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

Abstract

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.

Convolution Algebras: Relational Convolution, Generalised Modalities and Incidence Algebras

“…The generalised segment modalities from Section 10 can be adapted to an algebraic semantics for the interval temporal logic ITL [Mos12,CM16]. We ignore the next-step operator in our considerations, and our semantics is once again loose: it is not generated by a morphism from the ITL syntax.…”

“…Within this framework, different kinds of segments, with or without point segments and with different kinds of bounds or compositions, can be included in a uniform and modular way by setting up different kinds of partial semigroups or monoids. From that basis, a substantial part of the interval temporal logic ITL [Mos12] can be obtained, using a semigroup construction for stream functions to abstract from the dynamics of state spaces or program stores (Section 11), and by instantiating to a time domain of natural numbers. This semantics extends seamlessly to the duration calculus [ZH04] (Section 12) and one of its variants, the mean value calculus [PR98] (Section 13), by instantiating to a time domain of real numbers.…”

An Application of Temporal Projection to Interleaving Concurrency

“…For an in-depth presentation of PITL we refer the reader to [39]; see also [36], [31] and the ITL web pages [27]. The version of PITL used here has the syntax…”

“…The LTL operator U is also expressible using chop, and quantification. More details about PITL's expressiveness are found in [35], [38], [39].…”

Formal Semantics of English Sentences with Tense and Aspect

“…Many researchers have devised theories to explain tense and aspect in English (Bennett & Partee 1978;Davidson 2001;Parsons 1990;Prior 1967;Moszkowski 2012). All their ideas are couched in temporal logic, or I should say, in a time model (Konur 2008;Dowty 1979).…”