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Revisiting Space in Proof Complexity: Treewidth and Pathwidth

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Mathematical Foundations of Computer Science 2013 (MFCS 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8087))

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Abstract

So-called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs. The ordered pathwidth of a proof is shown to be roughly the same as its formula space. Length-space lower bounds for R(k)-refutations are generalized to arbitrary infinity axioms and strengthened in that the space measure is relaxed to ordered treewidth.

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References

  1. Alekhnovich, M., Ben-Sasson, E., Razborov, A.A., Wigderson, A.: Space complexity in propositional calculus. SIAM J. Comput. 31(4), 1184–1211 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atserias, A., Dalmau, V.: A combinatorial characterization of resolution width. J. of Computer and System Sciences 74(3), 323–334 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bang-Jensen, J., Gutin, G.: Digraphs, 2nd edn. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London (2009)

    Book  MATH  Google Scholar 

  4. Ben-Sasson, E.: Size-space tradeoffs for resolution. SIAM J. Comput. 38(6), 2511–2525 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Sasson, E., Nordström, J.: Understanding space in proof complexity: Separations and trade-offs via substitutions. ECCC 17, 125 (2010)

    Google Scholar 

  6. Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science 209(1-2), 1–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cook, S., Reckhow, R.: The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44, 36–50 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dantchev, S., Riis, S.: On relativisation and complexity gap for resolution-based proof systems. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 142–154. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  9. Esteban, J.L., Galesi, N., Messner, J.: On the complexity of resolution with bounded conjunctions. Theoretical Computer Science 321(2-3), 347–370 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Esteban, J.L., Torán, J.: Space bounds for resolution. Information and Computation 171(1), 84–97 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory, Series B 82(1), 138–154 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kinnersley, N.G.: The vertex separation number of a graph equals its path-width. Information Processing Letters 42(6), 345–350 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kleine Büning, H., Lettman, T.: Propositional logic: deduction and algorithms. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  14. Krajíček, J.: Bounded arithmetic, propositional logic, and complexity theory. Encyclopedia of Mathematics and its Applications, vol. 60. Cambridge Univ. Press (1995)

    Google Scholar 

  15. Krajíček, J.: On the weak pigeonhole principle. Fund. Math. 170(1-2), 123–140 (2001), Dedicated to the memory of Jerzy Łoś

    Google Scholar 

  16. Krajíček, J.: Combinatorics of first order structures and propositional proof systems. Archive for Mathematical Logic 43(4), 427–441 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Maciel, A., Pitassi, T., Woods, A.R.: A new proof of the weak pigeonhole principle. J. of Computer and System Sciences 64(4), 843–872 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mazala, R.: 2 Infinite games. In: Grädel, E., Thomas, W., Wilke, T. (eds.) Automata, Logics, and Infinite Games. LNCS, vol. 2500, pp. 23–38. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  19. Nordström, J.: Narrow proofs be spacious: separating space and width in resolution. SIAM J. Comput. 39(1), 59–121 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Paris, J., Wilkie, A.: Counting problems in bounded arithmetic. In: Methods in Mathematical Logic. LNM, vol. 1130, pp. 317–340 (1985)

    Google Scholar 

  21. Razborov, A.: Resolution lower bounds for the weak functional pigeonhole principle. Theoretical Computer Science 303(1), 233–243 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Riis, S.: A complexity gap for tree resolution. Comput. Compl. 10(3), 179–209 (2001)

    Article  MathSciNet  Google Scholar 

  23. Segerlind, N.: The complexity of propositional proofs. Bull. of Symbolic Logic 13(4), 417–481 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Segerlind, N., Buss, S., Impagliazzo, R.: A switching lemma for small restrictions and lower bounds for k-DNF resolution. SIAM J. Comput. 33(5), 1171–1200 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Stålmarck, G.: Short resolution proofs for a sequence of tricky formulas. Acta Informatica 33(3), 277–280 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Kurt Gödel Research Center, University of Vienna, Vienna, Austria

    Moritz Müller

  2. Institute of Information Systems, Vienna University of Technology, Vienna, Austria

    Stefan Szeider

Authors
  1. Moritz Müller
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  2. Stefan Szeider
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Editor information

Editors and Affiliations

  1. Institute for Science and Technology, 3400, Klosterneuburg, Austria

    Krishnendu Chatterjee

  2. Charles University, 11800, Prague, Czech Republic

    Jirí Sgall

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Müller, M., Szeider, S. (2013). Revisiting Space in Proof Complexity: Treewidth and Pathwidth. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_62

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  • DOI: https://doi.org/10.1007/978-3-642-40313-2_62

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40312-5

  • Online ISBN: 978-3-642-40313-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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