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this paper we present such a method. Applied to I \Sigma

This paper is part of a sequence of papers ([9],[10],[11],[12]) resulting from our Habilitation thesis [8] addressing the following question: What is the impact on the growth of extractable uniform bounds the use of various analytical principles \Gamma in a given proof of an 89--sentence might have? In particular we are interested in analyzing proofs of sentences having the form (1) 8u

Abstract. We present an extension of Heyting Arithmetic in finite types called Uniform Heyting Arithmetic (HA u) that allows for the extraction of optimized programs from constructive and classical proofs. The system HA u has two sorts of first-order quantifiers: ordinary quantifiers governed by the usual rules, and uniform quantifiers subject to stronger variable conditions expressing roughly that the quantified object is not computationally used in the proof. We combine a Kripke-style Friedman/Dragalin translation which is inspired by work of Coquand and Hofmann and a variant of the refined A-translation due to Buchholz, Schwichtenberg and the author to extract programs from a rather large class of classical first-order proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program. §1. Introduction. According to the Brouwer-Heyting-Kolmogorov interpretation of constructive logic a proof is a construction providing evidence for the proven formula [20]. Viewing this interpretation from a data-oriented perspective one arrives at the so-called proofs-as-programs paradigm associating a constructive proof with a program ‘realizing ’ the proven formula. This paradigm has been

Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1

In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are proof-theoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on non-collapsing hierarchies ( n -WKL+ ; n -WKL+ ) of principles which generalize (and for n = 0 coincide with) the so-called `weak' Konig's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context) Basic Research in Computer Science, Centre of the Danish National Research Foundation.

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in all finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a non-constructive -operator and his well-known results on the strength of this operator from the 70's. In this note we consider a weaker form LNOS (lesser numerical omniscience schema) of NOS which su#ces to derive the strong form of binary Konig's lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semi-classical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper. # Basic Research in Computer Science, Centre...

The so-called weak König's lemma WKL asserts the existence of an in nite path b in any in nite binary tree (given by a representing function f ). Based on this principle one can formulate subsystems of higher-order arithmetic which allow to carry out very substantial parts of classical mathematics but are 2 - conservative over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In [10] we established such conservation results relative to nite type extensions PRA of PRA (together with a quanti er-free axiom of choice schema). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional which selects uniformly in a given in nite binary tree f an in nite path f of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA only has a quanti er-free rule of extensionality QF-ER instead of the full axioms (E) of extensionality for all nite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be 2 -conservative over PRA, PRA + (E)+UWKL contains (and is conservative over) full Peano arithmetic PA.

In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to

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