The Use of a Logical Principle of Uniform Boundedness in Analysis

Kohlenbach, Ulrich

doi:10.1007/978-94-017-2109-7_7

Cited by 14 publications

(10 citation statements)

References 17 publications

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“…There are several further related approaches and variants of the aforementioned ones that we cannot list here completely. We just mention intuitionistic reverse mathematics as studied by Ishihara [26,27] and others and uniform reverse mathematics as proposed by Kohlenbach [31,30]. It is not too difficult to recognize that all these various approaches mentioned here are related in some form or the other, although they are expressed in terms that appear to be different on the first sight and they produce different and sometimes incompatible results.…”

Section: Realizability Of Theorems and Weihrauch Reducibilitymentioning

confidence: 99%

Effective Choice and Boundedness Principles in Computable Analysis

2011

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Abstract. In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn-Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example. §1. Introduction. The purpose of this paper is to propose a new approach to classify mathematical theorems according to their computational content and according to their logical complexity.2000 Mathematics Subject Classification. 03F60,03D30,03B30,03E15.

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Section: Realizability Of Theorems and Weihrauch Reducibilitymentioning

confidence: 99%

Effective Choice and Boundedness Principles in Computable Analysis

2011

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“…In particular, many more analytical principles than INF can directly be seen to have the form ∆ which avoids to have to analyse their proofs (say via WKL) in the proof mining process. Other WKL-related principles which do not have that form usually easily follow from a nonstandard principle of uniform boundedness (studied in [53,58]) which is allowed to be used in the meta-theorems and can be eliminated from the proof of the conclusion. In this way large parts of given proofs can simply be skipped in the process of proof mining.…”

Section: Theorem 33 ([46])mentioning

confidence: 99%

Proof Mining: A Systematic Way of Analysing Proofs in Mathematics

Kohlenbach¹,

Oliva²

2002

BRICS

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We call proof mining the process of logically analyzing proofs in mathematics with the aim of obtaining new information. In this survey paper we discuss, by means of examples from mathematics, some of the main techniques used in proof mining. We show that those techniques not only apply to proofs based on classical logic, but also to proofs which involve non-effective principles such as the attainment of the infimum of f ∈ C[0, 1] and the convergence for bounded monotone sequences of reals. We also report on recent case studies in approximation theory and fixed point theory where new results were obtained.

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“…While working on '2)' mentioned above and investigating what parts of analysis produce only provable recursive function(al)s which can be bounded by polynomials (see [24] for a survey) we faced the problem that already the formulation of WKL involves coding devices of exponential growth. That is why we introduced a non-standard axiom F which together with some form of quantifier-free choice proves a strong principle of uniform boundedness Σ 0 1 -UB which allows to give short proofs of the usual WKLapplications in analysis relative to very weak (polynomially bounded) systems (see [23], [25]) but does not contribute to the growth of provably recursive functionals. This axiom as well as the principle of uniform boundedness is 'non-standard' in the sense that it is not true in the full set-theoretic type structure.…”

Section: Introductionmentioning

confidence: 99%

Foundational and Mathematical Uses of Higher Types

Kohlenbach¹

1999

BRICS

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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 (Φ n -WKL + , Ψ n -WKL + ) of principles which generalize (and for n = 0 coincide with) the so-called 'weak' König'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 flexible 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 1 to be equivalent to the existence of a continuous modulus of pointwise continuity. For the direct representation of F the existence of such a modulus is independent even of full arithmetic in all finite types E-PA ω plus quantifierfree choice, as we show using a priority construction due to L. Harrington. The usual WKL-based proofs of properties of F given in reverse mathematics make use of the enrichment provided by codes of F , and WKL does not seem to be sufficient to obtain similar results for the direct representation of F in our setting. However, it turns out that Ψ 1 -WKL + is sufficient. Our conservation results for (Φ n -WKL + , Ψ n -WKL + ) are proved via a new elimination result for a strong non-standard principle of uniform Σ 0 1 -boundedness which we introduced in 1996 and which implies the WKL-extensions studied in this paper.

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The Use of a Logical Principle of Uniform Boundedness in Analysis

Cited by 14 publications

References 17 publications

Effective Choice and Boundedness Principles in Computable Analysis

Effective Choice and Boundedness Principles in Computable Analysis

Proof Mining: A Systematic Way of Analysing Proofs in Mathematics

Foundational and Mathematical Uses of Higher Types

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