research-article

Semantics Out of Context: Nominal Absolute Denotations for First-Order Logic and Computation

Author:

Murdoch J. GabbayAuthors Info & Claims

Journal of the ACM (JACM), Volume 63, Issue 3

Article No.: 25, Pages 1 - 66

https://doi.org/10.1145/2700819

Published: 28 June 2016 Publication History

Abstract

Call a semantics for a language with variables absolute when variables map to fixed entities in the denotation. That is, a semantics is absolute when the denotation of a variable a is a copy of itself in the denotation.

We give a trio of lattice-based, sets-based, and algebraic absolute semantics to first-order logic. Possibly open predicates are directly interpreted as lattice elements/sets/algebra elements, subject to suitable interpretations of the connectives and quantifiers. In particular, universal quantification ∀a.φ is interpreted using a new notion of “fresh-finite” limit Λ ^#a ⟦Φ⟧ and using a novel dual to substitution.

The interest in this semantics is partly in the nontrivial and beautiful technical details, which also offer certain advantages over existing semantics. Also, the fact that such semantics exist at all suggests a new way of looking at variables and the foundations of logic and computation, which may be well suited to the demands of modern computer science.

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Cited By

Gabbay MGabbay M(2017)Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completenessAnnals of Pure and Applied Logic10.1016/j.apal.2016.10.001168:3(501-621)Online publication date: Mar-2017
https://doi.org/10.1016/j.apal.2016.10.001

Index Terms

Semantics Out of Context: Nominal Absolute Denotations for First-Order Logic and Computation
1. Theory of computation

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cover image Journal of the ACM

Journal of the ACM Volume 63, Issue 3

September 2016

303 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2957788

Editor:
Éva Tardos
Cornell University

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Copyright © 2016 ACM.

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Publication History

Published: 28 June 2016

Accepted: 01 December 2015

Revised: 01 September 2015

Received: 01 August 2012

Published in JACM Volume 63, Issue 3

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Cited By

Gabbay MGabbay M(2017)Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completenessAnnals of Pure and Applied Logic10.1016/j.apal.2016.10.001168:3(501-621)Online publication date: Mar-2017
https://doi.org/10.1016/j.apal.2016.10.001

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