Fixed-parameter tractability of disjunction-free default reasoning

In this paper,the parameter which is the source of the complexity of disjunctionfree default reasoning is determined.It is shown that when the value of this parameter is fixed,the disjunction-free default reasoning can be solved time bounded by a polynomial whose degree does not depend on the parameter.Consequently,disjunction-free default reasoning is fixed parameter tractable.     

On a rule-based interpretation of default conditionals

In nonmonotonic reasoning, a default conditional α→β has most often been informally interpreted as a defeasible version of a classical conditional, usually the material conditional. There is however an alternative interpretation, in which a default is regarded essentially as a rule, leading from premises to conclusion. In this paper, we present a family of logics, based on this alternative interpretation. A general semantic framework under this rule-based interpretation is developed, and associated proof theories for a family of weak conditional logics is specified. Nonmonotonic inference is easily defined in these logics. Interestingly, the logics presented here are weaker than the commonly-accepted base conditional approach for defeasible reasoning. However, this approach resolves problems that have been associated with previous approaches.      

Default reasoning and belief revision: A syntax-independent approach

As an important variant of Reiter‘s default logic.Poole(1988) developed a nonmonotonic reasoning framework in the classical first-order language,Brewka and Nebel extended Poole‘s approach in order to enable a representation of priorities between defaults.In this paper a general framework for default reasoning is presented,which can be viewed as a generalization of the three approaches above.It is proved that the syntax-independent default reasoning in this framework is identical to the general belief revision operation introduced by Zhang et al.(1997).This esult provides a solution to the problem whether there is a correspondence between belief revision and default logic for the infinite case .As a by-product,an answer to the the question,raised by Mankinson and Gaerdenfors(1991),is also given about whether there is a counterpart contraciton in nonmonotonic logic.     

Embedding defaults into terminological knowledge representation formalisms

We consider the problem of integrating Reiter's default logic into terminological representation systems. It turns out that such an integration is less straightforward than we expected, considering the fact that the terminological language is a decidable sublanguage of first-order logic. Semantically, one has the unpleasant effect that the consequences of a terminological default theory may be rather unintuitive, and may even vary with the syntactic structure of equivalent concept expressions. This is due to the unsatisfactory treatment of open defaults via Skolemization in Reiter's semantics. On the algorithmic side, we show that this treatment may lead to an undecidable default consequence relation, even though our base language is decidable, and we have only finitely many (open) defaults. Because of these problems, we then consider a restricted semantics for open defaults in our terminological default theories: default rules are applied only to individuals that are explicitly present in the knowledge base. In this semantics it is possible to compute all extensions of a finite terminological default theory, which means that this type of default reasoning is decidable. We describe an algorithm for computing extensions and show how the inference procedures of terminological systems can be modified to give optimal support to this algorithm.This is a revised and extended version of a paper presented at the3rd International Conference on Principles of Knowledge Representation and Reasoning, October 1992, Cambridge, MA.     

Reasoning with Sets of Defaults in Default Logic

We present a general approach for representing and reasoning with sets of defaults in default logic, focusing 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 prioritized versions of default logic. In practical terms this means we can immediately use existing default logic theorem provers for an implementation. Also, 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 semimonotonic 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 a standard default logic. Hence one can encode notions such as skeptical and credulous conclusions, and can reason about such conclusions within a single extension.     

Modeling default reasoning using defaults

User modeling research can benefit from formal automated reasoning tools. However existing formal tools may need to be modified to suit the needs of user modeling. Theorist is a simple framework for default reasoning. It can be used as a tool for building and maintaining a user model, and as a model of a user's default reasoning. To apply Theorist to both tasks, we develop Nested Theorist (NT), a simple tool based on Theorist that allows default reasoning on arbitrarily-many levels. We extend NT in two ways: we allow prioritized defaults, and we allow reasoning about agents with limited reasoning capabilities. This paper focusses on applications, and uses wide-ranging examples from user-modeling literature to illustrate the usefulness of the tools presented.     

An Interpretation of Default Logic in Minimal Temporal Epistemic Logic

When reasoning about complex domains, where information available is usually only partial, nonmonotonic reasoning can be an important tool. One of the formalisms introduced in this area is Reiter's Default Logic (1980). A characteristic of this formalism is that the applicability of default (inference) rules can only be verified in the future of the reasoning process. We describe an interpretation of default logic in temporal epistemic logic which makes this characteristic explicit. It is shown that this interpretation yields a semantics for default logic based on temporal epistemic models. A comparison between the various semantics for default logic will show the differences and similarities of these approaches and ours.     

