A. E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Doklady Akademii Nauk SSSR, 282:5 (1985), pp. 10331037. English translation in: Soviet Mathematics Doklady, 31:3, 539-534  Home/Search   Document Not in Database   Summary   Related Articles   Check

....was only 4n (by Tiekenheinrich [14] A major breakthrough came in 1985, when Razborov [11] invented the method of approximation. It allowed him to prove a super polynomial lower bound as he showed that Clique requires monotone circuits of size n Omega Gamma1 1 n) Shortly thereafter, Andreev [3] applied Razborov s technique to another function and was thereby able to prove an exponential lower bound. Later, both these results were improved by Alon and Boppana [1] and in particular, they were the rst to prove an exponential lower bound for Clique. For a nice exposition of this result, ....

....circuit Psi computes the boolean function e o (x) Our proofs are based on the method of approximation which was invented by Razborov [11] It involves the use of a boolean function to approximate the output of every gate in a given circuit. The approximator functions used by Razborov, Andreev [3], and Alon and Boppana [1] are all monotone DNF formulas. The proofs in this paper, however, are inAEuenced by the work of Haken [7] and every gate e in a circuit Psi is approximated by two functions f e , the approximators for the gate e. The approximator f e has the form C 1 C 2 ....

[Article contains additional citation context not shown here]

Alexander E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Sov. Math. Dokl., 31:530534, 1985.

....best one was only 4n (by Tiekenheinrich [14] A major breakthrough came in 1985, when Razborov [11] invented the method of approximation. It allowed him to prove a super polynomial lower bound as he showed that Clique requires monotone circuits of size n 1 9 n) Shortly thereafter, Andreev [3] applied Razborov s technique to another function and was 2 Berg Ulfberg thereby able to prove an exponential lower bound. Later, both these results were improved by Alon and Boppana [1] and in particular, they were the rst to prove an exponential lower bound for Clique. For a nice exposition ....

....the circuit computes the boolean function e o (x) Our proofs are based on the method of approximation which was invented by Razborov [11] It involves the use of a boolean function to approximate the output of every gate in a given circuit. The approximator functions used by Razborov, Andreev [3], and Alon and Boppana [1] are all monotone DNF formulas. The proofs in this paper, however, are in uenced by the work of Haken [7] and every gate e in a circuit is approximated by two functions f D e and f C e , the approximators for the gate e. The approximator f D e has the form C 1 ....

[Article contains additional citation context not shown here]

Alexander E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Sov. Math. Dokl., 31:530534, 1985.

.... n) 2 = log log n) and in the case of polynomial size circuits the bound improves to optimal Omega Gammatima n) 2 ) The tool in [8] is a modification of the method of approximation, originally designed by Razborov [6, 5] to prove lower bounds on the size of monotone circuits (also used in [1, 2]) The method is roughly as follows. One considers some subset of the inputs, called test inputs. Given some monotone circuit C one replaces each gate g by an approximator g yielding a function C that approximates the function that C computes. In order to prove a lower bound on the size of C ....

A. E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Dokl. Akad. Nauk SSSR, 282(5):1033--1037, 1985. (In Russian); English translation in Soviet Math. Dokl. 31(3):530--534, 1985.

.... m 2 ) Omega Gamma n 2 = log 2 n) Another approach that has recently gained popularity is proving lower bounds for branching programs with bounds on various resources (width, multiplicity of reading) A similar approach to Boolean circuits has been quite successful recently [Ya2] [An], Ra] Ha] AB] Be] Our aim is to present two more results of this kind one under each type of restriction. 1.2. Bounded width branching programs for symmetric functions Bounded width branching programs have first been promoted by Borodin, Dolev, Fich and Paul [BDFP] Their main ....

A. E. Andreev, On a method of obtaining lower bounds for the complexity of individual monotone functions (in Russian), Dokl. Akad. Nauk SSSR 282/5 (1985), 1033-1037

....minimization and other hard problems: since the classical work by Quine then McCluskey[McC56] on boolean function minimization, this question has been abundantly studied from the theoretical point of view. We can refer for example to the work by the russian school: Neciporuk[Nec66] Andreev [And85] and many others (e.g. Mor71] have studied such aspects as minimal representation of a boolean formula under a certain basis. However, to our knowldge, this work has not lead to practical applications. Finding a minimum equivalent boolean expression, as stated earlier, is NP hard[Sto76] In ....

