On kernels, defaults and even graphs

Extensions in prerequisite-free, disjunction-free default theories have been shown to be in direct correspondence with kernels of directed graphs; hence default theories without odd cycles always have a ``standard'' kind of an extension. We show that, although all ``standard'' extensions can be enumerated explicitly, several other problems remain intractable for such theories: Telling whether a non-standard extension exists, enumerating all extensions, and finding the minimal standard extension. We also present a new graph-theoretic algorithm, based on vertex feedback sets, for enumerating all extensions of a general prerequisite-free, disjunction-free default theory (possibly with odd cycles). The algorithm empirically performs well for quite large theories

Graph theoretical structures in logic programs and default theories

In this paper we present a graph representation of logic programs and default theories. We show that many of the semantics proposed for logic programs can be expressed in terms of notions emerging from graph theory, establishing in this way a link between the fields. Namely the stable models, the partial stable models, and the well-founded semantics correspond respectively to the kernels, semikernels and the initial acyclic part of the associated graph. This link allows us to consider both theoretical problems (existence, uniqueness) and computational problems (tractability, algorithms, approximations) from a more abstract and rather combinatorial point of view. It also provides a clear and intuitive understanding about how conflicts between rules are resolved within the different semantics. Furthermore, we extend the basic framework developed for logic programs to the case of Default Logic by introducing the notions of partial, deterministic and well-founded extensions for default theories. These semantics capture different ways of reasoning with a default theory

Complexity of Prioritized Default Logics

In default reasoning, usually not all possible ways of resolving conflicts between default rules are acceptable. Criteria expressing acceptable ways of resolving the conflicts may be hardwired in the inference mechanism, for example specificity in inheritance reasoning can be handled this way, or they may be given abstractly as an ordering on the default rules. In this article we investigate formalizations of the latter approach in Reiter's default logic. Our goal is to analyze and compare the computational properties of three such formalizations in terms of their computational complexity: the prioritized default logics of Baader and Hollunder, and Brewka, and a prioritized default logic that is based on lexicographic comparison. The analysis locates the propositional variants of these logics on the second and third levels of the polynomial hierarchy, and identifies the boundary between tractable and intractable inference for restricted classes of prioritized default theories

More on Representation Theory for Default Logic

AbstractIn this paper, we investigate the representability of a family of theories as the set of extensions of a default theory. First, we present both new necessary conditions and sufficient ones for the representability by means of general default theories, which improves on similar results known before. Second, we show that one always obtains representable families by eliminating countably many theories from a representable family. Finally, we construct two examples of denumerable, representable families; one is not supercompactly nonincluding, and the other consists of mutually inconsistent theories but fails to be represented by a normal default theory

Well-Founded Semantics for Extended Logic Programs with Dynamic Preferences

The paper describes an extension of well-founded semantics for logic programs with two types of negation. In this extension information about preferences between rules can be expressed in the logical language and derived dynamically. This is achieved by using a reserved predicate symbol and a naming technique. Conflicts among rules are resolved whenever possible on the basis of derived preference information. The well-founded conclusions of prioritized logic programs can be computed in polynomial time. A legal reasoning example illustrates the usefulness of the approach.Comment: See http://www.jair.org/ for any accompanying file

Semantics of logic programs with explicit negation

After a historical introduction, the bulk of the thesis concerns the study of a declarative semantics for logic programs. The main original contributions are: ² WFSX (Well–Founded Semantics with eXplicit negation), a new semantics for logic programs with explicit negation (i.e. extended logic programs), which compares favourably in its properties with other extant semantics. ² A generic characterization schema that facilitates comparisons among a diversity of semantics of extended logic programs, including WFSX. ² An autoepistemic and a default logic corresponding to WFSX, which solve existing problems of the classical approaches to autoepistemic and default logics, and clarify the meaning of explicit negation in logic programs. ² A framework for defining a spectrum of semantics of extended logic programs based on the abduction of negative hypotheses. This framework allows for the characterization of different levels of scepticism/credulity, consensuality, and argumentation. One of the semantics of abduction coincides with WFSX. ² O–semantics, a semantics that uniquely adds more CWA hypotheses to WFSX. The techniques used for doing so are applicable as well to the well–founded semantics of normal logic programs. ² By introducing explicit negation into logic programs contradiction may appear. I present two approaches for dealing with contradiction, and show their equivalence. One of the approaches consists in avoiding contradiction, and is based on restrictions in the adoption of abductive hypotheses. The other approach consists in removing contradiction, and is based in a transformation of contradictory programs into noncontradictory ones, guided by the reasons for contradiction

Default reasoning and neural networks

In this dissertation a formalisation of nonmonotonic reasoning, namely Default logic, is discussed. A proof theory for default logic and a variant of Default logic - Prioritised Default logic - is presented. We also pursue an investigation into the relationship between default reasoning and making inferences in a neural network. The inference problem shifts from the logical problem in Default logic to the optimisation problem in neural networks, in which maximum consistency is aimed at The inference is realised as an adaptation process that identifies and resolves conflicts between existing knowledge about the relevant world and external information. Knowledge and data are transformed into constraint equations and the nodes in the network represent propositions and constraint equations. The violation of constraints is formulated in terms of an energy function. The Hopfield network is shown to be suitable for modelling optimisation problems and default reasoning.Computer ScienceM.Sc. (Computer Science