A decision procedure for linear "big O" equations

TL;DR

This paper presents a decision procedure for determining the validity of linear 'big O' equations within a formal logical framework, applicable across various underlying sets and ordered rings, with extensions for growth rate functions.

Contribution

It introduces a decidable logical system for linear 'big O' equations that is independent of the underlying set and ring, and extends to functions with increasing growth rates.

Findings

Decidability of quantifier-free formulas in the specified language.

Independence of validity from the underlying set and ring.

Extensions for functions with increasing growth rates.

Abstract

Let $F$ be the set of functions from an infinite set, $S$, to an ordered ring, $R$. For $f$, $g$, and $h$ in $F$, the assertion $f = g + O(h)$ means that for some constant $C$, $|f(x) - g(x)| \leq C |h(x)|$ for every $x$ in $S$. Let $L$ be the first-order language with variables ranging over such functions, symbols for $0, +, -, \min, \max$, and absolute value, and a ternary relation $f = g + O(h)$. We show that the set of quantifier-free formulas in this language that are valid in the intended class of interpretations is decidable, and does not depend on the underlying set, $S$, or the ordered ring, $R$. If $R$ is a subfield of the real numbers, we can add a constant 1 function, as well as multiplication by constants from any computable subfield. We obtain further decidability results for certain situations in which one adds symbols denoting the elements of a fixed sequence of functions of strictly increasing rates of growth.