A decidability theorem for a class of vector-addition systems (original) (raw)

The equality problem for vector addition systems is undecidable

Theoretical Computer Science, 1976

We demonstrate the usefulness of Petri nets for treating problems about vector addition systems by giving a simple exposition of Rabin's proof of the undecidability of the inclusion problem for v+ ctor addition system reachability sets, and then proceed to show that the inclusion problem can be reduced to the equality problem for reachability sets. * This relationship is also discussed in [ 11).

Reachability in Two-Dimensional Vector Addition Systems with States Is PSPACE-Complete

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, 2015

Determining the complexity of the reachability problem for vector addition systems with states (VASS) is a long-standing open problem in computer science. Long known to be decidable, the problem to this day lacks any complexity upper bound whatsoever. In this paper, reachability for two-dimensional VASS is shown PSPACE-complete. This improves on a previously known doubly exponential time bound established by Howell, Rosier, Huynh and Yen in 1986. The coverability and boundedness problems are also noted to be PSPACE-complete. In addition, some complexity results are given for the reachability problem in two-dimensional VASS and in integer VASS when numbers are encoded in unary.

The covering and boundedness problems for vector addition systems

Theoretical Computer Science, 1978

New decision proct; &,res for the covering and bcwdcdrxss pro Aems for cector addition systems are obtained. These procedures require at most space 2'" lop" t'or some constant c. The procedures nearly achieve recently established lower bounds on the amount of space inherently required to solve these problems, and so are much more efficienr than prt viously known non-primitive-recursive decision lxocedures.

Model Checking Vector Addition Systems with one zero-test

Logical Methods in Computer Science, 2012

We design a variation of the Karp-Miller algorithm to compute, in a forward manner, a finite representation of the cover (i.e., the downward closure of the reachability set) of a vector addition system with one zero-test. This algorithm yields decision procedures for several problems for these systems, open until now, such as place-boundedness or LTL model-checking. The proof techniques to handle the zero-test are based on two new notions of cover: the refined and the filtered cover. The refined cover is a hybrid between the reachability set and the classical cover. It inherits properties of the reachability set: equality of two refined covers is undecidable, even for usual Vector Addition Systems (with no zero-test), but the refined cover of a Vector Addition System is a recursive set. The second notion of cover, called the filtered cover, is the central tool of our algorithms. It inherits properties of the classical cover, and in particular, one can effectively compute a finite representation of this set, even for Vector Addition Systems with one zero-test.

Place-boundedness for vector addition systems with one zero-test

FSTTCS, 2010

Reachability and boundedness problems have been shown decidable for Vector Addition Systems with one zero-test. Surprisingly, place-boundedness remained open. We provide here a variation of the Karp-Miller algorithm to compute a basis of the downward closure of the reachability set which allows to decide place-boundedness. This forward algorithm is able to pass the zero-tests thanks to a finer cover, hybrid between the reachability and cover sets, reclaiming accuracy on one component. We show that this filtered cover is still recursive, but that equality of two such filtered covers, even for usual Vector Addition Systems (with no zero-test), is undecidable.

A new decidable problem, with applications

18th Annual Symposium on Foundations of Computer Science (sfcs 1977), 1977

STATEMENT OF THE THEOREM A number of decision problems that are unsolvable in general are solvable when restricted to systems with sufficiently simple "loop structure". Examples of such problems are the equivalence problems for flowchart schemata with nonintersecting loops and for the LOOP(l) programs of Meyer and Ritchie. We here present a theorem that gives a unifying view of the solvability of both of these problems, and also of a variety of other old and new solvable decision problems in automata theory, schematology, and logic.

The reachability problem for ground TRS and some extensions

Lecture Notes in Computer Science, 1989

The reachability problem for term rewriting systems (TRS) is the problem of deciding, for a given TRS S and two terms M and N, whether M can reduce to N by applying the rules of S. We show in this paper by some new methods based on algebraical tools of tree automata, the decidability of this problem for ground TRS's and, for every ground TRS S, we built a decision algorithm. In the order to obtain it, we compile the system S and the compiled algorithm works in a real time (as a fonction of the size of M and N). We establish too some new results for ground TRS modulo different sets of equations • modulo commutativity of an operator o, the reachability problem is shown decidable with technics of finite tree automata; modulo associativity, the problem is undecidable; modulo commutativity and associativity, it is decidable with complexity of reachability problem for vector addition systems.