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Consider a system with $n$ units where each unit is either working or failing. $x_j=1$ represents the condition that $j$-th unit is working. Suppose each unit is working with independent probability $p_j$, then show that the Boolean function $f(x_1, x_2, \dots x_n)$ is true with probability $F(p_1, p_2, \dots p_n)$ where $F$ is a polynomial in the variables in $p_1, p_2, \dots, p_n$.

I haven't understood the question properly. I understand what the Boolean variable $x_j$ means but what does the Boolean function $f$ mean and how do we prove anything about its probability ?

The solution uses the fact that $F$ is a multi linear integral representation of $f$, and that proves it. Now, I've read that a multilinear representation of a Boolean function is a sum of zero or more of its $2^n$ possible terms mod $2$. Now, an integral multilinear representation is this same multilinear representation such that each term is weighted with an integral coefficient so that $f(x_1,\dots x_n)$ has the correct value at all $2^n$ possible vectors without taking the reminder mod 2.But I don't understand how it relates. Please help.

My main questions are : 1. What is this Boolean function $f$ ? There can be any number of Boolean functions. How to prove something in general for all ?

  1. The concept of multilinear integral representation isn't clear. The author just said that a polynomial in its probabilities will be a multilinear integral representation of the Boolean function and that's that.
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I think it is a parrallel system, not a series system becauae the probability for each unit is independent of other units.

So, the problem changes into this simple question; for a parallel system with n units, find the probability that the system will not fail. (with the variables $x$ and $p$ given the same characteristic as mentioned above)

Then, $f=x_1+x_2+x_3+....\pmod2$. Assuming that + in this equation is or operation, if at least one $x$ can do work, that $x$ value becomes $1$, and $f$ has a value $1$ which means the system is working fine.

1 in probability means that it is true for any condition. So we minus the probability that all of its units fail to $1$. which results in this polynomial expression of all $p$ s. $F=1 - (1-p_1)\cdot(1-p_2)\cdot(1-p_3)....$ And that would be the answer for this question if it is a parallel system.

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  • $\begingroup$ The author clarified that the assumption that they're independent is a gross one to simplify the mathematics. $\endgroup$ – user230452 Jul 2 '16 at 9:48
  • $\begingroup$ I see you're new to the site and my question is your first post ! Thanks ! xD I edited your answer to put in $LaTeX$. This link was very helpful to me when I started out learning MathJax meta.math.stackexchange.com/questions/5020/… $\endgroup$ – user230452 Jul 2 '16 at 9:53
  • $\begingroup$ Thanks! I would try out the LaTeX later in my answer! $\endgroup$ – Chul Hyun Kim Jul 2 '16 at 10:23
  • $\begingroup$ Can you elaborate on why you chose $f$ the way you did ? If it's or, there's no need to put the modulo $2$ as that changes the meaning. (It becomes XOR). $\endgroup$ – user230452 Jul 2 '16 at 10:40
  • $\begingroup$ After scanning my answer, I realized that there is no need to put mod2 at there, and whether or not you put it the answer does not change because 0 or 1 is less than 2, and f can have only values 0 or 1. f tells in this case whether or not the system will work. $\endgroup$ – Chul Hyun Kim Jul 2 '16 at 11:07

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