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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
BPS counting for knots and combinatorics on words
We discuss relations between quantum BPS invariants defined in terms of a
product decomposition of certain series, and difference equations (quantum
A-polynomials) that annihilate such series. We construct combinatorial models
whose structure is encoded in the form of such difference equations, and whose
generating functions (Hilbert-Poincar\'e series) are solutions to those
equations and reproduce generating series that encode BPS invariants.
Furthermore, BPS invariants in question are expressed in terms of Lyndon words
in an appropriate language, thereby relating counting of BPS states to the
branch of mathematics referred to as combinatorics on words. We illustrate
these results in the framework of colored extremal knot polynomials: among
others we determine dual quantum extremal A-polynomials for various knots,
present associated combinatorial models, find corresponding BPS invariants
(extremal Labastida-Mari\~no-Ooguri-Vafa invariants) and discuss their
integrality.Comment: 41 pages, 1 figure, a supplementary Mathematica file attache
Compatible flat metrics
We solve the problem of description for nonsingular pairs of compatible flat
metrics in the general N-component case. The integrable nonlinear partial
differential equations describing all nonsingular pairs of compatible flat
metrics (or, in other words, nonsingular flat pencils of metrics) are found and
integrated. The integrating of these equations is based on reducing to a
special nonlinear differential reduction of the Lame equations and using the
Zakharov method of differential reductions in the dressing method (a version of
the inverse scattering method).Comment: 30 page
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