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sWe revisit the earliest temporal projection operator sssΠ and another standard PITL projection operator, both suitable for reasoning about different time granularities, are demonstrated by showing the two operators to be interdefinable. We briefly look at other (mostly interval-based) temporal logics with similar forms of projection, as well as some related applications and industrial standards.We revisit the earliest temporal projection operator ΠΠ in discrete-time Propositional Interval Temporal Logic (PITL) and use it to formalise interleaving concurrency. The logical properties of ΠΠ as a normal modality and a way to eliminate it in both PITL and conventional point-based Linear-Time Temporal Logic (LTL), which can be viewed as a PITL subset, are examined, as are stutter-invariant formulas. Striking similarities between the expressiveness of ΠΠ and the standard LTL operator UU (‘until’) are briefly illustrated. We also formalise concurrent imperative programming constructs with and without ΠΠ, and relate the two approaches. Peterson’s mutual exclusion algorithm is used to illustrate reasoning with ΠΠ about a concrete programming example. Projection with fairness and non-fairness assumptions are both discussed. This all illustrates an approach to the analysis of such concurrent interleaving finite-state systems using temporal logic formulas with projection constructs to reason about correctness properties. Unlike conventional LTL formulas about concurrency which normally largely focus on global time, properties expressed in LTL combined with ΠΠ help to reveal and analyse important differing viewpoints involving global time and the local projected time seen by each individual process. Links between ΠΠ and another standard PITL projection operator, both suitable for reasoning about different time granularities, are demonstrated by showing the two operators to be interdefinable. We briefly look at other (mostly interval-based) temporal logics with similar forms of projection, as well as some related applications and industrial standards.ΠΠ in discrete-time Propositional Interval Temporal Logic (PITL) and use it to formalise interleaving concurrency. The logical properties of ΠΠ as a normal modality and a way to eliminate it in both PITL and conventional point-based Linear-Time Temporal Logic (LTL), which can be viewed as a PITL subset, are examined, as are stutter-invariant formulas. Striking similarities between the expressiveness of ΠΠ and the standard LTL operator UU (‘until’) are briefly illustrated. We also formalise concurrent imperative programming constructs with and without ΠΠ, and relate the two approaches. Peterson’s mutual exclusion algorithm is used to illustrate reasoning with ΠΠ about a concrete programming example. Projection with fairness and non-fairness assumptions are both discussed. This all illustrates an approach to the analysis of such concurrent interleaving finite-state systems using temporal logic formulas with projection constructs to reason about correctness properties. Unlike conventional LTL formulas about concurrency which normally largely focus on global time, properties expressed in LTL combined with ΠΠ help to reveal and analyse important differing viewpoints involving global time and the local projected time seen by each individual process. Links between ΠΠ and another standard PITL projection operator, both suitable for reasoning about different time granularities, are demonstrated by showing the two operators to be interdefinable. We briefly look at other (mostly interval-based) temporal logics with similar forms of projection, as well as some related applications and industrial standards.
Author(s): Moszkowski B, Guelev DP
Publication type: Article
Publication status: Published
Journal: Formal Aspects of Computing
Year: 2017
Volume: 29
Issue: 4
Pages: 705-750
Print publication date: 03/07/2017
Online publication date: 01/07/2017
Acceptance date: 22/11/2016
ISSN (print): 0934-5043
ISSN (electronic): 1433-299X
Publisher: Springer Nature
URL: https://doi.org/10.1007/s00165-017-0417-3
DOI: 10.1007/s00165-017-0417-3
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