153,666 research outputs found
One-variable word equations in linear time
In this paper we consider word equations with one variable (and arbitrary
many appearances of it). A recent technique of recompression, which is
applicable to general word equations, is shown to be suitable also in this
case. While in general case it is non-deterministic, it determinises in case of
one variable and the obtained running time is O(n + #_X log n), where #_X is
the number of appearances of the variable in the equation. This matches the
previously-best algorithm due to D\k{a}browski and Plandowski. Then, using a
couple of heuristics as well as more detailed time analysis the running time is
lowered to O(n) in RAM model. Unfortunately no new properties of solutions are
shown.Comment: submitted to a journal, general overhaul over the previous versio
Recompression: a simple and powerful technique for word equations
In this paper we present an application of a simple technique of local
recompression, previously developed by the author in the context of compressed
membership problems and compressed pattern matching, to word equations. The
technique is based on local modification of variables (replacing X by aX or Xa)
and iterative replacement of pairs of letters appearing in the equation by a
`fresh' letter, which can be seen as a bottom-up compression of the solution of
the given word equation, to be more specific, building an SLP (Straight-Line
Programme) for the solution of the word equation.
Using this technique we give a new, independent and self-contained proofs of
most of the known results for word equations. To be more specific, the
presented (nondeterministic) algorithm runs in O(n log n) space and in time
polynomial in log N, where N is the size of the length-minimal solution of the
word equation. The presented algorithm can be easily generalised to a generator
of all solutions of the given word equation (without increasing the space
usage). Furthermore, a further analysis of the algorithm yields a doubly
exponential upper bound on the size of the length-minimal solution. The
presented algorithm does not use exponential bound on the exponent of
periodicity. Conversely, the analysis of the algorithm yields an independent
proof of the exponential bound on exponent of periodicity.
We believe that the presented algorithm, its idea and analysis are far
simpler than all previously applied. Furthermore, thanks to it we can obtain a
unified and simple approach to most of known results for word equations.
As a small additional result we show that for O(1) variables (with arbitrary
many appearances in the equation) word equations can be solved in linear space,
i.e. they are context-sensitive.Comment: Submitted to a journal. Since previous version the proofs were
simplified, overall presentation improve
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
Finding All Solutions of Equations in Free Groups and Monoids with Involution
The aim of this paper is to present a PSPACE algorithm which yields a finite
graph of exponential size and which describes the set of all solutions of
equations in free groups as well as the set of all solutions of equations in
free monoids with involution in the presence of rational constraints. This
became possible due to the recently invented emph{recompression} technique of
the second author.
He successfully applied the recompression technique for pure word equations
without involution or rational constraints. In particular, his method could not
be used as a black box for free groups (even without rational constraints).
Actually, the presence of an involution (inverse elements) and rational
constraints complicates the situation and some additional analysis is
necessary. Still, the recompression technique is general enough to accommodate
both extensions. In the end, it simplifies proofs that solving word equations
is in PSPACE (Plandowski 1999) and the corresponding result for equations in
free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As
a byproduct we obtain a direct proof that it is decidable in PSPACE whether or
not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk
at CSR 2014 in Moscow, June 7 - 11, 201
(Un)Decidability Results for Word Equations with Length and Regular Expression Constraints
We prove several decidability and undecidability results for the
satisfiability and validity problems for languages that can express solutions
to word equations with length constraints. The atomic formulas over this
language are equality over string terms (word equations), linear inequality
over the length function (length constraints), and membership in regular sets.
These questions are important in logic, program analysis, and formal
verification. Variants of these questions have been studied for many decades by
mathematicians. More recently, practical satisfiability procedures (aka SMT
solvers) for these formulas have become increasingly important in the context
of security analysis for string-manipulating programs such as web applications.
We prove three main theorems. First, we give a new proof of undecidability
for the validity problem for the set of sentences written as a forall-exists
quantifier alternation applied to positive word equations. A corollary of this
undecidability result is that this set is undecidable even with sentences with
at most two occurrences of a string variable. Second, we consider Boolean
combinations of quantifier-free formulas constructed out of word equations and
length constraints. We show that if word equations can be converted to a solved
form, a form relevant in practice, then the satisfiability problem for Boolean
combinations of word equations and length constraints is decidable. Third, we
show that the satisfiability problem for quantifier-free formulas over word
equations in regular solved form, length constraints, and the membership
predicate over regular expressions is also decidable.Comment: Invited Paper at ADDCT Workshop 2013 (co-located with CADE 2013
Context unification is in PSPACE
Contexts are terms with one `hole', i.e. a place in which we can substitute
an argument. In context unification we are given an equation over terms with
variables representing contexts and ask about the satisfiability of this
equation. Context unification is a natural subvariant of second-order
unification, which is undecidable, and a generalization of word equations,
which are decidable, at the same time. It is the unique problem between those
two whose decidability is uncertain (for already almost two decades). In this
paper we show that the context unification is in PSPACE. The result holds under
a (usual) assumption that the first-order signature is finite.
This result is obtained by an extension of the recompression technique,
recently developed by the author and used in particular to obtain a new PSPACE
algorithm for satisfiability of word equations, to context unification. The
recompression is based on performing simple compression rules (replacing pairs
of neighbouring function symbols), which are (conceptually) applied on the
solution of the context equation and modifying the equation in a way so that
such compression steps can be in fact performed directly on the equation,
without the knowledge of the actual solution.Comment: 27 pages, submitted, small notation changes and small improvements
over the previous tex
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