Time and G\"odel: Fuzzy temporal reasoning in PSPACE

TL;DR

This paper introduces a fuzzy temporal logic based on G"odel--Dummett logic, demonstrating its decidability and PSPACE-completeness through novel semantics and finite model properties.

Contribution

It defines a new G"odel-based temporal logic with real-valued and bi-relational semantics, proving its decidability and PSPACE-completeness.

Findings

Logic is decidable despite lack of finite model property.

Falsifiable formulas are falsifiable on finite quasimodels.

The logic is proven to be PSPACE-complete.

Abstract

We investigate a non-classical version of linear temporal logic whose propositional fragment is G\"odel--Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics, a real-valued semantics and a bi-relational semantics, and show that these indeed define one and the same logic. Although this G\"odel temporal logic does not have any form of the finite model property for these two semantics, we show that every falsifiable formula is falsifiable on a finite quasimodel, which yields decidability of the logic. We then strengthen this result by showing that this G\"odel temporal logic is PSPACE-complete.