Free Range Logic

AngleWyrm

Active Member
Here's the big three logic gates, that take as inputs a binary 0 or 1:

But what if those inputs were free to be any value across the range 0..1? For the sake of example let's say A=2/3 and B=1/4

I can surmise that NOT(A) = 1 - 2/3 = 1/3, and also that NOT(B) = 1 - 1/4 = 3/4
But what of AND and OR?

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fed2

Active Member
only one way to know, get some OR gates and feed in a 2/3, record the output. Mad Science!

katxt

Well-Known Member
Great practical solution, fed2.
Or ... inspecting the tables -
A AND B is the minimum of A and B
A OR B is the maximum of A and B
so that makes it easy.

AngleWyrm

Active Member
So then I should be able to work out the value of more complicated logic gate expressions and feed it some values.

Such as this one

Given: Y= (A OR B) AND (A OR (NOT C)) AND (B OR (NOT C)) and the set of inputs {A=2/3, B=1/4, C=1/2}

(A OR B) AND (A OR (1 - C)) AND (B OR (1 - C)) ; substituting NOT(X) -> 1-X
max(A,B) AND max(A, (1-C) AND max(B, (1-C)) ; substituting X OR Y -> max(X, Y)
min( max(A,B), max(A, (1-C), max(B, 1-C)) ; substituting X AND Y -> min(X,Y)
min( max(2/3,1/4), max(2/3, (1-1/2), max(1/4, 1-1/2)) ; substituting values for variables

y = min( max(2/3,1/4), max(2/3, (1-1/2), max(1/4, 1-1/2)) = 1/2

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AngleWyrm

Active Member
A Youtube series by a man who considers proper nouns very important

Another set of AND/OR operators is based on multiplication
NOT X -> 1 - X
X AND Y -> X * Y
X OR Y -> X+Y - X*Y
These look like functions that could describe and calculate probabilities, whereas the previous set of functions look more like they select between inputs based on logic.

(A AND B) OR (A AND (NOT C)) OR (B AND (NOT C)) where {A=2/3, B=1/4, C=1/2}

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katxt

Well-Known Member
Or ... inspecting the tables -
A AND B is the minimum of A and B
A OR B is the maximum of A and B
so that makes it easy.
This was just a glib throwaway based on a coincidence. Not something to initiate a major investigation.

AngleWyrm

Active Member
It's also the primary definition used in the linked article on Fuzzy Logic, for selecting among inputs based on logic.
You're not wrong, it's just useful for a different application.

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