Lifting for Constant-Depth Circuits and Applications to MCSP

Authors Marco Carmosino, Kenneth Hoover, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova

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Author Details

Marco Carmosino
  • Department of Computer Science, Boston University, MA, USA
Kenneth Hoover
  • Department of Computer Science, University of California, San Diego, CA, USA
Russell Impagliazzo
  • Department of Computer Science, University of California, San Diego, CA, USA
Valentine Kabanets
  • School of Computing Science, Simon Fraser University, Burnaby, Canada
Antonina Kolokolova
  • Department of Computer Science, Memorial University of Newfoundland, St. John’s, Canada


We thank the anonymous reviewers for their helpful and insightful comments.

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Marco Carmosino, Kenneth Hoover, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Lifting for Constant-Depth Circuits and Applications to MCSP. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Lifting arguments show that the complexity of a function in one model is essentially that of a related function (often the composition of the original function with a small function called a gadget) in a more powerful model. Lifting has been used to prove strong lower bounds in communication complexity, proof complexity, circuit complexity and many other areas. We present a lifting construction for constant depth unbounded fan-in circuits. Given a function f, we construct a function g, so that the depth d+1 circuit complexity of g, with a certain restriction on bottom fan-in, is controlled by the depth d circuit complexity of f, with the same restriction. The function g is defined as f composed with a parity function. With some quantitative losses, average-case and general depth-d circuit complexity can be reduced to circuit complexity with this bottom fan-in restriction. As a consequence, an algorithm to approximate the depth d (for any d > 3) circuit complexity of given (truth tables of) Boolean functions yields an algorithm for approximating the depth 3 circuit complexity of functions, i.e., there are quasi-polynomial time mapping reductions between various gap-versions of AC⁰-MCSP. Our lifting results rely on a blockwise switching lemma that may be of independent interest. We also show some barriers on improving the efficiency of our reductions: such improvements would yield either surprisingly efficient algorithms for MCSP or stronger than known AC⁰ circuit lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Circuit complexity
  • circuit complexity
  • constant-depth circuits
  • lifting theorems
  • Minimum Circuit Size Problem
  • reductions
  • Switching Lemma


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