Showing papers by "Alexander A. Razborov published in 1999"

Space Complexity in Propositional Calculus

Abstract: We study space complexity in the framework of propositional proofs. We consider a natural model analogous to Turing machines with a read-only input tape and such popular propositional proof systems as resolution, polynomial calculus, and Frege systems. We propose two different space measures, corresponding to the maximal number of bits, and clauses/monomials that need to be kept in the memory simultaneously. We prove a number of lower and upper bounds in these models, as well as some structural results concerning the clause space for resolution and Frege systems.

On P versus NP CO-NP for decision trees and read-once branching programs

Abstract: It is known that if a Boolean function f in n variables has a DNF and a CNF of size $ \le N $ then f also has a (deterministic) decision tree of size exp(O(log n log2 N)) We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp $ (\Omega({\rm log^2} N)) $ where N is the total number of monomials in minimal DNFs for f and ¬f Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs One example results from the Bruen—Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear Other examples have the additional property that f is in AC0