Finite LTL Synthesis is EXPTIME-complete

TL;DR

This paper proves that finite LTL synthesis, previously thought to be 2EXPTIME-complete, is actually EXPTIME-complete, aligning it with related planning problems and correcting earlier complexity claims.

Contribution

The paper establishes that finite LTL synthesis is EXPTIME-complete, contradicting prior reports of 2EXPTIME-completeness, and clarifies its relation to planning complexity.

Findings

Finite LTL synthesis is EXPTIME-complete.

Contradicts previous claims of 2EXPTIME-completeness.

Aligns finite LTL synthesis complexity with non-deterministic planning.

Abstract

LTL synthesis -- the construction of a function to satisfy a logical specification formulated in Linear Temporal Logic -- is a 2EXPTIME-complete problem with relevant applications in controller synthesis and a myriad of artificial intelligence applications. In this research note we consider De Giacomo and Vardi's variant of the synthesis problem for LTL formulas interpreted over finite rather than infinite traces. Rather surprisingly, given the existing claims on complexity, we establish that LTL synthesis is EXPTIME-complete for the finite interpretation, and not 2EXPTIME-complete as previously reported. Our result coincides nicely with the planning perspective where non-deterministic planning with full observability is EXPTIME-complete and partial observability increases the complexity to 2EXPTIME-complete; a recent related result for LTL synthesis shows that in the finite case with partial observability, the problem is 2EXPTIME-complete.