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Degrees of Freedom for Critical Random 2-SAT

Andreas Basse-O'Connor, Mette Skjøtt

TL;DR

This work analyzes the degrees of freedom for random 2-SAT at its critical phase, demonstrating that the weak degrees of freedom scale as $n^{1/3}$, which is considerably smaller than in under-constrained regimes ($n^{1/2}$) or in related $k$-SAT settings. The authors develop a probabilistic framework built on unit propagation, coupling arguments, and a two-regime decomposition (many vs few fixed variables) to quantify how many variables can be fixed without asymptotically eliminating satisfiability. Their main result shows that, at the critical point $\alpha=1$, fixing $f(n)$ variables with $f(n) \approx n^{1/3}$ leaves the SAT probability bounded away from 0 and 1, while fixing more drives it to 0, exposing a sharp structural transition and long-range dependencies among variables near criticality. This deepens the understanding of phase transitions in random SAT and has implications for the complexity and behavior of SAT solvers in the critical regime, highlighting a stark contrast to the under-constrained and highly constrained settings.

Abstract

The random $k$-SAT problem serves as a model that represents the 'typical' $k$-SAT instances. This model is thought to undergo a phase transition as the clause density changes, and it is believed that the random $k$-SAT problem is primarily difficult to solve near this critical phase. In this paper, we introduce a weak formulation of degrees of freedom for random $k$-SAT problems and demonstrate that the critical random $2$-SAT problem has $\sqrt[3]{n}$ degrees of freedom. This quantity represents the maximum number of variables that can be assigned truth values without affecting the formula's satisfiability. Notably, the value of $\sqrt[3]{n}$ differs significantly from the degrees of freedom in random $2$-SAT problems sampled below the satisfiability threshold, where the corresponding value equals $\sqrt{n}$. Thus, our result underscores the significant shift in structural properties and variable dependency as satisfiability problems approach criticality.

Degrees of Freedom for Critical Random 2-SAT

TL;DR

This work analyzes the degrees of freedom for random 2-SAT at its critical phase, demonstrating that the weak degrees of freedom scale as , which is considerably smaller than in under-constrained regimes () or in related -SAT settings. The authors develop a probabilistic framework built on unit propagation, coupling arguments, and a two-regime decomposition (many vs few fixed variables) to quantify how many variables can be fixed without asymptotically eliminating satisfiability. Their main result shows that, at the critical point , fixing variables with leaves the SAT probability bounded away from 0 and 1, while fixing more drives it to 0, exposing a sharp structural transition and long-range dependencies among variables near criticality. This deepens the understanding of phase transitions in random SAT and has implications for the complexity and behavior of SAT solvers in the critical regime, highlighting a stark contrast to the under-constrained and highly constrained settings.

Abstract

The random -SAT problem serves as a model that represents the 'typical' -SAT instances. This model is thought to undergo a phase transition as the clause density changes, and it is believed that the random -SAT problem is primarily difficult to solve near this critical phase. In this paper, we introduce a weak formulation of degrees of freedom for random -SAT problems and demonstrate that the critical random -SAT problem has degrees of freedom. This quantity represents the maximum number of variables that can be assigned truth values without affecting the formula's satisfiability. Notably, the value of differs significantly from the degrees of freedom in random -SAT problems sampled below the satisfiability threshold, where the corresponding value equals . Thus, our result underscores the significant shift in structural properties and variable dependency as satisfiability problems approach criticality.

Paper Structure

This paper contains 19 sections, 17 theorems, 232 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main result
  4. Related work
  5. Preliminaries
  6. Notation and conventions
  7. The random SAT-problem
  8. Fixing variables and the unit-propagation algorithm
  9. Sketch of proof
  10. Main decomposition of probability
  11. Technical lemmas
  12. Decomposition of probability when many variables are fixed
  13. Decomposition of probability when few variables are fixed
  14. Proofs
  15. Proof of Lemma \ref{['lemma over-constrained control size of M_1^r']}
  16. ...and 4 more sections

Key Result

Theorem 2

The random critical $2$-SAT problem has $n^{1/3}$ degrees of freedom weakly.

Figures (2)

  • Figure 1: Computational hardness of random $3$-SAT as a function of clause density. The y-axis displays the median resolution time of 10,000 instances solved using the DPLL algorithm. Credit: Fig. 8.2 in Biroli02
  • Figure 2: Satisfiability of random critical $2$-SAT as a function of the number of fixed variables. The different curves represent a varying number of input variables. Each data point is comprised of $2,000$ simulations. The vertical dotted line indicates $q=1/3$.

Theorems & Definitions (39)

  • Definition 1
  • Theorem 2
  • Remark 3
  • Lemma 4
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 10
  • proof : Proof of Definition \ref{['def degrees of freedom']} (2)
  • Lemma 11
  • ...and 29 more