## truth value, circular logic

Mechanics, discrete, statistics, ...
JamieCamardelle
Posts: 20
Joined: Wed May 14, 2008 5:34 am

### truth value, circular logic

Say I have all 64 candidate truth tables: {0,0,0,0,0,0} … {1,1,1,1,1,1}
I want to be sure I understand this formula that I will apply to each truth table and determine whether the output value is 0 or 1.

Is the formula equivalent to this?
[tv(a) AND tv(b XOR tv(b AND c))] AND
[tv(b) AND tv(c XOR tv(b AND c))] AND
[tv(c) AND tv(d XOR tv(b AND c))] AND
[tv(d) AND tv(e XOR tv(b AND c))] AND
[tv(e) AND tv(f XOR tv(b AND c))] AND
[tv(f) AND tv(a XOR tv(b AND c))]
(tv stands for the tau symbol you used for truth value.)

jaap
Posts: 550
Joined: Tue Mar 25, 2008 3:57 pm
Contact:

### Re: truth value, circular logic

I think you misunderstand what truth table means.

A 6-input truth table for τ looks like this:

Code: Select all

a b c d e f:τ
0 0 0 0 0 0:x
0 0 0 0 0 1:x
0 0 0 0 1 0:x
....
1 1 1 1 1 1:x

where each of the x's is a 0 or 1. It has 64 rows, one for each combination of values for a-f. So τ is a function of 6 binary variables.

We want to choose the 64 x values such that τ satisfies the condition stated in the problem:
τ(a, b, c, d, e, f) AND τ(b, c, d, e, f, a XOR (b AND c)) = 0

For any particular values of a,b,c,d,e,f the condition expresses a relation between two of the x values in the table.

JamieCamardelle
Posts: 20
Joined: Wed May 14, 2008 5:34 am

### Re: truth value, circular logic

I mis-spoke. I do understand what a truth table means, but what I meant to say is that I have all 64 possibilities for the truth values of (a,b,c,d,e,f)
Thanks for responding. Your response is enlightening. However, I am still confused about τ(a, b, c, d, e, f), for instance. Do I interpret the commas as ANDs, so that τ({1,1,1,1,1,1}) = 1 but τ of anything else = 0?

daniel.is.fischer
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Joined: Sun Sep 02, 2007 11:15 pm
Location: Bremen, Germany

### Re: truth value, circular logic

No, [tau] is a function of six (boolean) variables. It could be simply [tau](a,b,c,d,e,f) = a, or it could be [tau](a,b,c,d,e,f) = a AND (b OR c) OR (d XOR (e AND f)), ... There are 264 such functions. You have to determine how many of those satisfy the relation
[tau](a,b,c,d,e,f) AND [tau](b,c,d,e,f,a XOR (b AND c)) = 0 for all (a,b,c,d,e,f).
Il faut respecter la montagne -- c'est pourquoi les gypa&egrave;tes sont l&agrave;.

JamieCamardelle
Posts: 20
Joined: Wed May 14, 2008 5:34 am

### Re: truth value, circular logic

When you said "There are 264 such functions.", I got it. Now I understand the problem. Have no idea right now how to solve it with such a large number of possibilities, but I understand the question. Thanks a lot.

I understand that τ stands for the same six-input function on the left and right of that first AND, and that the input variable sextet for the second part is essentially a rotation of the sextet given the first part, but with the exception that the last variable of the sextet is a XOR (b AND c) instead of just a. N'est-ce pas?

daniel.is.fischer
Posts: 2400
Joined: Sun Sep 02, 2007 11:15 pm
Location: Bremen, Germany

### Re: truth value, circular logic

Oui.
Il faut respecter la montagne -- c'est pourquoi les gypa&egrave;tes sont l&agrave;.

JamieCamardelle
Posts: 20
Joined: Wed May 14, 2008 5:34 am

### Re: truth value, circular logic

I solved it!!
That was a very fun problem! Thanks for your help.