LTL with the Freeze Quantifier and Register Automata

TL;DR

This paper explores the complexity of linear temporal logic extended with a freeze quantifier and register automata, revealing decidability boundaries and computational complexity for various logical fragments and data word types.

Contribution

It provides a complete complexity classification for LTL with freeze quantifiers and registers, including decidability results and undecidability boundaries for different configurations.

Findings

Logic with future-time operators and 1 register is decidable but not primitive recursive over finite data words.

Adding past-time operators or more registers leads to undecidability.

Switching to infinite data words also causes undecidability.

Abstract

A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. To reason about data words, linear temporal logic is extended by the freeze quantifier, which stores the element at the current word position into a register, for equality comparisons deeper in the formula. By translations from the logic to alternating automata with registers and then to faulty counter automata whose counters may erroneously increase at any time, and from faulty and error-free counter automata to the logic, we obtain a complete complexity table for logical fragments defined by varying the set of temporal operators and the number of registers. In particular, the logic with future-time operators and 1 register is decidable but not primitive recursive over finite data words. Adding past-time operators or 1 more register, or switching to infinite data words, cause undecidability.