Embedding Non-Ground Logic Programs into Autoepistemic Logic for Knowledge Base Combination

In the context of the Semantic Web, several approaches to the combination of ontologies, given in terms of theories of classical first-order logic and rule bases, have been proposed. They either cast rules into classical logic or limit the interaction between rules and ontologies. Autoepistemic logic (AEL) is an attractive formalism which allows to overcome these limitations, by serving as a uniform host language to embed ontologies and nonmonotonic logic programs into it. For the latter, so far only the propositional setting has been considered. In this paper, we present three embeddings of normal and three embeddings of disjunctive non-ground logic programs under the stable model semantics into first-order AEL. While the embeddings all correspond with respect to objective ground atoms, differences arise when considering non-atomic formulas and combinations with first-order theories. We compare the embeddings with respect to stable expansions and autoepistemic consequences, considering the embeddings by themselves, as well as combinations with classical theories. Our results reveal differences and correspondences of the embeddings and provide useful guidance in the choice of a particular embedding for knowledge combination.Comment: 52 pages, submitte

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 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 wa

In search of a “true” logic of knowledge: the nonmonotonic perspective

AbstractModal logics are currently widely accepted as a suitable tool of knowledge representation, and the question what logics are better suited for representing knowledge is of particular importance. Usually, some axiom list is given, and arguments are presented justifying that suggested axioms agree with intuition. The question why the suggested axioms describe all the desired properties of knowledge remains answered only partially, by showing that the most obvious and popular additional axioms would violate the intuition.We suggest the general paradigm of maximal logics and demonstrate how it can work for nonmonotonic modal logics. Technically, we prove that each of the modal logics KD45, SW5, S4F and S4.2 is the strongest modal logic among the logics generating the same nonmonotonic logic. These logics have already found important applications in knowledge representation, and the obtained results contribute to the explanation of this fact

Strong-AI Autoepistemic Robots Build on Intensional First Order Logic

Neuro-symbolic AI attempts to integrate neural and symbolic architectures in a manner that addresses strengths and weaknesses of each, in a complementary fashion, in order to support robust strong AI capable of reasoning, learning, and cognitive modeling. In this paper we consider the intensional First Order Logic (IFOL) as a symbolic architecture of modern robots, able to use natural languages to communicate with humans and to reason about their own knowledge with self-reference and abstraction language property. We intend to obtain the grounding of robot's language by experience of how it uses its neuronal architectures and hence by associating this experience with the mining (sense) of non-defined language concepts (particulars/individuals and universals) in PRP (Properties/Relations/Propositions) theory of IFOL.\\ We consider the robot's four-levels knowledge structure: The syntax level of particular natural language (Italian, French, etc..), two universal language levels: its semantic logic structure (based on virtual predicates of FOL and logic connectives), and its corresponding conceptual PRP structure level which universally represents the composite mining of FOL formulae grounded on the last robot's neuro-system level. Finally, we provide the general method how to implement in IFOL (by using the abstracted terms) different kinds of modal logic operators and their deductive axioms: we present a particular example of robots autoepistemic deduction capabilities by introduction of the special temporal $Konow$ predicate and deductive axioms for it: reflexive, positive introspection and distributive axiom.Comment: 25 pages, 2 figure