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Fixed Points and Extensionality in Typed Functional Programming Languages
, 1992
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A Cube of Proof Systems for the Intuitionistic Predicate mu,nuLogic
 Dept. of Informatics, Univ. of Oslo
, 1997
"... This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight pr ..."
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Cited by 8 (5 self)
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This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight prooftheoretically interesting naturaldeduction calculi for this logic and propose a classification of these into a cube on the basis of the embeddibility relationships between these. 1 Introduction ¯,logics, i.e. logics with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers, have turned out to be a useful formalism in a number of computer science areas. The classical 1storder predicate ¯,logic can been used as a logic of (nondeterministic) imperative programs and as a database query language. It is also one of the relation description languages studied in descriptive complexity theory (finite model theory) (for a survey on this hi...
Inductive, projective, and retractive types
, 1993
"... We give an analysis of classes of recursive types by presenting two extensions of the simplytyped lambda calculus. The first language only allows recursive types with builtin principles of wellfounded induction, while the second allows more general recursive types which permit nonterminating com ..."
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We give an analysis of classes of recursive types by presenting two extensions of the simplytyped lambda calculus. The first language only allows recursive types with builtin principles of wellfounded induction, while the second allows more general recursive types which permit nonterminating computations. We discuss the expressive power of the languages, examine the properties of reductionbased operational semantics for them, and give examples of their use in expressing iteration over large ordinals and in simulating both callbyname and callbyvalue versions of the untyped lambda calculus. The motivations for this work come from category theoretic models. 1
Another iteration on “A synthesis of several sorting algorithms”
, 1994
"... In “A synthesis of several sorting algorithms”, Darlington showed how to use program transformation techniques to develop versions of six wellknown sorting algorithms. We provide more evidence for the naturalness of the resulting taxonomy of algorithms by showing how it follows almost immediately f ..."
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In “A synthesis of several sorting algorithms”, Darlington showed how to use program transformation techniques to develop versions of six wellknown sorting algorithms. We provide more evidence for the naturalness of the resulting taxonomy of algorithms by showing how it follows almost immediately from a consideration of the types of the objects involved. By exploiting the natural operations of iteration and coiteration over recursively defined data types, we may automatically derive the structure of each algorithm. 1
Fixpoint Computations and Coiteration (Extended Abstract)
"... ) Brian T. Howard Department of Computer and Information Sciences Kansas State University bhoward@cis.ksu.edu Abstract An extension of the simplytyped lambda calculus is presented which contains both wellstructured inductive and coinductive types, and which also identifies a class of types for wh ..."
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) Brian T. Howard Department of Computer and Information Sciences Kansas State University bhoward@cis.ksu.edu Abstract An extension of the simplytyped lambda calculus is presented which contains both wellstructured inductive and coinductive types, and which also identifies a class of types for which general recursion is possible. The motivations for this work are certain natural constructions in category theory, in particular the notion of an algebraically bounded functor, due to Freyd. We propose that this is a particularly elegant language in which to work with recursive objects, since the potential for general recursion is contained in a single operator which interacts well with the facilities for bounded iteration and coiteration. 1 Introduction In designing typed languages that include recursion, there has long been a tension between the structure provided by types based on wellfounded induction and the freedom permitted by types based on general recursion. Very few languages...
Least and Greatest Fixed Points in Intuitionistic Natural Deduction
, 2002
"... This paper is a comparative study of a number of (intensionalsemantically distinct) least and greatest fixed point operators that naturaldeduction proof systems for intuitionistic logics can be extended with in a prooftheoretically defendable way. Eight pairs of such operators are analysed. The e ..."
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This paper is a comparative study of a number of (intensionalsemantically distinct) least and greatest fixed point operators that naturaldeduction proof systems for intuitionistic logics can be extended with in a prooftheoretically defendable way. Eight pairs of such operators are analysed. The exposition is centered around a cubeshaped classification where each node stands for an axiomatization of one pair of operators as logical constants by intended proof and reduction rules and each arc for a proof and reductionpreserving encoding of one pair in terms of another. The three dimensions of the cube reflect three orthogonal binary options: conventionalstyle vs. Mendlerstyle, basic (``[co]iterative'') vs. enhanced (``primitive[co]recursive''), simple vs. courseofvalue [co]induction. Some of the axiomatizations and encodings are wellknown; others, however, are novel; the classification into a cube is also new. The differences between the least fixed point operators considered are illustrated on the example of the corresponding natural number types.
Another Iteration on Darlington's "A Synthesis of Several Sorting Algorithms"
, 1994
"... this paper was presented at California State University, Northridge. This work was partially supported by a grant from the Office of Naval Research. References ..."
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this paper was presented at California State University, Northridge. This work was partially supported by a grant from the Office of Naval Research. References