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Adversarial satisfiability (AdSAT), a complex problem related to Boolean formulas, shows a sharp transition. This study introduces algorithms and finds a stricter upper bound for the quantum satisfiability problem (QSAT).

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Area of Science:

  • Computational Complexity Theory
  • Boolean Satisfiability Problems
  • Quantum Computing

Background:

  • Adversarial Satisfiability (AdSAT) is a complex generalization of the Satisfiability (SAT) problem.
  • AdSAT resides in a higher complexity class than SAT, necessitating independent study.
  • AdSAT offers an upper bound for the transition threshold of the Quantum Satisfiability Problem (QSAT).

Purpose of the Study:

  • To present a complete algorithm for AdSAT.
  • To analyze the complexity of 2-AdSAT and stochastic algorithms for 3-AdSAT.
  • To investigate the critical region and transition phenomena in AdSAT.

Main Methods:

  • Development of a complete algorithm for AdSAT.
  • Complexity analysis, proving 2-AdSAT is in P.
  • Application and comparison of simulated annealing and its variant for 3-AdSAT.

Main Results:

  • A complete algorithm for AdSAT is presented.
  • 2-AdSAT is shown to be in P (polynomial time).
  • A sharp SAT-UNSAT transition is observed in 3-AdSAT at a critical clause density (αc≲1.5).

Conclusions:

  • The study provides a new upper bound for the QSAT transition threshold, tighter than previously known.
  • The findings highlight the distinct nature of AdSAT's critical region compared to SAT.
  • Stochastic algorithms demonstrate effectiveness in studying AdSAT transitions.