This paper of Tseitin is a landmark as the first to give non-trivial lower bounds for propositional proofs; although it pre-dates the first papers on

For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of sub-exponential algorithms for these problems. We introduce a generalized reduction which we call Sub-Exponential Reduction Family (SERF) that preserves sub-exponential complexity. We show that CircuitSAT is SERF-complete for all NP-search problems, and that for any fixed k, k-SAT, k-Colorability, k-Set Cover, Independent Set, Clique, Vertex Cover, are SERF--complete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, sub-exponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds for AC 0 ; that is, bounds of the form 2 \Omega\Gamma n) . This problem is even open for depth-3 circuits. In fact, such a bound for depth-3 circuits with even l...

We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment. We show that, for each k, the running time of ResolveSat on a k--CNF formula is significantly better than 2 n , even in the worst case. In particular, we show that the algorithm finds a satisfying assignment of a general satisfiable 3--CNF in time O(2 :448n ) with high probability; where the best previous algorithm [13] has running time O(2 :562n ). We obtain a better upper bound of 2 (2 ln 2\Gamma1)n+o(n) = O(2 0:387n ) for 3CNF that have exactly one satisfying assignment (unique k-SAT). For each k, the bounds for general k-CNF are the best currently known for ...

We present and analyze two simple algorithms for finding satisfying assignments of k-CNFs (Boolean formulae in conjunctive normal form with at most k literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a satisfiable k-CNF formula F in time O(n 2 jF j2 n\Gamman=k ). The second algorithm is deterministic, and its running time approaches 2 n\Gamman=2k for large n and k. The randomized algorithm is the best known algorithm for k ? 3; the deterministic algorithm is the best known deterministic algorithm for k ? 4. We also show an \Omega\Gamma n 1=4 2 p n ) lower bound on the size of depth 3 circuits of AND and OR gates computing the parity function. This bound is tight up to a constant factor. The key idea used in these upper and lower bounds is what we call the Satisfiability Coding Lemma. This basic lemma shows how to encode satisfying solutions of a k-CNF succinctly. 1 Introduction The problem of ...

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

We show that the permanent cannot be computed by uniform constant-depth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the uniform constant-depth threshold circuit model). In particular, this lower bound applies to any problem that is hard for the complexity classes PP or #P.

Russell Impagliazzo , Ramamohan Paturi , Michael E. Saks y Abstract The following size--depth trade--off for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least n 1+c` \Gammad edges, where constants c ? 0 and ` 3 are constants independent of n and d. Previously known constructions show that up to the choice of c and ` this bound is best possible. In particular, the lower bound implies an affirmative answer to the conjecture of Paturi and Saks that a bounded depth threshold circuit that computes parity requires a super--linear number of edges. This is the first super-- linear lower bound for an explicit function that holds for any fixed depth, and the first that applies to threshold circuits with unrestricted weights. The trade-off is obtained as a consequence of a general restriction theorem for threshold circuits with a small number of edges: For any threshold circuit with n inputs, de...

The correlation between two Boolean functions of n inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper, we compute, in closed form, the correlation between any two symmetric Boolean functions. As a consequence of our main result, we get that every symmetric Boolean function having an odd period has an exponentially small correlation (in n) with the parity function. This improves a result of Smolensky [12] restricted to symmetric Boolean functions: the correlation between parity and any circuit consisting of a Mod q gate over AND-gates of small fan-in, where q is odd and the function computed by the sum of the AND-gates is symmetric, is bounded by 2 \Gamma\Omega\Gamma n) . In addition, we find that for a large class of symmetric functions the correlation with parity is identically zero for infinitely many n. We characterize exactly those symmetric Boolean functions having this property.

We show that neural networks with three-times continuously differentiable activation functions are capable of computing a certain family of n-bit Boolean functions with two gates, whereas networks composed of binary threshold functions require at least \Omega\Gammaast n) gates. Thus, for a large class of activation functions, analog neural networks can be more powerful than discrete neural networks, even when computing Boolean functions. 1 Introduction. Artificial neural networks have become a popular model for machine learning and many results have been obtained regarding their application to practical problems. Typically, the network is trained to encode complex associations between inputs and outputs during supervised training cycles, where the associations are encoded by the weights of the network. Once trained, the network will compute an input/output mapping which (hopefully) is a good approximation of the original mapping. 1 Partially supported by NSF Grant CCR-9114545 In thi...

We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2 Ω(n). This is the first result of this type for general mod 3 subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a long-standing result of Smolensky, answering a question recently posed by Alon and Beigel. The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai, Green and Thierauf. The exponential sum and correlation bounds presented here are tight. 1