In orthogonal expression reduction systems, a common generalization of term rewriting and lambda calculus, we extend the concepts of normalization and needed reduction by considering, instead of the set of normal forms, a set S of "results".
When S satisfies some simple axioms, we prove the corresponding generalizations of some fundamental theorems: the existence of needed redexes, that needed reduction is normalizing, the existence of minimal normalizing reductions, and the optimality theorem.