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We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fan-in circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(Ο(n1/2k)) gates to calculate MODr functions for any r ≠ pm. This statement contains as special cases Yao's PARITY result [ Ya 85 ] and Razborov's new MAJORITY result [Ra 86] (MODm gate is an oracle which outputs zero, if the number of ones is divisible by m).
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Roman Smolensky (1987) studied this question.