Adolphus state expression

Good afternoon Professor Scott,

Regarding the state expression for the Bridge problem, I have sent you (a picture of) the correct derivation.  In essence, the formula on the website (and therefore the one on our published paper) is correct. 

An easy way to check is to add up the coefficients of the terms in the structure function, it should sum to 1 which is coherent with the theory, i.e., when all components are working, the structure function should return 1. The solution you obtained does not satisfy that condition.

No worries!

Best Regards, 

Adolphus Lye, Ph.D. (he/him) | Research Fellow

Adolphus:

I was surprised Friday when you explained that the reliability state expression (or whatever it's called) is just the arithmetic formulation of the logical expression.

I guess I must have forgotten this somehow, as I seem to have known it when I was in France, because the website https://sites.google.com/site/reliabilityuncertainty/code/8-bridge-relindep (which I remember writing) says it.

But in checking my understanding, I've stumbled onto something worrying.  Maybe it's simply another mistake.  I get a different expression for the bridge problem than that used in the website.  The website says it's

     ac+bd+ade+bce-abce-acde-abde-bcde-abcd+2abcde

but I get the formula

     ac+bd+ade+bce-abcd-acde-abde-abce-bcde+abcde

which is different in the last term.

Below in red is the step-by-step derivation I've used this morning.  Maybe it is obviously wrong.  But, if it's not, we've been using the wrong formulation for the bridge, haven't we?

Aghast,

Scott

To simplify the bridge's logical expression 

(a&c)|(b&d)|(a&e&d)|(b&e&c)

to its arithmetic form, let

W = ac

X = bd

Y = aed

Z = bec

then, because the conjunctions under independence are just the products W, X, Y, and Z, the logical expression is

W | X | Y | Z

but a four-fold application of the probabilistic sum (disjunction is the sum minus the product) means that this quadruple disjunction is

W + X + Y + Z - WX - WY - XY - WZ - XZ - YZ + WXZ + WYZ + XYZ - WXYZ

which means the logical expression is equal to this arithmetic expression

ac + bd + aed + bec - acbd - acaed - bdaed - acbec - bdbec - aedbec + acbdbec + acaedbec + bdaedbec - acbdaedbec

Remembering that, for Boolean values, xx = x and reordering the factors within terms yields

ac + bd + ade + bce - abcd - acde - abde - abce - bcde - abcde + abcde + abcde + abcde - abcde

of which four of the last five terms cancel, yielding

ac + bd + ade + bce - abcd - acde - abde - abce - bcde + abcde

Not sure where my error lies.