We prove that a function definable with parameters in an o-minimal structure is bounded away from ∞ as its argument goes to ∞ by a function definable without parameters, and that this new function can be chosen independently of the parameters in the original function. This generalizes a result in [1]. Moreover, this remains true if the argument is taken to approach any element of the structure , and the function has limit any element of the structure.

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.

In [U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 89–128], the second author obtained metatheorems for the extraction of effective bounds from classical, prima facie non-constructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT and normed linear spaces and guarantee the independence of the bounds from parameters ranging over metrically bounded spaces. Recently ]), the authors obtained generalizations of these (...) metatheorems which allow one to prove similar uniformities even for unbounded spaces as long as certain local boundedness conditions are satisfied. 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 semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to the classical case, there are practically no restrictions on the logical complexity of theorems for which bounds can be extracted. Again, our metatheorems guarantee very general uniformities, even in cases where the existence of uniform bounds is not obtainable by straightforward functional analytic means. Already in the purely intuitionistic case, where the existence of effective bounds is implicit, the metatheorems allow one to derive uniformities that may not be obvious at all from a given constructive proof. Finally, we illustrate our main metatheorem by an example from metric fixed point theory. (shrink)

A uniform upper bound on a class of Turing degrees is the Turing degree of a function which parametrizes the collection of all functions whose degree is in the given class. I prove that if a is a uniform upper bound on an ideal of degrees then a is the jump of a degree c with this additional property: there is a uniform bound b<a so that b V c < a.

In a previous paper we constructed a full and faithful functor ℳ from the category of locally compact metric spaces to the category of formal topologies . Here we show that for a real-valued continuous function f, ℳ factors through the localic positive reals if, and only if, f has a uniform positive lower bound on each ball in the locally compact space. We work within the framework of Bishop constructive mathematics, where the latter notion is strictly stronger than (...) point-wise positivity. (shrink)

In this note we show that proof-theoretic uniform boundedness or bounded collection principles which allow one to formalize certain instances of countable Heine–Borel compactness in proofs using abstract metric structures must be carefully distinguished from an unrestricted use of countable Heine–Borel compactness.

Let I be a countable jump ideal in $\mathscr{D} = \langle \text{The Turing degrees}, \leq\rangle$ . The central theorem of this paper is: a is a uniform upper bound on I iff a computes the join of an I-exact pair whose double jump a (1) computes. We may replace "the join of an I-exact pair" in the above theorem by "a weak uniform upper bound on I". We also answer two minimality questions: the class of uniform upper (...) bounds on I never has a minimal member; if ∪ I = L α [ A] ∩ ω ω for α admissible or a limit of admissibles, the same holds for nice uniform upper bounds. The central technique used in proving these theorems consists in this: by trial and error construct a generic sequence approximating the desired object; simultaneously settle definitely on finite pieces of that object; make sure that the guessing settles down to the object determined by the limit of these finite pieces. (shrink)

By suitably adapting an argument of Hirschfeld , we show that there is a single Δ1 formula that defeats “bounded collection” for any model of II2 Arithmetic that is either a recursive ultrapower or an existentially complete model. Some related facts are noted. MSC: 03F30, 03C62.

Proof uses forcing on perfect trees for 2-quantifier sentences in the language of arithmetic. The result extends to exact pairs for the hyperarithmetic degrees.

We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with MODm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MOD}_{m}$$\end{document} gates and a symmetric gate. The proof is based on a relativized, generalized form of Toda’s theorem expressed in terms of closure properties of formulas under bounded universal, existential and modular counting quantifiers. This allows the main proofs to be expressed in terms of formula classes instead of Boolean (...) circuits. The uniform version of the Beigel–Tarui theorem is then obtained automatically via the Furst–Saxe–Sipser and Paris–Wilkie translations. As a special case, we obtain a uniform version of Razborov and Smolensky’s representation of AC0[p]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {AC}^{0}[p]$$\end{document} circuits. The paper is partly expository, but is also motivated by the desire to recast Toda’s theorem, the Beigel–Tarui theorem, and their proofs into the language of bounded arithmetic. However, no knowledge of bounded arithmetic is needed. (shrink)