Reasoning situated in time I: basic concepts*

Abstract

The needs of a real-time reasoner situated in an environment may make it appropriate to view error-correction and non-monotonicity as much the same thing. This has led us to formulate situated (or step) logic, an approach to reasoning in which the formalism has a kind of real-time self-reference that affects the course of deduction itself. Here we seek to motivate this as a useful vehicle for exploring certain issues in commonsense reasoning. In particular, a chief drawback of more traditional logics is avoided: from a contradiction we do not have all wffs swamping the (growing) conclusion set. Rather, we seek potentially inconsistent, but nevertheless useful, logics where the real-time self-referential feature allows a direct contradiction to be spotted and corrective action taken, as part of the same system of reasoning. Some specific inference mechanisms for real-time default reasoning are suggested, notably a form of introspection relevant to default reasoning. Special treatment of ‘now’ and of contradictions are the main technical devices here. We illustrate this with a computer-implemented real time solution to R. Moore's Brother Problem.     

An approach to fuzzy default reasoning for function approximation

This paper discusses fuzzy reasoning for approximately realizing nonlinear functions by a small number of fuzzy if-then rules with different specificity levels. Our fuzzy rule base is a mixture of general and specific rules, which overlap with each other in the input space. General rules work as default rules in our fuzzy rule base. First, we briefly describe existing approaches to the handling of default rules in the framework of possibility theory. Next, we show that standard interpolation-based fuzzy reasoning leads to counterintuitive results when general rules include specific rules with different consequents. Then, we demonstrate that intuitively acceptable results are obtained from a non-standard inclusion-based fuzzy reasoning method. Our approach is based on the preference for more specific rules, which is a commonly used idea in the field of default reasoning. When a general rule includes a specific rule and they are both compatible with an input vector, the weight of the general rule is discounted in fuzzy reasoning. We also discuss the case where general rules do not perfectly but partially include specific rules. Then we propose a genetics-based machine learning (GBML) algorithm for extracting a small number of fuzzy if-then rules with different specificity levels from numerical data using our inclusion-based fuzzy reasoning method. Finally, we describe how our approach can be applied to the approximate realization of fuzzy number-valued nonlinear functions     

Regular disjunction-free default theories

In this paper, the class of regular disjunction-free default theories is introduced and investigated.A transformation from regular default theories to normal default theories is established. The initial theory andthe transformed theory have the same extensions when restricted to old variables. Hence, regular default theoriesenjoy some similar properties (e.g., existence of extensions, semi-monotonicity) as normal default theories. Then,a new algorithm for credulous reasoning of regular theories is developed. This algorithm runs in a time not morethan 0(1.45~n), where n is the number of defaults. In case of regular prerequisite-free or semi-2CNF defaulttheories, the credulous reasoning can be solved in polynomial time. However, credulous reasoning for semi-Horndefault theories is shown to be NP-complete although it is tractable for Horn default theories. Moreover, skepticalreasoning for regular unary default theories is co-NP-complete.     

Default reasoning by deductive planning

This paper deals with the automation of reasoning from incomplete information by means of default logics. We provide proof procedures for default logics' major reasoning modes, namely, credulous and skeptical reasoning. We start by reformulating the task of credulous reasoning in default logics as deductive planning problems. This interpretation supplies us with several interesting and valuable insights into the proof theory of default logics. Foremost, it allows us to take advantage of the large number of available methods, algorithms, and implementations for solving deductive planning problems. As an example, we demonstrate how credulous reasoning in certain variants of default logic is implementable by means of a planning method based on equational logic programming. In addition, our interpretation allows us to transfer theoretical results, such as complexity results, from the field of planning to that of default logics. In this way, we have isolated two yet unknown classes of default theories for which deciding credulous entailment is polynomial.Our approach to skeptical reasoning relies on an arbitrary method for credulous reasoning. It does not strictly require rather the inspection of all extensions, nor does it strictly require the computation of entire extensions to decide whether a formula is skeptically entailed. Notably, our approach abstracts from an underlying credulous reasoner. In this way, it can be used to extend existing formalisms for credulous reasoning to skeptical reasoning.This author was a visiting professor at the University of Darmstadt while parts of this work were being carried out. This author also acknowledges support from the Commission of the European Communities under grant no. ERB4001GT922433.     

What Should Default Reasoning be,by Default?

This is a position paper concerning the role of empirical studies of human default reasoning in the formalization of AI theories of default reasoning. We note that AI motivates its theoretical enterprise by reference to human skill at default reasoning, but that the actual research does not make any use of this sort of information and instead relies on intuitions of individual investigators. We discuss two reasons theorists might not consider human performance relevant to formalizing default reasoning: (a) that intuitions are sufficient to describe a model, and (b) that human performance in this arena is irrelevant to a competence model of the phenomenon. We provide arguments against both these reasons. We then bring forward three further considerations against the use of intuitions in this arena: (a) it leads to an unawareness of predicate ambiguity, (b) it presumes an understanding of ordinary language statements of typicality, and (c) it is similar to discredited views in other fields. We advocate empirical investigation of the range of human phenomena that intuitively embody default reasoning. Gathering such information would provide data with which to generate formal default theories and against which to test the claims of proposed theories. Our position is that such data are the very phenomena that default theories are supposed to explain.     

