## Representing Boolean Functions As Polynomials Modulo Composite Numbers (1994)

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Venue: | Computational Complexity |

Citations: | 53 - 6 self |

### BibTeX

@INPROCEEDINGS{Barrington94representingboolean,

author = {David A. Mix Barrington and Richard Beigel and Steven Rudich},

title = {Representing Boolean Functions As Polynomials Modulo Composite Numbers},

booktitle = {Computational Complexity},

year = {1994},

pages = {455--461}

}

### Years of Citing Articles

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### Abstract

. Define the MODm -degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 0-1 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm -degree of the OR of N variables is O( r p N ), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MODm -degree of both the MOD n and :MOD n functions is N\Omega\Gamma1/ exactly when there is a prime dividing n but not m. The MODm -degree of the MODm function is 1; we show that the MODm -degree of :MODm is N\Omega\Gamma30 if m is not a power of a prime, O(1) otherwise. A corollary is that there exists an oracle relative to which the MODmP classes (such as \PhiP) have this structure: MODmP is closed under complementation and union iff m is a prime power, and...

### Citations

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Citation Context ... 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 (Razborov 1987, =-=Smolensky 1987-=-). 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 MODm gates for any intege... |

222 |
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Citation Context ...at MODmP is closed under intersection for all m (Hertrampf, 1990); and that MODmP is closed under union if and only if MODmP is closed under complementation (Hertrampf, 1990). By standard techniques (=-=Furst et al. 1984-=-) it is possible to take circuit lower bounds and construct oracles that separate complexity classes. From our circuit lower bounds, we can construct an oracle relative to which no containment relatio... |

212 | Bounded-width polynomial-size branching programs recognize exactly those languages
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Citation Context ...m is divided by only two distinct primes, due to Tardos and Barrington (1994). This and 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 w... |

105 | Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs - Babai, Nisan, et al. - 1992 |

96 | On ACC and Threshold circuits - Yao - 1990 |

77 | Lower bounds for the size of circuits of bounded depth with basis
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Citation Context ... 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 (=-=Razborov 1987-=-, Smolensky 1987). 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 MODm gate... |

67 |
Counting classes are at least as hard as the polynomial-time hierarchy
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Citation Context ...PhiP. A language L belongs to MODmP if there exists a nondeterministic polynomialtime machine M such that x 2 L iff the number of accepting paths of M(x) is non-zero modulo m (Babai and Fortnow 1990, =-=Toda and Ogiwara 1992-=-, Tarui 1993). In Section 4, we use our lower bounds to construct an oracle such that MOD n P is closed under complementation and union iff n is a prime power, and MOD n P ` MODmP iff all prime diviso... |

61 | Counting classes: Thresholds, parity, mods, and fewness
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Citation Context ... degree polynomial which represents it. This model of boolean function complexity has been well explored in the case where m is a prime power (Smolensky 1987, Barrington 1992a, Beigel and Tarui 1991, =-=Beigel and Gill 1992-=-). It is known that ffi(OR; p) = dN=(p \Gamma 1)e (Smolensky 1987). It is also known that ffi(MOD n ; p) = \Omega\Gamma N) when n is not a power of p (Smolensky 1987). In the case of composite moduli,... |

50 | On the power of parity polynomial time - Cai, Hemachandra - 1990 |

47 | Non-uniform automata over groups - BARRINGTON, STRAUBING, et al. - 1990 |

43 |
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Citation Context ...1991, Barrington 1992a, Smolensky 1987). 4. An oracle for the conjectured relations among MODmP classes The class MODmP is a generalization of the counting class \PhiP (Papadimitriou and Zachos 1983, =-=Goldschlager and Parberry 1986-=-). First developed by Cai and Hemachandra (1990), these classes have since been studied by many others (Beigel 1991, Beigel and Gill 1992, Hertrampf 1990, Babai and Fortnow 1990, Toda and Ogiwara 1992... |

42 | Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan--in - Waack - 1991 |

40 | Finite monoids and the fine structure of NC - Barrington, Th'erien - 1988 |

23 | Relativized counting classes: relations among thresholds, parity, and mods
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Citation Context ...ation of the counting class \PhiP (Papadimitriou and Zachos 1983, Goldschlager and Parberry 1986). First developed by Cai and Hemachandra (1990), these classes have since been studied by many others (=-=Beigel 1991-=-, Beigel and Gill 1992, Hertrampf 1990, Babai and Fortnow 1990, Toda and Ogiwara 1992, Tarui 1993). It is known that MODmP = MODm 0 P where m 0 is the product of all distinct prime divisors of m (Hert... |

