Introduction to Read-Muller Logic

1. Introduction to Read-Muller Logic 2002. 10. 8. Ahn, Ki-yong

2. Positive Polarity Reed-Muller Form • f(x1,x2,…xn) = a0 ^ a1x1 ^ a2x2 ^…^ anxn^ … ^ amx1x2x3…xn • All variables are positive polarities • There is only 1 PPRM • EX) • F = 1 ^ x1 ^ x2 ^ x1x2

3. Fixed Polarity Reed-Muller Form • Each variable has positive or negative polarity • Polarity of variable is fixed • There is only 2n FPRMs • EX) • F = 1 ^ x1 ^ x2’ ^ x1x2’

4. Generalized Reed-Muller Forms • Each variable can have any polarity • Polarity of variable is not fixed • There is only 2n2n-1 GRMs • EX) • F = 1 ^ x1 ^ x2’ ^ x1’x2’

5. pD S 1 X1 X1 X1’ Fundamental Expansions • Shannon Expansion – S • f(x1,x2,…,xn) = x1’f0(x2,…xn) ^ x1f1(x2,…xn) • Positive Davio Expansion – pD • f(x1,x2,…,xn) = 1 f0(x2,…xn) ^ x1f2(x2,…xn) • f2 = f0 ^ f1

6. D nD 1 X1’ X1 1 Fundamental Expansions(2) • Negative Davio Expansion – nD • f(x1,x2,…,xn) = 1 f0(x2,…xn) ^ x1’f2(x2,…xn) • Generalized Davio Expansion – D • f(x1,x2,…,xn) = 1 f0(x2,…xn) ^ x1f2(x2,…xn) • x1 means both negative and positive

7. Kronecker Forms - KRO S • Using S, pD, nD for each variable • The same expansion must be used for the same variable a a’ pD pD b 1 b 1 nD nD nD nD 1 1 c’ 1 c’ c’ 1 c’

8. Pseudo-KRO Form - PDSKRO S • Using S, pD, nD for each variable a a’ S pD b 1 b b’ pD nD pD S 1 1 c’ 1 c c c’ c

9. S/D Tree S • Using Shannon(S) and Generalized Davio(D) a a’ D S b 1 b b’ D D S D 1 c 1 c’ 1 c c c a’ a’c ab abc a’bc’ ab’c a’bc ab’

10. D S D S D S D S a’ a’ a’ a’ 1 1 1 1 a a a a a a a a S S S D D S D D S S S D D S D D b b b b b b b b b’ b’ b’ b’ 1 1 1 1 b b b b b b b b b’ b’ b’ b’ 1 1 1 1 ab ab ab ab ab ab ab ab ab’ ab’ ab’ ab’ a a a a a’b’ a’b’ b’ b’ a’ a’ 1 1 a’b a’b a’b a’b b b b b Inclusive Forms for Two Variables N=1 N=2 N=2 N=4 • NIF = (1+2+2+4)+(4+8+8+16) = 45 N=4 N=8 N=8 N=16