991 research outputs found
Real-Time Vector Automata
We study the computational power of real-time finite automata that have been
augmented with a vector of dimension k, and programmed to multiply this vector
at each step by an appropriately selected matrix. Only one entry
of the vector can be tested for equality to 1 at any time. Classes of languages
recognized by deterministic, nondeterministic, and "blind" versions of these
machines are studied and compared with each other, and the associated classes
for multicounter automata, automata with multiplication, and generalized finite
automata.Comment: 14 page
The separation problem for regular languages by piecewise testable languages
Separation is a classical problem in mathematics and computer science. It
asks whether, given two sets belonging to some class, it is possible to
separate them by another set of a smaller class. We present and discuss the
separation problem for regular languages. We then give a direct polynomial time
algorithm to check whether two given regular languages are separable by a
piecewise testable language, that is, whether a sentence can
witness that the languages are indeed disjoint. The proof is a reformulation
and a refinement of an algebraic argument already given by Almeida and the
second author
On Nonnegative Integer Matrices and Short Killing Words
Let be a natural number and a set of -matrices
over the nonnegative integers such that the joint spectral radius of
is at most one. We show that if the zero matrix is a product
of matrices in , then there are with . This result has applications in
automata theory and the theory of codes. Specifically, if
is a finite incomplete code, then there exists a word of
length polynomial in such that is not a factor of any
word in . This proves a weak version of Restivo's conjecture.Comment: This version is a journal submission based on a STACS'19 paper. It
extends the conference version as follows. (1) The main result has been
generalized to apply to monoids generated by finite sets whose joint spectral
radius is at most 1. (2) The use of Carpi's theorem is avoided to make the
paper more self-contained. (3) A more precise result is offered on Restivo's
conjecture for finite code
The Non-Archimedean Theory of Discrete Systems
In the paper, we study behavior of discrete dynamical systems (automata)
w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be
behavior of the system w.r.t. variety of word transformations performed by the
system: We call a system completely transitive if, given arbitrary pair
of finite words that have equal lengths, the system , while
evolution during (discrete) time, at a certain moment transforms into .
To every system , we put into a correspondence a family of continuous maps of a suitable non-Archimedean metric space
and show that the system is completely transitive if and only if the family
is ergodic w.r.t. the Haar measure; then we find
easy-to-verify conditions the system must satisfy to be completely transitive.
The theory can be applied to analyze behavior of straight-line computer
programs (in particular, pseudo-random number generators that are used in
cryptography and simulations) since basic CPU instructions (both numerical and
logical) can be considered as continuous maps of a (non-Archimedean) metric
space of 2-adic integers.Comment: The extended version of the talk given at MACIS-201
Finite transducers for divisibility monoids
Divisibility monoids are a natural lattice-theoretical generalization of
Mazurkiewicz trace monoids, namely monoids in which the distributivity of the
involved divisibility lattices is kept as an hypothesis, but the relations
between the generators are not supposed to necessarily be commutations. Here,
we show that every divisibility monoid admits an explicit finite transducer
which allows to compute normal forms in quadratic time. In addition, we prove
that every divisibility monoid is biautomatic.Comment: 20 page
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