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On the upper bounds of (1,0)-super solutions for the regular balanced random (k,2s)-SAT problem

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Abstract

This paper explores the conditions which make a regular balanced random (k,2s)-CNF formula (1,0)-unsatisfiable with high probability. The conditions also make a random instance of the regular balanced (k − 1,2(k − 1)s)-SAT problem unsatisfiable with high probability, where the instance obeys a distribution which differs from the distribution obeyed by a regular balanced random (k − 1,2(k − 1)s)-CNF formula. Let F be a regular balanced random (k,2s)-CNF formula where k ⩾ 3, then there exists a number s0 such that F is (1,0)-unsatisfiable with high probability if s > s0. A numerical solution of the number s0 when k ∈ {5, 6,…, 14} is given to conduct simulated experiments. The simulated experiments verify the theoretical result. Besides, the experiments also suggest that F is (1,0)-satisfiable with high probability if s is less than a certain value.

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References

  1. Ginsberg M L, Parkes A J, Roy A. Supermodels and robustness. In: Proceedings of the 15th National Conference on Artificial Intelligence and 10th Innovative Applications of Artificial Intelligence Conference. 1998, 334–339

  2. Hebrard E, Hnich B, Walsh T. Super solutions in constraint programming. In: Proceedings of the 1st International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. 2004, 157–172

  3. Zhang P, Gao Y. A probabilistic study of generalized solution concepts in satisfiability testing and constraint programming. Theoretical Computer Science, 2017, 657: 98–110

    Article  MathSciNet  Google Scholar 

  4. Cook S A. The complexity of theorem-proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on Theory of Computing. 1971, 151–158

  5. Mitchell D, Selman B, Levesque H. Hard and easy distributions of SAT problems. In: Proceedings of the 10th National Conference on Artificial Intelligence. 1992, 459–465

  6. Friedgut E, Bourgain J. Sharp thresholds of graph properties, and the k-SAT problem. Journal of the American Mathematical Society, 1999, 12(4): 1017–1054

    Article  MathSciNet  Google Scholar 

  7. Chvátal V, Reed B. Mick gets some (the odds are on his side). In: Proceedings of the 33rd Annual Symposium on Foundations of Computer Science. 1992, 620–627

  8. Kaporis A C, Kirousis L M, Lalas E G. The probabilistic analysis of a greedy satisfiability algorithm. Random Structures & Algorithms, 2006, 28(4): 444–480

    Article  MathSciNet  Google Scholar 

  9. Díaz J, Kirousis L, Mitsche D, Pérez-Giménez X. On the satisfiability threshold of formulas with three literals per clause. Theoretical Computer Science, 2009, 410(30–32): 2920–2934

    Article  MathSciNet  Google Scholar 

  10. Chao M T, Franco J. Probabilistic analysis of two heuristics for the 3-satisfiability problem. SIAM Journal on Computing, 1986, 15(4): 1106–1118

    Article  MathSciNet  Google Scholar 

  11. Achioptas D, Sorkin G B. Optimal myopic algorithms for random 3-SAT. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science. 2000, 590–600

  12. Braunstein A, Mézard M, Zecchina R. Survey propagation: an algorithm for satisfiability. Random Structures & Algorithms, 2005, 27(2): 201–226

    Article  MathSciNet  Google Scholar 

  13. Zhou G, Kang R. On the lower bounds of (1,0)-super solutions for random k-SAT. International Journal of Foundations of Computer Science, 2019, 30(2): 247–254

    Article  MathSciNet  Google Scholar 

  14. Wang B, Zhou G. Super solutions of random (3+p)-SAT. Theoretical Computer Science, 2019, 793: 14–27

    Article  MathSciNet  Google Scholar 

  15. Zhou J C, Xu D Y, Lu Y J. Satisfiability threshold of the regular random (k, r)-SAT problem. Journal of Software, 2016, 27(12): 2985–2993

    MathSciNet  Google Scholar 

  16. Boufkhad Y, Dubois O, Interian Y, Selman B. Regular random k-SAT: properties of balanced formulas. Journal of Automated Reasoning, 2005, 35(1–3): 181–200

