Representations and evaluation strategies for feasibly approximable functions

TL;DR

This paper investigates classes of functions that can be feasibly integrated and maximized, proposing polynomial-time representations based on polynomial, rational, and piecewise polynomial approximations, reconciling theoretical hardness with practical efficiency.

Contribution

It introduces and compares polynomial-time representations for functions that enable efficient computation of integrals and maxima, resolving a paradox in computational analysis.

Findings

Piecewise polynomial and rational function representations are polytime equivalent.

All terms in a practical expressive language can be evaluated in polynomial time with the proposed representation.

Polynomial approximation and standard evaluation require exponential time for certain terms.

Abstract

A famous result due to Ko and Friedman (1982) 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 (2012). The representations we study are based on rigorous approximation by polynomials, piecewise polynomials, and rational functions. We compare these representations with respect to polytime reducibility as well as with respect to their ability to quickly evaluate symbolic expressions in a given language. We show that the representation based on rigorous approximation by piecewise polynomials is polytime equivalent to the representation based on rigorous 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.