An algebraic rewriting theorem of multiple linear recursions and its applications

Abstract

A Horn clause in linear recursive programs can be represented by a relational operator, and the set of all such operators can be constructed as an algebraic framework, called closed semiring. In the closed semiring, the evaluation of a linear recursive query can be reduced to solving a linear equation. This paper focuses on multiple linear (mL) recursions, which consist of m linear recursive rules and one or more exit rules. We first prove an algebraic rewriting theorem for 2L recursions, that is, (A+B)^*=B^*(A⁺B⁺)^*A^*, where A and B are two operators corresponding to the two recursive rules. Then two applications of this theorem are described. The first application is that a 2L recursion can be always transformed into a set of single linear (1L) recursions so that some existed algorithms for 1L recursions can be utilized to evaluate 2L recursions. The other one is to weaken existing sufficient conditions for testing rule sequencability. Sequencability is a very important semantic property of mL recursions, because a mL recursion satisfying rule sequencability can be evaluated sequentially, therefore, efficiently.