A decision procedure for linear "big O" equations
<|reference_start|>A decision procedure for linear "big O" equations: 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.<|reference_end|>
@article{avigad2007a,
    title={A decision procedure for linear "big O" equations},
    author={Jeremy Avigad and Kevin Donnelly},
    journal={arXiv preprint arXiv:cs/0701073},
    year={2007},
    archivePrefix={arXiv},
    eprint={cs/0701073},
    primaryClass={cs.LO}
}