We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a $\Sigma_1$ sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters is forceable, then (...) it is true. Further, if for every real x, $x^{\sharp}$ exists, and the second uniform indiscernible is less than $\omega_2$ , then the same holds for $\Sigma^1_4$ sentences. (shrink)

Allen, B., Arithmetizing Uniform NC, Annals of Pure and Applied Logic 53 1–50. We give a characterization of the complexity class Uniform NC as an algebra of functions on the natural numbers which is the closure of several basic functions under composition and a schema of recursion. We then define a fragment of bounded arithmetic, and, using our characterization of Uniform NC, show that this fragment is capable of proving the totality of all of the functions in (...) Uniform NC. Lastly, in the spirit of Buss, we show that any function which is definable by a ∑b1-formula in our theory is a function which is in Uniform NC. (shrink)

The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman's Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u with infinitely many zeros, the Blaschke sum of u provides the exact amount of (...) information necessary to compute the factorization of u. Along the way, we prove some uniform computability results for Blaschke products; these results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type-Two Effectivity as our foundation. (shrink)

We give an optimal lower bound in terms of large cardinal axioms for the logical strength of projective uniformization in conjuction with other regularity properties of projective sets of real numbers, namely Lebesgue measurability and its dual in the sense of category . Our proof uses a projective computation of the real numbers which code inital segments of a core model and answers a question in Hauser.

We consider a one-dimensional translation invariant point process of density one with uniformly bounded variance of the number NI of particles in any interval I. Despite this suppression of fluctuations we obtain a large deviation principle with rate function F(ρ) −L−1 log Prob(ρ) for observing a macroscopic density profile ρ(x), x ∈ [0, 1], corresponding to the coarse-grained and rescaled density of the points of the original process in an interval of length L in the limit L → ∞. F(ρ) (...) is not convex and is discontinuous at ρ ≡ 1, the typical profile. (shrink)

A fundamental question in causal inference is whether it is possible to reliably infer the manipulation effects from observational data. There are a variety of senses of asymptotic reliability in the statistical literature, among which the most commonly discussed frequentist notions are pointwise consistency and uniform consistency (see, e.g. Bickel, Doksum [2001]). Uniform consistency is in general preferred to pointwise consistency because the former allows us to control the worst case error bounds with a finite sample size. (...) In the sense of pointwise consistency, several reliable causal inference algorithms have been established under the Markov and Faithfulness assumptions [Pearl 2000, Spirtes et al. 2001]. In the sense of uniform consistency, however, reliable causal inference is impossible under the two assumptions when time order is unknown and/or latent confounders are present [Robins et al. 2000]. In this paper we present two natural generalizations of the Faithfulness assumption in the context of structural equation models, under which we show that the typical algorithms in the literature are uniformly consistent with or without modifications even when the time order is unknown. We also discuss the situation where latent confounders may be present and the sense in which the Faithfulness assumption is a limiting case of the stronger assumptions. (shrink)

We consider several puzzles of bounded rationality. These include the Allais- and Ellsberg paradox, the disjunction effect, and related puzzles. We argue that the present account of quantum cognition—taking quantum probabilities rather than classical probabilities—can give a more systematic description of these puzzles than the alternate treatments in the traditional frameworks of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always provide a deeper understanding and a true explanation of these puzzles. One reason is that quantum approaches introduce additional (...) parameters which possibly can be fitted to empirical data but which do not necessarily explain them. Hence, the phenomenological research has to be augmented by responding to deeper foundational issues. In this article, we make the general distinction between foundational and phenomenological research programs, explaining the foundational issue of quantum cognition from the perspective of operational realism. This framework is motivated by assuming partial Boolean algebras. They are combined into a uniform system via a mechanism preventing the simultaneous realization of perspectives. Gleason’s theorem then automatically leads to a distinction between probabilities that are defined by pure states and probabilities arising from the statistical mixture of pure states. This formal distinction relates to the conceptual distinction between risk and ignorance. Another outcome identifies quantum aspects in dynamic macro-systems using the framework of symbolic dynamics. Finally, we discuss several ideas that are useful for justifying complementarity in cognitive systems. (shrink)

We prove that the (non-intuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application. we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case.

