990 research outputs found
Synchronizing weighted automata
We introduce two generalizations of synchronizability to automata with
transitions weighted in an arbitrary semiring K=(K,+,*,0,1). (or equivalently,
to finite sets of matrices in K^nxn.) Let us call a matrix A
location-synchronizing if there exists a column in A consisting of nonzero
entries such that all the other columns of A are filled by zeros. If
additionally all the entries of this designated column are the same, we call A
synchronizing. Note that these notions coincide for stochastic matrices and
also in the Boolean semiring. A set M of matrices in K^nxn is called
(location-)synchronizing if M generates a matrix subsemigroup containing a
(location-)synchronizing matrix. The K-(location-)synchronizability problem is
the following: given a finite set M of nxn matrices with entries in K, is it
(location-)synchronizing?
Both problems are PSPACE-hard for any nontrivial semiring. We give sufficient
conditions for the semiring K when the problems are PSPACE-complete and show
several undecidability results as well, e.g. synchronizability is undecidable
if 1 has infinite order in (K,+,0) or when the free semigroup on two generators
can be embedded into (K,*,1).Comment: In Proceedings AFL 2014, arXiv:1405.527
Synchronizing Words for Weighted and Timed Automata
The problem of synchronizing automata is concerned with the existence of a word that sends all states of the automaton to one and the same state. This problem has classically been studied for complete deterministic finite automata, with the existence problem being NLOGSPACE-complete.
In this paper we consider synchronizing-word problems for weighted and timed automata. We consider the synchronization problem in several variants and combinations of these, including deterministic and non-deterministic timed and weighted automata, synchronization to unique location with possibly different clock valuations or accumulated weights, as well as synchronization with a safety condition forbidding the automaton to visit states outside a safety-set during synchronization (e.g. energy constraints). For deterministic weighted automata, the synchronization problem is proven PSPACE-complete under energy constraints, and in 3-EXPSPACE under general safety constraints. For timed automata the synchronization problems are shown to be PSPACE-complete in the deterministic case, and undecidable in the non-deterministic case
Reachability of Consensus and Synchronizing Automata
We consider the problem of determining the existence of a sequence of
matrices driving a discrete-time consensus system to consensus. We transform
this problem into one of the existence of a product of the transition
(stochastic) matrices that has a positive column. We then generalize some
results from automata theory to sets of stochastic matrices. We obtain as a
main result a polynomial-time algorithm to decide the existence of a sequence
of matrices achieving consensus.Comment: Update after revie
Synchronizing Data Words for Register Automata
Register automata (RAs) are finite automata extended with a finite set of
registers to store and compare data from an infinite domain. We study the
concept of synchronizing data words in RAs: does there exist a data word that
sends all states of the RA to a single state?
For deterministic RAs with k registers (k-DRAs), we prove that inputting data
words with 2k+1 distinct data from the infinite data domain is sufficient to
synchronize. We show that the synchronization problem for DRAs is in general
PSPACE-complete, and it is NLOGSPACE-complete for 1-DRAs. For nondeterministic
RAs (NRAs), we show that Ackermann(n) distinct data (where n is the size of the
RA) might be necessary to synchronize. The synchronization problem for NRAs is
in general undecidable, however, we establish Ackermann-completeness of the
problem for 1-NRAs.
Another main result is the NEXPTIME-completeness of the length-bounded
synchronization problem for NRAs, where a bound on the length of the
synchronizing data word, written in binary, is given. A variant of this last
construction allows to prove that the length-bounded universality problem for
NRAs is co-NEXPTIME-complete
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
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