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Journal of Logic and Computation 2005 15(5):787-813; doi:10.1093/logcom/exi042
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Vol. 15 No. 5, © The Author, 2005. Published by Oxford University Press. All rights reserved.

Original Articles

Ground Nonmonotonic Modal Logic S5: New Results

Mauricio Osorio Galindo1, Juan Antonio Navarro Pérez1, José R. Arrazola Ramírez2 and Verónica Borja Macías2

1 Universidad de las Américas - Puebla, Computer Science Department, Canada. Email: josorio{at}mail.udlap.mx, m10890{at}mail.udlap.mx, 2 Benemérita Universidad Autónoma de Puebla, Mathematics Department. Email: arrazola{at}fcfm.buap.mx, vero0304{at}fcfm.buap.mx

We study logic programs under Gelfond's translation in the context of modal logic S5. We show that for arbitrary logic programs (propositional theories where logic negation is associated with default negation) ground nonmonotonic modal logics between T and S5 are equivalent. Furthermore, we also show that these logics are equivalent to a nonmonotonic logic that we construct using the well known FOUR bilattice. We will call this semantic GNM-S5 as a reminder of its origin in the logic S5. Finally we show that, for normal programs, our approach is closely related to theWell-Founded-by-Cases Semantics introduced by Schlipf and the WFS+ proposed by Dix. We prove that GNM-S5 has the properties of classicality and extended cut. While WFS+ also supports classicality it fails to satisfy the extended cut principle, an important property available in other semantics such as stable models. Hence, we claim that GNM-S5 is a good candidate for defining a nonmonotonic semantics closer to the direction of classical logic.

Keywords: Modal logic S5, nonmonotonic reasoning, FOUR bilattice, logic programming, semantics of programming

Received 6 October 2004.


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