- 3 - Structural / Physical Level Level Procedural / Functional Level

We have developed an algebraic framework for the study of recursion. For immediate linear recursion, a Horn clause is represented by a relational algebra operator. We show that the set of all such operators forms a closed semiring. In this formalism, query answering corresponds to solving a linear equation. For the first time, we are able to have the query answer expressed in an explicit algebraic form within an algebraic structure. The manipulative power thus afforded has several implications on the implementation of recursive query processing algorithms. We present several possible decompositions of a given operator that improve the performance of the algorithms, as well as several transformations that give the ability to take into account any selections or projections that are present in a given query. In addition, we show that mutual linear recursion can also be studied within a closed semiring, by using relation vectors and operator matrices. Regarding nonlinear recursion, we first show that Horn clauses always give rise to multilinear recursion, which can always be reduced to bilinear recursion. We then show that bilinear recursion forms a nonassociative closed semiring. Finally, we give several sufficient and necessary & sufficient conditions for bilinear recursion to be equivalent to a linear one of a specific form. One of the sufficient conditions is derived by embedding bilinear recursion in an algebra. hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh 1 Some results of this work are contained in the paper "An Algebraic Approach to Recursive Inference", which appears in the Proc. of the 1st Inter. Conf. on Expert Database Systems, Charleston, SC, April 1986, and in the paper "Transforming Nonlinear Recursion into Linear Recursion", which appears in the Proc. of the 2nd Inter. Conf. on Expert Database Systems, Tysons Corner, VI, April 1988. 2 Partially supported by the National Science Foundation under Grant IRI-8703592.