Consider the bounded variable logics $L^k_{\infty\omega}$ (with k variable symbols), and $C^k_{\infty\omega}$ (with k variables in the presence of counting quantifiers $\exists^{\geq m}$ ). These fragments of infinitary logic $L_{\infty\omega}$ are well known to provide an adequate logical framework for some important issues in finite model theory. This paper deals with a translation that associates equivalence of structures in the k-variable fragments with bisimulation equivalence between derived structures. Apart from a uniform and intuitively appealing treatment of these equivalences, this (...) approach relates some interesting issues for the case of an arbitrary number of variables to the case of just three variables. Invertibility of the invariants for $\equiv_{C^3}$ , in particular, would imply a positive answer to the tempting conjecture that fixed-point logic with counting captures $\bigcup_k$ Ptime $\cap C^k_{\infty\omega}$. (shrink)

Starting in 2005, general logical metatheorems have been developed that guarantee the extractability of uniform effective bounds from large classes of proofs of theorems that involve abstract metric structures X. In this paper we adapt this to the class of CAT\)-spaces X for \ and establish a new metatheorem that explains specific bound extractions that recently have been achieved in this context as instances of a general logical phenomenon.

We develop an arithmetical theory and its variant , corresponding to “slightly nonuniform” . Our theories sit between and , and allow evaluation of log-depth bounded fan-in circuits under limited conditions. Propositional translations of -formulas provable in admit L-uniform polynomial-size Frege proofs.

We introduce and investigate the notion of uniform Lyndon interpolation property which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including \, \, \ and \ enjoy ULIP. Our proofs are modifications of Visser’s proofs of uniform interpolation property using layered bisimulations Gödel’96, logical foundations of mathematics, computer science and physics—Kurt Gödel’s legacy, Springer, Berlin, 1996). Also we give a new upper bound on the complexity of (...) class='Hi'>uniform interpolants for \ and \. (shrink)

The first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove that the first-order theory of a string automatic structure of bounded degree is decidable in doubly exponential space (for injective automatic presentations, this holds even uniformly). This result is shown to be optimal since we also present a string automatic structure of bounded degree whose first-order theory is hard for 2EXPSPACE. We prove similar (...) results also for tree automatic structures. These findings close the gaps left open in [28] by improving both the lower and the upper bounds. (shrink)

We estimate the derivation lengths of functionals in Gödel's system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies we define a (...) collapsing f unction\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal D}:T\to \omega$\end{document} so that for (closed) terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a,b$\end{document} of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} we have: If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a$\end{document} reduces to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $b$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\omega>{\cal D}(a)>{\cal D}(b).$\end{document} By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence new proof of strongly uniform termination of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of Troelstra's result on strong normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. Additionally, a slow growing analysis of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda$\end{document}-calculus. (shrink)

We discuss the two-dimensional harmonic oscillator in the presence of a uniform radial electric field around a cylindrical cavity. By including the Aharonov-Bohm flux and by assuming the existence and the absence of an infinity wall located at the radius of the cylindrical cavity, we show that bound states can be achieved around the cylindrical cavity in this two-dimensional charged harmonic oscillator in a uniform radial electric field.

We settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

We say a countable model ������ has a 0-basis if the types realized in ������ are uniformly computable. We say ������ has a (d-)decidable copy if there exists a model ������ ≅ ������ such that the elementary diagram of ������ is (d-)computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous model ������ with a 0-basis but no decidable copy. We extend this result here. Let d ≤ 0' be any low₂ degree. We show that there exists a homogeneous (...) model ������ with a 0-basis but no d-decidable copy. A degree d is 0-basis homogeneous bounding if any homogenous ������ with a 0-basis has a d-decidable copy. In previous work, we showed that the nonlow₂ $\Delta _{2}^{0}$ degrees are 0-basis homogeneous bounding. The result of this paper shows that this is an exact characterization of the 0-basis homogeneous bounding $\Delta _{2}^{0}$ degrees. (shrink)

Cimpian et al. observed that we accept generic statements of the form ‘Gs are f’ on relatively weak evidence, but that if we are unfamiliar with group G and we learn a generic statement about it, we still treat it inferentially in a much stronger way: all Gs are f. This paper makes use of notions like ‘representativeness’, ‘contingency’ and ‘relative difference’ from psychology to provide a uniform semantics of generics that explains why people accept generics based on weak (...) evidence. The spirit of the approach has much in common with Leslie’s cognition-based ideas about generics, but the semantics will be grounded on a strengthening of Cohen’s relative readings of generic sentences. In contrast to Leslie and Cohen, we propose a uniform semantic analysis of generics. The basic intuition is that a generic of the form ‘Gs are f’ is true because f is typical for G, which means that f is valuably associated with G. We will make use of Kahneman and Tversky’s Heuristics and Biases approach, according to which people tend to confuse questions about probability with questions about representativeness, to explain pragmatically why people treat many generic statements inferentially in a much stronger way. (shrink)

By exploring the hypothesis of magnetic monopoles, we consider the existence of electric fields produced by magnetic current densities. Then, we consider a uniformly rotating frame with the purpose of searching for effects of rotation on the interaction of axial electric fields with the magnetic quadrupole moment of a neutral particle. Our analysis is made through the WKB approximation. Therefore, by applying the WKB approximation, we search for bound state solutions to the Schrödinger equation in two particular cases.

If S ⊆{0,1}; * and S ′ = {0,1} * \sb S are both recognized within a certain nondeterministic time bound T then, in not much more time, one can write down tautologies A n → A′ n with unique interpolants I n that define S ∩{0,1} n ; hence, if one can rapidly find unique interpolants, then one can recognize S within deterministic time T p for some fixed p \s>0. In general, complexity measures for the problem of finding (...) unique interpolants in sentential logic yield new relations between circuit depth and nondeterministic Turing time, as well as between proof length and the complexity of decision procedures of logical theories. (shrink)

We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into R is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

In view of an enhancement of our implementation on the computer, we explore the possibility of an algorithmic optimization of the various proof-theoretic techniques employed by Kohlenbach for the synthesis of new effective uniform bounds out of established qualitative proofs in Numerical Functional Analysis. Concretely, we prove that the method of “colouring” some of the quantifiers as “non-computational” extends well to ε-arithmetization, elimination-of-extensionality and model-interpretation.

We study the Vapnik–Chervonenkis density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite $\mathrm {U}$-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.

Given a convergence theorem in analysis, under very general conditions a model-theoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory.

A nonlinear coupled wing model subject to unknown external disturbances is proposed in this paper. Since the model is modeled by partial differential equations, the traditional control design scheme based on the ordinary differential equation model is not applicable, and the control law design becomes very complex. In this paper, a new antidisturbance boundary control scheme based on a finite time convergent disturbance observer is proposed. The control laws are designed based on the new disturbance observers to make the external (...) disturbance errors converge to zero in a finite time and ensure the uniformly bounded stability of the controlled system. Finally, the effectiveness of the controllers and the finite-time convergence of disturbance errors are verified by the simulation and comparison. (shrink)

