The Complexity of Generalized Satisfiability for Linear Temporal Logic

TL;DR

This paper systematically analyzes the computational complexity of satisfiability problems in Linear Temporal Logic (LTL) across all combinations of propositional and temporal operators, revealing a spectrum of complexities including P, NP-complete, and PSPACE-complete.

Contribution

It provides a comprehensive classification of LTL satisfiability complexity for all operator combinations using Post's lattice, extending previous results.

Findings

Most problems are PSPACE-complete, NP-complete, or in P.

Complete complexity classification for all but two classes of propositional functions.

Identification of operator sets leading to lower complexity classes.

Abstract

In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.