253,345 research outputs found
Integer Vector Addition Systems with States
This paper studies reachability, coverability and inclusion problems for
Integer Vector Addition Systems with States (ZVASS) and extensions and
restrictions thereof. A ZVASS comprises a finite-state controller with a finite
number of counters ranging over the integers. Although it is folklore that
reachability in ZVASS is NP-complete, it turns out that despite their
naturalness, from a complexity point of view this class has received little
attention in the literature. We fill this gap by providing an in-depth analysis
of the computational complexity of the aforementioned decision problems. Most
interestingly, it turns out that while the addition of reset operations to
ordinary VASS leads to undecidability and Ackermann-hardness of reachability
and coverability, respectively, they can be added to ZVASS while retaining
NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure
Affine Extensions of Integer Vector Addition Systems with States
We study the reachability problem for affine -VASS, which are
integer vector addition systems with states in which transitions perform affine
transformations on the counters. This problem is easily seen to be undecidable
in general, and we therefore restrict ourselves to affine -VASS
with the finite-monoid property (afmp--VASS). The latter have the
property that the monoid generated by the matrices appearing in their affine
transformations is finite. The class of afmp--VASS encompasses
classical operations of counter machines such as resets, permutations,
transfers and copies. We show that reachability in an afmp--VASS
reduces to reachability in a -VASS whose control-states grow
linearly in the size of the matrix monoid. Our construction shows that
reachability relations of afmp--VASS are semilinear, and in
particular enables us to show that reachability in -VASS with
transfers and -VASS with copies is PSPACE-complete. We then focus
on the reachability problem for affine -VASS with monogenic
monoids: (possibly infinite) matrix monoids generated by a single matrix. We
show that, in a particular case, the reachability problem is decidable for this
class, disproving a conjecture about affine -VASS with infinite
matrix monoids we raised in a preliminary version of this paper. We complement
this result by presenting an affine -VASS with monogenic matrix
monoid and undecidable reachability relation
Semilinear Sets, Register Machines, and Integer Vector Addition (P) Systems
In this paper we consider P systems working with multisets with integer
multiplicities. We focus on a model in which rule applicability is not in
uenced by the
contents of the membrane. We show that this variant is closely related to blind register
machines and integer vector addition systems. Furthermore, we describe the computational
power of these models in terms of linear and semilinear sets of integer vectors
On the Size of Finite Rational Matrix Semigroups
Let be a positive integer and a set of rational -matrices such that generates a finite multiplicative semigroup.
We show that any matrix in the semigroup is a product of matrices in whose length is at most ,
where is the maximum order of finite groups over rational -matrices. This result implies algorithms with an elementary running time for
deciding finiteness of weighted automata over the rationals and for deciding
reachability in affine integer vector addition systems with states with the
finite monoid property
Reachability in Two-Dimensional Vector Addition Systems with States is PSPACE-complete
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.Comment: 27 pages, 8 figure
Scope-Bounded Reachability in Valence Systems
Multi-pushdown systems are a standard model for concurrent recursive programs, but they have an undecidable reachability problem. Therefore, there have been several proposals to underapproximate their sets of runs so that reachability in this underapproximation becomes decidable. One such underapproximation that covers a relatively high portion of runs is scope boundedness. In such a run, after each push to stack i, the corresponding pop operation must come within a bounded number of visits to stack i.
In this work, we generalize this approach to a large class of infinite-state systems. For this, we consider the model of valence systems, which consist of a finite-state control and an infinite-state storage mechanism that is specified by a finite undirected graph. This framework captures pushdowns, vector addition systems, integer vector addition systems, and combinations thereof. For this framework, we propose a notion of scope boundedness that coincides with the classical notion when the storage mechanism happens to be a multi-pushdown.
We show that with this notion, reachability can be decided in PSPACE for every storage mechanism in the framework. Moreover, we describe the full complexity landscape of this problem across all storage mechanisms, both in the case of (i) the scope bound being given as input and (ii) for fixed scope bounds. Finally, we provide an almost complete description of the complexity landscape if even a description of the storage mechanism is part of the input
Simulation Problems Over One-Counter Nets
One-counter nets (OCN) are finite automata equipped with a counter that can
store non-negative integer values, and that cannot be tested for zero.
Equivalently, these are exactly 1-dimensional vector addition systems with
states. We show that both strong and weak simulation preorder on OCN are
PSPACE-complete
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