Natural Proofs versus Derandomization

Holographic Proofs and Derandomization

Natural proofs

The ultimate goal we have is separating classes (or proving they are equal if they are). But our success so far seems to be limited: We know a (quite) tight time hierarchy, a tight space hierarchy, non-deterministic hierarchies, and some other (weaker) hierarchies. However, we are unable to even prove that LOG 6= P#P or that BPP 6= NEXP. In this lecture we seek some justification to this state ...

Natural Proofs Explained

The current inability to prove

Learning Algorithms from Natural Proofs

Based on Håstad’s (1986) circuit lower bounds, Linial, Mansour, and Nisan (1993) gave a quasipolytime learning algorithm for AC0 (constant-depth circuits with AND, OR, and NOT gates), in the PAC model over the uniform distribution. It was an open question to get a learning algorithm (of any kind) for the class of AC0[p] circuits (constant-depth, with AND, OR, NOT, and MODp gates for a prime p)....

Uniform proofs and natural deductions

Using some routine properties ([zu], [po]) of the Prawitz translation φ from (sequent calculus) derivations to (natural) deductions, and restricting ourselves to the language of first-order hereditary Harrop formulae, we show (i) that φ maps the simple uniform derivations of Miller et al onto the set of deductions in expanded normal form; and (ii) that φ identifies two such derivations iff they...