Nonmonotonic consequences in default domain theory

Default domain theory is a framework for representing and reasoning about commonsense knowledge. Although this theory is motivated by ideas in Reiter’s work on default logic, it is in some sense a dual framework. We make Reiter’s default extension operator into a constructive method of building models, not theories. Domain theory, which is a well established tool for representing partial information in the semantics of programming languages, is adopted as the basis for constructing partial models. This paper considers some of the laws of nonmonotonic consequence, due to Gabbay and to Kraus, Lehmann, and Magidor, in the light of default domain theory. We remark that in some cases Gabbay’s law of cautious monotony is open to question. We consider an axiomatization of the nonmonotonic consequence relation on prime open sets in the Scott topology – the natural logic – of a domain, which omits this law. We prove a representation theorem showing that such relations are in one to one correspondence with the consequence relations determined by extensions in Scott domains augmented with default sets. This means that defaults are very expressive: they can, in a sense, represent any reasonable nonmonotonic entailment. Results about what kind of defaults determine cautious monotony are also discussed. In particular, we show that the property of unique extensions guarantees cautious monotony, and we give several classes of default structures which determine unique extensions. This revised version was published online in June 2006 with corrections to the Cover Date.     

General default logic

In this paper, we propose a new nonmonotonic logic called general default logic. On the one hand, it generalizes Reiter’s default logic by adding to it rule-like operators used in logic programming. On the other hand, it extends logic programming by allowing arbitrary propositional formulas. We show that with this new logic, one can formalize naturally rule constraints, generalized closed world assumptions, and conditional defaults. We show that under a notion of strong equivalence, sentences of this new logic can be converted to a normal form. We also investigate the computational complexity of various reasoning tasks in the logic, and relate it to some other nonmonotonic formalisms such as Lin and Shoham’s logic of GK and Moore’s autoepistemic 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.     

On the intertranslatability of non‐monotonic logics

This paper concentrates on comparing the expressive powers of five non‐monotonic logics that have appeared in the literature. For this purpose, the concept of a polynomial, faithful and modular (PFM) translation function is adopted from earlier work by Gottlob, but a weaker notion of faithfulness is proposed. The existence of a PFM translation function from one non‐monotonic logic to another is interpreted to indicate that the latter logic is capable of expressing everything that the former logic does. Several translation functions are presented in the paper and shown to be PFM. Moreover, it is shown that PFM translation functions are impossible in certain cases, which indicates that the expressive powers of the logics involved differ strictly. The comparisons made in terms of PFM translation functions give rise to an exact classification of non‐monotonic logics, which is then named as the expressive power hierarchy (EPH) of non‐monotonic logics. Three syntactically restricted variants of default logic are also analyzed, and EPH is refined accordingly. Most importantly, the classes of EPH indicate some astonishing relationships in light of earlier results on the expressive power of non‐monotonic logics presented by Gottlob as well as Bonatti and Eiter: Moore’s autoepistemic logic and prerequisite‐free default logic are of equal expressive power and less expressive than Reiter’s default logic and Marek and Truszczyński’s strong autoepistemic logic. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tableau-based characterization and theorem proving for default logic

In this paper, we present a new method for computing extensions and for deriving formulae from a default theory. It is based on the semantic tableaux method and works for default theories with a finite set of defaults that are formulated over a decidable subset of first-order logic. We prove that all extensions (if any) of a default theory can be produced by constructing the semantic tableau ofone formula built from the general laws and the default consequences. This result allows us to describe an algorithm that provides extensions if there are any, and to decide if there are none. Moreover, the method gives a necessary and sufficient criterion for the existence of extensions of default theories with finitely many defaults provided they are formulated on a decidable subset of FOL.This work was completed while the author was at CNRS, Marseille.     

