September 2024
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4 Reads
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2 Citations
Journal of Logic and Computation
In the eighties, Immerman showed that the class of languages definable by first-order formulae coincides with the class of languages decidable by unbounded fan-in Boolean circuits of constant depth and polynomial size. We show an analogous result for real-valued computation, i.e. we define circuits of unbounded fan-in 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 real valued structures. Our characterization holds both non-uniformly as well as for many natural uniformity conditions.