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Inductive Families
 Formal Aspects of Computing
, 1997
"... A general formulation of inductive and recursive definitions in MartinLof's type theory is presented. It extends Backhouse's `DoItYourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. Th ..."
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Cited by 65 (13 self)
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A general formulation of inductive and recursive definitions in MartinLof's type theory is presented. It extends Backhouse's `DoItYourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalization to type theory of MartinLof's theory of iterated inductive definitions in predicate logic. Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalized induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion. The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applic...
The Wellfounded Semantics Is the Principle of Inductive Definition
 Logics in Arti Intelligence
, 1998
"... . Existing formalisations of (transfinite) inductive definitions in constructive mathematics are reviewed and strong correspondences with LP under least model and perfect model semantics become apparent. I point to fundamental restrictions of these existing formalisations and argue that the wellfou ..."
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Cited by 43 (26 self)
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. Existing formalisations of (transfinite) inductive definitions in constructive mathematics are reviewed and strong correspondences with LP under least model and perfect model semantics become apparent. I point to fundamental restrictions of these existing formalisations and argue that the wellfounded semantics (wfs) overcomes these problems and hence, provides a superior formalisation of the principle of inductive definition. The contribution of this study for LP is that it (re )introduces the knowledge theoretic interpretation of LP as a logic for representing definitional knowledge. I point to fundamental differences between this knowledge theoretic interpretation of LP and the more commonly known interpretations of LP as default theories or autoepistemic theories. The relevance is that differences in knowledge theoretic interpretation have strong impact on knowledge representation methodology and on extensions of the LP formalism, for example for representing uncertainty. Keywo...
A New Perspective on Integrating Functional and Logic Languages
 Languages, Proceedings of the 3rd International Conference on Fifth Generation Computer Systems
, 1992
"... Traditionally the integration of functional and logic languages is performed by attempting to integrate their semantic logics in some way. Many languages have been developed by taking this approach, but none manages to exploit fully the programming features of both functional and logic languages and ..."
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Cited by 14 (0 self)
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Traditionally the integration of functional and logic languages is performed by attempting to integrate their semantic logics in some way. Many languages have been developed by taking this approach, but none manages to exploit fully the programming features of both functional and logic languages and provide a smooth integration of the two paradigms. We propose that improved integrated systems can be constructed by taking a broader view of the underlying semantics of logic programming. A novel integrated language paradigm, Definitional Constraint Programming (DCP), is proposed. DCP generalises constraint logic programming by admitting userdefined functions via a purely functional subsystem and enhances it with the power to solve constraints over functional programs. This constraint approach to integration results in a homogeneous unified system in which functional and logic programming features are combined naturally. 1 Introduction During the past ten years the integration of funct...
Firstorder Lax Logic as a Framework for Constraint Logic Programming
, 1997
"... In this report we introduce a new prooftheoretic approach to the semantics of Constraint Logic Programming, based on an intuitionistic firstorder modal logic, called QLL. The distinguishing feature of this new approach is that the logic calculus of QLL is used not only to capture the usual exte ..."
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Cited by 12 (4 self)
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In this report we introduce a new prooftheoretic approach to the semantics of Constraint Logic Programming, based on an intuitionistic firstorder modal logic, called QLL. The distinguishing feature of this new approach is that the logic calculus of QLL is used not only to capture the usual extensional aspects of Logic Programming, i.e. "which queries are successful, " but also some of the intensional aspects, i.e. "what is the answer constraint and how is it constructed." It provides for a direct link between the modeltheoretic and the operational semantics following a formulasasprograms and proofsasconstraints principle. This approach makes use of logic in a different way than other approaches based on logic calculi. On the one side it is to be distinguished from the wellknown provability semantics which is concerned merely with what is derivable as opposed to how it is derivable, paying attention to the fact that it is the how that determines the answer constraint. ...
Logic Programming, Functional Programming, and Inductive Definitions
 In Extensions of Logic Programming, volume 475 of LNCS
, 1991
"... Machine. It is incomplete due to depthfirst search, but presumably there could be a version using iterative deepening. An ORparallel machine such as DelPhi [12] could support such languages in future. Functions make explicit the granularity for ORparallelism: evaluation is deterministic while sea ..."
