1,238 research outputs found
On Affine Reachability Problems
We analyze affine reachability problems in dimensions 1 and 2. We show that the reachability problem for 1-register machines over the integers with affine updates is PSPACE-hard, hence PSPACE-complete, strengthening a result by Finkel et al. that required polynomial updates. Building on recent results on two-dimensional integer matrices, we prove NP-completeness of the mortality problem for 2-dimensional integer matrices with determinants +1 and 0. Motivated by tight connections with 1-dimensional affine reachability problems without control states, we also study the complexity of a number of reachability problems in finitely generated semigroups of 2-dimensional upper-triangular integer matrices
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
Effective dimension of finite semigroups
In this paper we discuss various aspects of the problem of determining the
minimal dimension of an injective linear representation of a finite semigroup
over a field. We outline some general techniques and results, and apply them to
numerous examples.Comment: To appear in J. Pure Appl. Al
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