A Complete Axiom System for Propositional Interval Temporal Logic with Infinite Time

TL;DR

This paper presents a complete axiom system for propositional interval temporal logic with infinite time, enabling rigorous reasoning about nonterminating systems without complex automata-based methods.

Contribution

It introduces a novel completeness proof for propositional ITL with infinite time, reducing it to finite-time completeness and linear temporal logic, simplifying previous complex approaches.

Findings

Established a complete axiom system for infinite-time propositional ITL

Provided a semantic proof avoiding automata and tableau techniques

Demonstrated the naturalness of interval-based reasoning

Abstract

Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike completeness proofs of equally expressive logics with nonelementary computational complexity, our semantic approach does not use tableaux, subformula closures or explicit deductions involving encodings of omega automata and nontrivial techniques for complementing them. We believe that our result also provides evidence of the naturalness of interval-based reasoning.