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Computational and Proof Complexity of Partial String Avoidability

Published:21 January 2021Publication History
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Abstract

The partial string avoidability problem is stated as follows: given a finite set of strings with possible “holes” (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this article establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form satisfiability problem, where each clause has infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting proof lines (such as clauses, inequalities, polynomials, etc.). First, it is proved that there is a particular formula that has a short refutation in Resolution with a shift rule but requires classical proofs of exponential size. At the same time, it is shown that exponential lower bounds for classical proof systems can be translated for their shifted versions. Finally, it is shown that superpolynomial lower bounds on the size of shifted proofs would separate NP from PSPACE; a connection to lower bounds on circuit complexity is also established.

References

  1. Alfred V. Aho and Margaret J. Corasick. 1975. Efficient string matching: An aid to bibliographic search. Commun. ACM 18, 6 (June 1975), 333--340 Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Michael Alekhnovich and Alexander A. Razborov. 2001. Lower bounds for polynomial calculus: Non-binomial case. In Proceedings of the 42nd Annual Symposium on Foundations of Computer Science (FOCS’01). IEEE Computer Society, Las Vegas, NV, 190--199. DOI:https://doi.org/10.1109/SFCS.2001.959893 Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Valeriy Balabanov, Magdalena Widl, and Jie-Hong R. Jiang. 2014. QBF resolution systems and their proof complexities. In Proceedings of the 17th International Conference on Theory and Applications of Satisfiability Testing (SAT’14). Springer, Cham, 154--169.Google ScholarGoogle Scholar
  4. J. Berstel and D. Perrin. 2002. Finite and infinite words. In Algebraic Combinatorics on Words, M. Lothaire (Ed.). Cambridge University Press, Cambridge, 1--44.Google ScholarGoogle Scholar
  5. Olaf Beyersdorff, Leroy Chew, and Mikolás Janota. 2019. New resolution-based QBF calculi and their proof complexity. Trans. Comput. Theory 11, 4 (2019), 26:1--26:42. DOI:https://doi.org/10.1145/3352155 Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Olaf Beyersdorff, Luke Hinde, and Ján Pich. 2017. Reasons for hardness in QBF proof systems. In Proceedings of the 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’17) (LIPIcs), Satya V. Lokam and R. Ramanujam (Eds.), Vol. 93. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, 14:1--14:15. DOI:https://doi.org/10.4230/LIPIcs.FSTTCS.2017.14Google ScholarGoogle Scholar
  7. Brandon Blakeley, Francine Blanchet-Sadri, Josh Gunter, and Narad Rampersad. 2010. On the complexity of deciding avoidability of sets of partial words. Theor. Comput. Sci. 411, 49 (2010), 4263--4271. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Francine Blanchet-Sadri, Raphaël M. Jungers, and Justin Palumbo. 2009. Testing avoidability on sets of partial words is hard. Theor. Comput. Sci. 410, 8–10 (2009), 968--972. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Samuel R. Buss. 2012. Towards NP-P via proof complexity and search. Ann. Pure Appl. Logic 163, 7 (2012), 906--917.Google ScholarGoogle ScholarCross RefCross Ref
  10. Stephen A. Cook and Robert A. Reckhow. 1979. The relative efficiency of propositional proof systems. J. Symbol. Logic 44, 1 (Mar. 1979), 36--50.Google ScholarGoogle ScholarCross RefCross Ref
  11. Armin Haken. 1985. The intractability of resolution. Theor. Comput. Sci. 39 (1985), 297--308.Google ScholarGoogle ScholarCross RefCross Ref
  12. Russell Impagliazzo, Pavel Pudlák, and Jirí Sgall. 1999. Lower bounds for the polynomial calculus and the Gröbner basis algorithm. Comput. Complex. 8, 2 (1999), 127--144. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Mikolás Janota and Joao Marques-Silva. 2015. Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577 (2015), 25--42. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Hans Kleine Büning, Marek Karpinski, and Andreas Flögel. 1995. Resolution for quantified Boolean formulas. Info. Comput. 117, 1 (1995), 12--18. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. László Lovász, Moni Naor, Ilan Newman, and Avi Wigderson. 1995. Search problems in the decision tree model. SIAM J. Discrete Math. 8, 1 (1995), 119--132. DOI:https://doi.org/10.1137/S0895480192233867 Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Pavel Pudlak. 1997. Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symbol. Logic 62, 3 (1997), 981--998.Google ScholarGoogle ScholarCross RefCross Ref
  17. Alexander A. Razborov. 1998. Lower bounds for the polynomial calculus. Comput. Complex. 7, 4 (1998), 291--324. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Alasdair Urquhart. 1987. Hard examples for resolution. J. ACM 34, 1 (1987), 209--219. Google ScholarGoogle ScholarDigital LibraryDigital Library

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