CNF formula for exactly one variable is true

Question

Given a list (array) of $n$ variables, what's the algorithm that creates a CNF Formula that will make the SAT solver return assignments in which exactly one variable is true?

I know I'm not 100% with the terms here, but I just got this assignment, having SAT explained to me today.

To make things clear I'll translate the text of the exerise:

Code a function that, given a list of variable names, returns a CNF formula for which its "list of satisfactory assignments" matches all the instances for which only one variable gets the value TRUE.

The function signature is as follows

public static int[][] exactlyOne(int[] varNames) {}

score 2 · Accepted Answer · answered Nov 17 '17 at 02:56

2

You want at least one variable to be true but no pair to be both true. That is, one clause of the CNF is $$v_1 \text{ or } v_2 \text{ or } \ldots \text{ or } v_n$$ and the others are of the form $\text{not } v_i \text{ or } \text{not } v_j$.

answered Nov 17 '17 at 02:56

Robert Israel

448,999

Rob Arthan · Answer 2 · 2017-11-17T01:46:23.467

0

Hint: the DNF formula to assert this has one disjunct for each variable asserting that that variable is true and the other variables are false. Now put it into CNF.

edited Nov 17 '17 at 01:46

answered Nov 17 '17 at 01:16

Rob Arthan

48,577

say n=3 and a,b,c are my variables. (a or !b or !c) and (!a or b or !c) and (!a or !b or c). Is that what you mean? if yes, a,b,c=true will be a satisfactory assignment... – McLovin Nov 17 '17 at 01:22
Or is it (a or b or !c) and (a or !b or c) and (!a or b or c)? – McLovin Nov 17 '17 at 01:26
I misread your question and have now corrected my answer. – Rob Arthan Nov 17 '17 at 01:46

CNF formula for exactly one variable is true

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