No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Proof Mining in Topological Dynamics
Abstract
A famous theorem by van der Waerden states the following: Given any finite colouring of the integers, one colour contains arbitrar- ily long arithmetic progressions. Equivalently, for every q, k there is an N = N(q, k) such that for every q-colouring of an interval of length N one colour contains a progression of length k. An obvious question is: What is the growth rate of N(q, k)? Some proofs, like van der Waerden's combinatorial argument, answer this question directly, while the topo- logical proof by Furstenberg and Weiss does not. We present an analysis of (Girard's variant of) Furstenberg and Weiss' proof based on monotone functional interpretation, both yielding bounds and providing a general illustration of proof mining in topological dynamics. The bounds do not improve previous results by Girard, but only - as is also revealed by the analysis - because the combinatorial proof and the topological dynamics proof in principle are identical. "Proof mining" is the activity of extracting additional information from proofs in mathematics and computer science. The two main types of additional infor- mation that may be extracted are quantitative and qualitative information. An example of the former is extracting a rate of convergence from a proof that a certain iteration sequence in a compact metric space converges. An example of the latter is establishing that the convergence is uniform in the starting point of the iteration or that the result not only holds for compact metric spaces, but already for bounded ones. Naturally, when the proof to be analysed is constructive, we expect quantitative information - realizers or bounds - to be present explicitly in the proof, and qualitative strengthenings of the theorem may follow trivially from the exact structure of the realizers.
... Such a rate is readily transformed into a map A that validates (10). Secondly, the assumption that T is asymptotically regular can be dropped in the Hilbert space case. ...
... Overall, our aim is to construct maps Φ and Ψ as in (4) and (5). These will only depend on given maps A, η and numbers b, c, q as in (6)(7)(8)(9)(10). In addition to this quantitative data, we keep the assumption that T : C → C has a fixed point and is nonexpansive on the convex subset C of our uniformly convex Banach space. ...
... In view of this fact, it is reasonable to work with a rate of metastability from the outset, as this makes for a computationally weaker assumption. Concerning the next definition, recall that our standing assumptions provide a map (ε, g, h) → A(ε, g, h) that validates (10). The function γ is given by Definition 2.7. ...
We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Ces\`aro means of nonexpansive maps on Banach spaces.
... We now give an alternative characterization of total boundedness used in the context of proof mining first in [13]: ...
... Apply (13) with l := k and δ := g Φ ++ (k) and denote N 0 := ψ(k, g Φ ++ (k) ) to get that ...
We provide in a unified way quantitative forms of strong convergence results
for numerous iterative procedures which satisfy a general type of Fejer
monotonicity where the convergence uses the compactness of the underlying set.
These quantitative versions are in the form of explicit rates of so-called
metastability in the sense of T. Tao. Our approach covers examples ranging from
the proximal point algorithm for maximal monotone operators to various fixed
point iterations (x_n) for firmly nonexpansive, asymptotically nonexpansive,
strictly pseudo-contractive and other types of mappings. Many of the results
hold in a general metric setting with some convexity structure added (so-called
W-hyperbolic spaces). Sometimes uniform convexity is assumed still covering the
important class of CAT(0)-spaces due to Gromov.
... [13,14,21]) and paved the way for many new applications in fixed point theory, geodesic geometry, ergodic theory and topological dynamics that were guaranteed to be possible by these proof theoretic results (see e.g. [1,3,4,7,8,16,20,23,22]). For applications of related forms of 'proof mining' in proof theory itself see Mints [25,26]. ...
... Henceỹ = R 0 and so, by (8), y 1 − y = R 0, i.e. y = R y 1 = R y 2 . Again by (8) this gives A L (x 2 , y 2 ). ...
... Obviously, Argmin k+1 ( f ) ⊆ Argmin k ( f ) for every k ∈ N and Argmin( f ) = k∈N Argmin k ( f ). We recall [5,12] that a modulus of total boundedness for a metric space (X, d) is a function α : N → N such that for any k ∈ N and any sequence (x n ) in X, ...
We apply methods of proof mining to obtain uniform quantitative bounds on the strong convergence of the prox-imal point algorithm for finding minimizers of convex, lower semicontinuous, proper functions in CAT(0) spaces. Thus, for uniformly convex functions, we compute rates of convergence, while, for totally bounded CAT(0) spaces, we apply methods introduced by Kohlenbach, Leuştean and Nicolae to compute rates of metastability.
