The Gödel and the Splitting Translations

Pearce, David Andrew (2010). The Gödel and the Splitting Translations. In: "30 Years of Non-Monotonic Reasoning", 22/10/2010 - 25/10/2010, Kentucky, EEUU.


Title: The Gödel and the Splitting Translations
  • Pearce, David Andrew
Item Type: Presentation at Congress or Conference (Article)
Event Title: 30 Years of Non-Monotonic Reasoning
Event Dates: 22/10/2010 - 25/10/2010
Event Location: Kentucky, EEUU
Title of Book: Proceedings of the 30 Years of Non-Monotonic Reasoning
Date: 2010
Faculty: Facultad de Informática (UPM)
Department: Inteligencia Artificial
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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When the new research area of logic programming and non-monotonic reasoning emerged at the end of the 1980s, it focused notably on the study of mathematical relations between different non-monotonic formalisms, especially between the semantics of stable models and various non-monotonic modal logics. Given the many and varied embeddings of stable models into systems of modal logic, the modal interpretation of logic programming connectives and rules became the dominant view until well into the new century. Recently, modal interpretations are once again receiving attention in the context of hybrid theories that combine reasoning with non-monotonic rules and ontologies or external knowledge bases. In this talk I explain how familiar embeddings of stable models into modal logics can be seen as special cases of two translations that are very well-known in non-classical logic. They are, first, the translation used by Godel in 1933 to em- ¨ bed Heyting’s intuitionistic logic H into a modal provability logic equivalent to Lewis’s S4; second, the splitting translation, known since the mid-1970s, that allows one to embed extensions of S4 into extensions of the non-reflexive logic, K4. By composing the two translations one can obtain (Goldblatt, 1978) an adequate provability interpretation of H within the Goedel-Loeb logic GL, the system shown by Solovay (1976) to capture precisely the provability predicate of Peano Arithmetic. These two translations and their composition not only apply to monotonic logics extending H and S4, they also apply in several relevant cases to non-monotonic logics built upon such extensions, including equilibrium logic, non-monotonic S4F and autoepistemic logic. The embeddings obtained are not merely faithful and modular, they are based on fully recursive translations applicable to arbitrary logical formulas. Besides providing a uniform picture of some older results in LPNMR, the translations yield a perspective from which some new logics of belief emerge in a natural way

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Item ID: 9403
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Deposited by: Memoria Investigacion
Deposited on: 10 Nov 2011 12:24
Last Modified: 20 Apr 2016 17:49
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