Synchronizing Words and Monoid Factorization: A Parameterized Perspective

Synchronizing series-parallel deterministic finite automata with loops and related problems

“…This might yield an interesting (new?) parameterized complexity class between W [2] and WNL which we suggest to call W[sync] in [5]. Without going into details here, we mention that we could prove the further inclusions W[sync] ⊆ W[P] and W[sync] ⊆ A [2], which also suggests that W[sync] should be considered as a complexity class on its own.…”

“…This could also raise further interest in the parameterized complexity class WNL, defined by Guillemot in [25], as mostly formal language problems are known to be complete for this class. -In another paper [5], we studied other aspects of parameterized complexity in connection with DFA-SW, parameterized by a length upper bound k on the synchronizing word. For instance, we could prove FPT equivalence to Monoid Factorization (as defined in [7]: Given n mappings f i : X → X and a target mapping g : X → X, can g be represented by composing at most k mappings, i.e., is…”

Constrained Synchronization and Subset Synchronization Problems for Weakly Acyclic Automata

“…The problem Sync-Into-Subset is polynomial time solvable for weakly acyclic input automata. More generally 5 , given S, T Ď Q such that S contains all maximal states reachable from S, the existence of a word w P Σ such that δpS, wq Ď T could be decided in polynomial time.…”

“…In [4,5] the decision problem related to minimal synchronizing words was investigated for TTSPL automata. These are automata whose automaton graph, i.e., the multigraph resulting after forgetting about the labels, is a TTSPL graph, i.e, a two-terminal series-parallel graph with a start and sink node and where self-loops are allowed.…”

“…Many problems for series-parallel graphs are computationally easy [12], which partly motivated the aforementioned studies [4,5]. However, from a fixed parameter complexity perspective, for most parameters, synchronization problems remain hard on the corresponding automata class [4,5].…”