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Some recursive definitions in deductive database systems can be replaced by equivalent nonrecursive definitions. In this paper we give a linear-time algorithm that detects many such definitions, and specify a useful subset of recursive definitions for which the algorithm is complete. It is unlikely that our algorithm can be extended significantly, as recent results by Gaifman [5] and Vardi [19] show that the general problem is undecidable. We consider two types of initialization of the recursively defined relation: arbitrary initialization, and initialization by a given nonrecursive rule. This extends earlier work by Minker and Nicolas [10], and by Ioannidis [7], and is related to bounded tableau results by Sagiv [14]. Even if there is no equivalent equivalent nonrecursive definition, a modification of our algorithm can be used to optimize a recursive definition and improve the efficiency of the compiled evaluation algorithms proposed in Henschen and Naqvi [6] and in Bancilhon et al. [3].