# NOT satisfiable 3SAT instance certificate

Given a NOT satisfiable 3SAT instance, that we say $$S$$. Suppose that $$M$$ is a minimal subset of clauses of $$S$$ such that $$M$$ is NOT satisfiable. Say $$X$$ the subset of variables of $$S$$ that belong to the clause of $$M$$. For each Boolean values combination of the variables of $$X$$, we have a clause of $$M$$ that is false, because $$M$$ is NOT satisfiable.

The question is: is $$M$$ a polynomial certificate that $$S$$ is NOT satisfiable? I mean: if we have $$M$$, we can check that, for each possible Boolean value of the variables that are in $$X$$, there is a false clause in $$M$$. This means that $$M$$ is a certificate that $$S$$ is NOT satisfiable. Because each clause is false for only one Boolean value of the variables that are in the clause and becasue $$M$$ is minimal, the number of cases, that we have to check to verify that $$M$$ is NOT satisfiable, is the cardinality of $$M$$ (i.e. the number of clauses in $$M$$). Threfore the complexity of the certificate is linear on the number of clauses of $$M$$ (therefore linear on the number of clauses of $$S$$).

Here an example.
If $$X=\left\{a,b,c\right\}$$, $$M$$ contains all possibile combinations, three by three, of $$\left\{a,b,c,\bar{a},\bar{b},\bar{c}\right\}$$ ($$9$$ literals in all), without that a variable and its negation appear in the same clause (it would be a tautology). Therefore the number of clauses of $$M$$ is exponential on the number of the variables of $$X$$, but the number of clauses of $$M$$ is at most the number of clauses of $$S$$. In other word we have $$cardinality(X)=O(\log(numberClauses(S)))$$.

A CNF which is not satisfiable is usually called unsatisfiable. A CNF which is unsatisfiable but becomes satisfiable if we drop any clause is minimally unsatisfiable.

Papadimitrious and Wolfe constructed, in their paper The complexity of facets resolved (Lemma 1), a polynomial time reduction $$f$$ from CNFs to CNFs such that:

• If $$\varphi$$ is satisfiable then $$f(\varphi)$$ is satisfiable.
• If $$\varphi$$ is unsatisfiable then $$f(\varphi)$$ is minimally unsatisfiable.

It is likely that this can be extended to a reduction from 3CNFs to 3CNFs.

This reduction shows that the minimally unsatisfiable case is the "hardest". In particular, if there is an efficient way to check that a minimally unsatisfiable CNF is unsatisfiable, then P=NP.

• Ok, but what about my question? In particular, is the number of the variables in a minimally unsatisfiable 3CNF $O\log(numberClauses)$? Apr 5, 2021 at 14:59
• No. Take, for example, $x_1 \land (\lnot x_1 \lor x_2) \land (\lnot x_2 \lor x_3) \land \cdots \land (\lnot x_{n-1} \lor x_n) \land \lnot x_n$. Apr 5, 2021 at 15:02
• A more exciting example is Tsietin contradictions on 3-regular graphs, which are also minimally unsatisfiable. Apr 5, 2021 at 15:04