We study the class #AC 0 of functions computed by constantdepth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. No model-theoretic characterization for arithmetic circuit classes is known so far. Inspired by Immerman's characterization of the Boolean circuit class AC 0 , we remedy this situation and develop such a characterization of #AC 0 . Our characterization can be interpreted as follows: Functions in #AC 0 are exactly those functions counting winning strategies in first-order model checking games. A consequence of our results is a new model-theoretic characterization of TC 0 , the class of languages accepted by constant-depth polynomial-size majority circuits.
We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC 0 appear as classes of this hierarchy. In this way, we unconditionally place #AC 0 properly in a strict hierarchy of arithmetic classes within #P. Furthermore, we show that some of our classes admit efficient approximation in the sense of FPRAS. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al and argue that our approach is better suited to study arithmetic circuit classes such as #AC 0 which can be descriptively characterized as a class in our framework.
In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes NC 1 , SAC 1 and AC 1 as well as their arithmetic counterparts #NC 1 , #SAC 1 and #AC 1 . We build on Immerman's characterization of constant-depth polynomial-size circuits by formulae of first-order logic, i.e., AC 0 = FO, and augment the logical language with an operator for defining relations in an inductive way. Considering slight variations of the new operator, we obtain uniform characterizations of the three just mentioned Boolean classes. The arithmetic classes can then be characterized by functions counting winning strategies in semantic games for formulae characterizing languages in the corresponding Boolean class.
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