# Using narrowing to do isolation in symbolic equation solving — an experiment in automated reasoning

## Abstract

The PRESS symbolic equation solving system [STE82], and other algebraic manipulation packages, use a method known as *isolation* for equations containing only a single occurrence of an unknown. In effect, equations of the form *f (t)=t′* are rewritten by *t=f*^{−1}(*t′*), where *f*^{−1} is the inverse of *f* and *t* is a term containing the unknown. Meta-level inference is used to decide when isolation is applicable as opposed to other methods such as *attraction* and *collection* in which multiple occurrences of an unknown are drawn together.

This paper demonstrates how the technique of *narrowing* [RET85] implicitly performs isolation to solve equations. Narrowing involves the unification of the left-hand side of a rule with the equation to be solved, followed by rewriting by that rule (and others if applicable). Rewrite rules for isolation are provided, along with other properties of the functions involved, in the form *f*^{−1}(*f*(*v*))⇒*v*, where *v* is a simple variable.

The potential advantage of this is that it may be possible to avoid the need for meta-level inference by mixing rewrite rules for isolation with rules expressing other methods to form a single term rewriting system, in which the conditions for applicability of all rules are the same; namely, matching and unification.

A very preliminary investigation is made on the basis of a single example in which some of the limitations of narrowing are highlighted. In particular, it is not always possible to find a *finite confluent* term rewriting system containing all the necessary rules.

## Keywords

Solution Path Critical Pair Symbolic Equation Single Occurrence Recursive Decomposition## Preview

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