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A denotational semantics for equilibrium logic*

Published online by Cambridge University Press:  03 September 2015

FELICIDAD AGUADO ,
PEDRO CABALAR ,
DAVID PEARCE ,
GILBERTO PÉREZ  and
CONCEPCIÓN VIDAL
FELICIDAD AGUADO
Affiliation:
Department of Computer Science, University of Corunna, SPAIN (e-mail: aguado@udc.es, cabalar@udc.es, gperez@udc.es, eicovima@udc.es)
PEDRO CABALAR
Affiliation:
Department of Computer Science, University of Corunna, SPAIN (e-mail: aguado@udc.es, cabalar@udc.es, gperez@udc.es, eicovima@udc.es)
DAVID PEARCE
Affiliation:
Universidad Politécnica de Madrid, SPAIN (e-mail: david.pearce@upm.es)
GILBERTO PÉREZ
Affiliation:
Department of Computer Science, University of Corunna, SPAIN (e-mail: aguado@udc.es, cabalar@udc.es, gperez@udc.es, eicovima@udc.es)
CONCEPCIÓN VIDAL
Affiliation:
Department of Computer Science, University of Corunna, SPAIN (e-mail: aguado@udc.es, cabalar@udc.es, gperez@udc.es, eicovima@udc.es)
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Abstract

In this paper we provide an alternative semantics for Equilibrium Logic and its monotonic basis, the logic of Here-and-There (also known as Gödel's G3 logic) that relies on the idea of denotation of a formula, that is, a function that collects the set of models of that formula. Using the three-valued logic G3 as a starting point and an ordering relation (for which equilibrium/stable models are minimal elements) we provide several elementary operations for sets of interpretations. By analysing structural properties of the denotation of formulas, we show some expressiveness results for G3 such as, for instance, that conjunction is not expressible in terms of the other connectives. Moreover, the denotational semantics allows us to capture the set of equilibrium models of a formula with a simple and compact set expression. We also use this semantics to provide several formal definitions for entailment relations that are usual in the literature, and further introduce a new one called strong entailment. We say that α strongly entails β when the equilibrium models of α ∧ γ are also equilibrium models of β ∧ γ for any context γ. We also provide a characterisation of strong entailment in terms of the denotational semantics, and give an example of a sufficient condition that can be applied in some cases.

Keywords

Answer Set ProgrammingEquilibrium Logic
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 15 , Special Issue 4-5: 31st International Conference on Logic Programming , July 2015 , pp. 620 - 634
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

*

This research was partially supported by Spanish MEC project TIN2013-42149-P.

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