if c(u i ) = -2 n then f(x) has a decomposition

If C(ui) = -2n then f(X) has a decompositionf(X) = f*(X) xi where f*(X) is independent of xi.

If C(ui) = C(uj) = C(uij) = 0, i ≠ j then f(X) has a decomposition f(X) = f*(X) g(X) where g(X) = xi * xj,

* {∧,∨} and f*(X) is independent of both xi and xj.

THEOREM 1

THEOREM 2

EXAMPLE f(X) = (x1 ∨ x2x3) (x4x5)

first order coefficients

00001 0 00010 0 00100 16 01000 1610000 -16

00011 0 10001 000110 0 10010 0 01001 0 10100 -16 01100 16 11000 -16

second order coefficients

Work with the autocorrelation coefficients is continuing in many areas, including:

- a new classification method for switching functions- determining KDD decomposition tables- identification of symmetries- identification of degenerate and sparse functions

uses in three-level

decompositions

other uses

Autocorrelation Coefficients in the Representation and Classification of Switching Functions

RESULTSsuccesses avg. time

xor logic detected

AOXMIN 244/278 71.1 sec 54

our method 278/278 5.4 sec 59

- BDD-based techniques found to be very successful

computation techniques

x3x2x1 f(X)

0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

01100100

u3u2u1 B(u)

0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

30022002

what are autocorrelation coefficients?

J. E. Rice, University of Lethbridge

best BDD technique

best bruteforce technique

10 or fewer inputs

0.02 sec 0.78 sec

11 to 30 inputs

1.00 sec 67.6 sec

31 or greater inputs

4.58 sec 663 sec