Approximation Fixpoint Theory for Non-Deterministic Operators and Its Application in Disjunctive Logic Programming

Approximation fixpoint theory (AFT) constitutes an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper, we extend AFT to non-deterministic constructs such as disjunctive information. This is done by generalizing the main constructions and corresponding results to non-deterministic operators, whose ranges are sets of elements rather than single elements. The applicability and usefulness of this generalization is illustrated in the context of disjunctive logic programming.

A Unifying Approach for Nonmonotonic S4F, (Reflexive) Autoepistemic Logic, and Answer Set Programming

This paper presents a general strategy, bringing together some major types of nonmonotonic reasoning under a monotonic bimodal setting. Such formalisms are also of interest to the fields of knowledge representation and declarative programming. We exemplify the methodology, capturing minimal model reasoning that underlies nonmonotonicity over S4F first, but then we also show how to apply the technique to other nonmonotonic logics respectively based on the modal logics KD45 and SW5. We naturally succeed it, by modifying only the axioms of the underlying modal logic and show that it successfully works. The last two formalisms are also known as autoepistemic logic (AEL) and its reflexive extension (RAEL) in the given order: AEL is an important form of nonmonotonic reasoning, introduced by Robert C. Moore in order to allow an agent to reason about his own knowledge. Equilibrium logic (EL) is a general-purpose nonmonotonic reasoning formalism, proposed more recently by David Pearce as a semantical framework for answer set programming (ASP). The latter is an efficient declarative problem solving approach with lots of applications to science and technology. Fariñas et al. have embedded EL (and so ASP) into a monotonic bimodal logic. We take this work as an initiative and successfully apply a similar methodology to closely aligned nonmonotonic modal logics. We finally discuss the potential capability to subsume the epistemic extensions of ASP within our unified paradigm.

Safe Inductions: An Algebraic Study

In many knowledge representation formalisms, a constructive semantics is defined based on sequential applications of rules or of a semantic operator. These constructions often share the property that rule applications must be delayed until it is safe to do so: until it is known that the condition that triggers the rule will remain to hold. This intuition occurs for instance in the well-founded semantics of logic programs and in autoepistemic logic. In this paper, we formally define the safety criterion algebraically. We study properties of so-called safe inductions and apply our theory to logic programming and autoepistemic logic. For the latter, we show that safe inductions manage to capture the intended meaning of a class of theories on which all classical constructive semantics fail.

Knowledge compilation of logic programs using approximation fixpoint theory

Recent advances in knowledge compilation introduced techniques to compilepositivelogic programs into propositional logic, essentially exploiting the constructive nature of the least fixpoint computation. This approach has several advantages over existing approaches: it maintains logical equivalence, does not require (expensive) loop-breaking preprocessing or the introduction of auxiliary variables, and significantly outperforms existing algorithms. Unfortunately, this technique is limited tonegation-freeprograms. In this paper, we show how to extend it to general logic programs under the well-founded semantics.We develop our work in approximation fixpoint theory, an algebraical framework that unifies semantics of different logics. As such, our algebraical results are also applicable to autoepistemic logic, default logic and abstract dialectical frameworks.