A proof procedure for normal default theories (original) (raw)
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Alternative Approaches to Default Logic
Artificial Intelligence, 1994
Reiter's default logic has proven to be an enduring and versatile approach to nonmonotonic 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 Lukaszewicz and Brewka, and to the Theorist system.
A proof theory for Constructive Default Logic
Lecture Notes in Computer Science, 1993
We present what we call Constructive Default Logic (CDL) -a default logic in which the fixedpoint definition of extensions is replaced by a constructive definition which yield so-called constructive extensions. Selection functions are used to represent explicitly the control of the reasoning process in this default logic. It is well-known that Reiter's original default logic lacks, in general, a default proof theory. We will show that CDL does have a default proof theory, and we will also show that this is related to the fact that CDL has the existence property for constructive extensions and that it also has the semi-monotonicity property. Furthermore, we will also show that, with respect to some counter-examples that were suggested by Lukaszewicz, constructive extensions yield more intuitive conclusions than Reiter's extensions. Hence, constructive default logic does not only have heuristic advantages over Reiter's default theory from a computational point of view, but it is also more adequate with respect to knowledge representation.
Reasoning with Sets of Defaults in Default Logic
Computational Intelligence, 2004
We present a general approach for representing and reasoning with sets of defaults in default logic, focussing on reasoning about preferences among sets of defaults. First, we consider how to control the application of a set of defaults so that either all apply (if possible) or none do (if not). From this, an approach to dealing with preferences among sets of default rules is developed. We begin with an ordered default theory, consisting of a standard default theory, but with possible preferences on sets of rules. This theory is transformed into a second, standard default theory wherein the preferences are respected. The approach differs from other work, in that we obtain standard default theories and do not rely on prioritised versions of default logic. In practical terms this means we can immediately use existing default logic theorem provers for an implementation. As well, we directly generate just those extensions containing the most preferred applied rules; in contrast, most previous approaches generate all extensions, then select the most preferred. In a major application of the approach, we show how semi-monotonic default theories can be encoded so that reasoning can be carried out at the object level. With this, we can reason about default extensions from within the framework of standard default logic. Hence one can encode notions such as skeptical and credulous conclusions, and can reason about such conclusions within a single extension.
Default logic generalized and simplified
Annals of Mathematics and Artificial Intelligence, 2008
We provide a description of generalized default logic as a unified formalism for nonmonotonic reasoning. Special attention will be paid to the role of the monotonic logic underlying default reasoning, as well as to the representation opportunities created by the use of assumptions (justifications) in the heads of default rules. On the other hand, it will be shown that even the generalized default logic can be simplified to a formal system that involves only monotonic rules and default assumptions.
The need to make default assumptions is frequently encountered in reasoning'about incompletely specified worlds. Inferences sanctioned by default are best viewed as beliefs which may well be modified or rejected by subsequent observations. It is this property which leads to the non.monotonJcity of any logic of defaults. In this paper we propose a logic for default reasoning. We then specialize our treatment to a very large class of commonly occurring defaults. For this class we develop a complete proof theory and show how to interface it with a top down resolution theorem prover. Finally, we provide criteria under which the revision of derived beliefs must be effected.
Default Reasoning and Generics
Computational Intelligence, 1997
There are numerous logical formalisms capable of drawing conclusions using default rules. Such systems, however, do not normally determine where the default rules come from; i.e. what it is which makes Birds fly a good rule, but Birds drive trucks-a bad one. Generic sentences such as Birds fly are often used informally to describe default rules. I propose to take this characterization seriously, and claim that a default rule is adequate iff the corresponding generic sentence is true. Thus, if we know that Tweety is a bird, we may conclude, by default, that Tweety flies, just in case Birds fly is a true sentence. In this paper, a quantificational account of the semantics of generic sentences will be presented. It will be argued that a generic sentence is evaluated not in isolation, but with respect to a set of relevant alternatives. For example, Mammals bear live young is true because among mammals which bear live young, lay eggs, undergo mitosis or engage in some alternative form of procreation, the majority bear live young. Since male mammals do not procreate in any form, they do not count. Some properties of alternatives will be presented, and their interactions with the phenomena of focus and presupposition will be investigated. It will be shown how this account of generics can be used to characterize adequate default reasoning systems, and several desirable properties of such systems will be proved. The problems of the automatic acquisition of rules from natural language will be discussed. Since rules are often explicitly expressed as generics, it will be argued that the interpretation of generic sentences plays a crucial role in this endeavor, and it will be shown how the theory presented here can facilitate such interpretation.
An Abstract, Argumentation-Theoretic Approach to Default Reasoning
Artificial intelligence, 1997
We present an abstract framework for default reasoning, which includes Theorist, default logic, logic programming, autoepistemic logic, non-monotonic modal logics, and certain instances of circumscription as special cases. The framework can be understood as a generalisation of Theorist. The generalisation allows any theory formulated in a monotonic logic to be extended by a defeasible set of assumptions.
Optimality theory through default logic
2003
Optimality Theory is an approach to linguistic problems which is based on rules with exceptions, resorting to a ranking among rules to resolve conflicts arising from competing rules.
Expressing Default Logic Variants in Default Logic
Journal of Logic and Computation, 2005
Reiter's default logic is one of the best known and most studied of the approaches to nonmonotonic reasoning. Several variants of default logic have subsequently been proposed to give systems with properties differing from the original. In this paper, we examine the relationship between default logic and its major variants. We accomplish this by translating a default theory under a variant interpretation into a second default theory, under the original Reiter semantics, wherein the variant interpretation is respected. That is, in each case we show that, given an extension of a translated theory, one may extract an extension of the original variant default logic theory. We show how constrained, rational, justified, and cumulative default logic can be expressed in Reiter's default logic. As well, we show how Reiter's default logic can be expressed in rational default logic. From this, we suggest that any such variant can be similarly treated. Consequently, we provide a unification of default logics, showing how the original formulation of default logic may express its variants. Moreover, the translations clearly express the relationships between alternative approaches to default logic. The translations themselves are shown to generally have good properties. Thus, in at least a theoretical sense, we show that these variants are in a sense superfluous, in that for any of these variants of default logic, we can exactly mimic the behaviour of a variant in standard default logic. As well, the translations lend insight into means of classifying the expressive power of default logic variants; specifically we suggest that the property of semi-monotonicity represents a division with respect to expressibility, whereas regularity and cumulativity do not.