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Mathematical Logic: Proof Theory, Constructive Mathematics
[no abstract available
Strong normalisation for applied lambda calculi
We consider the untyped lambda calculus with constructors and recursively
defined constants. We construct a domain-theoretic model such that any term not
denoting bottom is strongly normalising provided all its `stratified
approximations' are. From this we derive a general normalisation theorem for
applied typed lambda-calculi: If all constants have a total value, then all
typeable terms are strongly normalising. We apply this result to extensions of
G\"odel's system T and system F extended by various forms of bar recursion for
which strong normalisation was hitherto unknown.Comment: 14 pages, paper acceptet at electronic journal LMC
Bibliography on Realizability
AbstractThis document is a bibliography on realizability and related matters. It has been collected by Lars Birkedal based on submissions from the participants in “A Workshop on Realizability Semantics and Its Applications”, Trento, Italy, June 30–July 1, 1999. It is available in BibTEX format at the following URL: http://www.cs.cmu.edu./~birkedal/realizability-bib.html
Polynomial time operations in explicit mathematics
In this paper we study (self-)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on = {0, 1}* are exactly the polynomial time computable functions. Our treatment of PTO is proof-theoretic and very much in the spirit of reductive proof theor
Consistency proof of a fragment of PV with substitution in bounded arithmetic
This paper presents proof that Buss's can prove the consistency of a
fragment of Cook and Urquhart's from which induction has been
removed but substitution has been retained.
This result improves Beckmann's result, which proves the consistency of such
a system without substitution in bounded arithmetic .
Our proof relies on the notion of "computation" of the terms of
.
In our work, we first prove that, in the system under consideration, if an
equation is proved and either its left- or right-hand side is computed, then
there is a corresponding computation for its right- or left-hand side,
respectively.
By carefully computing the bound of the size of the computation, the proof of
this theorem inside a bounded arithmetic is obtained, from which the
consistency of the system is readily proven.
This result apparently implies the separation of bounded arithmetic because
Buss and Ignjatovi\'c stated that it is not possible to prove the consistency
of a fragment of without induction but with substitution in
Buss's .
However, their proof actually shows that it is not possible to prove the
consistency of the system, which is obtained by the addition of propositional
logic and other axioms to a system such as ours.
On the other hand, the system that we have considered is strictly equational,
which is a property on which our proof relies.Comment: Submitted versio
On Bar Recursive Interpretations of Analysis.
PhDThis dissertation concerns the computational interpretation of analysis via proof interpretations,
and examines the variants of bar recursion that have been used to interpret the
axiom of choice. It consists of an applied and a theoretical component.
The applied part contains a series of case studies which address the issue of understanding
the meaning and behaviour of bar recursive programs extracted from proofs in analysis.
Taking as a starting point recent work of Escardo and Oliva on the product of selection
functions, solutions to Godel's functional interpretation of several well known theorems
of mathematics are given, and the semantics of the extracted programs described. In
particular, new game-theoretic computational interpretations are found for weak Konig's
lemma for 01
-trees and for the minimal-bad-sequence argument.
On the theoretical side several new definability results which relate various modes of
bar recursion are established. First, a hierarchy of fragments of system T based on finite
bar recursion are defined, and it is shown that these fragments are in one-to-one correspondence
with the usual fragments based on primitive recursion. Secondly, it is shown that
the so called `special' variant of Spector's bar recursion actually defines the general one.
Finally, it is proved that modified bar recursion (in the form of the implicitly controlled
product of selection functions), open recursion, update recursion and the Berardi-Bezem-
Coquand realizer for countable choice are all primitive recursively equivalent in the model
of continuous functionals.EPSR
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