Example of unique 3sat solution

Question

I am searching for 3sat problems having unique solution. Till now, I only found one site https://www.cs.ubc.ca/~hoos/SATLIB/benchm.html

There are good examples of 3sat instance on that site but the problem is they have more than 1 solution if it's satisfiable. I want such big instances of 3sat of having 20 40 variables and at least 100 clauses ( More will be good ) but having unique solution. Where can I get such list of questions Or is there any algorithm which can generate 3sat question having unique solution ?

Answer 1

https://it-in-industry.com/itii_papers/2019/7219itii03.pdf discusses generating unique 3-sat problems. Might be a good place to start.

Answer 2

To construct a 3SAT problem with unique solution, you could use an arithmetic circuit like an adder or a multiplier. The resulting output lines are then compared against the known result using a digital comparator.

The design flow could be as follows:

1. Write the arithmetic expression to be used.
   Examples: x * 7, x - 1, x * x
2. Translate the expression into a boolean circuit of logical gates.
   This can be done by a suitable python script.
3. Transform the circuit into Conjunctive Normal Form (CNF) using a tool like bc2cnf.
4. If the resulting number of clauses is too big or too small, go back to step 1. and adjust the bitlength of the expression operand.

The simplest variant of such a circuit would be a digital comparator without pre-attached arithmetic circuit.

To get a unique solution, the arithmetic expression must have a unique solution. Counterexamples include integer division and integer sqare root.