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...

We consider small-weight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of smallweight CP , our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems : (1) Tree-like CP proofs cannot polynomially simulate non-tree-like CP proofs. (2) Tree-like CP proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP proof system. In particular, they work for CP with a deduction rule, and also for proof systems that allow any formula with small communication complexity, and any set of sound rules of inference. 1 Introduction One of the most fundamental questions in pro...

To appear in Izvestiya of the RAN Abstract We show that if strong pseudorandom generators exist then the statement "ff encodes a circuit of size n(log * n) for SATISFIABILITY " is not refutable in S22 (ff). For refutation in S12 (ff), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constant-depth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolation-like theorems for certain "split versions " of classical systems of Bounded Arithmetic introduced in this paper.

Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although

We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC 6= monotone-P. 2. For every i 1, monotone-NC i 6= monotone-NC i+1 . 3. More generally: For any integer function D(n), up to n ffl (for some ffl ? 0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const \Delta D(n) (for some constant Const). Only a separation of monotone-NC 1 from monotone-NC 2 was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In...

Is it easier to solve two communication problems together than separately? This question is related to the complexity of the composition of boolean functions. Based on this relationship, an approach to separating NC¹ from P is outlined. Furthermore, it is shown that the approach provides a new proof of the separation of monotone NC¹ from monotone P.

We introduce a new Ehrenfeucht-Fra ss e game for proving lower bounds on the size of first-order formulas. Up until now such games have only been used to prove bounds on the operator depth of formulas, not their size. We use this game to prove that the CTL + formula Occurn # E[Fp 1 # Fp 2 # # Fpn ] which says that there is a path along which the predicates p 1 through pn occur in some order, requires size n! to express in CTL. Our lower bound is optimal. It follows that the succinctness of CTL + with respect to CTL is exactly #(n)!. Wilke had shown that the succinctness was at least exponential [Wil99]. We also use our games to prove an optimal# n) lower bound on the number of boolean variables needed for a weak reachability logic (RL w ) to polynomially embed the language LTL. The number of booleans needed for full reachability logic RL and the transitive closure logic FO 2 (TC) remain open [IV97, AI00]. 1

Using a notion of real communication complexity recently introduced by J. Krajcek, we prove a lower bound on the depth of monotone real circuits and the size of monotone real formulas for st-connectivity. This implies a super-polynomial speedup of dag-like over tree-like Cutting Planes proofs. Key words: computational complexity, monotone circuit, communication complexity, Cutting Planes proof Introduction A monotone real circuit is a circuit computing with real numbers in which every gate computes a nondecreasing binary real function. This class of circuits was introduced in [10]. We require that such a circuit outputs 0 or 1 on every input of 0's and 1's only. Hence, monotone real circuits are a generalization of monotone boolean circuits, which was shown to be strictly more powerful in [11]. The depth and size of a monotone real circuit are dened as usual, and we call it a formula if every gate has fan-out at most 1. We generalize the lower bounds on the depth of monotone boolea...

We prove that a monotone circuit of size n d recognizing connectivity must have depth ((log n) 2 = log d). For formulas this implies depth ((log n) 2 = log log n). For ((log n) 2)which is optimal up to a con-polynomial-size circuits the bound becomes stant. Warning: Essentially this paper has been published in SIAM Journal on Computing is hence subject to copyright restrictions. It is for personal use only. 1

y1) f(xk , yk )) communicating as few bits as possible. The Direct Sum Conjecture (henceforth DSC) of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing f(x1 , y1 ), then f(x2 , y2 ), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits since a variant of it implies NC 1 #= NC 2 . We consider two related problems. Enumeration: Alice and Bob output e # 2 k - 1 elements of {0, 1} k , one of which is f(x1 , y1) f(xk , yk ). Elimination: Alice and Bob output # b such that # b #= f(x1 , y1 ) f(xk , yk ). Selection: (k = 2) Al