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

翻译：本文给出了实数计算上的Imberman定理。我们定义了在实数上运行的电路，并证明：在Cucker和Meer的框架下，多项式规模且常数深度的此类电路族恰好能够判定那些在R结构上可用一阶逻辑定义的实数向量集合。我们的刻画在非均匀情形以及多种自然均匀性条件下均成立。