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Type Classes and Overloading in HigherOrder Logic
 Theorem Proving in Higher Order Logics: TPHOLs ’97, LNCS 1275
, 1997
"... Type classes and overloading are shown to be independent concepts that can both be added to simple higherorder logics in the tradition of Church and Gordon, without demanding more logical expressiveness. In particular, modeltheoretic issues are not affected. Our metalogical results may serve as a ..."
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Cited by 69 (7 self)
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Type classes and overloading are shown to be independent concepts that can both be added to simple higherorder logics in the tradition of Church and Gordon, without demanding more logical expressiveness. In particular, modeltheoretic issues are not affected. Our metalogical results may serve as a foundation of systems like Isabelle/Pure that offer the user Haskellstyle ordersorted polymorphism as an extended syntactic feature. The latter can be used to describe simple abstract theories with a single carrier type and a fixed signature of operations.
Completion of Rewrite Systems with Membership Constraints Part II: Constraint Solving
 J. Symbolic Computation
, 1992
"... this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an ordersorted logic with context variables and can be read independently of the first part. Thi ..."
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Cited by 66 (2 self)
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this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an ordersorted logic with context variables and can be read independently of the first part. This can also be compared with unification of term schemes of various kind (Chen & Hsiang, 1991; Salzer, 1992; Comon, 1995; R. Galbav'y and M. Hermann, 1992). Indeed,
Logic Programming over Polymorphically OrderSorted Types
, 1989
"... This thesis presents the foundations for relational logic programming over polymorphically ordersorted data types. This type discipline combines the notion of parametric polymorphism, which has been developed for higherorder functional programming, with the notion of ordersorted typing, which ha ..."
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Cited by 58 (0 self)
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This thesis presents the foundations for relational logic programming over polymorphically ordersorted data types. This type discipline combines the notion of parametric polymorphism, which has been developed for higherorder functional programming, with the notion of ordersorted typing, which has been developed for equational firstorder specification and programming. Polymorphically ordersorted types are obtained as canonical models of a class of specifications in a suitable logic accommodating sort functions. Algorithms for constraint solving, type checking and type inference are given and proven correct.
Refinement Types for Logical Frameworks
 Informal Proceedings of the Workshop on Types for Proofs and Programs
, 1993
"... We propose a refinement of the type theory underlying the LF logical framework by a form of subtypes and intersection types. This refinement preserves desirable features of LF, such as decidability of typechecking, and at the same time considerably simplifies the representations of many deductive s ..."
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Cited by 43 (9 self)
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We propose a refinement of the type theory underlying the LF logical framework by a form of subtypes and intersection types. This refinement preserves desirable features of LF, such as decidability of typechecking, and at the same time considerably simplifies the representations of many deductive systems. A subtheory can be applied directly to hereditary Harrop formulas which form the basis of Prolog and Isabelle. 1 Introduction Over the past two years we have carried out extensive experiments in the application of the LF Logical Framework [HHP93] to represent and implement deductive systems and their metatheory. Such systems arise naturally in the study of logic and the theory of programming languages. For example, we have formalized the operational semantics and type system of MiniML and implemented a proof of type preservation [MP91] and the correctness of a compiler to a variant of the Categorical Abstract Machine [HP92]. LF is based on a predicative type theory with dependent t...
Assumptions of ProblemSolving Methods
 LECTURE NOTES IN ARTIFICIAL INTELLIGENCE, 1076, 9TH EUROPEAN KNOWLEDGE ACQUISITION WORKSHOP, EKAW96
, 1996
"... Assumptions of problemsolving methods refer to necessary applicability conditions of problemsolving methods, indicating that a problemsolving method is only applicable to realize a task, if the assumptions are met. In principle, such assumptions may refer to any kind of condition involved in a ..."
