Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Satisfiability Threshold for K-XOR Games

Jared A. Hughes, J. William Helton

TL;DR

The paper establishes that the satisfiability threshold for uniformly tiled K-XORGAME instances coincides with the threshold for K-XORSAT, by linking solvability of random linear systems over ${\mathbb Z}_2$ to the structure of the 2-core of the associated hypergraph. The authors develop a second-moment framework centered on critical row sets, derive sharp large-deviation bounds via a key function $J_K$, and prove negativity of $J_K$ across relevant parameter ranges to bound the expected number of critical sets. They connect the 2-core analysis to the original problem through a maintenance-of-uniformity argument (borrowed from core theory), and use precise core-size asymptotics to translate core satisfiability into the full problem, yielding the threshold $c^*_K=h_K(Q^{-1}(K))$. This yields a rigorous, largely classical, combinatorial-probabilistic route to the threshold phenomenon and highlights the deep link between random linear systems and hypergraph cores. The results have implications for understanding phase transitions in XOR-type games and their connection to dually defined XORSAT thresholds, with exact threshold values computable via the given transcendental equations.

Abstract

A $K$-XORGAME system corresponds to a $K$-XORSAT system with the additional restriction that the variables divide uniformly into $K$ blocks. This forms a system of $m$ equations with $K n$ unknowns over $\mathbb{Z}_2$, and a perfect strategy corresponds to a solution to these equations. Equivalently, such equations correspond to colorings of a $K$-uniform $K$-partite hypergraph. This paper proves that the satisfiability threshold of $m/n$ for $K$-XORGAME problems exists and equals the satisfiability threshold for $K$-XORSAT.

The Satisfiability Threshold for K-XOR Games

TL;DR

The paper establishes that the satisfiability threshold for uniformly tiled K-XORGAME instances coincides with the threshold for K-XORSAT, by linking solvability of random linear systems over to the structure of the 2-core of the associated hypergraph. The authors develop a second-moment framework centered on critical row sets, derive sharp large-deviation bounds via a key function , and prove negativity of across relevant parameter ranges to bound the expected number of critical sets. They connect the 2-core analysis to the original problem through a maintenance-of-uniformity argument (borrowed from core theory), and use precise core-size asymptotics to translate core satisfiability into the full problem, yielding the threshold . This yields a rigorous, largely classical, combinatorial-probabilistic route to the threshold phenomenon and highlights the deep link between random linear systems and hypergraph cores. The results have implications for understanding phase transitions in XOR-type games and their connection to dually defined XORSAT thresholds, with exact threshold values computable via the given transcendental equations.

Abstract

A -XORGAME system corresponds to a -XORSAT system with the additional restriction that the variables divide uniformly into blocks. This forms a system of equations with unknowns over , and a perfect strategy corresponds to a solution to these equations. Equivalently, such equations correspond to colorings of a -uniform -partite hypergraph. This paper proves that the satisfiability threshold of for -XORGAME problems exists and equals the satisfiability threshold for -XORSAT.
Paper Structure (29 sections, 24 theorems, 110 equations, 3 figures, 1 table)

This paper contains 29 sections, 24 theorems, 110 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions of $k$-XORSAT, $K$-XORGAME, and satisfiability thresholds
  3. $K$-XORGAME, $k$-XORSAT, and 2-cores.
  4. Randomly-generated game linear equations
  5. Main theorems
  6. Hypergraph interpretation
  7. Definitions of random variables related to critical row sets
  8. Critical row sets vs second moment theorem
  9. Re-definition of game matrix in terms of $\mathcal{A}$
  10. Definition of $X$ and the random variables $Y_{m,n,k}^{(\ell)}$ and $Z_{m,{\bm{n}}}^{(\ell)}$
  11. Upper bounds on expected number of critical row sets in a 2-core
  12. Proof of this bound on $\mathop{\mathrm{\mathbb{E}}}\nolimits[Z_{m,{\bm{n}}}^{(\ell)}]$
  13. Proving that $J_{K}$ is negative
  14. General idea and Reader's Guide
  15. Outline of approach to bounding $J_{K}$
  16. ...and 14 more sections

Key Result

Theorem 1.2.1

Suppose $K \geq 3$. Let $m,n\to\infty$ such that $\lim m/n$ exists, and consider for each $(m,n)$ the probability that $\Gamma x = s$ is satisfiable when $(\Gamma,s)$ is generated uniformly at random on the space of uniformly-tiled $K$-XORGAME equations of this size ($m$ rows, $K$ blocks of $n$ colu The satisfiability and unsatisfiability probabilities can be obtained in the proof in sec:proofFina

Figures (3)

  • Figure 4.1: $J_{K}(\alpha,\boldsymbol{\zeta}_{lin};c) = \frac{1}{c} L_{K}(\alpha,c)$ plotted over $\alpha \in [\beta_K,1]$ for $c=K \in \{4,5,6\}$.
  • Figure 4.2: $J_{K}(\alpha,\boldsymbol{\zeta}_{lin}(\alpha);c)$ plotted for $c=2.02,2.1,2.3,2.6,2.99$ and $K=4$ over $\alpha \in [0.99\beta_K,1]$.
  • Figure 4.3: Regions used in gridding for $K=3$ and $\boldsymbol{\zeta}=\boldsymbol{\zeta}_{sqrt}$ to obtain $J_{K} < 0$ on the bold rectangle, which is the region $(\alpha,c) \in [0.99\beta_K,1/2] \times (2,3)$. Not to scale. The right edge is labeled with $\lambda \in [0,2.15]$ while the left edge is labeled with $Q(\lambda) \in [2,Q(2.15)]$. The top and bottom edges are both labeled with $\alpha \in [0,0.5]$.

Theorems & Definitions (48)

  • Theorem 1.2.1
  • proof
  • Theorem 1.2.2
  • proof : Proof Outline and Reader's Guide
  • Lemma 2.1.1
  • proof
  • Proposition 3.0.1
  • Lemma 3.1.1
  • proof
  • Lemma 4.2.1
  • ...and 38 more