# Is the counting version of 1-in-3 Sat #P-complete?

In the paper "Hard Tiling Problems with Simple Tiles", Moore and Robson prove that Cubic Planar Positive 1-in-3 Sat in NP-complete by a reduction from Positive 1-in-3 Sat.

Cubic Planar Positive 1-in-3 Satisfiability
Instance: A boolean formula $$B=(X,C)$$ where $$X$$ is a set of variables, $$C$$ a set of 3-element subsets over $$X$$ (clauses), and graph of B is a cubic planar graph.
Question: Does there exist a truth assignment for $$X$$ such that each clause is $$C$$ has exactly one true variable?

(see the note of the https://cstheory.stackexchange.com/a/42605/47855 regarding terminology)

Then they prove that a particular tiling problem is NP-complete by a reduction from Cubic Planar Positive 1-in-3 Sat. The following note is given after their reduction:

While the reduction from this version of SAT to the tiling problem we gave in the previous section is parsimonious, this does not show that the counting problem is #P-complete, since we do not have such a result for Cubic Planar Positive 1-in-3 SAT. We leave this as an open problem for the reader.

This gave me the impression that the counting version of 1-in-3 Sat is indeed #P-complete, but I couldn't find any source claiming this. Is the counting version of 1-in-3 Sat #P-complete?

• No, it is NP-complete. Do you mean its counting counterpart? Anyway, there is a reference right in the question that you linked to: sciencedirect.com/science/article/pii/S0890540196900164 . – Emil Jeřábek Jan 20 '20 at 9:41
• @Gamow, I don't understand why you remove the alternate names of the problem 1-in-3 Sat from the title. Many authors use the names Exactly one 3-sat and XSAT. If you are so sure it doesn't belong in the title, please consider adding these alternate names in the body of the question. – Cyriac Antony Jan 21 '20 at 16:32
• @EmilJeřábeksupportsMonica You are welcome to expand your comment into an answer. (I shall remove my answer after that.) I didn't realise that i forgot to explicitly state that it is the counting version i am talking about. – Cyriac Antony Jan 21 '20 at 16:33

Yes, the counting version of 1-in-3 Sat is $$\#P$$-complete. This is stated in "Complexity of Generalized Satisfiability Counting Problems" (Example 3.1), the reference pointed out by Emil in the comment.
Note: This leads us to the curious question: is the counting version of every NP-complete problem $$\#P$$-complete? The answers to this question (When does "X is NP-complete" imply "#X is #P-complete"?) suggest so!