Bernoulli variables again

• To: mathgroup at smc.vnet.net
• Subject: [mg46538] Bernoulli variables again
• From: jmyers6761 at aol.com (JMyers6761)
• Date: Sun, 22 Feb 2004 11:27:40 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Consider the following expresion where each of the xn are Bernoulli variables:

1 - (1 - (1 - (1 - x1) (1 - x2)) x4) (1 - (1 - (1 - x2) (1 - x3)) x5)

when this expression is Expanded we get:

x1 x4+x2 x4 - x1 x2 x4 + x2 x5 + x3 x5 - x2 x3 x5 - x1 x2 x4 x5 - x2^2 x4 x5 +
x1 x2^2 x4 x5 - x1 x3 x4 x5 - x2 x3 x4 x5 + 2 x1 x2 x3 x4 x5 + x2^2 x3 x4 x5 -
x1 x2^2 x3 x4 x5

but since a Bernoulli variable, x, can take on only values of 0 or 1 and x^n =
x this expression is subject to the transformation x^n-> x with the following
result:

x1 x4 + x2 x4 - x1 x2 x4 + x2 x5 + x3 x5 - x2 x3 x5 - x2 x4 x5 - x1 x3 x4 x5 +
x1 x2 x3 x4 x5

My question is does anyone know how to transform such an unexpanded expression
without having to first do an Expand? The actual expressions I am dealing with
have a very large number of fully expanded terms ( > 10^6) and as a result
Mathematica runs out of memory attemting to Expand the expressions. I know that
the resulting expressions, after the Bernoulli transformation, if they could be
expanded, would still be large but the number of terms would be much smaller,
on the order of 10^3 terms. Does anyone know of a technique that could used to
transform the unexpanded expression?