# Solving Term Inequalities

Gerald E Peterson

This work pertains to the Knuth-Bendix (KB) algorithm which tries to find a complete set of reductions from a given set of equations. In the KB algorithm a term ordering is employed and it is required that every equation be orientable in the sense that the left-hand side be greater than the right. The KB algorithm halts if a non-orientable equation is produced. A generalization of the KB algorithm has recently been developed in which every equation is orientable and which halts only when a complete set is generated. In the generalization a constraint is added to each equation. The constraint governs when the equation can be used as a reduction. The constraint is obtained from the equation by "solving" the term inequality left-hand side > right-hand side. To understand what it means to solve a term inequality, consider the analogy with algebra in which solving term equalities, i.e. unification, is analogous to solving algebraic equalities. Then solving term inequalities is analogous to solving algebraic inequalities. Thus, the solution of term inequalities relates to unification as the solution of algebraic inequalities relates to the solution of algebraic equalities. We show how to solve term inequalities when using the lexicographic path ordering.