Ben Moszkowski | Search arXiv

arXiv2012

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

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.

cs.LO↗

arXiv2006

A Hierarchical Analysis of Propositional Temporal Logic Based on Intervals

We present a hierarchical framework for analysing propositional linear-time temporal logic (PTL) to obtain standard results such as a small model property, decision procedures and axiomatic completeness. Both finite time and infinite time are considered and one consequent benefit of the framework is the ability to systematically reduce infinite-time reasoning to finite-time reasoning. The treatment of PTL with both the operator Until and past time naturally reduces to that for PTL without either one. Our method utilises a low-level normal form for PTL called a "transition configuration". In addition, we employ reasoning about intervals of time. Besides being hierarchical and interval-based, the approach differs from other analyses of PTL typically based on sets of formulas and sequences of such sets. Instead we describe models using time intervals represented as finite and infinite sequences of states. The analysis relates larger intervals with smaller ones. Steps involved are expressed in Propositional Interval Temporal Logic (PITL) which is better suited than PTL for sequentially combining and decomposing formulas. Consequently, we can articulate issues in PTL model construction of equal relevance in more conventional analyses but normally only considered at the metalevel. We also describe a decision procedure based on Binary Decision Diagrams.

cs.LO↗