Synchronizing Data Words for Register Automata

TL;DR

This paper investigates the problem of synchronizing register automata using data words, providing complexity results and bounds for both deterministic and nondeterministic cases, including decidability and computational hardness.

Contribution

It establishes new bounds and complexity classifications for synchronization problems in register automata, including PSPACE-completeness, Ackermann bounds, and NEXPTIME results.

Findings

2k+1 data sufficiency for deterministic RAs

PSPACE-complete for deterministic RAs

Ackermann complexity for nondeterministic RAs

Abstract

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.