This paper proposes a novel adaptive dynamic programming approach to address the optimal consensus control problem for discrete-time multiagent systems. Compared with the traditional optimal control algorithms for MASs, the proposed algorithm is designed on the basis of the event-triggered scheme which can save the communication and computation resources. First, the consensus tracking problem is transferred into the input-state stable problem. Based on this, the event-triggered condition for each agent is designed and the event-triggered ADP is presented. Second, neural networks (...) are introduced to simplify the application of the proposed algorithm. Third, the stability analysis of the MASs under the event-triggered conditions is provided and the estimate errors of the neural networks’ weights are also proved to be ultimately uniformly bounded. Finally, the simulation results demonstrate the effectiveness of the event-triggered ADP consensus control method. (shrink)

The dominated convergence theorem implies that if is a sequence of functions on a probability space taking values in the interval [0, 1], and converges pointwise a.e., then converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence and the (...) underlying space. We prove a slight strengthening of Tao’s theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorov’s theorem, and introduce a new mode of convergence related to these notions. (shrink)

Wilkie 5 397) proved a “theorem of the complement” which implies that in order to establish the o-minimality of an expansion of with C∞ functions it suffices to obtain uniform bounds on the number of connected components of quantifier free definable sets. He deduced that any expansion of with a family of Pfaffian functions is o-minimal. We prove an effective version of Wilkie's theorem of the complement, so in particular given an expansion of the ordered field with finitely (...) many C∞ functions, if there are uniform and computable upper bounds on the number of connected components of quantifier free definable sets, then there are uniform and computable bounds for all definable sets. In such a case the theory of the structure is effectively o-minimal: there is a recursively axiomatized subtheory such that each of its models is o-minimal. This implies the effective o-minimality of any expansion of with Pfaffian functions. We apply our results to the open problem of the decidability of the theory of the real field with the exponential function. We show that the decidability is implied by a positive answer to the following problem ): given a language L expanding the language of ordered rings, if an L-sentence is true in every L-structure expanding the ordered field of real numbers, then it is true in every o-minimal L-structure expanding any real closed field. (shrink)

Dozens of articles have addressed the challenge that gambles having undefined expectation pose for decision theory. This paper makes two contributions. The first is incremental: we evolve Colyvan's ``Relative Expected Utility Theory'' into a more viable ``conservative extension of expected utility theory" by formulating and defending emendations to a version of this theory proposed by Colyvan and H\'ajek. The second is comparatively more surprising. We show that, so long as one assigns positive probability to the theory that there is a (...) uniform bound on the utility of possible gambles (and assuming a uniform bound on the amount of utility that can accrue in a fixed amount of time), standard principles of anthropic reasoning (as formulated by Brandon Carter) place lower and upper bounds on the expected values of gambles advertised as having no expectation--even assuming that with positive probability, all gambles advertised as having infinite expected utility are administered faithfully. Should one accept the uniform bound premises, this reasoning thus dissolves (or nearly dissolves, in some cases) several puzzles in infinite decision theory. (shrink)

The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by ∝(f) the separation rank of a real-valued functionfinB1(X), withXa Polish space. Notice that in (...) [3], Bourgain has provided a positive answer to this problem in the case ofKsatisfyingwithXa compact metric space. The key ingredient in Bourgain's approach is that whenever a sequence of continuous functions pointwise converges to a functionf, then the possible discontinuities of the limit function reflect a local ℓ1-structure to the sequence (fn)n. More precisely the complexity of this ℓ1-structure increases as the complexity of the discontinuities offdoes. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A.S. Kechris and A. Louveau have invented the rank rND(f) which permits the link between thec0-structure of a sequence (fn)nof uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal'sc0-theorem [11] and thec0-index theorem [2] are consequences of this interaction.Passing to the case where either (fn)nare not continuous orXis a non-compact Polish space, this nice interaction is completely lost. (shrink)

