306 research outputs found
Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory
In this paper we characterize the congruence associated to the direct sum of
all irreducible representations of a finite semigroup over an arbitrary field,
generalizing results of Rhodes for the field of complex numbers. Applications
are given to obtain many new results, as well as easier proofs of several
results in the literature, involving: triangularizability of finite semigroups;
which semigroups have (split) basic semigroup algebras, two-sided semidirect
product decompositions of finite monoids; unambiguous products of rational
languages; products of rational languages with counter; and \v{C}ern\'y's
conjecture for an important class of automata
Matrix equations and Hilbert's tenth problem
We show a reduction of Hilbert's tenth problem to the solvability of
the matrix equation Xi1
1 Xi2
2 Xik
k = Z over non-commuting integral
matrices, where Z is the zero matrix, thus proving that the solvability
of the equation is undecidable. This is in contrast to the case whereby
the matrix semigroup is commutative in which the solvability of the
same equation was shown to be decidable in general.
The restricted problem where k = 2 for commutative matrices is
known as the \A-B-C Problem" and we show that this problem is
decidable even for a pair of non-commutative matrices over an algebraic
number field
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Given an integral matrix , the well-studied affine semigroup
\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be
stratified by the number of lattice points inside the parametric polyhedra
. Such families of parametric polyhedra appear in
many areas of combinatorics, convex geometry, algebra and number theory. The
key themes of this paper are: (1) A structure theory that characterizes
precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{
Sg}(A) such that has at least solutions. We
demonstrate that this set is finitely generated, it is a union of translated
copies of a semigroup which can be computed explicitly via Hilbert bases
computations. Related results can be derived for those right-hand-side vectors
for which has exactly solutions or fewer
than solutions. (2) A computational complexity theory. We show that, when
, are fixed natural numbers, one can compute in polynomial time an
encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function,
using a short sum of rational functions. As a consequence, one can identify all
right-hand-side vectors of bounded norm that have at least solutions. (3)
Applications and computation for the -Frobenius numbers. Using Generating
functions we prove that for fixed the -Frobenius number can be
computed in polynomial time. This generalizes a well-known result for by
R. Kannan. Using some adaptation of dynamic programming we show some practical
computations of -Frobenius numbers and their relatives
Polynomial Invariants for Affine Programs
We exhibit an algorithm to compute the strongest polynomial (or algebraic)
invariants that hold at each location of a given affine program (i.e., a
program having only non-deterministic (as opposed to conditional) branching and
all of whose assignments are given by affine expressions). Our main tool is an
algebraic result of independent interest: given a finite set of rational square
matrices of the same dimension, we show how to compute the Zariski closure of
the semigroup that they generate
Nonnegativity Problems for Matrix Semigroups
The matrix semigroup membership problem asks, given square matrices
of the same dimension, whether lies in the semigroup
generated by . It is classical that this problem is undecidable
in general but decidable in case commute. In this paper we
consider the problem of whether, given , the semigroup
generated by contains a non-negative matrix. We show that in
case commute, this problem is decidable subject to Schanuel's
Conjecture. We show also that the problem is undecidable if the commutativity
assumption is dropped. A key lemma in our decidability result is a procedure to
determine, given a matrix , whether the sequence of matrices is ultimately nonnegative. This answers a problem posed by S. Akshay
(arXiv:2205.09190). The latter result is in stark contrast to the notorious
fact that it is not known how to determine effectively whether for any specific
matrix index the sequence is ultimately nonnegative
(which is a formulation of the Ultimate Positivity Problem for linear
recurrence sequences)
Recent advances in algorithmic problems for semigroups
In this article we survey recent progress in the algorithmic theory of matrix
semigroups. The main objective in this area of study is to construct algorithms
that decide various properties of finitely generated subsemigroups of an
infinite group , often represented as a matrix group. Such problems might
not be decidable in general. In fact, they gave rise to some of the earliest
undecidability results in algorithmic theory. However, the situation changes
when the group satisfies additional constraints. In this survey, we give an
overview of the decidability and the complexity of several algorithmic problems
in the cases where is a low-dimensional matrix group, or a group with
additional structures such as commutativity, nilpotency and solvability.Comment: survey article for SIGLOG New
Vector Reachability Problem in
The decision problems on matrices were intensively studied for many decades
as matrix products play an essential role in the representation of various
computational processes. However, many computational problems for matrix
semigroups are inherently difficult to solve even for problems in low
dimensions and most matrix semigroup problems become undecidable in general
starting from dimension three or four.
This paper solves two open problems about the decidability of the vector
reachability problem over a finitely generated semigroup of matrices from
and the point to point reachability (over rational
numbers) for fractional linear transformations, where associated matrices are
from . The approach to solving reachability problems
is based on the characterization of reachability paths between points which is
followed by the translation of numerical problems on matrices into
computational and combinatorial problems on words and formal languages. We also
give a geometric interpretation of reachability paths and extend the
decidability results to matrix products represented by arbitrary labelled
directed graphs. Finally, we will use this technique to prove that a special
case of the scalar reachability problem is decidable
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
Semigroup intersection problems in the Heisenberg groups
We consider two algorithmic problems concerning sub-semigroups of Heisenberg
groups and, more generally, two-step nilpotent groups. The first problem is
Intersection Emptiness, which asks whether a finite number of given finitely
generated semigroups have empty intersection. This problem was first studied by
Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in
the Heisenberg groups over any algebraic
number field , as well as in direct products of Heisenberg groups.
We also extend our decidability result to arbitrary finitely generated 2-step
nilpotent groups.
The second problem is Orbit Intersection, which asks whether the orbits of
two matrices under multiplication by two semigroups intersect with each other.
This problem was first studied by Babai et al. (1996), who showed its
decidability within commutative matrix groups. We show that Orbit Intersection
is decidable within the Heisenberg group .Comment: 18 pages including appendix, 2 figure
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
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