Naive Evaluation of Recursively Defined Relations

Abstract

We address the problem of evaluating a recursively defined relation. We are given an equation of the form:

$$ R = f\left( R \right) $$

where f is a monotonie relational algebra expression and we want to evaluate the least fixpoint of that equation. We first recall that there is a simple naive evaluation of this solution and we argue that its performance is poor because it generates duplicate computations.

To solve this problem of duplication, we define a simple differential calculus on relational expressions. The goal is to see how a relational expression of the form f(R) increases when R is replaced by R+dR. The differential is thus defined as f(R+dR) - f(R). We give simple rules to compute the differential of relational expressions, using join, composition, union, intersection, projection, and selection. We also give optimized expressions for the differential when specific conditions, such as functional dependencies, hold. Finally, we show how the differential of an expression can be used to produce an “optimal” evaluation of a least fixpoint. This is done (i) by giving a logic-based definition of performance, and (ii) demonstrating a relational algebra program that computes the least fixpoint of a recursive equation and showing some optimal properties of that relational algebra program.