Similarity preservation in default logic

The paper identifies a problem in default reasoning in Reiter’s Default Logic and related systems: elements which are similar given the axioms only, become distinguishable in extensions. We explain why, sometimes, this is considered undesirable. Two approaches are presented for guaranteeing similarity preservation: One approach formalizes a way of uniformly applying the defaults to all similar elements by introducing generic extensions, which depend only on similarity types of objects. According to the second approach, for a restricted class of default theories, a default theory is viewed as a “shorthand notation” to what is “really meant” by its formulation. In this approach we propose a rewriting of defaults in a form that guarantees similarity preservation of the modified theory. It turns out that the above two approaches yield the same result. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tabling for non‐monotonic programming

Non‐monotonic extensions add power to logic programs. However, the main logic programming language, Prolog, is widely recognized as inadequate to implement these extensions due to its weak termination and complexity properties. By extending Prolog’s SLD resolution with tabling, Prolog can be improved in several ways. Tabling can allow a logic programming system to compute the well‐founded semantics for programs with bounded term depth, and to do so with polynomial data complexity. By exploiting these properties, tabling allows a variety of non‐monotonic extensions to be efficiently implemented, and used to solve practical problems. In this paper we describe tabling as it is implemented in the XSB system and show how it can be used to construct meta‐interpreters (or preprocessors) for two sample formalisms: the Well‐Founded Semantics with Explicit Negation, and Generalized Annotated Logic Programs. We also describe how non‐monotonic extensions are used in practical applications such as psychiatric diagnosis, extraction of information from poorly structured textual data, and model checking. This revised version was published online in June 2006 with corrections to the Cover Date.     

Uniform semantic treatment of default and autoepistemic logics

We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the lattice structure of the collection of all belief pairs. For each logic, we introduce a monotone operator on the lattice of belief pairs. We then show that a whole family of semantics can be defined in a systematic and principled way in terms of fixpoints of this operator (or as fixpoints of certain closely related operators). Our approach elucidates fundamental constructive principles in which agents form their belief sets, and leads to approximation semantics for autoepistemic and default logics. It also allows us to establish a precise one-to-one correspondence between the family of semantics for default logic and the family of semantics for autoepistemic logic. The correspondence exploits the modal interpretation of a default proposed by Konolige. Our results establish conclusively that default logic can be viewed as a fragment of autoepistemic logic, a result that has been long anticipated. At the same time, they explain the source of the difficulty to formally relate the semantics of default extensions by Reiter and autoepistemic expansions by Moore. These two semantics occupy different locations in the corresponding families of semantics for default and autoepistemic logics.     

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.     

Temporalizing Epistemic Default Logic

We present an epistemic default logic, based on the metaphore of a meta-level architecture. Upward reflection is formalized by a nonmonotonic entailment relation, based on the objective facts that are either known or unknown at the object level. Then, the meta (monotonic) reasoning process generates a number of default-beliefs of object-level formulas. We extend this framework by proposing a mechanism to reflect these defaults down. Such a reflection is seen as essentially having a temporal flavour: defaults derived at the meta-level are projected as facts in a next object level state. In this way, we obtain temporal models for default reasoning in meta-level formalisms which can be conceived as labeled branching trees. Thus, descending the tree corresponds to shifts in time that model downward reflection, whereas the branching of the tree corresponds to ways of combining possible defaults. All together, this yields an operational or procedural semantics of reasoning by default, which admits one to reason about it by means of branching-time temporal logic. Finally, we define sceptical and credulous entailment relations based on these temporal models and we characterize Reiter extensions in our semantics.     

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Priorities on defaults with prerequisites,and their application in treating specificity in terminological default logic

In a recent paper we have proposed terminological default logic as a formalism that combines means both for structured representation of classes and objects and for default inheritance of properties. The major drawback that terminological default logic inherits from general default logic is that it does not take precedence of more specific defaults over more general ones into account. This behavior has already been criticized in the general context of default logic, but it is all the more problematic in the terminological case where the emphasis lies on the hierarchical organization of concepts.The present paper addresses the problem of modifying terminological default logic such that more specific defaults are preferred. We assume that the specificity ordering is induced by the hierarchical organization of concepts, which means that default information is not taken into account when computing priorities. It turns out that the existing approaches for expressing priorities between defaults do not seem to be appropriate for defaults with prerequisites. Therefore we shall consider an alternative approach for dealing with prioritization in the framework of Reiter's default logic. The formalism is presented in the general setting of default logic where priorities are given by an arbitrary partial ordering on the defaults. We shall exhibit some interesting properties of the new formalism, compare it with existing approaches, and describe an algorithm for computing extensions. In the terminological case, we thus obtain an automated default reasoning procedure that takes specificity into account.This is an extended version of a paper presented at the13th International Joint Conference on Artificial Intelligence, August 1993, Chambery, France.     

