existence of such an inductive invariant is immediate. However, in this case, the computation of a Presburger formula denoting the reachability set is an open problem. Recently this problem got closed by proving that if the reachability set of a Petri net is definable in the Presburger arithmetic, then the Petri net is flatable, i.e. its reachability set can be obtained by runs labeled by words in a bounded language. As a direct consequence, classical algorithms based on acceleration techniques effectively compute a formula in the Presburger arithmetic denoting the reachability set.

CHARACTERIZATIONS OF BOUNDED SEMILINEAR LANGUAGES BY ONE-WAY AND TWO-WAY DETERMINISTIC MACHINES

A bounded language [Formula: see text] (for some k ≥ 1 and not-necessarily distinct nonempty words x1, …, xk) is bounded semilinear if the set [Formula: see text] is semilinear. We give characterizations of bounded semilinear languages in terms of one-way and two-way deterministic counter machines.