A mathematical logic puzzle

[uncool]

That's the conjunctive normal form version; if you want disjunctive normal form:

(A and C) or (A and not D) or (C and not B).

[fleet-footed Urban Youth]

Another, somewhat simpler expression:
(A or C) and (A or not B) and (C or not D)

yeah when i was working through the restrictions i labor under (the weird order of operations) i eventually arrived at this. because:

(A and (C or not D)) or (C and not B) =
(A OR C) AND ((C or !D) OR C) AND (A OR !B) AND ((C or !D) OR !B)

but, as it turns out,

((C OR !D) OR C) = (C OR !D)

and

(C OR !D) = (C OR !D) AND ((C OR !D) OR !B)

so i came to the same thing you did: (A OR C) AND (A OR !B) AND (C OR !D)

from there i started really looking at the system this represents because i had the feeling there was a circumstance under which (C or !D) would be sufficient, which would be great for efficiency. i think my method was sorta like what you were talking about in reverse, and this is what i came up with:

where X = "what i want"

A | B | C | D | X | (C OR !D)
------------------------------
1 | 1 | 1 | 1 | 1 |     1
1 | 1 | 1 | 0 | 1 |     1
1 | 1 | 0 | 1 | 0 |     0
1 | 1 | 0 | 0 | 1 |     1
1 | 0 | 1 | 1 | 1 |     1
1 | 0 | 1 | 0 | 1 |     1
1 | 0 | 0 | 1 | 0 |     0
1 | 0 | 0 | 0 | 1 |     1
0 | 1 | 1 | 1 | 0 |     1
0 | 1 | 1 | 0 | 0 |     1
0 | 1 | 0 | 1 | 0 |     0
0 | 1 | 0 | 0 | 0 |     1  
0 | 0 | 1 | 1 | 1 |     1
0 | 0 | 1 | 0 | 1 |     1 
0 | 0 | 0 | 1 | 0 |     0
0 | 0 | 0 | 0 | 0 |     1

in the first place, as i said, none of this conditional shit matters if the user does not have A and actually absolutely needs it. so i could ignore every row representing that (i.e. begins 0 and then 1). the user would never get this far if that's true. from there i realized that C OR !D is fine unless all the conditions are false. so what's the problem there? the user does not have A, does not need A, does not have C, and does not need both C and A. but i always want them to have at least one of the two. requiring both rather than just one or the other is an option the user can set. if i make sure that D is always true when B is not, by making it so the option to turn D off is only possible if the check represented by B is on, then i can ignore any situation where the user doesn't need A and doesn't need both A and C: any row where B and D are false.

in that case, C OR !D does exactly what i need, and the design of the program will preclude the user reaching any other state.

thank you for all your help mathemagicians!

[el jefe]

I was just about to say, in engineering we had to map this shit all the time.

Eta (what hh posted)

ditto. I also was thinking I vaguely remembered there being some nice, standard procedure for turning tables like this into boolean expressions. i think we learned it in digital logic.

and, yeah, i think I was thinking of the thing hh posted.

[el jefe]

That's the conjunctive normal form version; if you want disjunctive normal form:

(A and C) or (A and not D) or (C and not B).

is this the same as "product of sums" vs. "sum of products"? all of this understood to be strictly boolean arithmetic of course, with OR = + and AND = ×.

[uncool]

is this the same as "product of sums" vs. "sum of products"? all of this understood to be strictly boolean arithmetic of course, with OR = + and AND = ×.

Eh, it's close. "Sums" doesn't quite work for OR; if you want a full ring, you need to use XOR. But it's close - you do have a version of distributivity, which is what you need to turn an arbitrary expression into a "sum of products".

[Preno]

Mathematics is about reducing complicated problems to simpler ones.

A Boolean function which outputs 1 in one row (say, the row A = 1, B = 0, C = 1) and 0 everywhere else is easy to produce: A and not B and C. Likewise, a Boolean function which outputs 0 in one row (say, the row A = 1, B = 0, C = 1 again) and 1 everywhere else is easy to produce: not A or B or not C.

If you know how to produce a Boolean function of n variables which outputs 1 in one row and 0 everywhere else, then you know how to produce any Boolean function: take a suitable disjunction of such functions. Likewise, if you know how to produce a Boolean function of n variables which outputs 0 in one row and 0 everywhere else, then you also know how to produce any Boolean function: take a suitable conjunction of such functions.

[fleet-footed Urban Youth]

noted

[Heinz Hershold]

we have conditions A, B, and C. using only the operators AND, OR, and NOT, can we combine them in such as way as to produce this outcome:

A   B   C      Result desired
0   0   0      0
0   0   1      0
0   1   0      1
0   1   1      0
1   0   0      1
1   0   1      0
1   1   0      1
1   1   1      1

but perhaps more importantly, is there some program or tool that will automate the construction of of such a combination and spare us all more contact with the devil's language, mathematics, than needed? or is this way more obvious than i think and i'm just a moron? (yes)

You will like this!

The Product of Sums output is: y = (A + B) (A + !C) (B + !C)

same as BD's answer.