72 research outputs found
Vector Ambiguity and Freeness Problems in SL (2, ℤ).
We study the vector ambiguity problem and the vector freeness problem in SL(2,Z). Given a finitely generated n×n matrix semigroup S and an n-dimensional vector x, the vector ambiguity problem is to decide whether for every target vector y=Mx, where M∈S, M is unique. We also consider the vector freeness problem which is to show that every matrix M which is transforming x to Mx has a unique factorization with respect to the generator of S. We show that both problems are NP-complete in SL(2,Z), which is the set of 2×2 integer matrices with determinant 1. Moreover, we generalize the vector ambiguity problem and extend to the finite and k-vector ambiguity problems where we consider the degree of vector ambiguity of matrix semigroups
Decidability of the Membership Problem for integer matrices
The main result of this paper is the decidability of the membership problem
for nonsingular integer matrices. Namely, we will construct the
first algorithm that for any nonsingular integer matrices
and decides whether belongs to the semigroup generated
by .
Our algorithm relies on a translation of the numerical problem on matrices
into combinatorial problems on words. It also makes use of some algebraical
properties of well-known subgroups of and various
new techniques and constructions that help to limit an infinite number of
possibilities by reducing them to the membership problem for regular languages
On the decidability of semigroup freeness
This paper deals with the decidability of semigroup freeness. More precisely,
the freeness problem over a semigroup S is defined as: given a finite subset X
of S, decide whether each element of S has at most one factorization over X. To
date, the decidabilities of two freeness problems have been closely examined.
In 1953, Sardinas and Patterson proposed a now famous algorithm for the
freeness problem over the free monoid. In 1991, Klarner, Birget and Satterfield
proved the undecidability of the freeness problem over three-by-three integer
matrices. Both results led to the publication of many subsequent papers. The
aim of the present paper is three-fold: (i) to present general results
concerning freeness problems, (ii) to study the decidability of freeness
problems over various particular semigroups (special attention is devoted to
multiplicative matrix semigroups), and (iii) to propose precise, challenging
open questions in order to promote the study of the topic.Comment: 46 pages. 1 table. To appear in RAIR
The symmetric Post Correspondence Problem, and errata for the freeness problem for matrix semigroups
We define the symmetric Post Correspondence Problem (PCP) and prove that it
is undecidable. As an application we show that the original proof of
undecidability of the freeness problem for 3-by-3 integer matrix semigroups
works for the symmetric PCP, but not for the PCP in general.Comment: 12 pages. Small corrections and clarifications were added in version
The Context-Freeness Problem Is coNP-Complete for Flat Counter Systems
International audienceBounded languages have recently proved to be an important class of languages for the analysis of Turing-powerful models. For instance, bounded context-free languages are used to under-approximate the behav-iors of recursive programs. Ginsburg and Spanier have shown in 1966 that a bounded language L ⊆ a * 1 · · · a * d is context-free if, and only if, its Parikh image is a stratifiable semilinear set. However, the question whether a semilinear set is stratifiable, hereafter called the stratifiability problem, was left open, and remains so. In this paper, we give a partial answer to this problem. We focus on semilinear sets that are given as finite systems of linear inequalities, and we show that stratifiability is coNP-complete in this case. Then, we apply our techniques to the context-freeness problem for flat counter systems, that asks whether the trace language of a counter system intersected with a bounded regular language is context-free. As main result of the paper, we show that this problem is coNP-complete
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