Gödel’s first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs were obtained in the seventies by Jeff Paris [Some independence results for Peano arithmetic. J. Symbolic Logic 43 725–731] , Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977] and Laurie Kirby [L. Kirby, Jeff Paris, Accessible independence results for Peano Arithmetic, Bull. of (...) the LMS 14 285–293]) and Harvey Friedman [S.G. Simpson, Non-provability of certain combinatorial properties of finite trees, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 87–117; R. Smith, The consistency strength of some finite forms of the Higman and Kruskal theorems, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 119–136] who produced the first mathematically interesting independence results in Ramsey theory and well-order and well-quasi-order theory .In this article we investigate Friedman-style principles of combinatorial well-foundedness for the ordinals below ε0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.For these independence principles we classify their phase transitions, i.e. we classify exactly the bounding conditions which lead from provability to unprovability in the induced combinatorial well-foundedness principles.As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel’s coding of finite sequences from his classical 1931 paper on his incompleteness results.This choice makes the investigation highly non-trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions. For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem of Parameswaran which apparently is far remote from Gödel’s theorem. (shrink)

This paper addresses the strength of Ramsey's theorem for pairs ($RT^2_2$) over a weak base theory from the perspective of 'proof mining'. Let $RT^{2-}_2$ denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of $\Sigma^0_1$-induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). In the (...) resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of $\forall\exists$-theorems. There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles. So from the perspective of proof unwinding the computational content of concrete proofs based on $RT^2_2$ the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of $RT^2_2$ that are primitive recursive in f. (shrink)

Let H be a proof system for quantified propositional calculus (QPC). We define the Σqj-witnessing problem for H to be: given a prenex Σqj-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq1-witnessing problems for the systems G*1and G1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G*0 and G0, (...) in which cuts are restricted to quantifier-free formulas, and prove that the Σqj-witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G*0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb0-CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G*0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID. (shrink)

Journal of Mathematical Logic, Ahead of Print. We connect learning algorithms and algorithms automating proof search in propositional proof systems: for every sufficiently strong, well-behaved propositional proof system [math], we prove that the following statements are equivalent, (1) Provable learning. [math] proves efficiently that p-size circuits are learnable by subexponential-size circuits over the uniform distribution with membership queries. (2) Provable automatability. [math] proves efficiently that [math] is automatable by non-uniform circuits on propositional formulas expressing p-size circuit lower (...) class='Hi'>bounds. Here, [math] is sufficiently strong and well-behaved if I.–III. hold: I. [math] p-simulates Jeřábek’s system [math] (which strengthens the Extended Frege system [math] by a surjective weak pigeonhole principle); II. [math] satisfies some basic properties of standard proof systems which p-simulate [math]; III. [math] proves efficiently for some Boolean function [math] that [math] is hard on average for circuits of subexponential size. For example, if III. holds for [math], then Items 1 and 2 are equivalent for [math]. We use the following modified notion of automatability in Item 2, the automating circuits output a [math]-proof of a given formula (expressing a p-size circuit lower bound for a function [math]) in non-uniform p-time in the length of a shortest [math]-proof of a closely related but different formula (expressing an average-case subexponential-size circuit lower bound for the same function [math]). If there is a function [math] which is hard on average for circuits of size [math], for each sufficiently big [math], then there is an explicit propositional proof system [math] satisfying properties I.-III., i.e. the equivalence of Items 1 and 2 holds for [math]. (shrink)

