The expressive power of temporal and first-order metric logics

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Abstract

The First-Order Monadic Logic of Order ((FO[<])) is a prominent logic for the specification of properties of systems evolving in time. The celebrated result of Kamp [14] states that a temporal logic with just two modalities Until and Since has the same expressive power as (FO[<]) over the standard discrete time of naturals and continuous time of reals. An influential consequence of Kamp’s theorem is that this temporal logic has emerged as the canonical Linear Time Temporal Logic (LTL). Neither LTL nor (FO[<]) can express over the reals properties like P holds exactly after one unit of time. Such local metric properties are easily expressible in FO[<,+1] - the extension of (FO[<]) by +1 function. Hirshfeld and Rabinovich [10] proved that no temporal logic with a finite set of modalities has the same expressive power as FO[<,+1]. FO}[<,+1] lacks expressive power to specify a natural global metric property “the current moment is an integer.” Surprisingly, we show that the extension of FO[<,+1] by a monadic predicate “x is an integer” is equivalent to a temporal logic with a finite set of modalities.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer
Pages226-246
Number of pages21
DOIs
StatePublished - 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12180 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

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