The Gödel and the Splitting Translations

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 Gödel in 1933 to embed 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. ? This text forms the basis of an invited talk to be given by the author at the conference on 30 Years of Non-Monotonic Reasoning to be held in Lexington, Kentucky, USA in October 2010. The paper draws on some joint ongoing work with Pedro Cabalar and Mirek Truszczynski as well as on some recent and ongoing work with Levan Uridia. I am grateful for their permission to make use of this material. Additional thanks are due to Larisa Maksimova and Leo Esakia for help with some historical matters as well as to Misha Zakharyaschev who helped me a number of years ago to understand some aspects of the Goedel and splitting translations. This paper is dedicated to Viktor Marek and Mirek Truszczynski who have done so much to turn the area of non-monotonic logic into a systematic discipline. Work reported here was partially supported by the MICINN projects TIN2009-14562-C05 and CSD2007-00022.