Ulrich Kohlenbach

Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation

Abstract Kohlenbach, U., Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallee Poussins proof for Chebycheff approximation, Annals of Pure and Applied Logic 64 (1993) 27–94. We consider uniqueness theorems in classical analysis having the form (+) ⋀u ϵ U, v1, v2 ϵ Vu (G(u, v1)  0  G(u, v2)→v 1  v2), where U, V are complete separable metric spaces, Vu is compact in V and G:U x V → r is a constructive function. If (+) is proved by arithmetical means from analytical assumptions (++) ⋀x ϵ∈Xy ϵ Yx ∈z ϵ Z (F(x, y, z)  0) only (where X, Y, Z are complete separable metric spaces, Yx ⊂ Y is compact and F:XxYxZ→ r constructive), then we can extract from the proof of (++)→(+) an effective modulus of uniqueness, which depends on u, k but not on v1,v2, i.e., (+++) ⋀u ϵ U, v1, v2 ϵ Vu, k ϵ∈ n ⋚⋚G(u, v1)⋚, ⋚G(u, v 2)⋚ ⩽2-φuk→dv(v1,v2)⩽2-itk). Such a modulus φ can e.g. be used to give a finite algorithm which computes the (uniquely determined) zero of G(u,.) on Vu with prescribed precision if it exists classically. The extraction of φ uses a proof-theoretic combination of functional interpretation and pointwise majorization. If the proof of (++)→(+) uses only simple instances of induction, then φ is a simple mathematical operation (on a convenient standard representation of X, e.g. on f together with a modulus of uniform continuity for X  C[0,1]).

Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals

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∈IN 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 ω by quantifier–free choice AC–qf and analytical axioms Γ having the form ∀x∃y ≤ρ 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 + Γ ⊢ ∀u, k∀v ≤τ tuk∃w A0(u, k, v, w) one can extract a uniform bound Φ such that ∀u, k∀v ≤τ tuk∃w ≤ ΦukA0(u, k, v, w) holds in the full set–theoretic type structure. In case n = 2 (resp. n = 3) Φuk is a polynomial (resp. an elementary recursive function) in k, u := λx.max(u0, . . . , ux). In the present paper we show that for n ≥ 2, GnA +AC–qf+F proves a generalization of the binary Konig’s lemma yielding new conservation results since the conclusion of the above rule can be verified in Gmax(3,n)A ω in this case. In a subsequent paper we will show that many important ineffective analytical principles and theorems can be proved already in G2A +AC–qf+Γ for suitable Γ.

Effective Bounds from Ineffective Proofs in Analysis: An Application of Functional Interpretation and Majorization

We show how to extract effective bounds Φ for ∀u∀v ≤γ tu∃wG0–sentences which depend on u only (i.e. ∀u∀v ≤γ tu∃w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form (∗)∀x∃y ≤ρ sx∀zF0 (δ, ρ, τ are arbitrary finite types, η ≤ 2, G0, F0 are quantifier–free and s, t closed terms). If τ ≤ 2, (∗) can be weakened to ∀x, z∃y ≤ρ sx∀z ≤τ zF0. This is used to establish new conservation results about weak Knig’s lemma WKL. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers.

General logical metatheorems for functional analysis

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, Holder-Lipschitz, uniformly continuous, bounded and weakly quasi-nonexpansive 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.

An arithmetical hierarchy of the law of excluded middle and related principles

The topic of this paper is relative constructivism. We are concerned with classifying nonconstructive principles from the constructive viewpoint. We compare, up to provability in intuitionistic arithmetic, subclassical principles like Markovs principle, (a function-free version of) weak Konigs lemma, Posts theorem, excluded middle for simply existential and simply universal statements, and many others. Our motivations are rooted in the experience of one of the authors with an extended program extraction and of another author with bound extraction from classical proofs.

Mann iterates of directionally nonexpansive mappings in hyperbolic spaces

In a previous paper, the first author derived an explicit quantitative version of a theorem due to Borwein, Reich, and Shafrir on the asymptotic behaviour of Mann iterations of nonexpansive mappings of convex sets C in normed linear spaces. This quantitative version, which was obtained by a logical analysis of the ineffective proof given by Borwein, Reich, and Shafrir, could be used to obtain strong uniform bounds on the asymptotic regularity of such iterations in the case of bounded C and even weaker conditions. In this paper, we extend these results to hyperbolic spaces and directionally nonexpansive mappings. In particular, we obtain significantly stronger and more general forms of the main results of a recent paper by W. A. Kirk with explicit bounds. As a special feature of our approach, which is based on logical analysis instead of functional analysis, no functional analytic embeddings are needed to obtain our uniformity results which contain all previously known results of this kind as special cases.

Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces

This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the Krasnoselski-Mann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings.

New effective moduli of uniqueness and uniform a priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory

Let U and V be complete separable metric spacesVu compact in V and a continuous function. For a large class of (usually nonconstructive) proofs of uniqueness theorems one can extract an effective modulus of uniqueness φ by logical analysis, i.e. Since φ does not depend on v 1 v 2 it is an a priori estimate, which generalizes the notion of strong unicity in Chebycheff approximation theory. This applies to uniqueness proofs in Chebycheff approximation -approximation of [0, 1], and best uniform approximation by polynomials having bounded coefficients. Here we continue our proof-theoretic analysis started in [21]. A simplification of a proof by Young/Rice and a variant due to Borel are analyzed yielding explicit moduli φ and uniform a priori (lower) estimates for strong unicity which are significantly better than those obtained from de La Vallee Poussins uniqueness proof in [21]. It is explained how the numerical content of the three proofs depends on the logical form whereby certain analytical lemmas (e.g. ...

Uniform asymptotic regularity for Mann iterates

Abstract In Numer. Funct. Anal. Optim. 22 (2001) 641–656, we obtained an effective quantitative analysis of a theorem due to Borwein, Reich, and Shafrir on the asymptotic behavior of general Krasnoselski–Mann iterations for nonexpansive self-mappings of convex sets C in arbitrary normed spaces. We used this result to obtain a new strong uniform version of Ishikawas theorem for bounded C . In this paper we give a qualitative improvement of our result in the unbounded case and prove the uniformity result for the bounded case under the weaker assumption that C contains a point x whose Krasnoselski–Mann iteration ( x k ) is bounded. We also consider more general iterations for which asymptotic regularity is known only for uniformly convex spaces (Groetsch). We give uniform effective bounds for (an extension of) Groetschs theorem which generalize previous results by Kirk, Martinez-Yanez, and the author.

Proof theory and computational analysis

Abstract In this survey paper we start with a discussion how functionals of finite type can be used for the proof-theoretic extraction of numerical data (e.g. effective uniform bounds and rates of convergence) from non-constructive proofs in numerical analysis. We focus on the case where the extractability of polynomial bounds is guaranteed. This leads to the concept of hereditarily polynomial bounded analysis PBA. We indicate the mathematical range of PBA which turns out to be surprisingly large. Finally we discuss the relationship between PBA and so-called feasible analysis FA. It turns out that both frameworks are incomparable. We argue in favor of the thesis that PBA offers the more useful approach for the purpose of extracting mathematically interesting bounds from proofs. In a sequel of appendices to this paper we indicate the expressive power of PBA.