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

Using Temporal Logic to Analyse Temporal Logic: A Hierarchical Approach Based on Intervals

Temporal logic has been extensively utilized in academia and industry to formally specify and verify behavioural properties of numerous kinds of hardware and software. We present a novel way to apply temporal logic to the study of a version of itself, namely, propositional linear-time temporal logic (PTL). This involves a hierarchical framework for obtaining standard results for PTL, including a small model property, decision procedures and axiomatic completeness. A large number of the steps involved are expressed in a propositional version of Interval Temporal Logic (ITL) which is referred to as PITL. It is a natural generalization of PTL and includes operators for reasoning about periods of time and sequential composition. Versions of PTL with finite time and infinite time are both considered and one benefit of the framework is the ability to systematically reduce infinitetime reasoning to finite-time reasoning. The treatment of PTL with the operator until and past time naturally reduces to that for PTL without either one. The interval-oriented methodology differs from other analyses of PTL which typicall

Executing Temporal Logic Programs

Temporal logic is gaining recognition as an attractive and versatile formalism for rigorously specifying and reasoning about computer programs, digital circuits and message-passing systems. This book introduces Tempura, a programming language based on temporal logic. Tempura provides a way of directly executing suitable temporal logic specifications of digital circuits, parallel programs and other dynamic systems. Since every Tempura statement is also a temporal formula, the entire temporal logic formalism can be used as the assertion language and semantics. One result is that Tempura has the two seemingly contradictory properties of being a logic programming language and having imperative constructs such as assignment statements. The presentatio

From Petri Nets with Shared Variables to ITL

International audiencePetri nets and Interval Temporal Logic (ITL) are two formalisms for the specification and analysis of concurrent computing systems. Petri nets allow for a direct expression of causality aspects in system behaviour and in particular support system verification based on partial order reductions or invariant-based techniques. ITL' on the other hand' supports system verification by proving that the formula describing a system implies the formula describing a correctness requirement. It would therefore be desirable to establish a strong semantical link between these two models' thus allowing one to apply diverse analytical methods and techniques to a given system design. We have recently proposed such a semantical link between the propositional version of ITL (PITL) and Box Algebra (BA)' which is a compositional model of basic (low-level) Petri nets supporting handshake action synchronisation between concurrent processes. In this paper' we extend this result by considering a compositional model of (high-level) Petri nets where concurrent processes communicate through shared variables. The main result is a method for translating a design expressed using a high-level Petri net into a semantically equivalent ITL formula