Alexander A. Razborov

Natural Proofs

We introduce the notion ofnaturalproof. We argue that the known proofs of lower bounds on the complexity of explicit Boolean functions in nonmonotone models fall within our definition of natural. We show, based on a hardness assumption, that natural proofs can not prove superpolynomial lower bounds for general circuits. Without the hardness assumption, we are able to show that they can not prove exponential lower bounds (for general circuits) for the discrete logarithm problem. We show that the weaker class ofAC0-natural proofs which is sufficient to prove the parity lower bounds of Furst, Saxe, and Sipser, Yao, and Hastad is inherently incapable of proving the bounds of Razborov and Smolensky. We give some formal evidence that natural proofs are indeed natural by showing that every formal complexity measure, which can prove superpolynomial lower bounds for a single function, can do so for almost all functions, which is one of the two requirements of a natural proof in our sense.

Majority gates vs. general weighted threshold gates

In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth circuits. These results are the following:1.A single threshold gate with weights cannot in general be replaced by a polynomial fan-in unweighted threshold gate of parity gates.2.On the other hand it can be replaced by a depth 2 unweighted threshold circuit of polynomial size. An extension of this construction is used to prove that whatever can be computed by a depthd polynomial size threshold circuit with weights can be computed by a depthd+1 polynomial size unweighted threshold circuit, whered is an arbitrary fixed integer.3.A polynomial fan-in threshold gate (with weights) of parity gates cannot in general be replaced by a depth 2 unweighted threshold circuit of polynomial size.

Quantum communication complexity of symmetric predicates

We completely (that is, up to a logarithmic factor) characterize the bounded-error quantum communication complexity of every predicate ) depending only on . More precisely, given a predicate on , we put Then the bounded-error quantum communication complexity of is equal to (up to a logarithmic factor). In particular, the complexity of the set disjointness predicate is equal to . This result holds both in the model with prior entanglement and in the model without it.

On lower bounds for read- k -times branching programs

AbstractA syntactic read-k-times branching program has the restriction that no variable occurs more thank times on any path (whether or not consistent) of the branching program. We first extend the result in [31], to show that the “n/2 clique only function”, which is easily seen to be computable by deterministic polynomial size read-twice programs, cannot be computed by nondeterministic polynomial size read-once programs, although its complement can be so computed. We then exhibit an explicit Boolean functionf such that every nondeterministic syntactic read-k-times branching program for computingf has size exp

Lower Bounds for Deterministic and Nondeterministic Branching Programs

\left( {\Omega \left( {\frac{n}{{4^k k^3 }}} \right)} \right).

Why are there so many loop formulas

For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that