In certain complexity-theoretic settings, it is notoriously difficult to prove complexity
separations which hold almost everywhere, ie, for all but finitely many input lengths. For …

We consider the range avoidance problem (called Avoid): given the description of a circuit
with more output gates than input gates, find a string that is not in the range of the circuit …

We prove that for all constants a, NQP= NTIME [n polylog (n)] cannot be (1/2+ 2− log an)-
approximated by 2log an-size ACC 0∘ THR circuits (ACC 0 circuits with a bottom layer of …

We make the first progress on probabilistic-degree lower bounds and correlation bounds for
polynomials since the papers by Razborov and Smolensky in the 80's. The bounds hold for …

For any n∈ ℕ and d= o (loglog (n)), we prove that there is a Boolean function F on n bits and
a value γ= 2− Θ (d) such that F can be computed by a uniform depth-(d+ 1) AC 0 circuit with …

The inner product function⟨ x, y⟩=∑ _i x_i y_i mod 2 can be easily computed by a (linear-
size) AC⁰ (⊕) circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates …

It is shown that there exists f:{0, 1\}^ n/2 * {0, 1\}^ n/2 → {0, 1\} f: 0, 1 n/2× 0, 1 n/2→ 0, 1 in E^
NP NP such that for every 2^ n/2 * 2^ n/2 2 n/2× 2 n/2 matrix M of rank ≤ ρ≤ ρ we have P …

We consider Boolean exact threshold functions defined by linear equations and, more
generally, polynomials of degree $ d $. We give upper and lower bounds on the maximum …

In this thesis we study satisfiability algorithms and connections between algorithms and
circuit lower bounds. We give new results in the following three areas: Oracles and …

Northeastern University On Approximating by Polynomials and Several Related Models A
dissertation submitted in partial satisfact Page 1 Northeastern University On Approximating by …