Razborov, A. A. (1987) Lower bounds for the size of circuits of bounded depth with basis {#, #}, Math. Notes of the Academy of Sciences of the USSR 41:4 333338.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....circuit family as above yields a depth O(d(n) Boolean circuit family for transitive closure. Again, this entails no more than polynomial blow up of size. Our circuit result improves an earlier result of Borodin [Bo2] about the general relation of Boolean and arithmetic circuits. Razborov [Ra] showed that an m input can be well approximated by degree log m polynomials over finite fields. Using these polynomials and amplifying the approximation it is possible to get O(d log log n log n) depth and polynomial size semi unbounded arithmetic circuits that simulate semi unbounded (and ....

....the proof of Theorem 5. For proving Theorem 6 we construct the circuits C and transform their outputs to Boolean values as above. To achieve the O(d log k) depth for approximate simulations over a fixed finite field we use polynomials of degree k that approximate the of these values. By [Ra] (cf. Sm] Lemma 1) given any probability distribution on the 2 inputs there exist polynomials of degree k that compute the of T variables with probability (1 Gamma 2 ) over the input distribution. This concludes the proof of Theorem 6. Acknowledgements Venkateswaran independently ....

A. A. Razborov, "Lower bounds for the size of circuits of bounded depth with basis f; \Phig," Math. notes of the Academy of Sciences of the USSR, 41(4), (1987), pp. 333-338.

....(and similarly, every symmetric function) can be computed over fields by size O(n ) inhomogeneous Sigma Pi Sigma circuits, using one variable polynomial interpolation. This result shows the power of arithmetic circuits over Boolean circuits with MOD p gates, since as it was proved by Razborov [20] and Smolensky [23] that MAJORITY a symmetric function needs exponential size to be computed on any bounded depth Boolean circuit. Note, that our construction with homogeneous circuits modulo non prime power composites beats Ben Or s bound for k s less than c log log n (for some positive ....

A. Razborov. Lower bounds for the size of circuits of bounded depth with basis (AND, XOR). Mathematical Notes of the Academy of Science of the USSR, 41(4):333--338, 1987.

....be interested in are ae ACC [2] ae ACC Delta Delta Delta NC P It is believed, but not known, that P 6= NC: in other words, that there are inherently sequential problems in P that cannot be efficiently parallelized. Some small progress has been made towards proving this [1, 8, 21, 23]: parity is in [2] but not AC , ACC [p] and ACC [q] are incomparable if p and q are distinct primes, and majority is in TC but not ACC [2] Thus the first two inclusions in this series are proper, but ACC [6] and P (or even NP) could be identical for all anyone has been able to ....

A.A. Razborov, "Lower bounds for the size of circuits of bounded depth with basis f&; \Phig." Math. Notes Acad. Sci. USSR 41(4) (1987) 333--338.

....Tarui. Except for appendix 1, where we simplify his 0 sided error construction, all of our results were obtained independently. Supported in part by NSF grants CCR 8808949 and CCR 8958528. Supported in part by NSF grant CCR 8958528 under an REU supplement. of much recent research. Razborov [13] and Smolensky [14] showed that every AC predicate can be approximated by a low degree polynomial over any finite field. Their upper bounds yield circuit lower bounds related to Hastad s [10] Applying Hastad s lower bound for parity, Linial, Mansour, and Nisan [11] proved an important upper ....

A. A. Razborov. Lower bounds for the size of circuits of bounded depth with basis f; \Phig. Math. notes of the Academy of Science of the USSR, 41(4):333--338, Sept. 1987.

....Ajtai [Ajt83] first showed that constant depth circuits with NOT, OR, and AND gates require superpolynomial size to compute PARITY. Yao [Yao85] improved this and gave an exponential bound, and later Hastad [Has87] simplified a proof, and gave a further improved nearoptimal bound. Razborov [Raz87] showed that to compute MAJORITY, constant depth circuits with NOT, AND, OR, and PARITY gates require exponential size. Smolensky later extended this and showed that to compute the MOD q function, constant depth circuits with NOT, AND, OR, and MOD q gates require exponential size if q and q ....

....with error probability at most 1=4. It will be clear that the proof works for any error probability bounded away from 1=2. By using a standard probabilistic simulation of AND OR by MODm gates (we can either use Valiant and Vazirani s lemma [VV86] as in [AH90] or the RazborovSmolensky simulation [Raz87, Smo87]) and by the argument explained above, obtain N = O(n) deterministic ACC circuits C 1 ; CN consisting of MODm gates and fan in log n AND gates such that for every x 2 f0; 1g N ; Thus, if we let fg i ; g i g be the set of MODm gates that are connected to the ....

A. Razborov. Lower bounds for the size of circuits of bounded depth with basis f; \Phig. Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333--338, September 1987.

.... For instance, polynomial representations have been applied in the areas of circuit lower bounds, interactive proofs and PCP characterizations of NP, average case to worst case reductions for hard problems, and program checking (see [19] for a survey of some related work) The reader is referred to [28, 31, 24, 32, 3, 33, 26, 7] for results on polynomial representations and their applications in complexity theory. Our notion of randomizing polynomials may be viewed as another natural form of polynomial representation, with a different flavor of applications. The round complexity of interactive protocols is one of their ....

A. A. Razborov. Lower bounds for the size of circuits of bounded depth with basis {#, #}. Math. notes of the Academy of Science of the USSR, 41(4):333--338, 1987.

....0 ae ACC 0 [2] ae ACC 0 TC 0 NC 1 ACC 1 Delta Delta Delta NC P It is believed, but not known, that P 6= NC: in other words, that there are inherently sequential problems in P that cannot be efficiently parallelized. Some small progress has been made towards proving this [1, 8, 20, 22]: parity is in ACC 0 [2] but not AC 0 , ACC 0 [p] and ACC 0 [q] are incomparable if p and q are distinct primes, and majority is in TC 0 but not ACC 0 [2] Thus the first two inclusions in this series are proper, but ACC 0 [6] and P (or even NP) could be identical for all anyone has ....

A.A. Razborov, "Lower bounds for the size of circuits of bounded depth with basis f&; \Phig." Math. Notes Acad. Sci. USSR 41(4) (1987) 333--338.

....in are AC 0 ACC 0 [2] ACC 0 TC 0 NC 1 ACC 1 NC P It is believed, but not known, that P 6= NC: in other words, that there are inherently sequential problems in P that cannot be eciently parallelized. Some small progress has been made towards proving this [1, 13, 26, 29]: parity is in ACC 0 [2] but not AC 0 , ACC 0 [p] and ACC 0 [q] are incomparable if p and q are distinct primes, and majority is in TC 0 but not ACC 0 [2] Thus the rst two inclusions in this series are proper, but ACC 0 [6] and P (or even NP) could be identical for all anyone has ....

A.A. Razborov, \Lower bounds for the size of circuits of bounded depth with basis f&; g." Math. Notes Acad. Sci. USSR 41(4) (1987) 333-338.

....One of the central problems of theoretical computer science is the estimation 1 1 of the computational complexity of Boolean functions. One well studied measure of the complexity of Boolean function f is the degree of a polynomial P , which best approximates or represents f in some sense (see [Raz87], Smo87] NS94] ABFR91] or [Bei93] for a survey) According to this approach, a Boolean function is considered hard if the polynomial that represents or best approximates it has a high degree. Barrington, Beigel, and Rudich [BBR94] defined the mod m degree of 1 2 Boolean function f to be ....

Alexander Razborov. Lower bounds for the size of circuits of bounded depth with basis (AND, XOR). Mathematical Notes of the Academy of Science of the USSR, 41(4):333--338, 1987.

....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 fairly complicated functions (like division, implicit ....

A. A. Razborov. Lower bounds for the the size of circuits of bounded depth with basis , \Phi. Mathematical Notes of the Academy of Sciences of the USSR, pages 598--607, 1987. English translation in 41:4, pp. 333-338, 1987.

....automata theory [BT88] and formal logic [BIS88] We outline the methods and results in the case of three types of classes defined by circuits of constant depth, polynomial size, and unbounded fan in, for three different types of gates. These are AND and OR gates [FSS84] AND, OR, and MOD p gates [Ra87, Sm87], and modular gates alone with certain other restrictions [BST90] Most of this survey was prepared for the McGill University Workshop in Theoretical Computer Science, held in February 1989. 2. Introduction It would be nice to show that the CLIQUE problem is not solved by a family of boolean ....

....are one) and the majority operation (returning one iff at least half of the inputs are one) It is not hard to show [FSS84] that parity is in the AC 0 closure of majority and thus that majority is itself not in AC 0 . We shall see below that majority is not in the AC 0 closure of parity [Ra87] we understand this class well enough to show this. In fact, if we define a modulo p operation appropriately for any prime p, we understand the AC 0 closure of modulo p well enough to exclude majority (and modulo q for any integer q not divisible by p) from it [Sm87] This understanding ....

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A. A. Razborov, "Lower bounds for the size of circuits of bounded depth with basis f&; \Phig", Mathematicheskie Zametki 41:4 (April 1987), 598-607 (in Russian). English translation Math. Notes Acad. Sci. USSR 41:4 (Sept. 1987), 333-338.

....and allowing the gate functions to vary. If we take the ordinary AND and OR functions we get the class AC 0 . If we gates which calculate the exclusive OR function we get a class which we may call AC 0 [2] this class is strictly larger than AC 0 , by the result quoted above) Razborov [Ra87] has shown that AC 0 [2] does not contain the language fx 2 f0; 1g : x has more ones than zeroes g. Smolensky [Sm87] has considered the classes AC 0 [q] defined similarly to AC 0 [2] but with gates which test whether the number of inputs which are one is divisible by some number q. He ....

A. A. Razborov, "Lower bounds for the size of circuits of bounded depth with basis f&; \Phig", Mathematicheskie Zametki 41:4 (April 1987), 598-607 (in Russian). English translation Math. Notes Acad. Sci. USSR 41:4 (Sept. 1987), 333-338.

....area where proving lower bounds has been successful is considering bounded depth circuits. Improving the lower bounds of [1, 34, 100] Hastad [43] proved exponential lower bounds for computing the parity function by bounded depth circuits with unbounded fan in gates over the basis f; g. Razborov [75] proved that computing the majority function by bounded depth circuits requires exponential size even allowing the use of parity gates. Smolensky [86] extended the above results, proving exponential lower bounds for computing the MOD r function by bounded depth circuits over f; MOD p g, if p ....

....by such a polynomial. Similarly, bounded fan in Boolean circuits are easy to simulate by bounded fan in depth O(d) arithmetic circuits. For semi unbounded fan in circuits, replacing each gate by the corresponding polynomial we obtain Omega Gamma d log n) depth arithmetic circuits. Razborov [75] showed that an m input can be well approximated by degree log m polynomials over finite fields. Using these polynomials and amplifying the approximation it is possible to get O(d log log n log n) depth and polynomial size semi unbounded arithmetic circuits that simulate semi unbounded (and ....

[Article contains additional citation context not shown here]

A. A. Razborov, "Lower bounds for the size of circuits of bounded depth with basis f; \Phig," Math. notes of the Academy of Sciences of the USSR, 41(4), 1987, pp. 333-338.

....: 60 4 Sitharam 1. Introduction In the context of complexity lower bounds, the approximation method usually refers to the method originated by Razborov in [58] and [60] for proving monotone lower bounds. The approach was continued by [59] and [67] and several others including [6] 68] 9] 69] 72] 78] 40] for general lower bounds, and further used in monotone lower bounds such as [2] 79] and [10] Other complexity lower bounds that can be generally classified as being based on nonapproximability by low degree or sparse ....

....D is fixed to be the uniform distribution and the subset S is sometimes fixed to be the entire domain, i.e, oe = 1. Sometimes, S is existentially quantified, with oe bounded away from 1=2, which is often essential when the issue is interpolability, i. e, when fl = 1, as, for example, in the case of [59] and [67] When the approximation of interest is in the 1 norm, as, for example, for threshold circuit complexity lower bounds, the universal quantifier is implicit for the distribution D: the approximation h in A must have the same sign as g everywhere. This fact has been extensively used in ....

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A.A. Razborov, "Lower bounds on the size of circuits of bounded depth with basis f; \Phig," Math. notes of the Aca. of Science of the USSR, 41 (4), pp. 333338, 1987.

....and, thus, illustrates the power of alternation with respect to unbounded weight threshold operations. 1 Introduction In the last decade a lot of research has been done on the computational power of small depth circuits of unbounded fan in over AND , OR , XOR , modular and threshold gates (e.g. [21][22] 8] 23] 12] 1] to give at least a sample of the most important results) A large number of results achieved in this area shows that attempts to analyze the computational power of various architectures of small depth circuits often lead to another fundamental question in Boolean complexity ....

.... low degree polynomials over the Boolean semiring [13] Another example is that all functions having polynomial size constant depth circuits over AND , OR and MOD p gates, p prime (for short: AC 0 [p] functions) can be computed by randomized low degree ZZ p polynomials with bounded error [21][22] This leads to the important result that for all q 6= p prime, MOD q is not an AC 0 [p] function [22] Another interesting characterization of the complexity of AC 0 functions is that they can be efficiently approximated by 2 low degree integer polynomials [4] Moreover, using Yao s ....

Razborov, A.A: Lower bounds for the size of circuits of bounded depth with basis f\Phi; g, Journal Math. Zametki. 41, 1987, 598--607.

....m. 1 Introduction Lower bounds in circuit complexity are currently hindered by what at first glance appears to be a small technical point. It is known that AC 0 circuits which also allow mod p gates for some fixed prime, p, can t compute the mod q function for any q which is not a power of p [18, 19]. In contrast, it is not known if AC 0 circuits which also allow mod 6 gates can compute every function in NP . It is conjectured that (as with the case of mod p) AC 0 with mod m gates for any integer m can t compute the mod n function when there is a prime dividing n but not m [19] Indeed, ....

....all 0 1 valued assignments x, F ( x) 0 iff P ( x) 0. In other words, we interpret the output of P to be the boolean value 1 if P ( x) 6= 0 mod m, and 0 otherwise. This is very similar to the standard definition of a mod m gate which outputs 1 iff the number of input 1s is non zero modulo m [18, 19, 3]. The MODm degree of F , denoted ffi (F; m) is the degree of the lowest degree polynomial which represents it. This model of boolean function complexity has been well explored in the case where m is a prime power [19, 8, 11, 10] It is known that for the OR of N variables ffi (OR; p) dv= p ....

A. A. Razborov. Lower bounds for the size of circuits of bounded depth with basis f; \Phig. Math. notes of the Academy of Science of the USSR, 41(4):333--338, Sept. 1987.

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Razborov A.A., Lower bounds for the size of circuits of bounded depth with basis f&; \Phig. Mathematicheskie Zametki 41:4 (April 1987), 598607 (in Russian). English translation Math. Notes Acad. Sci. USSR 41:4 (Sept. 1987), 333-338.

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Razborov, A. A. (1987) Lower bounds for the size of circuits of bounded depth with basis {#, #}, Math. Notes of the Academy of Sciences of the USSR 41:4 333338.

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A. A. Razborov, Lower bounds for the size of circuits of bounded depth with basis f; \Phig: Math. Notes of the Academy of Sciences of the USSR 41:4 (1987), 333--338.

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Alexander Razborov. Lower bounds for the size of circuits of bounded depth with basis f^; g. Methematical Notes of the Academy of Science of the USSR, (41):333-338, 1987.

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A. Razborov. Lower bounds for the size of circuits of bounded depth with basis (AND, XOR). Mathematical Notes of the Academy of Science of the USSR, 41(4):333--338, 1987.

No context found.

A. Razborov. Lower bounds for the size of circuits of bounded depth with basis (AND, XOR). Mathematical Notes of the Academy of Science of the USSR, 41(4):333--338, 1987.

No context found.

Alexander Razborov. Lower bounds for the size of circuits of bounded depth with basis f^; g. Methematical Notes of the Academy of Science of the USSR, (41):333-338, 1987.

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A. Razborov, Lower bounds for the size of circuits of bounded depth with basis f; \Phig. In Math. Notes Acad. Sci. USSR 41(4), (1987), 333-338.

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A. Razborov, Lower bounds for the size of circuits of bounded depth with basis {#, #}, Math. notes of the Academy of Sciences of the USSR, 41 (1987), pp. 333-- 338.

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