We say that a circuit C over a field F {functionally} computes a polynomial P in F[x_1, x_2, ..., x_n] if for every x in {0,1}^n we have that C(x) = P(x). This is in contrast to syntactically computing P, when C = P as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-3 and depth-4 arithmetic circuits for functional computation. We prove the following results: 1. Exponential lower bounds for homogeneous depth-3 arithmetic circuits for a polynomial in VNP. 2. Exponential lower bounds for homogeneous depth-4 arithmetic circuits with bounded individual degree for a polynomial in VNP. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-4 arithmetic circuits for the Permanent imply a separation between #P and ACC0. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [Kumar/Saptharishi, ECCC 2015] that over constant sized finite fields, strong enough {average case} functional lower bounds for homogeneous depth-4 circuits imply superpolynomial lower bounds for homogeneous depth-5 circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest.
@InProceedings{forbes_et_al:LIPIcs.CCC.2016.33, author = {Forbes, Michael A. and Kumar, Mrinal and Saptharishi, Ramprasad}, title = {{Functional Lower Bounds for Arithmetic Circuits and Connections to Boolean Circuit Complexity}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {33:1--33:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.33}, URN = {urn:nbn:de:0030-drops-58266}, doi = {10.4230/LIPIcs.CCC.2016.33}, annote = {Keywords: boolean circuits, arithmetic circuits, lower bounds, functional computation} }
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