8,514 research outputs found
The Hardness of Solving Simple Word Equations
We investigate the class of regular-ordered word equations. In such equations, each variable occurs at most once in each side and the order of the variables occurring in both left and right hand sides is preserved (the variables can be, however, separated by potentially distinct constant factors). Surprisingly, we obtain that solving such simple equations, even when the sides contain exactly the same variables, is NP-hard. By considerations regarding the combinatorial structure of the minimal solutions of the more general quadratic equations we obtain that the satisfiability problem for regular-ordered equations is in NP. The complexity of solving such word equations under regular constraints is also settled. Finally, we show that a related class of simple word equations, that generalises one-variable equations, is in P
Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility
Word equations are a crucial element in the theoretical foundation of
constraint solving over strings, which have received a lot of attention in
recent years. A word equation relates two words over string variables and
constants. Its solution amounts to a function mapping variables to constant
strings that equate the left and right hand sides of the equation. While the
problem of solving word equations is decidable, the decidability of the problem
of solving a word equation with a length constraint (i.e., a constraint
relating the lengths of words in the word equation) has remained a
long-standing open problem. In this paper, we focus on the subclass of
quadratic word equations, i.e., in which each variable occurs at most twice. We
first show that the length abstractions of solutions to quadratic word
equations are in general not Presburger-definable. We then describe a class of
counter systems with Presburger transition relations which capture the length
abstraction of a quadratic word equation with regular constraints. We provide
an encoding of the effect of a simple loop of the counter systems in the theory
of existential Presburger Arithmetic with divisibility (PAD). Since PAD is
decidable, we get a decision procedure for quadratic words equations with
length constraints for which the associated counter system is \emph{flat}
(i.e., all nodes belong to at most one cycle). We show a decidability result
(in fact, also an NP algorithm with a PAD oracle) for a recently proposed
NP-complete fragment of word equations called regular-oriented word equations,
together with length constraints. Decidability holds when the constraints are
additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page
Equations over free inverse monoids with idempotent variables
We introduce the notion of idempotent variables for studying equations in
inverse monoids.
It is proved that it is decidable in singly exponential time (DEXPTIME)
whether a system of equations in idempotent variables over a free inverse
monoid has a solution. The result is proved by a direct reduction to solve
language equations with one-sided concatenation and a known complexity result
by Baader and Narendran: Unification of concept terms in description logics,
2001. We also show that the problem becomes DEXPTIME hard , as soon as the
quotient group of the free inverse monoid has rank at least two.
Decidability for systems of typed equations over a free inverse monoid with
one irreducible variable and at least one unbalanced equation is proved with
the same complexity for the upper bound.
Our results improve known complexity bounds by Deis, Meakin, and Senizergues:
Equations in free inverse monoids, 2007.
Our results also apply to larger families of equations where no decidability
has been previously known.Comment: 28 pages. The conference version of this paper appeared in the
proceedings of 10th International Computer Science Symposium in Russia, CSR
2015, Listvyanka, Russia, July 13-17, 2015. Springer LNCS 9139, pp. 173-188
(2015
Complexity transitions in global algorithms for sparse linear systems over finite fields
We study the computational complexity of a very basic problem, namely that of
finding solutions to a very large set of random linear equations in a finite
Galois Field modulo q. Using tools from statistical mechanics we are able to
identify phase transitions in the structure of the solution space and to
connect them to changes in performance of a global algorithm, namely Gaussian
elimination. Crossing phase boundaries produces a dramatic increase in memory
and CPU requirements necessary to the algorithms. In turn, this causes the
saturation of the upper bounds for the running time. We illustrate the results
on the specific problem of integer factorization, which is of central interest
for deciphering messages encrypted with the RSA cryptosystem.Comment: 23 pages, 8 figure
Non-Abelian Analogs of Lattice Rounding
Lattice rounding in Euclidean space can be viewed as finding the nearest
point in the orbit of an action by a discrete group, relative to the norm
inherited from the ambient space. Using this point of view, we initiate the
study of non-abelian analogs of lattice rounding involving matrix groups. In
one direction, we give an algorithm for solving a normed word problem when the
inputs are random products over a basis set, and give theoretical justification
for its success. In another direction, we prove a general inapproximability
result which essentially rules out strong approximation algorithms (i.e., whose
approximation factors depend only on dimension) analogous to LLL in the general
case.Comment: 30 page
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
- …