On the Satisfaction Probabilities of $k$-CNF Formulas

TL;DR

This work provides a complete classification of the complexity of threshold satisfaction problems for $k$-CNF formulas, $k\text{SAT-} > p$, by developing a novel order-theoretic framework around the spectra of satisfaction probabilities. The key technical advances are the Spectral Well-Ordering Theorem, sunflower-based kernelization with safe plucking and pruning rules, and threshold locality driven by backdoor-like structures. These tools yield a sharp trichotomy: for fixed $k$ and $p$, $k\text{SAT-} > p$ is NP-complete, NL-complete, or in AC^0, with the case distinctions captured by whether the input class has room for 3-SAT or 2-SAT at $p$. The paper also provides practical algorithmic variants, including a gap-size oblivious algorithm, and shows consequences for related problems such as maj-maj-sat, as well as broader implications for CSPs and algebraic formulations of satisfaction probabilities.

Abstract

The satisfaction probability Pr[$φ$] := Pr$_{β:vars(φ) \to \{0,1\}}[β\models φ]$ of a propositional formula $φ$ is the likelihood that a random assignment $β$ makes the formula true. We study the complexity of the problem $k$SAT-Pr$_{>p}$ = {$φ$ is a $k$CNF formula | Pr[$φ$] > p} for fixed $k$ and $p$. While 3SAT-Pr$_{>0}$ = 3SAT is NP-complete and SAT-Pr$_{>1/2}$ is PP-complete, Akmal and Williams recently showed that 3SAT-Pr$_{>1/2}$ lies in P and 4SAT-Pr$_{>1/2}$ is NP-complete; but the methods used to prove these striking results stay silent about, say, 4SAT-Pr$_{>3/4}$, leaving the computational complexity of $k$SAT-Pr$_{>p}$ open for most $k$ and $p$. In the present paper we give a complete characterization in the form of a trichotomy: $k$SAT-Pr$_{>p}$ lies in AC$^0$, is NL-complete, or is NP-complete. 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 statement allows us to (1) kernelize $k$SAT-Pr$_{\ge p}$ for the joint parameters $k$ and $p$, (2) show that the variables of the kernel form a backdoor set when the trichotomy states membership in AC$^0$ or NL, and (3) prove locality properties for $k$CNF formulas $φ$, by which Pr[$φ$] < $p$ implies that Pr[$ψ$] < $p$ holds already for a subset $ψ$ of $φ$'s clauses whose size depends only on $k$ and $p$, and Pr[$φ$] = $p$ implies $φ\equiv ψ$ for some $k$CNF formula $ψ$ whose size once more depends only on $k$ and $p$.

Figures (7)