... [75][76][77]), and it is wellknown that nets and filters provide an equivalent framework (see [7] (f) Domain theory and associated topologies are studied in RM ( [64,78]), and nets take central stage in domain theory in [44,45,47]. (g) Nets are used in topological dynamics ( [41]), which is studied in the proof mining program ( [43,60]). More generally, ergodic theory is also studied in RM ( [32]) and proof theory ( [6]), and it is therefore a natural question how strong basic results regarding nets are. ...
Nets are generalisations of sequences involving possibly uncountable index sets; this notion was introduced about a century ago by Moore and Smith. They also established the generalisation to nets of various basic theorems of analysis due to Bolzano-Weierstrass, Dini, Arzela, and others. More recently, nets are central to the development of domain theory, providing intuitive definitions of the associated Scott and Lawson topologies, among others. This paper deals with the Reverse Mathematics study of basic theorems about nets. We restrict ourselves to nets indexed by subsets of Baire space, and therefore third-order arithmetic, as such nets suffice to obtain our main results. Over Kohlenbach's base theory of higher-order Reverse Mathematics, the Bolzano-Weierstrass theorem for nets implies the Heine-Borel theorem for uncountable covers. We establish similar results for other basic theorems about nets and even some equivalences, e.g. for Dini's theorem for nets. Finally, we show that replacing nets by sequences is hard, but that replacing sequences by nets can obviate the need for the Axiom of Choice, a foundational concern in domain theory. In an appendix, we study the power of more general index sets, establishing that the 'size' of a net is directly proportional to the power of the associated convergence theorem.
... Obviously, Argmin k+1 ( f ) ⊆ Argmin k ( f ) for every k ∈ N and Argmin( f ) = k∈N Argmin k ( f ). We recall [5,12] that a modulus of total boundedness for a metric space (X, d) is a function α : N → N such that for any k ∈ N and any sequence (x n ) in X, ...
We apply methods of proof mining to obtain uniform quantitative bounds on the strong convergence of the proximal point algorithm for finding minimizers of convex, lower semicontinuous proper functions in CAT(0) spaces. Thus, for uniformly convex functions we compute rates of convergence, while, for totally bounded CAT(0) spaces we apply methods introduced by Kohlenbach, the first author and Nicolae to compute rates of metastability.
... This notion was first used in [4] to analyze, using proof mining methods, the Furstenberg-Weiss proof of the Multiple Birkhoff Recurrence Theorem. One can easily see that X is totally bounded if and only if it has a modulus of total boundedness. ...
We compute, using techniques originally introduced by Kohlenbach, the first author and Nicolae, uniform rates of metastability for the proximal point algorithm in the context of CAT(0) spaces (as first considered by Bacak), specifically for the case where the ambient space is totally bounded. This result is part of the program of proof mining, which aims to apply methods of mathematical logic with the purpose of extracting quantitative information out of ordinary mathematical proofs, which may not be necessarily constructive.
... As pointed out in [10], where two different moduli are considered, C is totally bounded if and only if C has a modulus of total boundedness. This quantitative version of total boundedness was used in [5] to obtain, also using proof mining, quantitative results in topological dynamics. ...
We compute uniform rates of metastability for the Ishikawa iteration of a Lipschitz pseudo-contractive self-mapping of a compact convex subset of a Hilbert space. This extraction is an instance of the proof mining program that aims to apply tools from mathematical logic in order to extract the hidden quantitative content of mathematical proofs. We prove our main result by applying methods developed by Kohlenbach, the first author and Nicolae for obtaining quantitative versions of strong convergence results for generalized Fej\'er monotone sequences in compact subsets of metric spaces.
... First of all, we would like to see if the interpretations that we have developed in this paper could be used to " unwind " or " proof-mine " nonstandard arguments. Nonstandard arguments have been used in areas where proof-mining techniques have also been successful, such as metric fixed point theory (for methods of nonstandard analysis applied to metric fixed point theory, see [1, 17] ; for application of proof-mining to metric fixed point theory, see [7, 9, 18, 22, 24, 29]) and ergodic theory (for a nonstandard proof of an ergodic theorem, see [15] ; for applications of proof-mining to ergodic theory , see [3, 4, 10, 11, 21, 23, 39]), therefore this looks quite promising. For the former type of applications to work in full generality, one would have to extend our functional interpretation to include types for abstract metric spaces, as in [12, 19]. ...
