65 research outputs found
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
A decidable policy language for history-based transaction monitoring
Online trading invariably involves dealings between strangers, so it is
important for one party to be able to judge objectively the trustworthiness of
the other. In such a setting, the decision to trust a user may sensibly be
based on that user's past behaviour. We introduce a specification language
based on linear temporal logic for expressing a policy for categorising the
behaviour patterns of a user depending on its transaction history. We also
present an algorithm for checking whether the transaction history obeys the
stated policy. To be useful in a real setting, such a language should allow one
to express realistic policies which may involve parameter quantification and
quantitative or statistical patterns. We introduce several extensions of linear
temporal logic to cater for such needs: a restricted form of universal and
existential quantification; arbitrary computable functions and relations in the
term language; and a "counting" quantifier for counting how many times a
formula holds in the past. We then show that model checking a transaction
history against a policy, which we call the history-based transaction
monitoring problem, is PSPACE-complete in the size of the policy formula and
the length of the history. The problem becomes decidable in polynomial time
when the policies are fixed. We also consider the problem of transaction
monitoring in the case where not all the parameters of actions are observable.
We formulate two such "partial observability" monitoring problems, and show
their decidability under certain restrictions
Integer Polynomial Optimization in Fixed Dimension
We classify, according to their computational complexity, integer
optimization problems whose constraints and objective functions are polynomials
with integer coefficients and the number of variables is fixed. For the
optimization of an integer polynomial over the lattice points of a convex
polytope, we show an algorithm to compute lower and upper bounds for the
optimal value. For polynomials that are non-negative over the polytope, these
sequences of bounds lead to a fully polynomial-time approximation scheme for
the optimization problem.Comment: In this revised version we include a stronger complexity bound on our
algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time
approximation scheme) to maximize a non-negative integer polynomial over the
lattice points of a polytop
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
Quantum hypercomputation based on the dynamical algebra su(1,1)
An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is
presented. The method that was used was to replace the Weyl-Heisenberg algebra
by other dynamical algebra of low dimension that admits infinite-dimensional
irreducible representations with naturally defined generalized coherent states.
We have selected the Lie algebra , due to that this algebra
posses the necessary characteristics for to realize the hypercomputation and
also due to that such algebra has been identified as the dynamical algebra
associated to many relatively simple quantum systems. In addition to an
algebraic adaptation of KHQA over the algebra , we
presented an adaptations of KHQA over some concrete physical referents: the
infinite square well, the infinite cylindrical well, the perturbed infinite
cylindrical well, the P{\"o}sch-Teller potentials, the Holstein-Primakoff
system, and the Laguerre oscillator. We conclude that it is possible to have
many physical systems within condensed matter and quantum optics on which it is
possible to consider an implementation of KHQA.Comment: 25 pages, 1 figure, conclusions rewritten, typing and language errors
corrected and latex format changed minor changes elsewhere and
Computing the Noncomputable
We explore in the framework of Quantum Computation the notion of
computability, which holds a central position in Mathematics and Theoretical
Computer Science. A quantum algorithm that exploits the quantum adiabatic
processes is considered for the Hilbert's tenth problem, which is equivalent to
the Turing halting problem and known to be mathematically noncomputable.
Generalised quantum algorithms are also considered for some other mathematical
noncomputables in the same and of different noncomputability classes. The key
element of all these algorithms is the measurability of both the values of
physical observables and of the quantum-mechanical probability distributions
for these values. It is argued that computability, and thus the limits of
Mathematics, ought to be determined not solely by Mathematics itself but also
by physical principles.Comment: Extensively revised and enlarged with: 2 new subsections, 4 new
figures, 1 new reference, and a short biography as requested by the journal
edito
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
Scalar Ambiguity and Freeness in Matrix Semigroups over Bounded Languages
There has been much research into freeness properties of
finitely generated matrix semigroups under various constraints, mainly
related to the dimensions of the generator matrices and the semiring over
which the matrices are defined. A recent paper has also investigated freeness
properties of matrices within a bounded language of matrices, which
are of the form M1M2 · · · Mk ⊆ F
n×n
for some semiring F [9]. Most freeness
problems have been shown to be undecidable starting from dimension
three, even for upper-triangular matrices over the natural numbers.
There are many open problems still remaining in dimension two.
We introduce a notion of freeness and ambiguity for scalar reachability
problems in matrix semigroups and bounded languages of matrices.
Scalar reachability concerns the set {ρ
TMτ |M ∈ S}, where ρ, τ ∈ F
n
are vectors and S is a finitely generated matrix semigroup. Ambiguity
and freeness problems are defined in terms of uniqueness of factorizations
leading to each scalar. We show various undecidability results
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