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Cited by 10 (0 self)
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Machine. It is incomplete due to depthfirst search, but presumably there could be a version using iterative deepening. An ORparallel machine such as DelPhi [12] could support such languages in future. Functions make explicit the granularity for ORparallelism: evaluation is deterministic while search is not.
Program Development Schemata as Derived Rules
, 2000
"... This paper makes several contributions towards a clarified view of schemabased program development. First, we propose that schemata can be understood, formalized, and used in a simple way: program development schemata are derived rules. We mean this in the standard sense of a derived rule of infere ..."
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Cited by 9 (2 self)
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This paper makes several contributions towards a clarified view of schemabased program development. First, we propose that schemata can be understood, formalized, and used in a simple way: program development schemata are derived rules. We mean this in the standard sense of a derived rule of inference in logic. A schema like Figure i can be formulated as a rule stating that the conclusion follows from the premises defining F, G, and the applicability conditions. By deriving the rule in an axiomatic theory, we validate a semantic statement about it: the conclusion of the rule holds in every model where both the axioms of the theory and the premises of the rule are true. Hence, by selecting a language to work in we control which development schemata are formalizable, and by selecting a theory we determine which schemata are derivable
Deriving and Applying Logic Program Transformers
 In Algorithms, 30 P. Anderson and D. Basin Concurrency and Knowledge (1995 Asian Computing Science Conference), volume 1023 of LNCS
, 1995
"... We present a methodology for logic program development based on the use of verified transformation templates. We use the Isabelle Logical Framework to formalize transformation templates as inference rules. We derive these rules in higherorder logic and afterwards use higherorder unification to app ..."
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Cited by 5 (4 self)
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We present a methodology for logic program development based on the use of verified transformation templates. We use the Isabelle Logical Framework to formalize transformation templates as inference rules. We derive these rules in higherorder logic and afterwards use higherorder unification to apply them to develop programs in a deductive synthesis style. Our work addresses the pragmatics of template formalization and application as well as which theories and semantics of programs and data we require to derive templates. Key words: program transformation, logic programs, higherorder unification, logical frameworks. 1 Introduction We investigate the transformation of logic programs based on the use of transformation templates which formalize equivalences between schemata representing logic programs. Our focus and contributions center on three problems: 1. How may we acquire and formalize useful templates? 2. What foundations are appropriate for formally demonstrating the correctn...
Semantics of Constraint Logic Programs with Bounded Quantifiers
, 1998
"... We survey the areas of Constraint programming, Bounded Quantifiers and Collection Types, then we describe an extension of constraint logic programming by bounded quantifiers. Bounded quantifiers provide the support for finite domain constraint programming in a natural way. We define several semantic ..."
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Cited by 3 (0 self)
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We survey the areas of Constraint programming, Bounded Quantifiers and Collection Types, then we describe an extension of constraint logic programming by bounded quantifiers. Bounded quantifiers provide the support for finite domain constraint programming in a natural way. We define several semantics for constraint logic programs with bounded quantifiers and prove their equivalence. Our results can be used to define semantics for some existing constraint logic programming languages, like cc(FD).
FALCON: Functional and Logic Language with Constraints
, 1993
"... Introduction During the past ten years, integration of functional and logic programming languages has attracted much research. An extensive survey and classification of their results can be found in [GLDD90]. Traditionally this integration is performed by attempting to integrate the respective sema ..."
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Cited by 3 (2 self)
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Introduction During the past ten years, integration of functional and logic programming languages has attracted much research. An extensive survey and classification of their results can be found in [GLDD90]. Traditionally this integration is performed by attempting to integrate the respective semantic logics of functional and logic languages in some way, resulting in a "super logic language". As we presented in Chapter ??, the conventional understanding of logic programming is that a logic program defines a logical theory and computation is attempting to prove that a query is a logical consequence of this theory. Taking this view, integration is regarded as enhancing the original logic underlying a logic programming system to cope with functional programming features and results in a new logic programming system. Most approaches take firstorder equational logic as the semantic logic of functional languages and combine it with Horn clause logic. The following