We connect learning algorithms and algorithms automating proof search in propositional proof systems: for every sufficiently strong, well-behaved propositional proof system [Formula: see text], we prove that the following statements are equivalent, (1) Provable learning. [Formula: see text] proves efficiently that p-size circuits are learnable by subexponential-size circuits over the uniform distribution with membership queries. (2) Provable automatability. [Formula: see text] proves efficiently that [Formula: see text] is automatable by non-uniform circuits on propositional formulas expressing p-size circuit lower (...) bounds. Here, [Formula: see text] is sufficiently strong and well-behaved if I.–III. hold: I. [Formula: see text] p-simulates Jeřábek’s system [Formula: see text] (which strengthens the Extended Frege system [Formula: see text] by a surjective weak pigeonhole principle); II. [Formula: see text] satisfies some basic properties of standard proof systems which p-simulate [Formula: see text]; III. [Formula: see text] proves efficiently for some Boolean function [Formula: see text] that [Formula: see text] is hard on average for circuits of subexponential size. For example, if III. holds for [Formula: see text], then Items 1 and 2 are equivalent for [Formula: see text]. We use the following modified notion of automatability in Item 2, the automating circuits output a [Formula: see text]-proof of a given formula (expressing a p-size circuit lower bound for a function [Formula: see text]) in non-uniform p-time in the length of a shortest [Formula: see text]-proof of a closely related but different formula (expressing an average-case subexponential-size circuit lower bound for the same function [Formula: see text]). If there is a function [Formula: see text] which is hard on average for circuits of size [Formula: see text], for each sufficiently big [Formula: see text], then there is an explicit propositional proof system [Formula: see text] satisfying properties I.-III., i.e. the equivalence of Items 1 and 2 holds for [Formula: see text]. (shrink)

We study effectively inseparable (abbreviated as e.i.) prelattices (i.e., structures of the form$L = \langle \omega, \wedge, \vee,0,1,{ \le _L}\rangle$whereωdenotes the set of natural numbers and the following four conditions hold: (1)$\wedge, \vee$are binary computable operations; (2)${ \le _L}$is a computably enumerable preordering relation, with$0{ \le _L}x{ \le _L}1$for everyx; (3) the equivalence relation${ \equiv _L}$originated by${ \le _L}$is a congruence onLsuch that the corresponding quotient structure is a nontrivial bounded lattice; (4) the${ \equiv _L}$-equivalence classes of 0 and 1 (...) form an effectively inseparable pair of sets). Solving a problem in (Montagna & Sorbi, 1985) we show (Theorem 4.2), that ifLis an e.i. prelattice then${ \le _L}$is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relationRthere exists a computable functionfreducingRto${ \le _L}$, i.e.,$xRy$if and only if$f\left( x \right){ \le _L}f\left( y \right)$, for all$x,y$. In fact (Corollary 5.3)${ \le _L}$is locally universal, i.e., for every pair$a{ < _L}b$and every c.e. preordering relationRone can find a reducing functionffromRto${ \le _L}$such that the range offis contained in the interval$\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also (Theorem 5.7)${ \le _L}$is uniformly dense, i.e., there exists a computable functionfsuch that for every$a,b$if$a{ < _L}b$then$a{ < _L}f\left( {a,b} \right){ < _L}b$, and if$a{ \equiv _L}a\prime$and$b{ \equiv _L}b\prime$then$f\left( {a,b} \right){ \equiv _L}f\left( {a\prime,b\prime } \right)$. Some consequences and applications of these results are discussed: in particular (Corollary 7.2) for$n \ge 1$the c.e. preordering relation on${{\rm{\Sigma }}_n}$sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s systemRorQis locally universal and uniformly dense; and (Corollary 7.3) the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)

Let ≤r and ≤sbe two binary relations on 2ℕ which are meant as reducibilities. Let both relations be closed under finite variation and consider the uniform distribution on 2ℕ, which is obtained by choosing elements of 2ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤rX, differs from the lower ≤s-cone of X. By c (...) osure under finite variation, the Kolmogorov 0-1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2ℕ which can be defined by a countable set of total continuous functionals on 2ℕ in the same way as the usual resource-bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations. (shrink)

We consider a uniform r-bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al.. In (...) particular, if X is a generalized Grassmannian \ and the rank r is less than or equal to some number that depends only on X, then E splits as a direct sum of line bundles. So we improve the main theorem of Muñoz et al. 664:141–162, 2012, Theorem 3.1) when X is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert–Mülich–Barth theorem on rational homogeneous spaces. (shrink)