Abstract not found

This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifier--free choice AC--qf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non-- standard' axiom F - which does not hold in the full set--theoretic model but in the strongly majorizable functionals): From a proof GnA # +AC--qf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full set--theoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +AC--qf+F - proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +AC--qf+# for suitable #. 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.

this paper and discuss now only (PCM) in order to motivate the results of the present paper which is the second one in a sequence of papers resulting from the authors Habilitationsschrift [12]. All undefined notions are used in the sense of [14] on which this paper relies. A 0 , B 0 , C 0 , . . . always denote quantifier-free formulas. Using a convenient representation of real numbers, (PCM) can be formalized as follows: (PCM) : 0 hk(|a m )). (PCM) immediately follows from its arithmetical weakening (PCM - ) : # #k 0 n(|a m by an application of AC ar to n(|a m k + 1 1 (# IR 1 follows from the fact that real numbers are given as Cauchy sequences of rationals with fixed rate of convergence in our theories). It is well--known that a constructive functional interpretation of the negative translation of AC ar requires so--called bar-recursion and cannot be caried out e.g. in Godel's term calculus T (see [21] and [15] ). AC ar is (using classical logic) equivalent to CA ar +AC --qf, where CA ar : (g(x) = 0 0 A(x)) with A (and AC --qf is the restriction of AC ar to quantifier-free formulas) and therefore causes an immense rate of growth (when added to e.g. G 2 A ). From the work in the context of `reverse mathematics' (see e.g. [3],[20]) it is known that 1)--5) imply CA ar relatively to (a second-order version of) \ +AC --qf (see [1] for the definition of \ ). In [12] it is shown that this holds even relatively to G 2 A . In contrast to these general facts we prove in this paper a meta--theorem which in particular implies that if (PCM) is applied in a proof only to sequences (a n ) which are given explicitely in the parameters of the proposition (which is proved) then this pr...

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...

Let U and V be complete separable metric spaces, Vu compact in V and G : U IR a continuous function. For a large class of (usually non--constructive) proofs of uniqueness theorems Vu G(u, v1 ) = inf G(u, v) = G(u, v2) v1 = v2 one can extract an e#ective modulus of uniqueness # by logical analysis, i.e.

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 this paper we present another case study in the general project of proof mining which means the logical analysis of prima facie non-effective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation (developed in [17]) to analyze Cheney's simplification [6] of Jackson's original proof [10] from 1921 of the uniqueness of the best L 1 -approximation of continuous functions f # C[0, 1] by polynomials p # Pn of degree # n. Cheney's proof is non-effective in the sense that it is based on classical logic and on the non-computational principle WKL (binary Konig lemma). The result of our analysis provides the first e#ective (in all parameters f, n and #) uniform modulus of uniqueness (a concept which generalizes `strong uniqueness' studied extensively in approximation theory). Moreover, the extracted modulus has the optimal #-dependency as follows from Kroo [21]. The paper also describes how the uniform modulus of uniqueness can be used to compute the best L 1 -approximations of a fixed f # C[0, 1] with arbitrary precision. We use this result to give a complexity upper bound on the computation of the best L 1 -approximation in [24].

It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano-Weierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper

Abstract not found