# A Circuit-Based Proof of Toda's Theorem

@article{Kannan1993ACP, title={A Circuit-Based Proof of Toda's Theorem}, author={Ravi Kannan and H. Venkateswaran and V. Vinay and Andrew Chi-Chih Yao}, journal={Inf. Comput.}, year={1993}, volume={104}, pages={271-276} }

- Published in Inf. Comput. 1993
DOI:10.1006/inco.1993.1033

Abstract We present a simple proof of Toda′s result (Toda (1989), in "Proceedings, 30th Annual IEEE Symposium on Foundations of Computer Science," pp. 514-519), which states that ⊕ P is hard for the Polynomial Hierarchy under randomized reductions. Our approach is circuit-based in the sense that we start with uniform circuit definitions of the Polynomial Hierarchy and apply the Valiant-Vazirani lemma on these circuits (Valiant and Vazirani (1986), Thoeret. Comput. Sci. 47, 85-93).

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## Uniform proofs of ACC representations

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## Nonuniform ACC Circuit Lower Bounds

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## Depth Reduction for Circuits of Unbounded Fan-In

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