20 | A lower bound on the mod 6 degree of the OR function
- Tardos, Barrington
- 1998
(Show Context)
Citation Context ...s have failed to find a counterexample to the N = 10 conjecture. However, actually confirming the conjecture directly by computer search seems so far to be infeasible. It has been shown analytically (=-=Tardos and Barrington, 1994-=-) that ffi(OR; 6) =\Omega\Gamma449 (N)) in general, and ffi(OR; 6) ? 2 for N ? 18, but this is not a satisfying answer. Acknowledgements Many thanks to David Applegate who wrote the program which foun... |

17 | Randomized polynomials, threshold circuits and polynomial hierarchy - Tarui - 1993 |

15 | Relations among Mod-classes - Hertrampf - 1990 |

12 | On interpolation by analytic functions with special properties and some weak lower bounds on the size of circuits with symmetric gates - Smolensky - 1990 |

12 |
Algebraic Methods in Lower Bounds for Computational Models with Limited Communication
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- 1989
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Citation Context ...n that ffi(MOD n ; p) = \Omega\Gamma N) when n is not a power of p (Smolensky 1987). In the case of composite moduli, there have been very few results in this model (see, e.g., Krause and Waack 1991, =-=Szegedy 1989-=-, which we review below) . The obvious reason for this technical gap is that the techniques in the case of a prime modulus p have heavily relied on the fact that Z p is a field. We prove results, modu... |

11 | Some problems involving Razborov-Smolensky polynomials,” in Boolean Function - Barrington |

11 | Automata theory meets circuit complexity - McKenzie, Barrington - 1989 |

9 |
A note on a theorem of Razborov
- Barrington
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Citation Context ...1 if P (~x) 6= 0 mod m, and 0 otherwise. This is very similar to the standard definition of a MODm gate which outputs 1 iff the number of input 1s is non-zero modulo m (Razborov 1987, Smolensky 1987, =-=Barrington 1986-=-). 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... |

8 | Th' erien, Finite monoids and the fine structure - Barrington, D - 1988 |

7 | Expanded Edition. The first edition appeared in - Perceptrons - 1988 |

6 |
Width-3 permutation branching programs
- Barrington
- 1985
(Show Context)
Citation Context ...f 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 ... |

4 |
A characterization of #P by arithmetic straight-line programs
- Babai, Fortnow
- 1990
(Show Context)
Citation Context ...lize the definition of \PhiP. A language L belongs to MODmP if there exists a nondeterministic polynomialtime machine M such that x 2 L iff the number of accepting paths of M(x) is non-zero modulo m (=-=Babai and Fortnow 1990-=-, Toda and Ogiwara 1992, Tarui 1993). In Section 4, we use our lower bounds to construct an oracle such that MOD n P is closed under complementation and union iff n is a prime power, and MOD n P ` MOD... |

3 |
Th' erien, Non-uniform automata over groups
- Barrington, Straubing, et al.
- 1990
(Show Context)
Citation Context ...hat for non-symmetric polynomials the MODm -degree of OR might be as low as O(log N ), the lower bound proved after our work by Tardos and Barrington (1994)---previously the best bound was only !(1) (=-=Barrington et al. 1990-=-). We show that a low degree or sparse sub-linear degree polynomial for OR would have as a consequence the existence of small, low-depth MODm circuits for the AND function. Define the N-variable boole... |

2 | lower bound on the size of CC 2 (q)-circuits computing the AND function - Th'erien, Linear - 1991 |

1 | The current state of circuit lower bounds
- Barrington
- 1990
(Show Context)
Citation Context ...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... |

1 | 783--792. Revised version in this volume - Sci - 1991 |

1 | On the Weak Mod-m Degree of the GIP Function - Grolmusz - 1994 |

1 |
Relations among MOD-classes. Theoret
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(Show Context)
Citation Context ...t an N\Omega\Gamma23 lower bound on the MODm -degree of the :MODm function when m is not a prime power. This lower bound contrasts sharply with the corresponding lower bounds when m is a prime power (=-=Hertrampf 1990-=-, Beigel and Gill 1992, Beigel and Tarui 1991, Barrington 1992a, Smolensky 1987). If the set of prime divisors of n is contained in the set of prime divisors of m, then the MODm -degrees of :MOD n and... |

1 | P' eladeau and D. Th' erien, NC : the automata-theoretic viewpoint - McKenzie, P - 1991 |

1 | The first edition appeared in 1968. 16 - Edition |

1 | 167--183. D. Th' erien, Circuits of MOD m gates cannot compute AND in sublinear size - Sci - 1993 |