The Complexity of the Haj os Calculus

The Hajj os Calculus is a simple, nondeterministic procedure which generates the class of non-3-colorable graphs. Mansseld and Welsch MW] posed the question of whether there exist graphs which require exponential-sized Hajj os constructions. Unless NP 6 = coNP, there must exist graphs which require exponential-sized constructions, but to date, little progress has been made on this question, despite considerable eeort. In this paper, we prove that the Hajj os Calculus generates polynomial-sized constructions for all non-3-colorable graphs if and only if Extended Frege systems are polynomially bounded. Extended Frege systems are a very powerful family of proof systems for proving tautologies, and proving superpolynomial lower bounds for these systems is a long-standing, important problem in logic and complexity theory. We also establish a relationship between a complete subsystem of the Hajj os Calculus, and bounded-depth Frege systems; this enables us to prove exponential lower bounds on this subsystem of the Hajj os Calculus.