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.     

Bilattices and Reasoning in Artificial Intelligence: Concepts and Foundations

The past few decades have seen a resurgence ofreasoning techniques in artificial intelligenceinvolving both classical and non-classical logics. Inhis paper, ``Multi-valued Logics: A Uniform Approach toReasoning in Artificial Intelligence', Ginsberg hasshown that through the use of bilattices,several reasoning techniques can be unified under asingle framework. A bilattice is a structure that canbe viewed as a class of truth values that canaccommodate incomplete and inconsistent informationand in certain cases default information. Inbilattice theory, knowledge is ordered along twodimensions: truth/falsity and certainty/uncertainty. By defining the corresponding bilattices as truthspaces, Ginsberg has shown that the same theoremprover can be used to simulate reasoning in firstorder logic, default logic, prioritized default logicand assumption truth maintenance system. Although thisis a significant contribution, Ginsberg's paper waslengthy and involved. This paper summarizes some ofthe essential concepts and foundations of bilatticetheory. Furthermore, it discusses the connections ofbilattice theory and several other existingmulti-valued logics such as the various three-valuedlogics and Belnap's four-valued logic. It is notedthat the set of four truth values in Belnap's logicform a lattice structure that is isomorphic to thesimplest bilattice. Subsequently, Fitting proposed aconflation operation that can be used to selectsub-sets of truth values from this and otherbilattices. This method of selecting sub-sets oftruth values provides a means for identifyingsub-logic in a bilattice.     

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.     

Predicate Logic Based Image Grammars for Complex Pattern Recognition

Predicate logic based reasoning approaches provide a means of formally specifying domain knowledge and manipulating symbolic information to explicitly reason about different concepts of interest. Extension of traditional binary predicate logics with the bilattice formalism permits the handling of uncertainty in reasoning, thereby facilitating their application to computer vision problems. In this paper, we propose using first order predicate logics, extended with a bilattice based uncertainty handling formalism, as a means of formally encoding pattern grammars, to parse a set of image features, and detect the presence of different patterns of interest. Detections from low level feature detectors are treated as logical facts and, in conjunction with logical rules, used to drive the reasoning. Positive and negative information from different sources, as well as uncertainties from detections, are integrated within the bilattice framework. We show that this approach can also generate proofs or justifications (in the form of parse trees) for each hypothesis it proposes thus permitting direct analysis of the final solution in linguistic form. Automated logical rule weight learning is an important aspect of the application of such systems in the computer vision domain. We propose a rule weight optimization method which casts the instantiated inference tree as a knowledge-based neural network, interprets rule uncertainties as link weights in the network, and applies a constrained, back-propagation algorithm to converge upon a set of rule weights that give optimal performance within the bilattice framework. Finally, we evaluate the proposed predicate logic based pattern grammar formulation via application to the problems of (a) detecting the presence of humans under partial occlusions and (b) detecting large complex man made structures as viewed in satellite imagery. We also evaluate the optimization approach on real as well as simulated data and show favorable results.     

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.     

Evidential bilattice logic and lexical inference

This paper presents an information-based logic that is applied to the analysis of entailment, implicature and presupposition in natural language. The logic is very fine-grained and is able to make distinctions that are outside the scope of classical logic. It is independently motivated by certain properties of natural human reasoning, namely partiality, paraconsistency, relevance, and defeasibility: once these are accounted for, the data on implicature and presupposition comes quite naturally.The logic is based on the family of semantic spaces known as bilattices, originally proposed by Ginsberg (1988), and used extensively by Fitting (1989, 1992). Specifically, the logic is based on a subset of bilattices that I call evidential bilattices, constructed as the Cartesian product of certain algebras with themselves. The specific details of the epistemic agent approach of the logical system is derived from the work of Belnap (1975, 1977), augmented by the use of evidential links for inferencing. An important property of the system is that it has been implemented using an extension of Fitting's work on bilattice logic programming (1989, 1991) to build a model-based inference engine for the augmented Belnap logic. This theorem prover is very efficient for a reasonably wide range of inferences.A shorter version of this material was originally presented at the Fifth International Symposium on Logic and Language, Noszvaj, Hungary, 1994. The author is now in the Mathematical Reasoning Group, Department of Artificial Intelligence, University of Edinburgh, 80 South Bridge, Edinburgh EH1 1HN, U.K.