# A non-probabilistic switching lemma for the Sipser function

## Abstract

Valiant [12] showed that the clique function is structurally different than the majority function by establishing the following “switching lemma”: Any function *f* whose set of prime implicants is a large enough subset of the set of cliques (and thus requiring big Σ_{2}-circuits), has a large set of prime clauses (i.e., big Π_{2}-circuits). As a corollary, an exponential lower bound was obtained for monotone ΣΠΣ-circuits computing the clique function. The proof technique is the only non-probabilistic super polynomial lower bound method from the literature. We prove, by a non-probabilistic argument as well, a similar switching lemma for the NC^{1}-complete Sipser function. Using this we then show that a monotone depth-3 (i.e., ΣΠΣ or ΠΣΠ) circuit computing the Sipser function must have super quasipolynomial size. Moreover, any depth-*d* quasipolynomial size non-monotone circuit computing the Sipser function has a depth-(*d*—1) gate computing a function with exponentially many both prime implicants and (monotone) prime clauses. These results are obtained by a top-down analysis of the circuits.

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