    MathSciNet  Google Scholar 

  17. Rathi V, Aurell E, Rasmussen L, Skoglund M. Bounds on threshold of regular random k-SAT. In: Proceedings of the 13th International Conference on Theory and Applications of Satisfiability Testing. 2010, 264–277

  18. Sumedha, Krishnamurthy S, Sahoo S. Balanced K-satisfiability and biased random K-satisfiability on trees. Physical Review E, 2013, 87(4): 042130

    Article  Google Scholar 

  19. Zhou J, Xu D, Lu Y, Dai C. Strictly regular random (3,s)-SAT model and its phase transition phenomenon. Journal of Beijing University of Aeronautics and Astronautics, 2016, 42(12): 2563–2571

    Google Scholar 

  20. Zhou J, Xu D, Lu Y. Satisfiability threshold of regular (k, r)-SAT problem via 1RSB theory. Journal of Huazhong University of Science and Technology: Natural Science Edition, 2017, 45(12): 7–13

    Google Scholar 

  21. Coja-Oghlan A, Wormald N. The number of satisfying assignments of random regular k-SAT formulas. Combinatorics, Probability and Computing, 2018, 27(4): 496–530

    Article  MathSciNet  Google Scholar 

  22. Achlioptas D, Peres Y. The threshold for random k-SAT is 2klog2-O(k). Journal of the American Mathematical Society, 2004, 17(4): 947–973

    Article  MathSciNet  Google Scholar 

  23. Mahajan Y S, Fu Z, Malik S. Zchaff2004: an efficient SAT solver. In: Proceedings of the 7th International Conference on Theory and Applications of Satisfiability Testing. 2005, 360–375

  24. Wang Y, Xu D. Properties of the satisfiability threshold of the strictly d-regular random (3,2s)-SAT problem. Frontiers of Computer Science, 2020, 14(6): 146404

    Article  Google Scholar 

  25. Gardy D. Some results on the asymptotic behaviour of coefficients of large powers of functions. Discrete Mathematics, 1995, 139(1–3): 189–217

    Article  MathSciNet  Google Scholar 

  26. Richardson T, Urbanke R. Modern Coding Theory. London: Cambridge University Press, 2008

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Scientific Research Project for Introduced Talents of Guizhou University of Finance and Economics (No. 2021YJ007), the National Natural Science Foundation of China (Grant Nos. 61862051, 61762019, 62241206), the Top-notch Talent Program of Guizhou Province (No. KY[2018]080), the Science and Technology Foundation of Guizhou Province (No. 20191299), and the foundation of Qiannan Normal University for Nationalities (Nos. QNSYRC201715, QNSY2018JS013).

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Authors and Affiliations

  1. School of Information, Guizhou University of Finance and Economics, Guiyang, 550025, China

    Yongping Wang

  2. College of Computer Science and Technology, Guizhou University, Guiyang, 550025, China

    Daoyun Xu

  3. School of Computer and Information, Qiannan Normal University for Nationalities, Duyun, 558000, China

    Jincheng Zhou

  4. Key Laboratory of Complex Systems and Intelligent Optimization of Guizhou Province, Duyun, 558000, China

    Jincheng Zhou

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  1. Yongping Wang
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  2. Daoyun Xu
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Correspondence to Jincheng Zhou.

Additional information

Yongping Wang received the PhD degree from Guizhou University, China in 2020. He is an associate professor at School of Information, Guizhou University of Finance and Economics, China. His research interests include computability and computational complexity, and algorithm design and analysis.

Daoyun Xu received the PhD degree from Nanjing University, China in 2002. He is a professor and PhD supervisor at College of Computer Science and Technology, Guizhou University, China, and the senior member of CCF. His research interests include computability and computational complexity, and algorithm design and analysis.

Jincheng Zhou received the PhD degree from Guizhou University, China in 2016. He is a professor and master supervisor at School of Computer and Information, Qiannan Normal University for Nationalities, China, and the senior member of CCF. His research interests include computational complexity, and algorithm design and analysis.

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Wang, Y., Xu, D. & Zhou, J. On the upper bounds of (1,0)-super solutions for the regular balanced random (k,2s)-SAT problem. Front. Comput. Sci. 18, 184403 (2024). https://doi.org/10.1007/s11704-023-2752-2

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