Hierarchical Inequality Reasoning

Abstract

This paper describes a program called BOUNDER that proves inequalities between elementary functions over finite sets of constraints. Previous inequality algorithms perform well on some subset of the elementary functions, but poorly elsewhere. Although complex algorithms perform better than simple ones for most functions, exceptions exist. To overcome these problems, BOUNDER maintains a hierarchy of increasingly complex algorithms. When on fails to resolve an inequality, it tries the next. This strategy resolves more inequalities than any single algorithm. It also performs well on hard problems without wasting time on easier ones. The current hierarchy consists of four algorithms: bounds propogation, substitution, derivative inspection, and iterative approximation. Propogation is an extension of interval arithmetic that takes linear time, but ignores constraints between variables and multiple occurences of variables. The remaining algorithms consider these factors, but require exponential time. Substitution is a new, provably correct, algorithm for utilizing constraints between variables. An earlier attempt by Brooks does not terminate on all inputs and exploits fewer constraints. The final two algorithms analyze constraints between variables. Inspection examines the signs of partial derivatives. Iteration is based on several earlier algorithms from interval arithmetic.