7 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
Using Sat solvers for synchronization issues in partial deterministic automata
We approach the task of computing a carefully synchronizing word of minimum
length for a given partial deterministic automaton, encoding the problem as an
instance of SAT and invoking a SAT solver. Our experimental results demonstrate
that this approach gives satisfactory results for automata with up to 100
states even if very modest computational resources are used.Comment: 15 pages, 3 figure
On the interplay between Babai and Černý’s conjectures
Motivated by the Babai conjecture and the Černý 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 n states in this class, we prove that the reset thresholds are upperbounded by 2n2 -6n + 5 and can attain the value (Formula presented). In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter (formula presented). © Springer International Publishing AG 2017
An improved upper bound for the length of preset distinguishing sequences of distinguished merging finite state machines
In an earlier work, we have studied a special class of Finite
State Machines (FSMs) called Distinguished Merging FSMs (DMFSMs)
and showed that one can construct a Preset Distinguishing Sequence
(PDS) for a DMFSM with n states, p input symbols, and r output symbols
in time O(n^4 + pn^2) of length no longer than O(n^3). In this work,
we improve the upper bound for the length of a PDS to (n-1)^2, and
present an algorithm to construct such a PDS for a DMFSM in time
O(n^4 + pn^2) or in time O(rn^3 + pn^2)
A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices
A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries (the k-RT). We prove that this value is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We then report numerical results comparing our upper bound on the k-RT with heuristic approximation methods