The Complexity of Finding Read-Once NAE-Resolution Refutations

  • Hans Kleine Büning
  • Piotr Wojciechowski
  • K. Subramani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10119)


In this paper, we analyze boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation. A read-once (resolution) refutation is one in which each input clause is used at most once. It is well-known that read-once resolution is not complete, i.e., there exist unsatisfiable formulas for which no read-once resolution exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called Not-All-Equal Satisfiability (NAE-Satisfiability). NAE-Satisfiability is the problem of checking whether an arbitrary CNF formula has a satisfying assignment in which at least one literal in each clause is set to false. It is well-known that NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of the class of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution, which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. We focus on a variant of NAE-resolution called read-once NAE-resolution, in which each input clause can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for checking the NAE-satisfiability of 2CNF formulas; we also provide a polynomial time algorithm to determine the shortest read-once NAE-resolution of a 2CNF formula. Finally, we establish that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.


Read-once NAE-SAT Refutation Optimal length refutation 


  1. 1.
    Beame, P., Pitassi, T.: Propositional proof complexity: past, present, future. Bull. EATCS 65, 66–89 (1998)MathSciNetMATHGoogle Scholar
  2. 2.
    Buss, S.R.: Propositional proof complexity: an introduction.
  3. 3.
    Cook, S.A., Reckhow, R.A.: On the lengths of proofs in the propositional calculus (preliminary version). In: Proceedings of the 6th Annual ACM Symposium on Theory of Computing, Seattle, Washington, USA, 30 April – 2 May 1974, pp. 135–148 (1974)Google Scholar
  4. 4.
    Harrison, J.: Handbook of Practical Logic and Automated Reasoning, 1st edn. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  5. 5.
    Iwama, K., Miyano, E.: Intractability of read-once resolution. In: Proceedings of the 10th Annual Conference on Structure in Complexity Theory (SCTC 1995), CA, USA, pp. 29–36. IEEE Computer Society Press, Los Alamitos, June 1995Google Scholar
  6. 6.
    Kleine Büning, H., Wojciechowski, P., Subramani, K.: On the computational complexity of read once resolution decidability in 2CNF formulas.
  7. 7.
    Moore, C., Mertens, S.: The Nature of Computation, 1st edn. Oxford University Press, Oxford (2011)CrossRefMATHGoogle Scholar
  8. 8.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, New York (1994)MATHGoogle Scholar
  9. 9.
    Robinson, J.A.: A machine-oriented logic based on the resolution principle. J. ACM 12(1), 23–41 (1965)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Schaefer, T.: The complexity of satisfiability problems. In: Aho, A. (ed.) Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 216–226. ACM Press, New York City (1978)Google Scholar
  11. 11.
    Urquhart, A.: The complexity of propositional proofs. Bull. Symbolic Logic 1(4), 425–467 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Hans Kleine Büning
    • 1
  • Piotr Wojciechowski
    • 2
  • K. Subramani
    • 2
  1. 1.Universität PaderbornPaderbornGermany
  2. 2.LCSEEWest Virginia UniversityMorgantownUSA

Personalised recommendations