We establish several ``sharp threshold'' results for computational complexity. For certain tasks, we can prove a resource lower bound of $n^c$ for $c \geq 1$ (or obtain an efficient circuit-analysis algorithm for $n^c$ size), there is strong intuition that a similar result can be proved for larger functions of $n$, ... more >>>

We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula $F$, it is NP-hard to find a refutation of $F$ in the Nullstellensatz, Polynomial Calculus, or Sherali--Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a ... more >>>

For every constant c > 0, we show that there is a family {P_{N,c}} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, and that satisfies the following properties:
* For every family {f_n} of polynomials in VP, where f_n is an n ... more >>>