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Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four Chillara, Suryajith Functional Lower Bounds Boolean Circuit Lower Bounds Depth Four Connections to Boolean Complexity Iterated Matrix Multiplication Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit d^{O(1)}-variate and degree d polynomial P_{d} ∈ VNP such that if any depth four circuit C of bounded formal degree d which computes a polynomial of bounded individual degree O(1), that is functionally equivalent to P_d, then C must have size 2^Ω(√dlog{d}). The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for ACC⁰ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in ACC⁰ can also be computed by algebraic Σ∧ΣΠ circuits (i.e., circuits of the form - sums of powers of polynomials) of 2^(log^O(1) n) size. Thus they argued that a 2^{ω(polylog n)} "functional" lower bound for an explicit polynomial Q against Σ∧ΣΠ circuits would imply a lower bound for the "corresponding Boolean function" of Q against non-uniform ACC⁰. In their work, they ask if their lower bound be extended to Σ∧ΣΠ circuits. In this paper, for large integers n and d such that ω(log²n) ≤ d ≤ n^{0.01}, we show that any Σ∧ΣΠ circuit of bounded individual degree at most O(d/k²) that functionally computes Iterated Matrix Multiplication polynomial IMM_{n,d} (∈ VP) over {0,1}^{n²d} must have size n^Ω(k). Since Iterated Matrix Multiplication IMM_{n,d} over {0,1}^{n²d} is functionally in GapL, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of ACC⁰ from GapL. For the sake of completeness, we also show a syntactic size lower bound against any Σ∧ΣΠ circuit computing IMM_{n,d} (for the same regime of d) which is tight over large fields. Like Forbes, Kumar and Saptharishi [CCC, 2016], we too prove lower bounds against circuits of bounded formal degree which functionally compute IMM_{n,d}, for a slightly larger range of individual degree. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Suryajith Chillara 2021 Is Part Of LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSTTCS.2021.14 urn:nbn:de:0030-drops-155251 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.14 eng https://creativecommons.org/licenses/by/4.0/legalcode