Bounded Second-Order Unification Is NP-Complete

  • Jordi Levy
  • Manfred Schmidt-Schauß
  • Mateu Villaret
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4098)


Bounded Second-Order Unification is the problem of deciding, for a given second-order equation \({t {\stackrel{_?}=} u}\) and a positive integer m, whether there exists a unifier σ such that, for every second-order variable F, the terms instantiated for F have at most m occurrences of every bound variable.

It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-Order Unification is undecidable. We prove that Bounded Second-Order Unification is NP-complete, provided that m is given in unary encoding, by proving that a size-minimal solution can be represented in polynomial space, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time.


Equivalence Class Main Path Surface Position Tree Automaton Tree Grammar 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jordi Levy
    • 1
  • Manfred Schmidt-Schauß
    • 2
  • Mateu Villaret
    • 3
  1. 1.IIIA, CSICBarcelonaSpain
  2. 2.Institut für InformatikJohann Wolfgang Goethe-UniversitätFrankfurtGermany
  3. 3.IMA, UdGGironaSpain

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