Results 1  10
of
11
A predicative strong normalisation proof for a λcalculus with interleaving inductive types
 TYPES FOR PROOF AND PROGRAMS, INTER40 A. ABEL AND T. ALTENKIRCH NATIONAL WORKSHOP, TYPES '99, SELECTED PAPERS. LECTURE NOTES IN COMPUTER SCIENCE
, 1999
"... We present a new strong normalisation proof for a λcalculus with interleaving strictly positive inductive types λ^µ which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metaleve ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
We present a new strong normalisation proof for a λcalculus with interleaving strictly positive inductive types λ^µ which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based  a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone
Functional interpretation and inductive definitions
 Journal of Symbolic Logic
"... Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1. ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
(Show Context)
Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1.
The metamathematics of ergodic theory
 THE ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theo ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how prooftheoretic methods can be used to locate their “constructive content.”
Induction and inductive definitions in fragments of second order arithmetic
 The Journal of Symbolic Logic
"... A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way, that no second order var ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae increases the strength by one inductive definition. 1 Introduction and Related Work The study of subsystems of second order arithmetic (“Analysis”) has a long tradition in proof theory. Here we investigate a fragment that is defined by a restriction of the language. By allowing quantification of a second order variable only for formulae with at most this second order variable free, we obtain a proof
Cutelimination for the mucalculus with one variable
"... We establish syntactic cutelimination for the onevariable fragment of the modal mucalculus. Our method is based on a recent cutelimination technique by Mints that makes use of Buchholz ’ Ωrule. 1 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
We establish syntactic cutelimination for the onevariable fragment of the modal mucalculus. Our method is based on a recent cutelimination technique by Mints that makes use of Buchholz ’ Ωrule. 1
A Buchholz Rule for Modal Fixed Point Logics
"... Buchholz’s Ωµ+1rules provide a major tool for the prooftheoretic analysis of arithmetical inductive definitions. The aim of this paper is to put this approach into the new context of modal fixed point logic. We introduce a deductive system based on an Ωrule tailored for modal fixed point logic an ..."
Abstract
 Add to MetaCart
(Show Context)
Buchholz’s Ωµ+1rules provide a major tool for the prooftheoretic analysis of arithmetical inductive definitions. The aim of this paper is to put this approach into the new context of modal fixed point logic. We introduce a deductive system based on an Ωrule tailored for modal fixed point logic and develop the basic techniques for establishing soundness and completeness of the corresponding system. In the concluding section we prove a cut elimination and collapsing result similar to that of Buchholz [3].
Intuitionistic Fixed Point Theories for Strictly Positive Operators
, 2001
"... In this paper it is shown that the intuitionistic fixed point theory # ID i # (strict) for # times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and # 0 2 sentences over the theory ACA i # for # times iterated arithmetic comprehension without ..."
Abstract
 Add to MetaCart
In this paper it is shown that the intuitionistic fixed point theory # ID i # (strict) for # times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and # 0 2 sentences over the theory ACA i # for # times iterated arithmetic comprehension without set parameters. This generalizes results previously due to Buchholz [5] and Arai [2].