We introduce constructive and classical systems for nonstandard arithmetic
and show how variants of the functional interpretations due to Goedel and
Shoenfield can be used to rewrite proofs performed in these systems into
standard ones. These functional interpretations show in particular that our
nonstandard systems are conservative extensions of extensional Heyting and
Peano arithmetic in all finite types, strengthening earlier results by
Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting
algorithm can be used for term extraction purposes. To conclude the paper, we
will point out some open problems and directions for future research and
mention some initial results on saturation principles.
... Unwinding of proofs has had applications in number theory [104,117], algebra [31,29,27,28,114] and combinatorics [6,50,69,154]. However, the most systematic development of proof mining took place in connection with applications to approximation theory [79,80,81,123,100], metric fixed point theory [85,86,87,95,94,89,45,12,13,14,96,109,110,15,97,111,17,16], as well as ergodic theory and topological dynamics [2,46,97,47]. ...
In this survey we present some recent applications of proof mining to the fixed point theory of (asymptotically) nonexpansive mappings and to the metastability (in the sense of Terence Tao) of ergodic averages in uniformly convex Banach spaces.
We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Cesàro means of non-expansive maps on Banach spaces.
Nets are generalisations of sequences involving possibly uncountable index sets; this notion was introduced about a century ago by Moore and Smith. They also established the generalisation to nets of various basic theorems of analysis due to Bolzano–Weierstrass, Dini, Arzelà, and others. More recently, nets are central to the development of domain theory, providing intuitive definitions of the associated Scott and Lawson topologies, among others. This paper deals with the Reverse Mathematics study of basic theorems about nets. We restrict ourselves to nets indexed by subsets of Baire space, and therefore third-order arithmetic, as such nets suffice to obtain our main results. Over Kohlenbach’s base theory of higher-order Reverse Mathematics, the Bolzano–Weierstrass theorem for nets implies the Heine–Borel theorem for uncountable covers. We establish similar results for other basic theorems about nets and even some equivalences, e.g. for Dini’s theorem for nets. Finally, we show that replacing nets by sequences is hard, but that replacing sequences by nets can obviate the need for the Axiom of Choice, a foundational concern in domain theory. In an appendix, we study the power of more general index sets, establishing that the ‘size’ of a net is directly proportional to the power of the associated convergence theorem.
2009 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium '09 - Volume 16 Issue 1
0.1 What is Ramsey Theory and why did we write this book?.. 7 0.2 What is a purely combinatorial proof?............. 8 0.3 Who could read this book?.................... 9 0.4 Abbreviations used in this book................. 9
In previous papers we have developed proof-theoretic techniques for ex- tracting eective uniform bounds from large classes of ineective existence proofs in functional analysis. 'Uniform' here means independence from pa- rameters in compact spaces. A recent case study in xed point theory sys- tematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to xed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting eective uniform bounds and trans- forming the original proof into one for the stronger uniformity result. Our
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.
This tutorial gives an introduction to an applied form of proof theory that has its roots in G. Kreisel's pioneering ideas on 'unwinding proofs' but has evolved into a systematic activity only during that last 10 years. The general approach is to apply proof theoretic techniques (notably specially designed proof interpretations) to concrete mathematical proofs with the aim of extracting new quantitative results (such as eective bounds) as well as new qualitative uniformity results from (prima facie ineective) proofs. During the last years logical metatheorems have been developed which guarantee the extractability of highly uniform bounds from large classes of proofs in nonlinear analysis. 'Uniform' here refers to the fact that the bounds are independent from parameters in abstract classes of spaces as long as some local bounds on certain metric distances are given. In recent years interesting conections between this type of applied proof theory and Terence Tao's program of 'finite analysis' emerged. Part I outlines the general program of this kind of applied proof theory motivated by examples from dierent parts of mathematics and emphasizes some of the problems one has to address. Part II develops some of the logical machinery that guarantees the extractability of uniform bounds from large classes of proofs and outlines a few typical applications in nonlinear analysis, ergodic theory and topological dynamics. The tutorial does not require specific proof-theoretic prerequisites.
Intuitionistic formal systems.- Models and computability.- Realizability and functional interpretations.- Normalization theorems for systems of natural deduction.- Applications of Kripke models.- Iterated inductive definitions, trees and ordinals.- Erratum.
A new proof of Szemeredi's Theorem. Geometric and Functional Analaysis 11
- T Gowers
T. Gowers. A new proof of Szemeredi's Theorem. Geometric and Functional
Analaysis 11 2001, 465-588.
- B M Landman
- A Robertson