Results 1  10
of
31
Analyzing Proofs in Analysis
 LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE
, 1993
"... ..."
Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... 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 ..."
Abstract

Cited by 34 (21 self)
 Add to MetaCart
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 quantifierfree choice ACqf 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 settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #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 settheoretic 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 # +ACqf+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 # +ACqf+# for suitable #. 1
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
Abstract

Cited by 31 (18 self)
 Add to MetaCart
In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
Pointwise hereditary majorization and some applications
 Arch. Math. Logic
, 1992
"... A pointwise version of the Howard–Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel’s T as well as Kleene/Feferma ..."
Abstract

Cited by 14 (11 self)
 Add to MetaCart
A pointwise version of the Howard–Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel’s T as well as Kleene/Feferman’s P R) is applied to systems of intuitionistic and classical arithmetic (H and Hc) in all finite types with full induction as well as to the corresponding systems with restricted induction ˆ H  \ and ˆ H  \c 1) H and ˆ H  \ are closed under a generalized fan–rule. For a restricted class of formulae this also holds for H c and ˆ H  \c 2) We give a new and very perspicuous proof that for each Φ 2 ∈ T ( P R) one can construct a functional ˜ Φ 2 ∈ T ( P R) such that ˜ Φα is a modulus of uniform continuity for Φ on {β 1 ∀n(βn ≤ αn)}. Such a modulus can also be obtained by majorizing any modulus of pointwise continuity for Φ. 3) The type structure M of all pointwise majorizable set–theoretical functionals of finite type is used to give a short proof that quantifier–free “choice ” with uniqueness (AC!) 1,0 –qf. is not provable within classical arithmetic in all finite types plus comprehension (given by the schema (C) ρ: ∃y0ρ∀xρ (yx = 0 ↔ A(x)) for arbitrary A), dependent ω–choice and bounded choice. Furthermore M separates several µ–operators. 1
Relative Constructivity
 J. Symbolic Logic
, 1996
"... This paper essentially contains material from chapter 8 of the author's Habilitationschrift. Some of the results were presented at the Logic Colloquium 94 at ClermontFerrand (see [7]). 1 As is wellknown (cf. the discussion at the end of x3 of [10]), the use of classical logic (on which the syste ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
This paper essentially contains material from chapter 8 of the author's Habilitationschrift. Some of the results were presented at the Logic Colloquium 94 at ClermontFerrand (see [7]). 1 As is wellknown (cf. the discussion at the end of x3 of [10]), the use of classical logic (on which the systems GnA ! are based) has the consequence that the extractability of an effective (and for n = 2 polynomial) bound from a proof of an 89Asentence is (in general) guaranteed only if A is quantifier free (or purely existential). In the present paper we study proofs which may use mathematically strong nonconstructive analytical principles as e.g. 1) Attainment of the maximum of f 2 C([0; 1] d ; IR) 2) Mean value theorem for integrals 3) CauchyPeano existence theorem 4) Brouwer's fixed point theorem for continuous functions f : [0; 1] d ! [0; 1] d 5) A generalization WKL 2 seq of the binary Konig's lemma WKL 6) Comprehension for negated formulas: CA ae : : 9\Phi 0ae x ae :1 0 8y ae \Gamma \Phiy = 0 0 $ :A(y) \Delta ; where A is arbitrary: as well as the nonintuitionistic logical principles 7) The `double negation shift' DNS : 8x ae ::A ! ::8x ae A for arbitrary types ae and formulas A 8) The `lesser limited principle of omniscience' LLPO : 8x 1 ; y 1 9k 0 1([k = 0 ! x IR y] [k = 1 ! y IR x]) 9) The independence of premise principle for negated formulas IP: : (:A ! 9y ae B) ! 9y ae (:A ! B); where y is not free in A, plus the schema AC of full choice but apply these principles only in the context of the intuitionistic versions (E)GnA ! i of the theories (E)G n A ! . The restriction to intuitionistic logic guarantees the extractability of (uniform) effective bounds for arbitrary 89Asentences (see theorem 4.1 below). Indeed w...
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 ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
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.
Strongly Uniform Bounds from SemiConstructive Proofs
, 2004
"... 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 nor ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
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 semiintuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain nonconstructive principles. Contrary to
The Use of a Logical Principle of Uniform Boundedness in Analysis
, 1996
"... 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 89sentence might have? ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
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 89sentence might have? In particular we are interested in analyzing proofs of sentences having the form (1) 8u