705 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
Groups and Semigroups Defined by Colorings of Synchronizing Automata
In this paper we combine the algebraic properties of Mealy machines
generating self-similar groups and the combinatorial properties of the
corresponding deterministic finite automata (DFA). In particular, we relate
bounded automata to finitely generated synchronizing automata and characterize
finite automata groups in terms of nilpotency of the corresponding DFA.
Moreover, we present a decidable sufficient condition to have free semigroups
in an automaton group. A series of examples and applications is widely
discussed, in particular we show a way to color the De Bruijn automata into
Mealy automata whose associated semigroups are free, and we present some
structural results related to the associated groups
Finitely generated ideal languages and synchronizing automata
We study representations of ideal languages by means of strongly connected
synchronizing automata. For every finitely generated ideal language L we
construct such an automaton with at most 2^n states, where n is the maximal
length of words in L. Our constructions are based on the De Bruijn graph.Comment: Submitted to WORDS 201
Computational Complexity of Synchronization under Regular Commutative Constraints
Here we study the computational complexity of the constrained synchronization
problem for the class of regular commutative constraint languages. Utilizing a
vector representation of regular commutative constraint languages, we give a
full classification of the computational complexity of the constraint
synchronization problem. Depending on the constraint language, our problem
becomes PSPACE-complete, NP-complete or polynomial time solvable. In addition,
we derive a polynomial time decision procedure for the complexity of the
constraint synchronization problem, given some constraint automaton accepting a
commutative language as input.Comment: Published in COCOON 2020 (The 26th International Computing and
Combinatorics Conference); 2nd version is update of the published version and
1st version; both contain a minor error, the assumption of maximality in the
NP-c and PSPACE-c results (propositions 5 & 6) is missing, and of
incomparability of the vectors in main theorem; fixed in this version. See
(new) discussion after main theore
On the interplay between Babai and Cerny's conjectures
Motivated by the Babai conjecture and the Cerny conjecture, we study the
reset thresholds of automata with the transition monoid equal to the full
monoid of transformations of the state set. For automata with states in
this class, we prove that the reset thresholds are upper-bounded by
and can attain the value . In addition, we study diameters
of the pair digraphs of permutation automata and construct -state
permutation automata with diameter .Comment: 21 pages version with full proof
Reset Complexity of Ideal Languages over a Binary Alphabet
We prove PSPACE-completeness of checking whether a given ideal language serves as the language of reset words for some automaton with at most four states over a binary alphabet. We compare the reset complexity and the state complexity for languages related to slowly synchronizing automata. © 2019 World Scientific Publishing Company.Russian Foundation for Basic Research, RFBR: 16-01-00795Ministry of Education and Science of the Russian Federation, Minobrnauka: 1.3253.2017Ural Federal University, UrFUThe author acknowledges anonymous reviewers for comments and suggestions. Also the author acknowledges support by the Russian Foundation for Basic Research, Grant No. 16-01-00795, the Ministry of Education and Science of the Russian Federation, Project No. 1.3253.2017, and the Competitiveness Enhancement Program of Ural Federal University
Principal ideal languages and synchronizing automata
We study ideal languages generated by a single word. We provide an algorithm
to construct a strongly connected synchronizing automaton for which such a
language serves as the language of synchronizing words. Also we present a
compact formula to calculate the syntactic complexity of this language.Comment: 15 pages, 9 figure
A Theory of Transformation Monoids: Combinatorics and Representation Theory
The aim of this paper is to develop a theory of finite transformation monoids
and in particular to study primitive transformation monoids. We introduce the
notion of orbitals and orbital digraphs for transformation monoids and prove a
monoid version of D. Higman's celebrated theorem characterizing primitivity in
terms of connectedness of orbital digraphs. A thorough study of the module (or
representation) associated to a transformation monoid is initiated. In
particular, we compute the projective cover of the transformation module over a
field of characteristic zero in the case of a transitive transformation or
partial transformation monoid. Applications of probability theory and Markov
chains to transformation monoids are also considered and an ergodic theorem is
proved in this context. In particular, we obtain a generalization of a lemma of
P. Neumann, from the theory of synchronizing groups, concerning the partition
associated to a transformation of minimal rank
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