A. E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Soviet Math. Doklady, 31(3), 1985.

....The study of circuit complexity has in one sense been successful and in another not so successful. While there are still no non linear lower bounds on circuit size for any function in NP , several interesting results have been shown for restricted circuit classes e.g. monotone circuits [4, 16, 3, 13, 14, 15] and circuits of bounded depth [1, 9, 10, 17, 19, 21] The smallest natural circuit class that is not known to be strictly contained in NP is TC 0 , the set of functions computable by constant depth polynomialsize circuits containing threshold gates. Threshold gates are quite powerful and many ....

A. E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Dokl. Ak. Nauk. SSSR 282, pages 1033-- 1037, 1985. English translation in Sov. Math. Dokl., 31:530--534, 1985.

....a fact that emphasizes the significance of Razborov s result. The new bound also introduced the method of approximations, later used also to prove other lower bounds for monotone size as well as lower bounds in other models (e.g. bounded depth circuits) Shortly after Razborov s result, Andreev [An85] used the method of approximations to prove the first exponential lower bound for a somewhat unnatural function, known as Andreev s function. Later, Alon and Boppana [AlBo87] Department of Applied Mathematics and Computer Science, Weizmann Institute, Rehovot, 76100 Israel. Work supported by an ....

.... f doesn t always output 1, but rather f outputs a 1 only on a large enough fraction of the inputs in the set (and analogously for negative test inputs) In our result, for simplicity, we use a slightly more general technique, used before for choosing the negative test inputs in Andreev s function [An85]. Instead of looking at a set of inputs and counting the number of errors on such inputs, we take a distribution over the test inputs and ask what is the probability of error on such an input. We look at two distributions: ffl Positive test distribution: A distribution on the inputs, such that ....

A. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Dolk. Akad. Nauk. SSSR 282(5) (1985), pp. 1033--1037 (in Russian). English translation in: Soviet Math. Dokl. 31(3) (1985), pp. 530--534.

....Razborov [Ra85a] proved a super polynomial lower bound for the monotone size of the Clique function, and as a conclusion obtained the separation of monotone P from monotone NP. Using Razborov s technique, exponential lower bounds for the monotone size of other functions were proved by Andreev [An85], and an exponential lower bound for the monotone size of the Clique function was finally proved by Alon and Boppana [AlBo87] A simpler proof for that lower bound was recently presented by Haken [Ha95] Those lower bounds, and other lower bounds for the monotone size of functions, immediately ....

A. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Dolk. Akad. Nauk. SSSR 282(5) (1985), 1033--1037 (in Russian). English translation in: Soviet Math. Dokl. 31(3) (1985), 530--534.

.... of Chicago Chicago, IL 60637, USA simon cs.uchicago.edu Shi Chun Tsai Information Management Department National Chi Nan University Pu Li, Nan Tou 545, TAIWAN tsai csie.ncnu.edu.tw January 26, 1999 Abstract Both the bottleneck counting argument [7, 8] and Razborov s approximation method [1, 4, 12] have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and vice versa. We also illustrate the elegance of the bottleneck counting technique with a ....

....Haken [8] has applied this argument to prove an exponential lower bound on the size of monotone circuits for the Broken Mosquito Screen problem, which is a special version of the CLIQUE problem. Similar lower bounds have been proven earlier by Razborov [12] and later strengthened by Andreev [4], and Alon and Boppana [1] using the method of approximation. While the result is old, the bottleneck method provides a new simple proof for this strong lower bound. Chronologically, the two methods appeared in publications about the same time, 1985. Since then the approximation method has drawn ....

