Free Range Logic

[AngleWyrm]

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?

 

[fed2]

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

 

[katxt]

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]

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

 

[Dason]

It looks like you're just giving a different name to an already established idea: Fuzzy Logic

 

[AngleWyrm]

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


From the linked article

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}

 

[katxt]

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]

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.

 

[fed2]

good job katxt, you passed AngleWyrm's test.

 

[Dason]

AND OR min(max( did, they))?