40 research outputs found
Many bounded versions of undecidable problems are NP-hard
Several physically inspired problems have been proven undecidable; examples
are the spectral gap problem and the membership problem for quantum
correlations. Most of these results rely on reductions from a handful of
undecidable problems, such as the halting problem, the tiling problem, the Post
correspondence problem or the matrix mortality problem. All these problems have
a common property: they have an NP-hard bounded version. This work establishes
a relation between undecidable unbounded problems and their bounded NP-hard
versions. Specifically, we show that NP-hardness of a bounded version follows
easily from the reduction of the unbounded problems. This leads to new and
simpler proofs of the NP-hardness of bounded version of the Post correspondence
problem, the matrix mortality problem, the positivity of matrix product
operators, the reachability problem, the tiling problem, and the ground state
energy problem. This work sheds light on the intractability of problems in
theoretical physics and on the computational consequences of bounding a
parameter.Comment: 10 pages and 7 pages of appendices, 8 figures; v2,v3: minor change
Decidability of the Membership Problem for integer matrices
The main result of this paper is the decidability of the membership problem
for nonsingular integer matrices. Namely, we will construct the
first algorithm that for any nonsingular integer matrices
and decides whether belongs to the semigroup generated
by .
Our algorithm relies on a translation of the numerical problem on matrices
into combinatorial problems on words. It also makes use of some algebraical
properties of well-known subgroups of and various
new techniques and constructions that help to limit an infinite number of
possibilities by reducing them to the membership problem for regular languages
ASP(AC): Answer Set Programming with Algebraic Constraints
Weighted Logic is a powerful tool for the specification of calculations over
semirings that depend on qualitative information. Using a novel combination of
Weighted Logic and Here-and-There (HT) Logic, in which this dependence is based
on intuitionistic grounds, we introduce Answer Set Programming with Algebraic
Constraints (ASP(AC)), where rules may contain constraints that compare
semiring values to weighted formula evaluations. Such constraints provide
streamlined access to a manifold of constructs available in ASP, like
aggregates, choice constraints, and arithmetic operators. They extend some of
them and provide a generic framework for defining programs with algebraic
computation, which can be fruitfully used e.g. for provenance semantics of
datalog programs. While undecidable in general, expressive fragments of ASP(AC)
can be exploited for effective problem-solving in a rich framework. This work
is under consideration for acceptance in Theory and Practice of Logic
Programming.Comment: 32 pages, 16 pages are appendi
On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond
We consider the following variant of the Mortality Problem: given matrices , does there exist nonnegative integers such that the product is equal to the zero matrix? It is known that this problem is decidable when for matrices over algebraic numbers but becomes undecidable for sufficiently large and even for integral matrices. In this paper, we prove the first decidability results for . We show as one of our central results that for this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for and for matrices over algebraic numbers and for and for matrices over real algebraic numbers. Another consequence is that the set of triples for which the equation equals the zero matrix is equal to a finite union of direct products of semilinear sets. For we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations
On Reachability Problems for Low-Dimensional Matrix Semigroup
We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers
Analysis of instance hardness for the maximally diverse grouping problem and the iterated maxima search heuristic
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