eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-07-02
44:1
44:20
10.4230/LIPIcs.ICALP.2021.44
article
Lifting for Constant-Depth Circuits and Applications to MCSP
Carmosino, Marco
1
Hoover, Kenneth
2
Impagliazzo, Russell
2
Kabanets, Valentine
3
Kolokolova, Antonina
4
Department of Computer Science, Boston University, MA, USA
Department of Computer Science, University of California, San Diego, CA, USA
School of Computing Science, Simon Fraser University, Burnaby, Canada
Department of Computer Science, Memorial University of Newfoundland, St. John’s, Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol198-icalp2021/LIPIcs.ICALP.2021.44/LIPIcs.ICALP.2021.44.pdf
circuit complexity
constant-depth circuits
lifting theorems
Minimum Circuit Size Problem
reductions
Switching Lemma