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exercise sheet 2 question 2:

how can we Convert the formula ϕ to Shannon normal form?

I don't understand the meaning of Trues and Falses!

How do we get them?

in * TF "Emb. Sys. and Rob." by (1.7k points)

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The Shannon Normal Form (SNF) is a formula built from if-then-else-operators as defined on slide 4 of the BDD-Chapter. We defined (α⇒β|γ) as α∧β∨¬α∧γ. That means „if α is satisfied by the current assignment, then our formula has the truth value of β, otherwise it has the truth value of γ“

On the same slide, there are some examples. φ∧ψ is the same as (φ⇒ψ|0). Why? If an assignment satisfies φ, then the value of φ∧ψ depends precisely on the value of ψ. If it doesn't satisfy φ, then φ∧ψ is not satisfied by the assignment, and we write “0”.

The idea of the Shannon Normal Form is that we not only chaotically pick variables for the case distinction (in the previous example, we could as well just have written (ψ⇒φ|0)) but we fix a variables order. This is shown on slide 5. There, we have the formula x1∧x2∨x3∧x4, we start with variable x1 and do a case-distinction on this variable. We get (x1⇒x2∨x3∧x4|x3∧x4), which says that the formula evaluates to the same as x2∨x3∧x4 if x1 is true, and to the same as x3∧x4 if x1 is false. One can get to this result by thinking about it. But one can also plug in 1 for x1 on the left side of |, and 0 on the right side and then just apply the simplifications that are shown on slide 4 (last bullet point). After that, we continue with variable x2 both on the left and the right side of |. We continue recursively.

The true or false basically mean that we are assigning true to that variable in the formula to obtain the positive co factor and false to the variable to obtain negative co factor. We recursively do this with respect to the given variable ordering.

by (1k points)
edited by
@amala Paulson: Thanks for the explanation here. while solving the equation
ϕ= a&c&d|c&!a&!d|d&!c|!c&!d with ordering a⪯b⪯c⪯d, I didn't understand how should we go further after taking the case distinction of d variable. The next step I got is (d=> !a&!d|d&!c|!c&d | c&!a&!d|d&!c|!c&d). Sorry for long question. Can you please help.
This answer is incorrect (d=> !a&!d|d&!c|!c&d | c&!a&!d|d&!c|!c&d).
So if we have the formula x1∧x2∨x3∧x4 and the ordering is  x4⪯x3⪯x2⪯x1,, then you take x1 do the case distinction on it. That is if x1 is true then we have  x2∨x3∧x4, else if x1 is false then x3∧x4. So we can write it as

(x1 => x2∨x3∧x4 | x3∧x4).

So basically you are writing the if-else condition of the variables with respect to the given ordering.

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