Grigni: [4] Monotone Complexity

We give a general complexity classi cation scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes includingmAC , mNC , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We de ne 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 , motivated by Barrington's result [Bar89] that BWBP = NC . Although we cannot answer this question, we show two preliminary results: every monotone branching program for majority has size (n) with no width restriction, and no monotone analogue of Barrington's gadget exists.