Annotated logics were introduced in [43] and later studied in [5, 7, 31, 32]. In [31], annotations were extended to allow variables and functions, and it was argued that such logics can be used to provide a formal semantics for rule-based expert systems with uncertainty. In this paper we continue to investigate the power of this approach. First, we introduce a new semantics for such programs based on ideals of lattices. Subsequently, some proposals for multivalued logic programming [5, 7, 32, 47, 40, 18] as well as some formalisms for temporal reasoning [1, 3, 42] are shown to fit into this framework. As an interesting by-product of this investigation, we obtain a new result concerning multivalued logic programming: a model theory for Fitting's bilattice-based logic programming, which until now has not been characterized model-theoretically. This is accompanied by a corresponding proof theory. 1 Introduction Large knowledge bases can be inconsistent in many ways. Nevertheless, certain...

Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1

Abstract: Early attempts at combining multiple inheritance with nonmonotonic reasoning were based on straight forward extensions of tree-structured inheritance systems, and were theoretically unsound. In The Mathematics of Inheritance Systems, or TMOIS, Touretzky described two problems these systems cannot handle: reasoning in the presence of true but redundant assertions, and coping with ambiguity. TMOIS provided a definition and analysis of a theoretically sound multiple inheritance system, accompanied by inference algorithms. Other definitions for inheritance have since been proposed that are equally sound and intuitive, but do not always agree with TMOIS. At the heart of the controversy is a clash of intuitions about certain fundamental issues such as skepticism versus credulity, the direction in which inheritance paths are extended, and classical versus intuitive notions of consistency. Just as there are alternative logics, there may be no single "best" approach to nonmonotonic multiple inheritance. 1.

The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies.

Most known computational approaches to reasoning have problems when facing inconsistency, so they assume that a given logical system is consistent. Unfortunately, the latter is difficult to verify and very often is not true. It may happen that addition of data to a large system makes it inconsistent, and hence destroys the vast amount of meaningful information. We present a logic, called APC (annotated predicate calculus; cf. annotated logic programs of [3], that treats any set of clauses, either consistent or not, in a uniform way. In this logic, consequences of a contradiction are not nearly as damaging as in the standard predicate calculus, and meaningful information can still be extracted from an inconsistent set of formulae. APC has a resolution-based sound and complete proof procedure. We also introduce a novel notion of "epistemic entailment" and show its importance for investigating inconsistency in predicate calculus as well as its application to nonmonotonic reasoning. Most importantly, our claim that a logical theory is an adequate model of human perception of inconsistency, is actually backed by rigorous arguments.

In this paper we further develop the methodology of temporal logic as an executable imperative language, presented by Moszkowski [Mos86] and Gabbay [Gab87, Gab89] and present a concrete framework, called METATEM for executing (modal and) temporal logics. Our approach is illustrated by the development of an execution mechanism for a propositional temporal logic and for a restricted first order temporal logic.

In valid-time indeterminacy it is known that an event stored in a database did in fact occur, but it is not known exactly when. In this paper we extend the SQL data model and query language to support valid-time indeterminacy. We represent the occurrence time of an event with a set of possible instants, delimiting when the event might have occurred, and a probability distribution over that set. We also describe query language constructs to retrieve information in the presence of indeterminacy. These constructs enable users to specify their credibility in the underlying data and their plausibility in the relationships among that data. A denotational semantics for SQL’s select statement with optional credibility and plausibility constructs is given. We show that this semantics is reliable, in that it never produces incorrect information, is maximal, in that if it were extended to be more informative, the results may not be reliable, and reduces to the previous semantics when there is no indeterminacy. Although the extended data model and query language provide needed modeling capabilities, these extensions appear initially to carry a significant execution cost. A contribution of this paper is to demonstrate that our approach is useful and practical. An efficient representation of valid-time indeterminacy and efficient query processing algorithms are provided. The cost of

This article is a revised and extended version of our earlier work which appeared in Proceedings of the 3rd International Symposium on Requirements Engineering (1997), pages 78 -- 86; Authors' addresses: A. Hunter, Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK; email: A.Hunter@cs.ucl.ac.uk; B. Nuseibeh, Department of Computing, Imperial College, 180 Queen's Gate, London, SW7 2BZ, UK; email: ban@doc.ic.ac.uk.

In requirements elicitation, different stakeholders often hold different views of how a proposed system should behave, resulting in inconsistencies between their descriptions. Consensus may not be needed for every detail, but it can be hard to determine whether a particular disagreement affects the critical properties of the system. In this paper, we describe the # bel framework for merging and reasoning about multiple, inconsistent state machine models. # bel permits the analyst to choose how to combine information from the multiple viewpoints, where each viewpoint is described using an underlying multi-valued logic. The different values of our logics typically represent different levels of agreement. Our multi-valued model checker, # chek, allows us to check the merged model against properties expressed in a temporal logic. The resulting framework can be used as an exploration tool to support requirements negotiation, by determining what properties are preserved for various combinations of inconsistent viewpoints.

The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a so-called knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P ---it is the well-founded model. There is also a dual largest stable model, S k P , which has not been considered before. There is another ordering based on degree of truth. Taking the meet and the join, in the truth ordering, of the two extreme stable models s k P and S k P just mentioned, yields the alternating fixed points of [29], denoted s t P and S t P here. From s t P and S t P in turn, s k P and S k P can be produced again, using the meet and join of the knowledge ordering. All stable models are bounded by these four valuations. Further, the methods of proof apply not just to logic programs considered classically, but to logic programs over any bilattice meeting certain co...