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On Tractability of Disjoint AND-Decomposition of Boolean Formulas

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Perspectives of System Informatics (PSI 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8974))

Abstract

Disjoint AND-decomposition of a boolean formula means its representation as a conjunction of two (or several) formulas having disjoint sets of variables. We show that deciding AND-decomposability is intractable in general for boolean formulas given in CNF or DNF and prove tractability of computing AND-decompositions of boolean formulas given in positive DNF, Full DNF, and ANF. The results follow from tractability of multilinear polynomial factorization over the finite field of order 2, for which we provide a polytime factorization algorithm based on identity testing for partial derivatives of multilinear polynomials.

An extended version of the paper containing proofs is available from http://persons.iis.nsk.su/files/persons/pages/and-decomp-full.pdf.

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Notes

  1. 1.

    Existence and complexity of decomposition algorithms in various logics have been studied in [6, 10, 12, 13].

  2. 2.

    We note that besides the factors of the form \(x\) and \(x+1\), there is a number of other simple cases of factorization that can be recognized easily.

References

  1. Bengtsson, T., Martinelli, A., Dubrova, E.: A fast heuristic algorithm for disjoint decomposition of Boolean functions. In: Notes of the 11th IEEE/ACM International Workshop on Logic & Synthesis (IWLS 2002), pp. 51–55 (2002)

    Google Scholar 

  2. Bioch, J.C.: Decomposition of boolean functions. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering. Encyclopedia of Mathematics and its Applications, vol. 134, pp. 39–78. Cambridge University Press, New York (2010)

    Chapter  Google Scholar 

  3. Chen, H., Janota, M., Marques-Silva, J.: QBF-based boolean function bi-decomposition. In: Proceedings of the Design, Automation & Test in Europe Conference (DATE 2012), pp. 816–819. IEEE (2012)

    Google Scholar 

  4. Choudhury, M., Mohanram, K.: Bi-decomposition of large boolean functions using blocking edge graphs. In: Proceedings of the 2010 IEEE/ACM International Conference on Computer-Aided Design (ICCAD 2010), pp. 586–591. IEEE Press, Piscataway (2010)

    Google Scholar 

  5. Khatri, S.P., Gulati, K. (eds.): Advanced Techniques in Logic Synthesis, Optimizations and Applications. Springer, New York (2011)

    Google Scholar 

  6. Konev, B., Lutz, C., Ponomaryov, D., Wolter, F.: Decomposing description logic ontologies. In: Proceedings of the Twelfth International Conference on Principles of Knowledge Representation and Reasoning (KR 2010). AAAI Press, Palo Alto (2010)

    Google Scholar 

  7. Kuon, I., Tessier, R., Rose, J.: FPGA architecture: survey and challenges. Now Publishers Inc, Boston - Delft (2008)

    Google Scholar 

  8. Mishchenko, A., Sasao, T.: Large-scale SOP minimization using decomposition and functional properties. In: Proceedings of the 40th ACM/IEEE Design Automation Conference (DAC 2003), pp. 149–154. ACM, New York (2003)

    Google Scholar 

  9. Mishchenko, A., Steinbach, B., Perkowski, M.A.: An algorithm for bi-decomposition of logic functions. In: Proceedings of the 38th ACM/IEEE Design Automation Conference (DAC 2001), pp. 103–108. ACM, New York (2001)

    Google Scholar 

  10. Morozov, A., Ponomaryov, D.: On decidability of the decomposability problem for finite theories. Siberian Math. J. 51(4), 667–674 (2010)

    Article  MATH  Google Scholar 

  11. Perkowski, M.A., Grygiel, S.: A survey of literature on function decomposition, Version IV. PSU Electrical Engineering Department report, Department of Electrical Engineering, Portland State University, Portland, Oregon, USA, November 1995

    Google Scholar 

  12. Ponomaryov, D.: On decomposability in logical calculi. Bull. Novosib. Comput. Cent. 28, 111–120 (2008). http://persons.iis.nsk.su/files/persons/pages/delta-decomp.pdf

    MATH  Google Scholar 

  13. Ponomaryov, D.: The algorithmic complexity of decomposability in fragments of first-order logic, Research Note. In: Abstract appears in Proceedings Logic Colloquium (2014). http://persons.iis.nsk.su/files/persons/pages/sigdecomp.pdf

  14. Shpilka, A., Volkovich, I.: On the relation between polynomial identity testing and finding variable disjoint factors. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 408–419. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  15. Steinbach, B., Lang, C.: Exploiting functional properties of Boolean functions for optimal multi-level design by bi-decomposition. Artif. Intell. Rev. 20(3–4), 319–360 (2003)

    Article  MATH  Google Scholar 

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Acknowledgements

The first author was supported by the Russian Foundation for Humanities, grant No. 13-01-12003B. The second author was supported by the German Research Foundation within the Transregional Collaborative Research Center SFB/TRR 62 “Companion-Technology for Cognitive Technical Systems”.

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Authors and Affiliations

  1. Institute of Informatics Systems, Novosibirsk, Russia

    Pavel Emelyanov & Denis Ponomaryov

  2. Institute of Artificial Intelligence, Ulm, Germany

    Denis Ponomaryov

Authors
  1. Pavel Emelyanov
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  2. Denis Ponomaryov
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Correspondence to Denis Ponomaryov .

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Editors and Affiliations

  1. School of Computer Science, University of Manchester, Manchester, United Kingdom

    Andrei Voronkov

  2. A.P. Ershov Institute of Informatics Systems, Novosibirsk, Russia

    Irina Virbitskaite

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Emelyanov, P., Ponomaryov, D. (2015). On Tractability of Disjoint AND-Decomposition of Boolean Formulas. In: Voronkov, A., Virbitskaite, I. (eds) Perspectives of System Informatics. PSI 2014. Lecture Notes in Computer Science(), vol 8974. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46823-4_8

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  • DOI: https://doi.org/10.1007/978-3-662-46823-4_8

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-46822-7

  • Online ISBN: 978-3-662-46823-4

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