2,038 research outputs found
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
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. © IFIP International Federation for Information Processing 2017
Complexity of checking whether two automata are synchronized by the same language
A deterministic finite automaton is said to be synchronizing if it has a
reset word, i.e. a word that brings all states of the automaton to a particular
one. We prove that it is a PSPACE-complete problem to check whether the
language of reset words for a given automaton coincides with the language of
reset words for some particular automaton.Comment: 12 pages, 4 figure
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
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
Ideal regular languages and strongly connected synchronizing automata
We introduce the notion of a reset left regular decomposition of an ideal regular language, and we prove that the category formed by these decompositions with the adequate set of morphisms is equivalent to the category of strongly connected synchronizing automata. We show that every ideal regular language has at least one reset left regular decomposition. As a consequence, every ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton. Furthermore, this one-to-one correspondence allows us to introduce the notion of reset decomposition complexity of an ideal from which follows a reformulation of Černý's conjecture in purely language theoretic terms. Finally, we present and characterize a subclass of ideal regular languages for which a better upper bound for the reset decomposition complexity holds with respect to the general case
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
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