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In this paper we give an Immerman's Theorem for real-valued computation. Wedefine circuits operating over real numbers and show that families of suchcircuits of polynomial size and constant depth decide exactly those sets ofvectors of reals that can be defined in first-order logic on R-structures inthe sense of Cucker and Meer. Our characterization holds both non-uniformily aswell as for many natural uniformity conditions.

A Logical Characterization of Constant-Depth Circuits over the Reals