Sketch an IP proof for minSAT

Question

I have to sketch an IP proof for the minSAT problem. The minSAT problem is define in this way:

• For a given formulae find a satisfying assignment with a min subset of variables assigned to True;

My sketched IP protocol is:

1. Prover and Verifier share the input formula

2. Prover sends to the Verifier a set of variables assigned to True, which should be the minimum

3. Verifier checks if the given set indeed satisfies the formula. If it is the case, it chooses one variable, re-computes the formula and sends the new formula to the Prover

I would iterate this process untill the Prover gives an empty set or I reach a contradiction.

My questions are:

• Is this a right protocol? If not, how can I exploit the cases in which the protocol doesn't work?

• How can I evaluate the probability of completness and soundness for this protocol?

Answer 1

Your protocol is not very clear. What do you mean at step 3 by "it chooses one variable, re-computes the formula"?

I assume the verifier just sets that variable to 1 and re-computes the formula without that variable.

Then, this protocol still has some problems. Let's say that there are two sets of variables that satisfy the formula: $\{x\}$ and $\{y,z\}$. The minSat is $x$. But what happens if the prover says it is $y,z$? the verifier checks this proposed set, and sees that the set satisfies the formula; then the verifier removes one variable, say $y$, and sends the formula back to the prover. The prover will then reply with $\{z\}$ and the verifier will accept it.

Even if the verifier renames the variables in a random way, (say, replace $x$ with $a$ and $z$ with $b$), it's not clear that this protocol works. Still, maybe the prover is able to identify the variable that previously was $z$ from the structure of the formula, and repeat the faulty behaviour described above.

So although in the suggested protocol the prover cannot reply with a set that doesn't satisfy the formula, the prover can give a non-minimal set, and the verifier will (wrongly) accept it.