On the Satisfaction Probability of $k$-CNF Formulas

The satisfaction probability σ(φ) := Prβ: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 ksat-prob>δ = {φ is a kcnf formula | σ(φ) > δ} for fixed k and δ. 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 ∈ P and 4sat-prob>1/2 ∈ NP-complete; but the methods used to prove these striking results stay silent about, say, 4sat-prob>1/3, leaving the computational complexity of ksat-prob>δ open for most k and δ. In the present paper we give a complete characterization in the form of a trichotomy: ksat-prob>δ lies in AC , is NL-complete, or is NP-complete; and given k and δ we can decide which of the three applies. The proof of the trichotomy hinges on a new order-theoretic insight: Every set of kcnf formulas contains a formula of maximum satisfaction probability. This deceptively simple result allows us to (1) kernelize ksat-prob≥δ, (2) show that the variables of the kernel form a strong backdoor set when the trichotomy states membership in AC or NL, and (3) prove a new locality property for the models of second-order formulas that describe problems like ksat-prob≥δ. The locality property will allow us to prove a conjecture of Akmal and Williams: The majority-of-majority satisfaction problem for kcnfs lies in P for all k. 2012 ACM Subject Classification Theory of computation→ Problems, reductions and completeness

[1]  Gerry Leversha,et al.  Set theory: the third millennium edition , by Thomas Jech. Pp. 769. £77. 2003. ISBN 3 540 44085 2 (Springer). , 2005, The Mathematical Gazette.

[2]  Max Bannach,et al.  Fast Parallel Fixed-parameter Algorithms via Color Coding , 2015, IPEC.

[3]  Max Bannach,et al.  Computing Kernels in Parallel: Lower and Upper Bounds , 2018, IPEC.

[4]  Yoshio Okamoto,et al.  On Problems as Hard as CNF-SAT , 2011, 2012 IEEE 27th Conference on Computational Complexity.

[5]  Max Bannach,et al.  Computing Hitting Set Kernels By AC0-Circuits , 2019, Theory of Computing Systems.

[6]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[7]  René van Bevern.  Towards Optimal and Expressive Kernelization for d-Hitting Set , 2011, Algorithmica.

[8]  Bart Selman,et al.  Backdoors To Typical Case Complexity , 2003, IJCAI.

[9]  Neil Immerman,et al.  Descriptive Complexity , 1999, Graduate Texts in Computer Science.

[10]  Shyan Akmal,et al.  MAJORITY-3SAT (and Related Problems) in Polynomial Time , 2021, 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS).

[11]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[12]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[13]  Adnan Darwiche,et al.  Same-decision probability: A confidence measure for threshold-based decisions , 2012, Int. J. Approx. Reason..

[14]  Umut Oztok,et al.  Solving PPPP-Complete Problems Using Knowledge Compilation , 2016, KR.

[15]  Janos Simon.  On some central problems in computational complexity , 1975 .

[16]  Max Bannach,et al.  Computing Hitting Set Kernels By AC0-Circuits , 2018, Theory of Computing Systems.

[17]  John T. Gill,et al.  Computational complexity of probabilistic Turing machines , 1974, STOC '74.

[18]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[19]  Michael Elberfeld,et al.  On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness , 2015, Algorithmica.

[20]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[21]  René van Bevern.  Towards Optimal and Expressive Kernelization for d-Hitting Set , 2012, COCOON.

[22]  H. L. Abbott,et al.  Intersection Theorems for Systems of Sets , 1972, J. Comb. Theory, Ser. A.