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15
Primitive Recursion for HigherOrder Abstract Syntax
 Theoretical Computer Science
, 1997
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Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Primitive recursion for higher order abstract syntax
 Carnegie Mellon University
, 1996
"... Higherorder abstract syntax is a central representation technique in logical frameworks which maps variables of the object language into variables in the metalanguage. It leads to concise encodings, but is incompatible with functions defined by primitive recursion or proofs by induction. In this p ..."
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Higherorder abstract syntax is a central representation technique in logical frameworks which maps variables of the object language into variables in the metalanguage. It leads to concise encodings, but is incompatible with functions defined by primitive recursion or proofs by induction. In this paper we propose an extension of the simplytyped lambdacalculus with iteration and case constructs which preserves the adequacy of higherorder abstract syntax encodings. The wellknown paradoxes are avoided through the use of a modal operator which obeys the laws of S4. In the resulting calculus many functions over higherorder representations can be expressed elegantly. Our central technical result, namely that our calculus is conservative over the simplytyped lambdacalculus, is proved by a rather complex argument using logical relations. We view our system as an important first step towards allowing the methodology of LF to be employed effectively in systems based on induction principles such as ALF, Coq, or Nuprl, leading to a synthesis of currently incompatible paradigms.
A most artistic package of a jumble of ideas
"... In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defen ..."
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In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel’s recasting of intuitionistic arithmetic via the “Dialectica ” interpretation, discuss the extra principles that the interpretation validates, and comment on extensionality and higher order equality. The latter sections focus on the role of majorizability considerations within the “Dialectica ” and related interpretations for extracting computational information from ordinary proofs in mathematics. I Kurt Gödel’s realism, a stance “against the current ” of his time, is now wellknown
Injecting uniformities into Peano arithmetic
, 2008
"... We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic. As a consequence, some uniform boundedness principles (not necessarily settheoretically true) are interpre ..."
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We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic. As a consequence, some uniform boundedness principles (not necessarily settheoretically true) are interpreted while maintaining unmoved the Π0 2sentences of arithmetic. We explain why this interpretation is taylored to yield conservation results.
Gödel's Dialectica interpretation and its twoway stretch
 in Computational Logic and Proof Theory (G. Gottlob et al eds.), Lecture Notes in Computer Science 713
, 1997
"... this article has appeared in Computational Logic and Proof Theory (Proc. 3 ..."
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this article has appeared in Computational Logic and Proof Theory (Proc. 3
PROOF INTERPRETATIONS AND MAJORIZABILITY
"... Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpret ..."
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Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents