D. M. Barrington. Some problems involving Razborov-Smolensky polynomials. In M. S. Paterson, editor, London Mathematical Society Lecture Note Series 169: Boolean Function Complexity, pp. 109--128. Cambridge University Press, 1992. Home/Search Document Details and Download
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The Polynomial Method in Circuit Complexity - Beigel (1995) (26 citations) (Correct)
....special case, where x 6= 0 ( 1. Henceforth use the standard representation for outputs. Thus a polynomial g over F represents a function f if and only if Henceforth let p and q denote distinct primes. A polynomial g over Z p represents a function f if and only if It is well known [24, 49, 15, 12] that the class of Boolean functions represented by low degree polynomials over Z p is closed under Boolean operations. Theorem 17 (Folklore) Let g and h be represented by degree d polynomials over Z p . Then i. g is represented by a degree ( p Gamma 1)d) polynomial over Z p . ii. g h is ....
....by a degree 2d polynomial over Z p . iii. g h is represented by a degree (2(p Gamma 1)d) polynomial over Z p . Proof: Let G represent g and H represent h. i. 1 Gamma G represents :g. ii. GH represents g h. 15 iii. G H Gamma (GH) represents g h. Another folk theorem [24, 49, 15, 12] says that for every constant k the class of Boolean functions represented by low degree polynomials over Z p is the same as the class of Boolean functions represented by low degree polynomials over Z p k . Theorem 18 (Folklore) Let m and k be positive integers. i. If g is represented by a ....
D. M. Barrington. Some problems involving Razborov-Smolensky polynomials. In M. S. Paterson, editor, London Mathematical Society Lecture Note Series 169: Boolean Function Complexity, pp. 109--128. Cambridge University Press, 1992.
The Current State of Circuit Lower Bounds - Barrington (1993) (1 citation) Self-citation (Barrington) (Correct)
....depth with only modular gates) require exponential length (size) to do AND. This would appear to require modifying the techniques to work with Razborov Smolensky polynomials over general finite rings as well as just fields. Some preliminary work along these lines has been reported by Barrington [Ba90]. 7. ....
D. A. M. Barrington, "Some problems involving Razborov-Smolensky polynomials ", COINS Technical Report 90-59, University of Massachusetts.
Representing Boolean Functions As Polynomials Modulo.. - Barrington, Beigel.. (1994) (20 citations) Self-citation (Barrington) (Correct)
.... related questions came up in the study of permutation branching programs, or non uniform automata over groups ( Barrington 1989, Barrington and Th erien 1988, Barrington et al. 1990) This model of computation is closely related both to polynomials over finite rings and to circuits of MODm gates (Barrington 1990, 1992a) It was here, in the study of width three permutation branching programs (Barrington 1985) that an important distinction was noticed. With MODm calculations, it is difficult or impossible to force a computation to always give one of two output values (e.g. to compute the characteristic ....
D. A. M. Barrington, Some problems involving Razborov-Smolensky polynomials. In Boolean Function Complexity, ed. M. S. Patterson, London Mathematical Society Lecture Note Series 169, Cambridge University Press, 1992a, 109--128.
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