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Criticality of AC0-formulae

Series: Bangalore Theory Seminars

Speaker: Prahladh Harsha,Tata Institute of Fundamental Research, Mumbai

Date/Time: Mar 20 16:00:00

Location: CSA Seminar Hall (Room No. 254, First Floor)

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Hastads celebrated switching lemma gives inverse-exponential bounds (In terms of t) on the probability that an k-DNF when hit by a p-restriction requires decision-trees of depth larger than t. The switching lemma has proved to be extremely powerful, since its discovery, leading to optimal size lower bounds for AC0-circuits [Hastad 1986] and AC0 formulae [Rossman 2015] against the parity function.
More recently, the search for optimal correlation bounds against parity led to the notion of criticality [Rossman 2019]. The criticality of a Boolean function f:{0,1}n → {0,1} is the minimum λ≥1 such that for all positive integers t, we have Prρ∼Rp[DTdepth(f|ρ)≥t]≤(pλ)t.
Hastad (2014) proved that size S and depth (d+1) AC0-circuits have criticiality at most O((logS)d) leading to optimal correlation bounds of AC0-circuits against parity. Rossman (2019) subsequently proved that size S and depth (d+1) AC0-formulae, which are regular (i.e., all gates of same depth have fan-in) having criticality at most O((logS/d)d).
In this work, we strengthen and unify all the above results by proving that any (not necessarily regular) AC0-formula of size S and depth (d+1) have criticality at most O((logS/d)d). The criticality bound implies tight correlation bounds against parity, tight Fourier concentration results and improved #SAT algorithm for AC0-formulae.
The talk is based on joint works with Tulasimohan Molli and Ashutosh Shankar.
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We are grateful to the Kirani family for generously supporting the theory seminar series

Hosts: Rameesh Paul, Aditya Subramanian, Aditya Abhay Lonkar and Rahul Madhavan

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