Nonlinear versus linear recursion: a perspective from computing transitive closures of a binary relation by the join domain nested loops approach

The author considers a more general class of transitive closure of a binary relation by relaxing the linear variable binding of the original transitive closure. A syntactical (i.e. grammatical) approach is used to analyze the properties of this class of recursive rules. The existing directed algorithms for computing the (standard) transitive closure of a binary relation actually compute a nonlinear recursion. In recursive query processing, linear recursion was 'preferred to' nonlinear recursion probably because of the lack of efficient algorithms for computing nonlinear recursion rather than because linear recursion is more likely to occur than nonlinear recursion. A linear transitive closure is semantically subsumed by its corresponding nonlinear transitive closure. The author provides counter-evidence through the worst case analysis of computing nonstandard transitive closures to show that computing nonlinear recursive rules is simpler than computing linear ones by using the join domain nested loops approach, even though linear recursive transitive closure is semantically subsumed by its corresponding nonlinear recursive transitive closure.<>