. 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...
|
233
|
Algebraic methods in the theory of lower bounds for boolean circuit complexity
– Smolensky
- 1987
|
|
174
|
circuits, and the polynomial-time hierarchy
– FURST, SAXE, et al.
- 1984
|
|
171
|
Bounded-width polynomial-size branching programs recognize exactly those languages
– Barrington
- 1989
|
|
82
|
Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs
– Babai, Nisan, et al.
- 1992
|
|
74
|
On ACC and threshold circuits
– Yao
- 1990
|
|
68
|
Lower Bounds for the the Size of Circuits of Bounded Depth with Basis
– Razborov
- 1987
|
|
55
|
Counting classes: Thresholds, parity, mods, and fewness
– Beigel, Gill
- 1992
|
|
55
|
Counting classes are at least as hard as the polynomial-time hierarchy
– Toda, Ogiwara
- 1992
|
|
49
|
On the power of parity polynomial time
– Cai, Hemachandra
- 1990
|
|
45
|
Variation Ranks of Communication Matrices and Lower Bounds for Depth Two Circuits having Symmetric Gates and Unbounded Fan-in
– Krause, Waack
- 1991
|
|
40
|
Finite monoids and the fine structure of NC
– Barrington, Therien
- 1988
|
|
38
|
On the Construction of Parallel Computers from various bases of Boolean Circuits
– Goldschlager, Parberry
- 1986
|
|
38
|
Non-Uniform Automata Over Groups
– Straubing, Th'erien
- 1990
|
|
20
|
Relativized counting classes: Relations among thresholds, parity, and mods
– Beigel
|
|
20
|
A lower bound on the MOD 6 degree of the OR function
– Tardos, Barrington
- 1995
|
|
17
|
Randomized polynomials, threshold circuits, and the polynomial hierarchy
– Tarui
- 1990
|
|
14
|
Relations among MOD-classes
– Hertrampf
- 1990
|
|
13
|
Algebraic Methods in Lower Bounds for Computational Models with Limited Communication
– Szegedy
- 1989
|
|
11
|
On interpolation by analytic functions with special properties and some weak lower bounds on the size of circuits with symmetric gates
– Smolensky
- 1990
|
|
11
|
Automata theory meets circuit complexity
– McKenzie, Barrington
- 1989
|
|
9
|
A note on a theorem of Razborov
– Barrington
- 1986
|
|
8
|
Th' erien, Finite monoids and the fine structure of NC
– Barrington, D
- 1988
|
|
8
|
Some problems involving Razborov-Smolensky polynomials
– Barrington
- 1992
|
|
6
|
Expanded Edition. The first edition appeared in
– Perceptrons
- 1988
|
|
4
|
A characterization of #P by arithmetic straight-line programs
– Babai, Fortnow
- 1990
|
|
3
|
Width 3 permutation branching programs
– Barrington
- 1985
|
|
3
|
Th' erien, Non-uniform automata over groups
– Barrington, Straubing, et al.
- 1990
|
|
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
|
|
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
– Hertrampf
- 1990
|
|
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
|
|
