Synchronizing Deterministic PushDown Automata Can Be Really Hard
Abstract
The question if a deterministic finite automaton admits a software reset in the form of a socalled synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic onecounter automata. This is also true for another classical mild extension of regularity, namely that of deterministic oneturn pushdown automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACEcomplete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata. There are several interpretations of what synchronizability should mean for deterministic pushdown automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.01381
 Bibcode:
 2020arXiv200501381F
 Keywords:

 Computer Science  Formal Languages and Automata Theory;
 Computer Science  Computational Complexity;
 68Q45;
 68Q17
 EPrint:
 arXiv admin note: text overlap with arXiv:2005.01374