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Rewriting Logic as a Logical and Semantic Framework
, 1993
"... 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 und ..."
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Cited by 165 (55 self)
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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...
Fundamentals Of Deductive Program Synthesis
 IEEE Transactions on Software Engineering
, 1992
"... 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 construct ..."
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Cited by 74 (1 self)
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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 deductivetableau system, a theoremproving framework particularly suitable for program synthesis. The system includes a nonclausal resolution rule, facilities for reasoning about equality, and a wellfounded 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 ...
Temporal Deductive Databases
, 1992
"... 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 expressivenes ..."
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Cited by 69 (10 self)
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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.
The Substitutional Framework for Sorted Deduction: Fundamental Results on Hybrid Reasoning
 Artificial Intelligence
, 1990
"... Researchers in artificial intelligence have recently been taking great interest in hybrid representations, among them sorted logicslogics that link a traditional logical representation to a taxonomic (or sort) representation such as those prevalent in semantic networks. This paper introduces a ge ..."
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Cited by 53 (9 self)
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Researchers in artificial intelligence have recently been taking great interest in hybrid representations, among them sorted logicslogics that link a traditional logical representation to a taxonomic (or sort) representation such as those prevalent in semantic networks. This paper introduces a general frameworkthe substitutional frameworkfor 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 ...
A Calculus of Substitutions for IncompleteProof Representation in Type Theory
, 1997
"... : In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the CurryHoward isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambdaterms. In ..."
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Cited by 18 (1 self)
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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 CurryHoward isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambdaterms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambdacalculus with new operators. First, we consider typed metavariables 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 metavariables. Unfortunately, the theory of explicit substitution calculi with typed metavariables is more complex than that of lambdacalculus. 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...
Ontological Knowledge Base Reasoning with SortHierarchy and Rigidity
 In Proceedings of Knowledge Representation (KR2004
, 2004
"... Although sorts and unary predicates are semantically identical in ordersorted logic, they are classified as different kinds of properties in formal ontology (e.g. sortal and nonsortal). This ontological analysis is an essential notion to deal with properties (or sorts) of objects in knowledge re ..."
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Cited by 14 (10 self)
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Although sorts and unary predicates are semantically identical in ordersorted logic, they are classified as different kinds of properties in formal ontology (e.g. sortal and nonsortal). 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 ordersorted logic with the ontological property classification. This logic contains types (rigid sorts), nonrigid sorts and unary predicates to distinguishably express the properties: substantial sorts, nonsubstantial sorts and nonsortal properties. We define a sorted Hornclause 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).
An ordersorted resolution with implicitly negative sorts
 In Proceedings of the 2001 Int. Conf. on Logic Programming
, 2001
"... Abstract. We usually use natural language vocabulary for sort names in ordersorted logics, and some sort names may contradict other sort names in the sorthierarchy. These implicit negations, called lexical negations in linguistics, are not explicitly prefixed by the negation connective. In this pa ..."
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Cited by 12 (10 self)
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Abstract. We usually use natural language vocabulary for sort names in ordersorted logics, and some sort names may contradict other sort names in the sorthierarchy. 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 sorthierarchy. 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
Predicate logic with sequence variables and sequence function symbols
 Proc. of the 3rd Int. Conference on Mathematical Knowledge Management. Vol. 3119 of LNCS
, 2004
"... Abstract. We describe an extension of firstorder logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, LöwenheimSkolem, and Model Existence theorems remain valid. The obtained logic can be encoded ..."
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Cited by 11 (7 self)
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Abstract. We describe an extension of firstorder logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, LöwenheimSkolem, and Model Existence theorems remain valid. The obtained logic can be encoded as a special ordersorted firstorder theory. We also define an inductive theory with sequence variables and formulate induction rules. The calculus forms a basis for the topdown systematic theory exploration paradigm. 1