Expressive Completeness for Metric Temporal Logic

Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. Hirshfeld and Rabinovich have shown that over the reals, firstorder logic with binary order relation <; and unary function...

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Published in:2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 349 - 357
Main Authors: Hunter, Paul, Ouaknine, Joel, Worrell, James
Format: Conference Proceeding
Language:English
Published: IEEE 01.06.2013
Subjects:
Computer science
Expressive Completeness
First-Order Logic
Linear Temporal Logic
Metric Temporal Logic
Semantics
Silicon
Syntactics
Time-domain analysis
Timing
ISBN:1479904139, 9781479904136
ISSN:1043-6871
Online Access:Get full text
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Abstract Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. Hirshfeld and Rabinovich have shown that over the reals, firstorder logic with binary order relation <; and unary function +1 is strictly more expressive than MTL with integer constants. Indeed they prove that no temporal logic whose modalities are definable by formulas of bounded quantifier depth can be expressively complete for FO(<;, +1). In this paper we show that if we allow unary functions +q, q ∈ Q, in first-order logic and correspondingly allow rational constants in MTL, then the two logics have the same expressive power. This gives the first generalisation of Kamp's theorem on the expressive completeness of LTL for FO(<;) to the quantitative setting. The proof of this result involves a generalisation of Gabbay's notion of separation to the metric setting.
AbstractList Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. Hirshfeld and Rabinovich have shown that over the reals, firstorder logic with binary order relation <; and unary function +1 is strictly more expressive than MTL with integer constants. Indeed they prove that no temporal logic whose modalities are definable by formulas of bounded quantifier depth can be expressively complete for FO(<;, +1). In this paper we show that if we allow unary functions +q, q ∈ Q, in first-order logic and correspondingly allow rational constants in MTL, then the two logics have the same expressive power. This gives the first generalisation of Kamp's theorem on the expressive completeness of LTL for FO(<;) to the quantitative setting. The proof of this result involves a generalisation of Gabbay's notion of separation to the metric setting.
Author Hunter, Paul
Worrell, James
Ouaknine, Joel
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  givenname: Joel
  surname: Ouaknine
  fullname: Ouaknine, Joel
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  givenname: James
  surname: Worrell
  fullname: Worrell, James
  email: james.worrell@cs.ox.ac.uk
  organization: Dept. of Comput. Sci., Univ. of Oxford, Oxford, UK
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Snippet Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express...
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StartPage 349
SubjectTerms Computer science
Expressive Completeness
First-Order Logic
Linear Temporal Logic
Metric Temporal Logic
Semantics
Silicon
Syntactics
Time-domain analysis
Timing
Title Expressive Completeness for Metric Temporal Logic
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