On the synthesis of function inverses

Abstract

We present a method for synthesising recursive inverses for first-order functions. Since inverse functions are not, in general, single-valued, we introduce a powerdomain to define their semantics, in terms of which we express their transformation into recursive form. First, inverses that require unification at run-time are synthesised and these are then optimised by term-rewriting based on a set of axioms that facilitates a form of compile-time unification. The optimisations reduce the dependency on run-time unification, in many instances removing it entirely to give a recursive inverse. The efficiency of the use of relations in two modes is thereby improved, so enhencing extended functional languages endowed with logical variables and narrowing semantics. Our function-level, axiomatised system is more generally applicable than previous approaches to this type of optimis tion, and in general induces more mechanisable transformation systems.