Yet some more complexity results for default logic

We identify several new tractable subsets and several new intractable simple cases for reasoning in the propositional version of Reiter's default logic. The majority of our findings are related to brave reasoning. By making some intuitive observations, most classes that we identify can be derived quite easily from some subsets of default logic already known in the literature. Some of the subsets we discuss are subclasses of the so-called “extended logic programs”. All the tractable subsets presented in this paper can be recognized in linear time.     

Towards a formal theory of defeasible deontic conditionals

The paper outlines the approach to the analysis of deontic conditionals taken in earlier work of Jones and Pörn, comparing it briefly with two main trends within dyadic deontic logic, and discussing problems associated with the augmentation principle and the factual detachment principle. A modification of the Jones and Pörn system is then presented, using a classical but not normal (in the sense of Chellas) deontic modality, to provide the basis for an alternative analysis of deontic conditionals. This new analysis validates neither the factual detachment nor the augmentation principles. However, influenced by the approach of Delgrande to default reasoning, it is shown how a restricted form of factual detachment might be accommodated within the revised system.     

Default Reasoning and Generics

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 that 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 if 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 is presented. It is 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 that 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 are presented, and their interactions with the phenomena of focus and presupposition is investigated.
It is shown how this account of generics can be used to characterize adequate default reasoning systems, and several desirable properties of such systems are proved. The problems of the automatic acquisition of rules from natural language are discussed. Because rules are often explicitly expressed as generics, it is argued that the interpretation of generic sentences plays a crucial role in this endeavor, and it is shown how the theory presented here can facilitate such interpretation.     

Default logic and specification of nonmonotonic reasoning

In this paper constructions leading to the formation of belief sets by agents are studied. The focus is on the situation when possible belief sets are built incrementally in stages. An infinite sequence of theories that represents such a process is called a reasoning trace. A set of reasoning traces describing all possible reasoning scenarios for the agent is called a reasoning frame. Default logic by Reiter is not powerful enough to represent reasoning frames. In the paper a generalization of default logic of Reiter is introduced by allowing infinite sets of justifications. This formalism is called infinitary default logic. In the main result of the paper it is shown that every reasoning frame can be represented by an infinitary default theory. A similar representability result for antichains of theories (belief frames) is also presented.     

Identification of inductive properties during verification of synchronous sequential circuits

A backward reasoning approach is described to show that the behavioral specification of a finite-state machine satisfies some high-level input-output specification. It is based on the principles of theorem proving. The proof may require inductive reasoning. The theorem prover seeks to find out the properties on which induction needs to be carried out and then proves such properties. Goal-reduction-based backward reasoning is employed. A brief account of the proof procedure is presented. This is followed by an example illustrating the approach. Finally, the approach is presented in an algorithmic form.     

Probabilistic Argumentation Systems and Abduction

Probabilistic argumentation systems are based on assumption-based reasoning for obtaining arguments supporting hypotheses and on probability theory to compute probabilities of supports. Assumption-based reasoning is closely related to hypothetical reasoning or inference through theory formation. The latter approach has well known relations to abduction and default reasoning. In this paper assumption-based reasoning, as an alternative to theory formation aiming at a different goal, will be presented and its use for abduction and model-based diagnostics will be explained. Assumption-based reasoning is well suited for defining a probability structure on top of it. On the base of the relationships between assumption-based reasoning on the one hand and abduction on the other hand, the added value introduced by probability into model based diagnostics will be discussed. Furthermore, the concepts of complete and partial models are introduced with the goal to study the quality of inference procedures. In particular this will be used to compare abductive to possible explanations.     

DEFAULT LOGICS: A UNIFIED VIEW

Default logic has been introduced for handling reasoning with incomplete knowledge. It has been widely studied, and various definitions have been proposed for it. Most of the variants have been defined by means of fixed points of some operator. We propose here another approach, which is based on a study of the way in which general rules with exceptions, used in a default reasoning process, can contradict one another. We then isolate sets of noncontradicting rules, as large as possible in order to exploit as much information as possible, and construct, for each of these sets of rules, the set of conclusions that can be deduced from it. We show that our framework encompasses most of the existing variants of default logic, allowing those variants to be compared from a knowledge representation point of view. Our approach also enables us to provide an operational definition of extensions in some interesting cases. Proof-theoretical and semantical aspects are investigated.     

子句型缺省逻辑中的分情形推理

引进一种树型方法以研究缺省逻辑中分情形推理下的Roos扩张,深入讨论了Roos扩张的计算,并分析了Roos扩张与Reiter扩张的关系.为计算Roos扩张,引入了从子句集分解最小文字集的算法.方法对于在缺省逻辑中计算Roos扩张以及分析分情形推理的计算复杂性是有用的.