James P. Delgrande

A first-order conditional logic for prototypical properties

Abstract An approach for representing knowledge about defaults and prototypical properties is presented. This is accomplished by adding to first-order logic a “variable conditional” operator to express relations between entities and prototypical properties of such entities. Truth conditions for this operator are based on a possible-worlds semantics; a proof theory is provided, and the logic is shown to be sound and complete. Properties of the resultant formal system are argued to correspond to common intuitions concerning defaults and prototypical properties. Moreover the system is argued to provide a more appropriate basis for representing knowledge about such entities than other existing approaches.

A Classification and Survey of Preference Handling Approaches in Nonmonotonic Reasoning

In recent years, there has been a large amount of disparate work concerning the representation and reasoning with qualitative preferential information by means of approaches to nonmonotonic reasoning. Given the variety of underlying systems, assumptions, motivations, and intuitions, it is difficult to compare or relate one approach with another. Here, we present an overview and classification for approaches to dealing with preference. A set of criteria for classifying approaches is given, followed by a set of desiderata that an approach might be expected to satisfy. A comprehensive set of approaches is subsequently given and classified with respect to these sets of underlying principles.

A framework for compiling preferences in logic programs

We introduce a methodology and framework for expressing general preference information in logic programming under the answer set semantics. An ordered logic program is an extended logic program in which rules are named by unique terms, and in which preferences among rules are given by a set of atoms of form s p t where s and t are names. An ordered logic program is transformed into a second, regular, extended logic program wherein the preferences are respected, in that the answer sets obtained in the transformed program correspond with the preferred answer sets of the original program. Our approach allows the specification of dynamic orderings, in which preferences can appear arbitrarily within a program. Static orderings (in which preferences are external to a logic program) are a trivial restriction of the general dynamic case. First, we develop a specific approach to reasoning with preferences, wherein the preference ordering specifies the order in which rules are to be applied. We then demonstrate the wide range of applicability of our framework by showing how other approaches, among them that of Brewka and Eiter, can be captured within our framework. Since the result of each of these transformations is an extended logic program, we can make use of existing implementations, such as dlv and smodels. To this end, we have developed a publicly available compiler as a front-end for these programming systems.

Alternative approaches to default logic

Abstract Reiters default logic has proven to be an enduring and versatile approach to non-monotonic reasoning. Subsequent work in default logic has concentrated in two major areas. First, modifications have been developed to extend and augment the approach. Second, there has been ongoing interest in semantic foundations for default logic. In this paper, a number of variants of default logic are developed to address differing intuitions arising from the original and subsequent formulations. First, we modify the manner in which consistency is used in the definition of a default extension. The idea is that a global rather than local notion of consistency is employed in the formation of a default extension. Second, we argue that in some situations the requirement of proving the antecedent of a default is too strong. A second variant of default logic is developed where this requirement is dropped; subsequently these approaches are combined, leading to a final variant. These variants then lead to default systems which conform to alternative intuitions regarding default reasoning. For all of these approaches, a fixed-point and a pseudo-iterative definition are given; as well a semantic characterisation of these approaches is provided. In the combined approach we argue also that one can now reason about a set of defaults and can determine, for example, if a particular default in a set is redundant. We show the relation of this work to that of Łukaszewicz and Brewka, and to the Theorist system.

Expressing preferences in default logic

We address the problem of reasoning about preferences among properties (outcomes, desiderata, etc.) in Reiters default logic. Preferences are expressed using an ordered default theory, consisting of default rules, world knowledge, and an ordering, reflecting preference, on the default rules. In contrast with previous work in the area, we do not rely on prioritised versions of default logic, but rather we transform an ordered default theory into a second, standard default theory wherein the preferences are respected, in that defaults are applied in the prescribed order. This translation is accomplished via the naming of defaults, so that reference may be made to a default rule from within a theory. In an elaboration of the approach, we allow an ordered default theory where preference information is specified within a default theory. Here one may specify preferences that hold by default, in a particular context, or give preferences among preferences. In the approach, one essentially axiomatises how different orderings interact within a theory and need not rely on metatheoretic characterisations. As well, we can immediately use existing default logic theorem provers for an implementation. From a theoretical point of view, this shows that the explicit representation of priorities among defaults adds nothing to the overall expressibility of default logic.

