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...

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

We prove that monotone circuits computing the perfect matching function on n-vertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits.

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.

We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and informal arguments supporting the claim that a conceivable answer is V 1 (which, in view of RSUV -isomorphism, is equivalent to S 2 ), although some major results about the complexity of Boolean functions can be proved in (presumably) weaker subsystems like U 1 . As a by-product of this analysis, we give a more constructive version of the proof of Hastad Switching Lemma which probably is interesting in its own right.

this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies

In this paper we study the pairs (U; V ) of disjoint NP-sets representable in a theory T of Bounded Arithmetic in the sense that T proves U " V = ;. For a large variety of theories T we exhibit a natural disjoint NP-pair which is complete for the class of disjoint NP-pairs representable in T . This allows us to clarify the approach to showing independence of central open questions in Boolean complexity from theories of Bounded Arithmetic initiated in [11]. Namely, in order to prove the independence result from a theory T , it is sufficient to separate the corresponding complete NP-pair by a (quasi)poly-time computable set. We remark that such a separation is obvious for the theory S(S 2 ) + S \Sigma 2 \Gamma PIND considered in [11], and this gives an alternative proof of the main result from that paper.

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...

this paper we shall assume that S is a commutative ring with identity. Then each instruction f i can be identified with the polynomial that is computed at f i , if\Omega and \Phi are interpreted as the ring operations in the polynomial ring S[x 1 ; \Delta \Delta \Delta ; x n ]. Among natural multivariate polynomials whose complexity in this model is of interest are Hamiltonian circuits (HC), the permanent (PERM) and the determinant (DET). These are defined over a matrix X of indeterminates fx 11 ; \Delta \Delta \Delta ; x nn g where x ij

this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64].