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Rewriting logic [72] is proposed as a logical framework in which other logics can be represented, and as a semantic framework for the specification of languages and systems. Using concepts from the theory of general logics [70], representations of an object logic L in a framework logic F are understood as mappings L ! F that translate one logic into the other in a conservative way. The ease with which such maps can be defined for a number of quite different logics of interest, including equational logic, Horn logic with equality, linear logic, logics with quantifiers, and any sequent calculus presentation of a logic for a very general notion of "sequent," is discussed in detail. Using the fact that rewriting logic is reflective, it is often possible to reify inside rewriting logic itself a representation map L ! RWLogic for the finitely presentable theories of L. Such a reification takes the form of a map between the abstract data types representing the finitary theories of...

An informal tutorial is presented for program synthesis, with an emphasis on deductive methods. According to this approach, to construct a program meeting a given specification, we prove the existence of an object meeting the specified conditions. The proof is restricted to be sufficiently constructive, in the sense that, in establishing the existence of the desired output, the proof is forced to indicate a computational method for finding it. That method becomes the basis for a program that can be extracted from the proof. The exposition is based on the deductive-tableau system, a theorem-proving framework particularly suitable for program synthesis. The system includes a nonclausal resolution rule, facilities for reasoning about equality, and a well-founded induction rule. INTRODUCTION This is an introduction to program synthesis, the derivation of a program to meet a given specification. It focuses on the deductive approach, in which the derivation task is regarded as a problem of ...

We survey a number of approaches to the problem of finite representation of infinite temporal extensions. Two of them, Datalog 1S and Templog, are syntactical extensions of Datalog; the third is based on repetition and arithmetic constraints. We provide precise characterizations of the expressiveness and the computational complexity of these languages. We also describe query evaluation methods.

Researchers in artificial intelligence have recently been taking great interest in hybrid representations, among them sorted logics---logics that link a traditional logical representation to a taxonomic (or sort) representation such as those prevalent in semantic networks. This paper introduces a general framework---the substitutional framework---for integrating logical deduction and sortal deduction to form a deductive system for sorted logic. This paper also presents results that provide the theoretical underpinnings of the framework. A distinguishing characteristic of a deductive system that is structured according to the substitutional framework is that the sort subsystem is invoked only when the logic subsystem performs unification, and thus sort information is used only in determining what substitutions to make for variables. Unlike every other known approach to sorted deduction, the substitutional framework provides for a systematic transformation of unsorted deductive systems ...

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: In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the Curry-Howard isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambda-terms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambda-calculus with new operators. First, we consider typed meta-variables to represent the parts of a proof that are under construction, and second, we make explicit the substitution mechanism in order to deal with capture of variables that are bound in terms containing meta-variables. Unfortunately, the theory of explicit substitution calculi with typed meta-variables is more complex than that of lambda-calculus. And worse, in general they do not share the same properties, notably with respect to confluence and strong normalization. A contribution of this thesis is to show that the pr...

Abstract. We usually use natural language vocabulary for sort names in order-sorted logics, and some sort names may contradict other sort names in the sort-hierarchy. These implicit negations, called lexical negations in linguistics, are not explicitly prefixed by the negation connective. In this paper, we propose the notions of structured sorts, sort relations, and the contradiction in the sort-hierarchy. These notions specify the properties of these implicit negations and the classical negation, and thus, we can declare the exclusivity and the totality between two sorts, one of which is affirmative while the other is negative. We regard the negative affix as a strong negation operator, and the negative lexicon as an antonymous sort that is exclusive to its counterpart in the hierarchy. In order to infer from these negations, we integrate a structured sort constraint system into a clausal inference system. 1

Although sorts and unary predicates are semantically identical in order-sorted logic, they are classified as different kinds of properties in formal ontology (e.g. sortal and non-sortal). This ontological analysis is an essential notion to deal with properties (or sorts) of objects in knowledge representation and reasoning. In this paper, we propose an extension of an order-sorted logic with the ontological property classification. This logic contains types (rigid sorts), non-rigid sorts and unary predicates to distinguishably express the properties: substantial sorts, non-substantial sorts and non-sortal properties. We define a sorted Horn-clause calculus for such property expressions in a knowledge base. Based on the calculus, we develop a reasoning algorithm for many separated knowledge bases where each knowledge base can extract rigid property information from other knowledge bases (called rigid property derivation).

The notions of meta-ontology enhance the ability to process knowledge in information systems; in particular, ontological property classification deals with the kinds of properties in taxonomic knowledge based on a philosophical analysis. The goal of this paper is to devise a reasoning mechanism to check the ontological and logical consistency of knowledge bases, which is important for reasoning services on taxonomic knowledge. We first consider an ontological property classification that is extended to capture individual existence and time and situation dependencies. To incorporate