[Article contains additional citation context not shown here]

A. E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Soviet Math. Dokl, 31:530--534, 1985. 7

....by monotone circuits. He proved n Omega Gamma1 6 n) lower bounds for the monotone circuit complexity of the clique and perfect matching functions on n node graphs. Based on Razborov s method exponential lower bounds were obtained for the monotone circuit complexity of several functions from NP [8, 7]. The above results can be used to derive lower bounds for the depth of monotone circuits, but the depth lower bounds obtained this way will be logarithmic in the size bound. Karchmer and Wigderson [48] introduced a technique for proving lower bounds on the depth of monotone circuits that are ....

A. E. Andreev: "On a method for obtaining lower bounds for the complexity of individual monotone functions", Sov. Math. Dokl., 31, 1985 pp. 530-534.

....In sharp contrast, there has been substantial progress in obtaining strong lower bounds on the size of a monotone circuit, that consists of AND and OR gates. Razborov[13] obtained superpolynomial lower bounds on the size of a monotone circuit for the clique function. Shortly thereafter, Andreev [1] and Alon and Boppana[3] obtained exponential lower bounds on the size of a monotone circuit for the clique function and for some other functions. In short, we have strong lower bounds on circuit size when we restrict ourselves to circuits without NOT gates, whereas we have very weak lower bounds ....

A.E. Andreev, "On a Method for Obtaining Lower Bounds for the Complexity of Individual Monotone Functions", Soviet Math. Dokl., Vol. 31, No. 3, pp. 530--534, 1985.

....The generalization was proved independently by Pudl ak using a different method, who also showed that the result can be used to obtain an exponential lower bound on the size of unrestricted cutting plane proofs in the propositional calculus. 1 Introduction The Razborov Andreev [Raz85] AB87] [And85] exponential lower bound on the size of monotone Boolean circuits which detect cliques represented a breakthrough in the theory of monotone circuit complexity. The proof introduced the method of approximation, which has been used for other important lower bounds (see [BS90] and [Weg87] for ....

A. E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Doklady Akad. Nauk SSSR 282, 5, pages 1033--1037, 1985. English translation in Soviet Math. Dokl. 31, pages 530--534, 1985.

....understood. On the other hand, if we a priori know that our circuit does not have such chains, then the task becomes more tractable. First exponential lower bounds for different models of circuits without null chains were proved by Pulatov [10] for DeMorgan formulas) Razborov [11] and Andreev [1] (for monotone circuits) and Jukna [4, 5] for switching and rectifier networks. All these proofs use different arguments. The optimal versus stable phenomenon described above, leads to one more argument for null chain free formulas. Moreover, this argument works also for Boolean functions (for ....

....function. Let q be a prime power, and f = POLY(q; s) be a monotone Boolean function in n = q 2 variables x a;b , indexed by the pairs of elements a; b 2 GF (q) which accepts an inputs iff there is a polynomial p(z) of degree s over GF (q) such that x a;p(a) 1 for all a 2 GF (q) Andreev [1] has proved that this function requires monotone circuits (and hence, also monotone formulas) of exponential size. Monotone formulas do not have negated variables x i at all. Using Theorem 4.1 we can prove a similar lower bound also when negations are allowed important is that they do not ....

A.E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Doklady Akademii Nauk SSSR 282 : 5 (1985), 1033-1037.

....Trier, Germany Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. E mail: jukna ti.uni trier.de. 2 monotone circuit complexity of the clique function was proved. Shortly after, such (and even exponential) lower bounds were obtained for different Boolean functions [21, 2, 1, 26, 27], including those whose non monotone circuits are polynomial [21, 26] After this impressing progress one principal question still remained unclear: is there a tractable lower bounds criterion for monotone circuits Razborov raised this problem as a candidate for a final chord in that direction ....

....of degree at most v Gamma 1. For every 1 6 t v, this is a partial t (n; k; design with n = q 2 , k = q and = q v Gammat ; the number of blocks in this design is jD v j = q v . The corresponding monotone Boolean function f Dv , denoted also POLY(q; v) was investigated by Andreev [2] who proved that any circuit with fanin 2 And and Or gates computing this function (for appropriate values of v) requires size exponential in Omega Gamma n 1=8 Gamma ) Using Razborov s method of approximations, Alon and Boppana [1] were able to essentially improve this bound until q ....

