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How can a Boolean representation be displayed as symbolic algebra?

Posted 11 years ago
Can Mathematica 9 convert a Boolean representation of an RBD to its symbolic algebraic form?

If I have a simple series configuration made up of 2 components
and the corresponding symbolic representation for the system would be RS1 = R1*R2

The system I am interested in is more complex and contains 7 components in a 4 path architecture:
Of course, other systems can have higher complexity. 

How can the algebraic form of the reliability function be obtained for bexpr2?

Thank you. 
5 Replies
There is also a more direct way to do this using a Library function, which will probably scale better for large problems:
POSTED BY: Malte Lenz
Posted 11 years ago

You produced an impressive solution, thank you.

I verified that your code in fact produces the result shown in Eq.(3) and that it works on other cases as well.
Your solution is exactly what I was looking for, nice piece of work indeed !

Also, kudos to the Mathematica Development Team for building such a brilliant computing environment ; do keep up the good work and keep pushing the boundaries !
For example,
dists = Table[{ToExpression["c" ~~ ToString[i]],
    BernoulliDistribution[ToExpression["r" ~~ ToString[i]]]}, {i, 7}];

Expand[PDF[ReliabilityDistribution[bexpr2, dists], 1]]

(* Out[7]= r1 r2 r5 r7 + r1 r3 r5 r7 - r1 r2 r3 r5 r7 + r1 r3 r6 r7 +
r1 r4 r6 r7 - r1 r3 r4 r6 r7 - r1 r3 r5 r6 r7 - r1 r2 r4 r5 r6 r7 +
r1 r2 r3 r4 r5 r6 r7 *)
This is the same as Eqn. (3):
% === r1 r4 r6 r7 + r1 r3 r6 r7 - r1 r3 r4 r6 r7 +
  r1 r2 r3 r4 r5 r6 r7 + r1 r3 r4 r5 r6 r7 - r1 r2 r3 r4 r5 r6 r7 -
  r1 r2 r3 r5 r6 r7 - r1 r3 r5 r6 r7 + r1 r2 r3 r5 r6 r7 -
  r1 r2 r4 r5 r6 r7 - r1 r3 r4 r5 r6 r7 + r1 r2 r3 r4 r5 r6 r7 +
  r1 r2 r5 r7 + r1 r3 r5 r7 - r1 r2 r3 r5 r7

(* Out[8]= True *)
POSTED BY: Ilian Gachevski
Posted 11 years ago

Thank you for the reply. I think I should have been more descriptive in my question.

The solution I’m trying to confirm can be seen in Eqn. (3) on page 5 in the PDF file at the following link. 
Are you just looking to use the engineering (+,*) notation for boolean logic?

One quick solution is to replace And with Times and Or with Plus:
bexpr2 /. {And -> Times, Or -> Plus}
c1 c2 c5 c7 + c1 c3 c5 c7 + c1 c3 c6 c7 + c1 c4 c6 c7
There are better ways of doing this using the Notation package that will allow Mathematica to still interpret this as a boolean logic statement.
POSTED BY: Sean Clarke
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