458 research outputs found
Deciding the finiteness of the number of simple permutations contained in a wreath-closed class is polynomial
We present an algorithm running in time O(n ln n) which decides if a
wreath-closed permutation class Av(B) given by its finite basis B contains a
finite number of simple permutations. The method we use is based on an article
of Brignall, Ruskuc and Vatter which presents a decision procedure (of high
complexity) for solving this question, without the assumption that Av(B) is
wreath-closed. Using combinatorial, algorithmic and language theoretic
arguments together with one of our previous results on pin-permutations, we are
able to transform the problem into a co-finiteness problem in a complete
deterministic automaton
Combinatorial specification of permutation classes
This article presents a methodology that automatically derives a
combinatorial specification for the permutation class C = Av(B), given its
basis B of excluded patterns and the set of simple permutations in C, when
these sets are both finite. This is achieved considering both pattern avoidance
and pattern containment constraints in permutations.The obtained specification
yields a system of equations satisfied by the generating function of C, this
system being always positiveand algebraic. It also yields a uniform random
sampler of permutations in C. The method presentedis fully algorithmic
Deciding the finiteness of simple permutations contained in a wreath-closed class is polynomial
International audienceWe present an algorithm running in time O(n log n) which decides if a wreath-closed permutation class Av(B) given by its finite basis B contains a finite number of simple permutations
The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable
We prove that a semigroup generated by a reversible two-state Mealy automaton
is either finite or free of rank 2. This fact leads to the decidability of
finiteness for groups generated by two-state or two-letter
invertible-reversible Mealy automata and to the decidability of freeness for
semigroups generated by two-state invertible-reversible Mealy automata
Algebraic hierarchical decomposition of finite state automata : a computational approach
The theory of algebraic hierarchical decomposition of finite state automata
is an important and well developed branch of theoretical computer science
(Krohn-Rhodes Theory). Beyond this it gives a general model for some
important aspects of our cognitive capabilities and also provides possible
means for constructing artificial cognitive systems: a Krohn-Rhodes decomposition
may serve as a formal model of understanding since we comprehend
the world around us in terms of hierarchical representations. In order to
investigate formal models of understanding using this approach, we need
efficient tools but despite the significance of the theory there has been no
computational implementation until this work.
Here the main aim was to open up the vast space of these decompositions
by developing a computational toolkit and to make the initial steps of the
exploration. Two different decomposition methods were implemented: the
VuT and the holonomy decomposition. Since the holonomy method, unlike
the VUT method, gives decompositions of reasonable lengths, it was chosen
for a more detailed study.
In studying the holonomy decomposition our main focus is to develop
techniques which enable us to calculate the decompositions efficiently, since
eventually we would like to apply the decompositions for real-world problems.
As the most crucial part is finding the the group components we
present several different ways for solving this problem. Then we investigate
actual decompositions generated by the holonomy method: automata with
some spatial structure illustrating the core structure of the holonomy decomposition,
cases for showing interesting properties of the decomposition
(length of the decomposition, number of states of a component), and the
decomposition of finite residue class rings of integers modulo n.
Finally we analyse the applicability of the holonomy decompositions as
formal theories of understanding, and delineate the directions for further
research
Computational Group Theory
This sixth workshop on Computational Group Theory proved that its main themes “finitely presented groups”, “-groups”, “matrix groups” and “representations of groups” are lively and active fields of research. The talks also presented applications to number theory, invariant theory, topology and coding theory
A decidable subclass of finitary programs
Answer set programming - the most popular problem solving paradigm based on
logic programs - has been recently extended to support uninterpreted function
symbols. All of these approaches have some limitation. In this paper we propose
a class of programs called FP2 that enjoys a different trade-off between
expressiveness and complexity. FP2 programs enjoy the following unique
combination of properties: (i) the ability of expressing predicates with
infinite extensions; (ii) full support for predicates with arbitrary arity;
(iii) decidability of FP2 membership checking; (iv) decidability of skeptical
and credulous stable model reasoning for call-safe queries. Odd cycles are
supported by composing FP2 programs with argument restricted programs
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