Fixed points of logical functions

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Assume I have a function $F: \{0,1\}^n \to \{0,1\}^n$. I would like to analyse the fixed points of such function. This is a difficult problem, determining if there is a fixed point it's NP-complete if I am not mistaken. But I thought it might be doable if I put further restrictions on $F$. For example, I could assume that $F$ must be represented by a CNF formula with a most $k < n$ literals, or by some other representation with limited expressiveness. Is this something anybody has already worked on? I would be glad for any hints.