Proof Theory and Computational Analysis

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. effectiveuniform 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 analysisFA. 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

A proof-theoretic bound extraction theorem for CAT(κ)-spaces

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(κ)-spaces X for κ > 0 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.German Science FoundationDirección General de Enseñanza Superio

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity

The Computational Strength of Extensions of Weak König’s Lemma

The weak König's lemma WKL is of crucial significance in the study of fragments of mathematics which on the one hand are mathematically strong but on the other hand have a low proof-theoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKLis also `weak' in that the tree predicate is quantifier-free. Whereas in general the computational and proof-theoretic strength increases when logically more complex trees are allowed, we show that this is not the case for trees which aregiven by formulas in a class Phi where we allow an arbitrary function quantifier prefix over bounded functions in front of a Pi^0_1-formula. This results in a schema Phi-WKL.Another way of looking at WKL is via its equivalence to the principle For all x there exists y there exists

Things that can and things that can’t be done in PRA

It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano-Weierstrass principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmeticalcomprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper we determine precisely the arithmetical and computational strength (in terms of optimal conservation results and subrecursive characterizations of provably recursive functions) of weaker function parameter-free schematic versions S− of S, therebyexhibiting different levels of strength between these principles as well as a sharp borderline between fragments of analysis which are still conservative over PRA and extensions which just go beyond the strength of PRA

Recent Advances in Σ-definability over Continuous Data Types

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas

Representations and evaluation strategies for feasibly approximable functions

A famous result due to Ko and Friedman (Theoretical Computer Science 20 (1982) 323–352) asserts that the problems of integration and maximisation of a univariate real function are computationally hard in a well-defined sense. Yet, both functionals are routinely computed at great speed in practice. We aim to resolve this apparent paradox by studying classes of functions which can be feasibly integrated and maximised, together with representations for these classes of functions which encode the information which is necessary to uniformly compute integral and maximum in polynomial time. The theoretical framework for this is the second-order complexity theory for operators in analysis which was introduced by Kawamura and Cook (ACM Transactions on Computation Theory 4(2) (2012) 5). The representations we study are based on approximation by polynomials, piecewise polynomials, and rational functions. We compare these representations with respect to polytime reducibility. We show that the representation based on approximation by piecewise polynomials is polytime equivalent to the representation based on approximation by rational functions. With this representation, all terms in a certain language, which is expressive enough to contain the maximum and integral of most functions of practical interest, can be evaluated in polynomial time. By contrast, both the representation based on polynomial approximation and the standard representation based on function evaluation, which implicitly underlies the Ko-Friedman result, require exponential time to evaluate certain terms in this language. We confirm our theoretical results by an implementation in Haskell, which provides some evidence that second-order polynomial time computability is similarly closely tied with practical feasibility as its first-order counterpart