64 research outputs found
All functions g:N-->N which have a single-fold Diophantine representation are dominated by a limit-computable function f:N\{0}-->N which is implemented in MuPAD and whose computability is an open problem
Let E_n={x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For any
integer n \geq 2214, we define a system T \subseteq E_n which has a unique
integer solution (a_1,...,a_n). We prove that the numbers a_1,...,a_n are
positive and max(a_1,...,a_n)>2^(2^n). For a positive integer n, let f(n)
denote the smallest non-negative integer b such that for each system S
\subseteq E_n with a unique solution in non-negative integers x_1,...,x_n, this
solution belongs to [0,b]^n. We prove that if a function g:N-->N has a
single-fold Diophantine representation, then f dominates g. We present a MuPAD
code which takes as input a positive integer n, performs an infinite loop,
returns a non-negative integer on each iteration, and returns f(n) on each
sufficiently high iteration.Comment: 17 pages, Theorem 3 added. arXiv admin note: substantial text overlap
with arXiv:1309.2605. text overlap with arXiv:1404.5975, arXiv:1310.536
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
The Identity Correspondence Problem and its Applications
In this paper we study several closely related fundamental problems for words
and matrices. First, we introduce the Identity Correspondence Problem (ICP):
whether a finite set of pairs of words (over a group alphabet) can generate an
identity pair by a sequence of concatenations. We prove that ICP is undecidable
by a reduction of Post's Correspondence Problem via several new encoding
techniques.
In the second part of the paper we use ICP to answer a long standing open
problem concerning matrix semigroups: "Is it decidable for a finitely generated
semigroup S of square integral matrices whether or not the identity matrix
belongs to S?". We show that the problem is undecidable starting from dimension
four even when the number of matrices in the generator is 48. From this fact,
we can immediately derive that the fundamental problem of whether a finite set
of matrices generates a group is also undecidable. We also answer several
question for matrices over different number fields. Apart from the application
to matrix problems, we believe that the Identity Correspondence Problem will
also be useful in identifying new areas of undecidable problems in abstract
algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the
published journal version of this article, see footnote 3 on page 1
Reachability problems for products of matrices in semirings
We consider the following matrix reachability problem: given square
matrices with entries in a semiring, is there a product of these matrices which
attains a prescribed matrix? We define similarly the vector (resp. scalar)
reachability problem, by requiring that the matrix product, acting by right
multiplication on a prescribed row vector, gives another prescribed row vector
(resp. when multiplied at left and right by prescribed row and column vectors,
gives a prescribed scalar). We show that over any semiring, scalar reachability
reduces to vector reachability which is equivalent to matrix reachability, and
that for any of these problems, the specialization to any is
equivalent to the specialization to . As an application of this result and
of a theorem of Krob, we show that when , the vector and matrix
reachability problems are undecidable over the max-plus semiring
. We also show that the matrix, vector, and scalar
reachability problems are decidable over semirings whose elements are
``positive'', like the tropical semiring .Comment: 21 page
Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers
We prove decidability of univariate real algebra extended with predicates for
rational and integer powers, i.e., and . Our decision procedure combines computation over real algebraic
cells with the rational root theorem and witness construction via algebraic
number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated
Deduction, 2015. Proceedings to be published by Springer-Verla
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
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