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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.

Cite AsGet BibTex

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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