93 research outputs found
Synchronization Problems in Automata without Non-trivial Cycles
We study the computational complexity of various problems related to
synchronization of weakly acyclic automata, a subclass of widely studied
aperiodic automata. We provide upper and lower bounds on the length of a
shortest word synchronizing a weakly acyclic automaton or, more generally, a
subset of its states, and show that the problem of approximating this length is
hard. We investigate the complexity of finding a synchronizing set of states of
maximum size. We also show inapproximability of the problem of computing the
rank of a subset of states in a binary weakly acyclic automaton and prove that
several problems related to recognizing a synchronizing subset of states in
such automata are NP-complete.Comment: Extended and corrected version, including arXiv:1608.00889.
Conference version was published at CIAA 2017, LNCS vol. 10329, pages
188-200, 201
On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata
Cerny's conjecture is a longstanding open problem in automata theory. We
study two different concepts, which allow to approach it from a new angle. The
first one is the triple rendezvous time, i.e., the length of the shortest word
mapping three states onto a single one. The second one is the synchronizing
probability function of an automaton, a recently introduced tool which
reinterprets the synchronizing phenomenon as a two-player game, and allows to
obtain optimal strategies through a Linear Program.
Our contribution is twofold. First, by coupling two different novel
approaches based on the synchronizing probability function and properties of
linear programming, we obtain a new upper bound on the triple rendezvous time.
Second, by exhibiting a family of counterexamples, we disprove a conjecture on
the growth of the synchronizing probability function. We then suggest natural
follow-ups towards Cernys conjecture.Comment: A preliminary version of the results has been presented at the
conference LATA 2015. The current ArXiv version includes the most recent
improvement on the triple rendezvous time upper bound as well as formal
proofs of all the result
Strongly transitive automata and the Cerny conjecture
The synchronization problem is investigated for a new class of deterministic automata called strongly transitive. An extension to unambiguous automata is also considered
Complexity of problems concerning reset words for some partial cases of automata
A word w is called a reset word for a deterministic finite automaton A if it maps all states of A to one state. A word w is called a compressing to M states for a deterministic finite automaton A if it maps all states of A to at most M states. We consider several subclasses of automata: aperiodic, D-trivial, monotonic, partially monotonic automata and automata with a zero state. For these subclasses we study the computational complexity of the following problems. Does there exist a reset word for a given automaton? Does there exist a reset word of given length for a given automaton? What is the length of the shortest reset word for a given automaton? Moreover, we consider complexity of the same problems for compressing words
Synchronizing automata with a letter of deficiency 2
AbstractWe present two infinite series of synchronizing automata with a letter of deficiency 2 whose shortest reset words are longer than those for synchronizing automata obtained by a straightforward modification of Černý’s construction
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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