Determinisability of register and timed automata

TL;DR

This paper characterizes the decidability and complexity of the deterministic membership problem for register and timed automata, identifying precise conditions under which the problem is decidable or undecidable.

Contribution

It provides a complete decidability and complexity landscape for the deterministic membership problem in both register and timed automata, including optimal decidability results.

Findings

Decidable for one-register nondeterministic register automata with fixed output registers.

Decidable for one-clock nondeterministic timed automata with fixed output clocks.

Undecidable in all other cases involving multiple registers or clocks, or when using epsilon transitions.

Abstract

The deterministic membership problem for timed automata asks whether the timed language given by a nondeterministic timed automaton can be recognised by a deterministic timed automaton. An analogous problem can be stated in the setting of register automata. We draw the complete decidability/complexity landscape of the deterministic membership problem, in the setting of both register and timed automata. For register automata, we prove that the deterministic membership problem is decidable when the input automaton is a nondeterministic one-register automaton (possibly with epsilon transitions) and the number of registers of the output deterministic register automaton is fixed. This is optimal: We show that in all the other cases the problem is undecidable, i.e., when either (1) the input nondeterministic automaton has two registers or more (even without epsilon transitions), or (2) it uses guessing, or (3) the number of registers of the output deterministic automaton is not fixed. The landscape for timed automata follows a similar pattern. We show that the problem is decidable when the input automaton is a one-clock nondeterministic timed automaton without epsilon transitions and the number of clocks of the output deterministic timed automaton is fixed. Again, this is optimal: We show that the problem in all the other cases is undecidable, i.e., when either (1) the input nondeterministic timed automaton has two clocks or more, or (2) it uses epsilon transitions, or (3) the number of clocks of the output deterministic automaton is not fixed.