On the Satisfaction Probability of $k$-CNF Formulas

The satisfaction probability $\sigma(\phi) := \Pr_{\beta:\mathrm{vars}(\phi) \to \{0,1\}}[\beta\models \phi]$ of a propositional formula $\phi$ is the likelihood that a random assignment $\beta$ makes the formula true. We study the complexity of the problem $k$sat-prob$_{>\delta} = \{ \phi$ is a $k\mathrm{cnf}$ formula $\mid \sigma(\phi) > \delta\}$ for fixed $k$ and $\delta$. While 3sat-prob$_{>0}$ = 3sat is NP-complete and sat-prob$_{>1/2}$ is PP-complete, Akmal and Williams recently showed 3sat-prob$_{>1/2} \in$ P and 4sat-prob$_{>1/2} \in$ NP-complete; but the methods used to prove these striking results stay silent about, say, 4sat-prob$_{>1/3}$, leaving the computational complexity of $k$sat-prob$_{>\delta}$ open for most $k$ and $\delta$. In the present paper we give a complete characterization in the form of a trichotomy: $k$sat-prob$_{>\delta}$ lies in AC$^0$, is NL-complete, or is NP-complete; and given $k$ and $\delta$ we can decide which of the three applies. The proof of the trichotomy hinges on a new order-theoretic insight: Every set of $k$cnf formulas contains a formula of maximum satisfaction probability. This deceptively simple result allows us to (1) kernelize $k$sat-prob$_{\ge \delta}$, (2) show that the variables of the kernel form a strong backdoor set when the trichotomy states membership in AC$^0$ or NL, and (3) prove a new locality property for the models of second-order formulas that describe problems like $k$sat-prob$_{\ge \delta}$. The locality property will allow us to prove a conjecture of Akmal and Williams: The majority-of-majority satisfaction problem for $k$cnfs lies in P for all $k$.