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Cited by 42 (14 self)
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Assumptions of problemsolving methods refer to necessary applicability conditions of problemsolving methods, indicating that a problemsolving method is only applicable to realize a task, if the assumptions are met. In principle, such assumptions may refer to any kind of condition involved in a problemsolving method's applicability, including its required domain knowledge. In this paper, we propose a conceptual organization for assumptions of problemsolving methods and suggest a formal language to describe them. For illustration we take examples from the Propose & Revise problemsolving method and from diagnosis.
(ML)²: A formal language for KADS models of expertise
, 1993
"... This paper reports on an investigation into a formal language for specifying kads models of expertise. After arguing the need for and the use of such formal representations, we discuss each of the layers of a kads model of expertise in the subsequent sections, and define the formal constructions tha ..."
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Cited by 35 (9 self)
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This paper reports on an investigation into a formal language for specifying kads models of expertise. After arguing the need for and the use of such formal representations, we discuss each of the layers of a kads model of expertise in the subsequent sections, and define the formal constructions that we use to represent the kads entities at every layer: ordersorted logic at the domain layer, metalogic at the inference layer, and dynamiclogic at the task layer. All these constructions together make up (ml) 2 , the language that we use to represent models of expertise. We illustrate the use of (ml) 2 in a small example model. We conclude by describing our experience to date with constructing such formal models in (ml) 2 , and by discussing some open problems that remain for future work. 1 Introduction One of the central concerns of "knowledge engineering" is the construction of a model of some problem solving behaviour. This model should eventually lead to the construction of a...
A Mechanization of Strong Kleene Logic for Partial Functions
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
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Cited by 28 (11 self)
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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.
AgentOriented Integration of Distributed Mathematical Services
 Journal of Universal Computer Science
, 1999
"... Realworld applications of automated theorem proving require modern software environments that enable modularisation, networked interoperability, robustness, and scalability. These requirements are met by the AgentOriented Programming paradigm of Distributed Artificial Intelligence. We argue that ..."
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Cited by 19 (10 self)
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Realworld applications of automated theorem proving require modern software environments that enable modularisation, networked interoperability, robustness, and scalability. These requirements are met by the AgentOriented Programming paradigm of Distributed Artificial Intelligence. We argue that a reasonable framework for automated theorem proving in the large regards typical mathematical services as autonomous agents that provide internal functionality to the outside and that, in turn, are able to access a variety of existing external services. This article describes...
Planning Mathematical Proofs with Methods
 JOURNAL OF INFORMATION PROCESSING AND CYBERNETICS, EIK
, 1994
"... In this article we formally describe a declarative approach for encoding plan operators in proof planning, the socalled methods. The notion of method evolves from the much studied concept tactic and was first used by Bundy. While significant deductive power has been achieved with the planning appro ..."
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Cited by 15 (5 self)
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In this article we formally describe a declarative approach for encoding plan operators in proof planning, the socalled methods. The notion of method evolves from the much studied concept tactic and was first used by Bundy. While significant deductive power has been achieved with the planning approach towards automated deduction, the procedural character of the tactic part of methods, however, hinders mechanical modification. Although the strength of a proof planning system largely depends on powerful general procedures which solve a large class of problems, mechanical or even automated modification of methods is nevertheless necessary for at least two reasons. Firstly methods designed for a specific type of problem will never be general enough. For instance, it is very difficult to encode a general method which solves all problems a human mathematician might intuitively consider as a case of homomorphy. Secondly the cognitive ability of adapting existing methods to suit novel situa...
DynamicallyTyped Computations for OrderSorted Equational Presentations (Extended Abstract)
 Proc. 21st International Colloquium on Automata, Languages, and Programming, volume 820 of Lecture Notes in Computer Science
, 1994
"... Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in ordersorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework w ..."
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Cited by 10 (8 self)
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Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in ordersorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework where equational, membership and existence formulas can be expressed. A complete deduction calculus is provided to incorporate the interaction between all these formulas. The notion of decorated terms is proposed to memorize local sort information, dynamically changed by a rewriting process. A completion procedure for equational presentations with ordered sorts computes a set of rewrite rules with which not only equational theorems of the form (t = t 0 ), but also typing theorems of the for...