贪婪缺省逻辑

提出一种贪婪缺省逻辑,旨在构造扩展的过程中尽可能的保留缺省规则当中的信息.给出了贪婪缺省逻辑的推演系统-GD系统和贪婪缺省的GD-扩展的定义.并且证明了对于缺省理论 的一个扩展,必定存在一个贪婪缺省理论的GD-扩展使得缺省逻辑的扩展是贪婪缺省逻辑扩展的子集.并且存在贪婪缺省理论 的某一GD-扩展,该GD-扩展不包含缺省理论的任一扩展.因此缺省逻辑和贪婪缺省逻辑是两种不同的逻辑.     

Restricted semantics for default reasoning

In nonmonotonic reasoning there are the problems of inconsistency and incoherence in general, and in default reasoning there may be only one trivial extension or no extension in special. We propose the restricted semantics of four-valued logic for default reasoning to resolve the problems of inconsistency and incoherence and in the meantime retain classical extensions in the presence of consistency and coherency. The restricted semantics can maintain both the expressiveness and reasoning ability of default logic. We provide a transformation approach to compute the restricted extensions by reducing them into classical ones.     

A Standard Model—Theoretic Approach to Operational Semmantics of Recursiv Programs

In this paper we try to introduce a new approach to operational semantics of recursive programs by using ideas in the“priority method”which is a fundamental tool in Recursion Theory.In lieu of modelling partial functions by introducing undefined values in a traditional approach,we shall define a priority derivation tree for every term,and by respecting thr rule“attacking the subtem of the highest priority first”we define transition relations,computation sequences etc.directly based on a standard interpretation whic includes no undefined value in its domain,Finally,we prove that our new approach generates the same opeational semantics as the traditional one.It is also pointed out that we can use our strategy oto refute a claim of Loeckx and Sieber that the opperational semantics of recursive programs cannot be built based on predicate logic.     

Nonmonotonic reasoning as prioritized argumentation

This paper proposes a formalism for nonmonotonic reasoning based on prioritized argumentation. We argue that nonmonotonic reasoning in general can be viewed as selecting monotonic inferences by a simple notion of priority among inference rules. More importantly, these types of constrained inferences can be specified in a knowledge representation language where a theory consists of a collection of rules of first order formulas and a priority among these rules. We recast default reasoning as a form of prioritized argumentation and illustrate how the parameterized formulation of priority may be used to allow various extensions and modifications to default reasoning. We also show that it is possible, but more difficult, to express prioritized argumentation by default logic: Even some particular forms of prioritized argumentation cannot be represented modularly by defaults under the same language     

Operational concepts of nonmonotonic logics part 1: Default logic

We give an introduction to default logic, one of the most prominent nonmonotonic logics. Emphasis is given to providing an operational interpretation for the semantics of default logic that is usually defined by fixed-point concepts (extensions). We introduce a process model that allows to exactly calculate the extensions of a default theory in a quite easy way. We give a prototypical implementation of processes in Prolog able to handle the examples that can be found in literature. Finally, we develop some theoretical results about default logic and give new simple proofs using the process model as a theoretical tool.     

Using default and causal reasoning in diagnosis

We present a theory of default reasoning that is specifically targeted to causal domains. These domains encompass a wide variety of current applications of default reasoning, but here we concentrate on model-based diagnosis. The theory is unique in that it integrates a formal notion of causality with nonmonotonic reasoning techniques based on default logic and abduction. The main structure of the theory is a default causal net (DCN) representing the causal connections among propositions in the domain. The causal net provides a framework for the two nonmonotonic reasoning techniques of assuming defaults and generating explanations for observations, allowing them to be combined in a principled way.     

Consequence finding and computing answers with defaults

Consequence finding has been recognized as an important technique in many intelligent systems involving inference. In previous work, propositional or first-order clausal theories have been considered for consequence finding. In this paper, we consider consequence finding from a default theory, which consists of a first-order clausal theory and a set of normal defaults. In an extension of a default theory, consequence finding can be done with the generating defaults for the extension. Alternatively, all extensions can be represented at once with the conditional answer format, which represents how a conclusion depends on which defaults. We also propose a procedure for consequence finding and query answering in a default theory using the first-order consequence-finding procedure SOL. In computing consequences from default theories efficiently, the notion of TCS-freeness is most important to prune a large number of irrational tableaux induced by the generating defaults for an extension. In order to simulate the TCS-freeness, the refined SOL calculus called SOL-S(Γ) is adopted using skip preference and complement checking. This research is supported in part by JSPS Grant-in-Aid for Scientific Research (No. 17300051)