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Presented here are an algebra and a logic characterizing the complexity class NC', which consists of functions computed by uniform families of polynomial size, log depth circuits. In both characterizations, NC' functions are regarded as functions from one class of finite relational structures to another. In the algebraic characterization a recursion scheme called upward tree recursion is applied to a class of simple functions. In the logical characterization, first-order logic is augmented bydoi:10.1016/0890-5401(90)90063-n fatcat:fn746ffmdzco5gaq7p2tlcsek4