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A denotational semantics for equilibrium logic*

Published online by Cambridge University Press:  03 September 2015

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)

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 G 3 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 G 3 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 G 3 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.

Type
Regular Papers
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

*

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

References

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