• Figure 1: Visualization of the complexity of $k\text{\upshape\normalfontsat-pr}_{>p}$ for $k=1$, $k=2$, $k=3$, and $k=4$. Each green triangle represents a value of $p$ for which the characterization from Theorem \ref{['thm-main']} states $\text{\normalfont\small$\mathrm{NL}$}$-completeness, while for each red triangle it states $\text{\normalfont\small$\mathrm{NP}$}$-completeness (there are no green triangles directly above red triangles, but this is not possible to visualize as these are highly intertwined). Each gray cross is an element of $k\text{\upshape\normalfontcnfs-pr-spectrum}$. For them, $k\text{\upshape\normalfontsat-pr}_{>p}$ lies in $\text{\normalfont\small$\mathrm{AC}$}^0$ according to Theorem \ref{['thm-main']}. For "white" values of $p$, which lie outside the spectra, $k\text{\upshape\normalfontsat-pr}_{>p}$ also lies in $\text{\normalfont\small$\mathrm{AC}$}^0$. Note that while the visualization may suggest that the spectra become dense close to $0$, they are in fact nowhere dense (by Theorem \ref{['thm-order']}, the Spectral Well-Ordering Theorem). For a discussion of the marked specific values like $p = 63/128$, please see the conclusion.
• Figure 2: The long line visualizes the set $[0,1]$ of possible satisfaction probabilities with a gray spectral gap below some value $p$, meaning that $\Pr[\phi]$ cannot lie in this gap for any $\phi \in k\text{\upshape\normalfontcnfs}$. In particular, for two formulas $\psi,\psi' \in k\text{\upshape\normalfontcnfs}$ the values $\Pr[\psi]$ and $\Pr[\psi']$ must either lie in the light red part (as is the case for $\Pr[\psi] \le p - \operatorname{spectral-gap}_{k\text{\upshape\normalfontcnfs}}(p) < p$) or in the light green part (as is the case for $\Pr[\psi'] \ge p$). The open interval $I_{\psi}$ is centered on $\Pr[\psi]$ and protrudes to the left and to the right by $\epsilon = \operatorname{spectral-gap}_{k\text{\upshape\normalfontcnfs}}(p)/2$; likewise for $I_{\psi'}$. Suppose that for an arbitrary value $f \in [0,1]$ we (just) know that $f$ must lie in (some) interval $I_{\phi}$, centered on (an unknown) $\Pr[\phi]$ and having the size of the spectral gap. Then $f < p - \epsilon$ implies that $\Pr[\phi]$ must "lie in the red part" and $\Pr[\phi]<p$ holds, whereas $f > p - \epsilon$ implies that $\Pr[\phi]$ "lies in the green part" and $\Pr[\phi] \ge p$ holds.
• Figure 3: Different ways how $\Pr[\phi_i]$ can change in a sequence $(\phi_1,\phi_2,\dots,\phi_q)$, ending with some $\phi^* = \phi_q$, when $\phi_i$ is always gap-close to $\phi_{i-1}$. In each row, the crosses in the red lines are elements of $k\text{\upshape\normalfontcnfs-pr-spectrum}$ smaller than $p$, while the crosses on the green lines are at least $p$. In the first line, $\Pr[\phi_1] \ge p$ holds and each $\phi_{i+1}$ has smaller or equal satisfaction probability, but the gap-closeness ensures that $\Pr[\phi^*]$ gets "stuck" at $p$ as it cannot "tunnel through" the spectral gap by Lemma \ref{['lem-tunnel']} and the dashed arrow is an impossible change in the satisfaction probability. In the second line, the satisfaction probabilities increase, but also get stuck, only now at the lower end of the spectral gap. In the third line, $\Pr[\phi_i]$ is stuck at a much larger value than $p$.
• Figure 4: Left, a formula $\phi \in \text{\upshape\normalfont5cnfs}$ is visualized by drawing, for each clause in $\phi$, a line that "touches" exactly the clause's literals; so the upper dashed line represents the clause $\{f,x,y,z\}$. The solid(-line) clauses form a sunflower $\psi \subseteq \phi$ with core $c = \{x,\neg y,z\}$. Although the dotted clauses also contain the core, they are not part of the sunflower: The upper dotted clause $\{x,\neg y,z,l,\neg e\}$ shares the literal "$l$" with the petal $\{x,\neg y, z,l,m\}$ of the sunflower, while the second dotted clause shares the variable "$l$" (though not the literal) with this petal. The dashed clauses are not part of the sunflower as they do not contain all of the literals of the core (containing the variables is not enough). A key property of a sunflower is that it is "unlikely that an assignment makes the sunflower true, but not its core": For $\phi$, this happens only when $g$, $j$, and $\neg k$ are all set to true as well as at least one of $\neg h$ or $i$, and one of $l$ or $m$. The probability that all of this happens is just $\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2} \cdot \frac{3}{4}\cdot\frac{3}{4} = \frac{9}{128}$. In particular, for $\phi' = (\phi \setminus \psi) \cup \{c\}$ shown right, we have $\Pr[\phi] - \Pr[\phi'] \le \frac{9}{128}$.
• Figure 5: Left, an irredundant formula $\omega' \in 5\text{\upshape\normalfontcnfs}$ is shown that has room for $2\text{\upshape\normalfontsat}$ since it contains the green clause $c_* = \{x,\neg y, z\} \in \omega'$, which is "small," meaning of size $5-2 = 3$ and implying that it "has room" for adding up to two literals in a reduction from $\text{\upshape\normalfont2sat}$ to $5\text{\upshape\normalfontsat-pr}_{>p'}$ for $p' = \Pr[\omega']$. How the reduction works is shown right: The clauses of a formula $\psi \in \text{\upshape\normalfont2cnfs}$ with "fresh" variables $v_i$ are added to $c_*$, which will result in a 5cnf formula since there is always enough room to add only two literals to a size-3 clause. The resulting formula $\rho$ has a satisfaction probability strictly larger than that of $\omega'$ iff $\psi$ is satisfiable.

Theorems & Definitions (62)

• Theorem 1: Spectral Well-Ordering Theorem
• Definition 2
• Corollary 3
• Theorem 4
• Theorem 5: Threshold Locality Theorem
• Theorem 6: Spectral Trichotomy Theorem
• Lemma 7: Packing Probability Lemma
• Definition 8: Gap-Close Formulas
• Lemma 9: No Tunneling Lemma
• Definition 10