A. E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Doklady Akademii Nauk SSSR, 282:5 (1985), pp. 1033-1037. English translation in: Soviet Mathematics Doklady, 31 (1985), 530--534.

....usual Boolean ordering. It is easy to see that monotone circuits compute exactly the class of monotone functions. The first strong lower bound concerning this model is due to Razborov [Ra85a] who proved that the clique function has superpolynomial monotone complexity. Shortly thereafter Andreev [An85], using similar methods, proved an exponential lower bound, further tightened by Alon and Boppana [AB87] Razborov s theorems on monotone and bounded depth circuits, as well as the aforementioned proof of Ajtai, rely upon a technique which has come to be called the approximation method. One of ....

A. E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Doklady Akademii Nauk SSSR 282:5, 1033--1037 (in Russian). English translation in Soviet Mathematics Doklady 31:3, 530--534, 1985.

....Razborov [32] showed that the clique function, which returns 1 iff the input graph contains a k clique, requires superpolynomial size monotone circuits for appropriately chosen k. This lower bound was made properly exponential (i.e. exp(n c ) for some c 0) by Alon and Boppana [5] and Andreev [7], for this and several other mNP functions. mP 6= P mono: By the same method Razborov also showed that the matching function, which returns 1 iff the input (bipartite) graph contains a perfect matching, requires monotone circuits of superpolynomial size (i.e. of size n Omega Gamma42 n) In ....

A. E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Doklady Akademii Nauk SSSR, 282(5):1033--1037, 1985. In Russian; English translation appears in Soviet Math. Dokl. 31(3):530--534, 1985.

....in a computational model at least as strong as Boolean circuits (any circuit of size m is the unique interpolant of an implication with resolution proof of size O(m) As no non trivial lower bounds are known for the size of general circuits, it makes no sense to use any stronger model. However, [23, 2, 1] provide lower bounds for monotone Boolean circuits and in the applications of effective interpolation to lower bounds monotone effective interpolation is thus used (cf. 4, 16, 21] It says that if A is closed upwards (or B downwards) then a separating set exists that is closed upwards. It is ....

....W ) where H is the W Theta N matrix with rows h i; Gamma . Thus determines a semantic refutation X 1 ; X k of E 1 ; F satisfying the hypothesis of Theorem 5.2. This proves the theorem. q.e.d. Theorem 6. 1 yields a lower bound for R(CP) using the strong lower bound of [23, 2, 1] for the size of monotone circuits separating graphs on n vertices with a clique of size 1 from those that are colorable, for particular . Clique n; and Color n; are the sets of CP inequalities defined before [16, Cor. 7.3] Namely, let n; 1 be natural numbers, and let Gamma n ....

Andreev, A. E. (1985) On a method for obtaining lower bounds for the complexity of individual monotone functions (in Russian), Doklady AN SSSR, 282(5) : 1033-1037.

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A. E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Doklady Akademii Nauk SSSR, 282:5 (1985), pp. 10331037. English translation in: Soviet Mathematics Doklady, 31:3, 539-534

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A. E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Doklady Akademii Nauk SSSR, 282:5 (

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A.E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions. Soviet Math. Dokl. 31(3):530-534, 1985.

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Andreev A.E., On a method for obtaining lower bounds for the complexity of individual monotone functions. Dokl. Ak. Nauk. SSSR 282 (

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A. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions, Dolk. Akad. Nauk. SSSR 282(5) (

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Andreev, A.E. (1985). On a method for obtaining lower bounds for the complexity of individual monotone functions. Soviet Math. Doklady 31, 530--534.

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Andreev, A. E. (1985) On a method for obtaining lower bounds for the complexity of individual mnotone functions (in Russian), Doklady AN SSSR, 282(5) , pp.1033-1037.

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Andreev A., On a method for obtaining lower bounds for the complexity of individual monotone functions. Doklady Akademii Nauk SSSR, Vol. 282 (1985), pp. 1033--1037. English translation in Soviet Mathematics Doklady, Vol. 31 (1985), pp. 530--534.

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