Consider, over a finite set of boolean variables $X$, a Boolean system in CNF (conjunctive normal form) whose clauses only contain non-negated literals.

For every assignment of the variables which satisfies the system, one may consider the set of variables equal to true : these form a subposet of the poset of all variables.

**QA**) I would like to know the complexity (NP-complete, #P-complete etc.) of the following problems: find a satisfying assignment of the variables such that the set of variables to true is

- minimal with respect to inclusion
- of minimum cardinal among all solutions

**QB**) Can one construct such a system with both several inclusion-minimal solutions of non minimal cardinal and several cardinal-minimal solutions?

I am especially interested in those systems with the following "planarity / connectedness restrictions": the variables are the vertices of a planar graph whose faces have degree 4 and the variables in any clause must form a connected subgraph.

What do questions **A** and **B** become in this context?

PS : I am rather new to these satisfiability questions, I consulted a few references online (relevant Wikipédia pages, the "Handbook of satisfiability", courses etc) without seeing any mention of this question. There are some papers about "MIN-SAT" and "MAX-SAT" but these address a different question, namely extremize the number of satisfied clauses, whereas i'm interested in extremizing the number of variables.

Thanks you for your help,