We give a general complexity classification scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic log-space) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = co-NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st-connectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...

One way to address the NP = co - NP question is to consider the length of proofs of tautologies in various proof systems. In this work we consider proof systems defined by appropriate classes of automata. In general, starting from a given class of automata we can define a corresponding proof system in a natural way. An interesting new proof system that we consider is based on the class of push down automata. We present an exponential lower bound for oblivious read-once branching programs which implies that the new proof system based on push down automata is, in a certain sense, more powerful than oblivious regular resolution. 1 Introduction One of the famous open questions of complexity theory is: does NP equal co-NP ? Put another way do tautologies always have "short" proofs? If proof is taken in its most general form, i.e. does some nondeterministic polynomial time Turing Machine correctly accept exactly the class of tautologies, then the question seems to be completely beyond ou...