Abstract
The satisfaction probability σ(ϕ) := Pr_{β:vars(ϕ) → {0,1}}[β ⊧ 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 kSATPROB_{> δ} = {ϕ is a kCNF formula ∣ σ(ϕ) > δ} for fixed k and δ. While 3SATPROB_{> 0} = 3SAT is NPcomplete and SATPROB}_{> 1/2} is PPcomplete, Akmal and Williams recently showed 3SATPROB_{> 1/2} ∈ P and 4SATPROB_{> 1/2} ∈ NPcomplete; but the methods used to prove these striking results stay silent about, say, 4SATPROB_{> 3/4}, leaving the computational complexity of kSATPROB_{> δ} open for most k and δ. In the present paper we give a complete characterization in the form of a trichotomy: kSATPROB_{> δ} lies in AC⁰, is NLcomplete, or is NPcomplete; and given k and δ we can decide which of the three applies. The proof of the trichotomy hinges on a new ordertheoretic insight: Every set of kCNF formulas contains a formula of maximal satisfaction probability. This deceptively simple result allows us to (1) kernelize kSATPROB_{≥ δ}, (2) show that the variables of the kernel form a strong backdoor set when the trichotomy states membership in AC⁰ or NL, and (3) prove a locality property by which for every kCNF formula ϕ we have σ(ϕ) ≥ δ iff σ(ψ) ≥ δ for every fixedsize subset ψ of ϕ’s clauses. The locality property will allow us to prove a conjecture of Akmal and Williams: The majorityofmajority satisfaction problem for kCNFS lies in P for all k.
BibTeX  Entry
@InProceedings{tantau:LIPIcs.CCC.2022.2,
author = {Tantau, Till},
title = {{On the Satisfaction Probability of kCNF Formulas}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {2:12:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772419},
ISSN = {18688969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16564},
URN = {urn:nbn:de:0030drops165648},
doi = {10.4230/LIPIcs.CCC.2022.2},
annote = {Keywords: Satisfaction probability, majority it\{k\}sat, kernelization, well orderings, locality}
}
Keywords: 

Satisfaction probability, majority it{k}sat, kernelization, well orderings, locality 
Collection: 

37th Computational Complexity Conference (CCC 2022) 
Issue Date: 

2022 
Date of publication: 

11.07.2022 