Exact synchronization in partial deterministic automata

Abstract

We approach the task of computing a carefully synchronizing word of optimum length for a given partial deterministic automaton, encoding the problem as an instance of SAT and invoking a SAT solver. Our experiments demonstrate that this approach gives satisfactory results for automata with up to 100 states even if very modest computational resources are used. We compare our results with the ones obtained by the first author for exact synchronization, which is another version of synchronization studied in the literature, and draw some theoretical conclusions.

Careful synchronization of partial deterministic finite automata

“…For studying synchronization of CFAs, the SAT-solver approach was initiated by Skvortsov and Tipikin [39] and Günic ¸en, Erdem, and Yenigün [15]. The present authors extended the approach to careful synchronization of PFAs in the conference paper [38], while the paper [36] by the first author dealt with exact synchronization. We mention also our earlier articles [35,37], where the SAT-solver approach was applied for a study of synchronizing nondeterministic automata.…”

“…Remark 2. In [36], the first author has constructed a SAT-encoding for the problem ESW (the existence of a exactly synchronizing word of a given length), which is defined analogously to CSW. We will not reproduce this encoding here; it uses the same set of variables as the above encoding for CSW but the set of clauses is essentially different.…”

“…, x ℓ } and the words of length ℓ over any fixed 2-letter alphabet. In more formal terms, we can modify the above encoding of CSW and the corresponding encoding of ESW from [36], substituting x t for x 1,t and ¬x t for x 2,t for all t = 1, 2, . .…”

Computational complexity of problems for deterministic presentations of sofic shifts

“…A deterministic presentation can be seen as a DFA with a partially defined transition function (a function whose domain is a subset of Q × Σ); call a DFA with a partial transition function a partial DFA. There are multiple ways to generalize the notion of a synchronizing word to partial DFAs, for example, a carefully synchronizing word [24] and a exact synchronizing word [23,25]. The former is a word whose transition is defined at all states and sends all states to a single state; the latter is a word whose transition is defined at least one state and sends every state (where it is defined) to the same state.…”

Synchronizing Strongly Connected Partial DFAs

“…In particular, the observer knows the state of an ε-machine precisely if and only if a reset word for the underlying partial DFA was applied. Some experimental results on finding shortest reset words for partial DFAs were recently presented in [31].…”