On first-order conditional logics

Abstract Conditional logics have been developed as a basis from which to investigate logical properties of “weak” conditionals representing, for example, counterfactual and default assertions. This work has largely centred on propositional approaches. However, it is clear that for a full account a first-order logic is required. Existing or obvious approaches to first-order conditional logics are inadequate; in particular, various representational issues in default reasoning are not addressed by extant approaches. Further, these problems are not unique to conditional logic, but arise in other nonmonotonic reasoning formalisms. I argue that an adequate first-order approach to conditional logic must admit domains that vary across possible worlds; as well the most natural expression of the conditional operator binds variables (although this binding may be eliminated by definition). A possible worlds approach based on Kripke structures is developed, and it is shown that this approach resolves various problems that arise in a first-order setting, including specificity arising from nested quantifiers in a formula and an analogue of the lottery paradox that arises in reasoning about default properties.

A General Approach to Specificity in Default Reasoning

We present an approach addressing the notion of specificity, or of preferring a more specific default sentence over a less specific one, in commonsense reasoning. Historically, approaches have either been too weak to provide a full account of defeasible reasoning while accounting for specificity, or else have been too strong and fail to enforce specificity. Our approach is to use the techniques of a weak system, as exemplified by System Z, to isolate minimal sets of conflicting defaults. From the specificity information intrinsic in these sets, a default theory in a target language is specified. In this paper we primarily deal with theories expressed (ultimately) in Default Logic. However other approaches would do just as well, as we illustrate by also considering Autoepistemic Logic and variants of Default Logic. In our approach, the problems of weak systems, such as lack of adequate property inheritance and (occasional) unwanted specificity relations, are avoided. Also, difficulties inherent with stronger systems, in particular, lack of specificity are addressed. This work differs from previous work in specifying priorities in Default Logic, in that we obtain a theory expressed in Default Logic, rather than ordered sets of rules requiring a modification to Default Logic.

Expressing Time Intervals and Repetition Within a Formalization of Calendars

We investigate a formal representation of time units, calendars, and time unit instances as restricted temporal entities for reasoning about repeated events. We generalize Allens interval relations to a class level, and based on interval classes we define time units. We examine characteristics of time units, and provide a categorization of the hierarchical relations among them. Hence we define an abstract hierarchical unit structure (a calendar structure) that expresses specific relations and properties among the units that compose it. Specific objects in the time line are represented based on this formalism, including nonconvex intervals corresponding to repeated events. A goal of this research is to be able to represent and reason efficiently about repetition in time.

A Model-Theoretic Approach to Belief Change in Answer Set Programming

We address the problem of belief change in (nonmonotonic) logic programming under answer set semantics. Our formal techniques are analogous to those of distance-based belief revision in propositional logic. In particular, we build upon the model theory of logic programs furnished by SE interpretations, where an SE interpretation is a model of a logic program in the same way that a classical interpretation is a model of a propositional formula. Hence we extend techniques from the area of belief revision based on distance between models to belief change in logic programs. We first consider belief revision: for logic programs P and Q, the goal is to determine a program R that corresponds to the revision of P by Q, denoted P * Q. We investigate several operators, including (logic program) expansion and two revision operators based on the distance between the SE models of logic programs. It proves to be the case that expansion is an interesting operator in its own right, unlike in classical belief revision where it is relatively uninteresting. Expansion and revision are shown to satisfy a suite of interesting properties; in particular, our revision operators satisfy all or nearly all of the AGM postulates for revision. We next consider approaches for merging a set of logic programs, P1, ..., Pn. Again, our formal techniques are based on notions of relative distance between the SE models of the logic programs. Two approaches are examined. The first informally selects for each program Pi those models of Pi that vary the least from models of the other programs. The second approach informally selects those models of a program P0 that are closest to the models of programs P1, ..., Pn. In this case, P0 can be thought of as a set of database integrity constraints. We examine these operators with regards to how they satisfy relevant postulate sets. Last, we present encodings for computing the revision as well as the merging of logic programs within the same logic programming framework. This gives rise to a direct implementation of our approach in terms of off-the-shelf answer set solvers. These encodings also reflect the fact that our change operators do not increase the complexity of the base formalism.

A preference-based framework for updating logic programs

We present a framework for updating logic programs under the answer-set semantics that builds on existing work on preferences in logic programming. The approach is simple and general, making use of two distinct complementary techniques: defaultification and preference. While defaultification resolves potential conflicts by inducing more answer sets, preferences then select among these answer sets, yielding the answer sets generated by those rules that have been added more recently.We examine instances of the framework with respect to various desirable properties; for the most part, these properties are satisfied by instances of our framework. Finally, the proposed framework is also easily implementable by off-the-shelf systems.