Complexity of DNF and Isomorphism of Monotone Formulas

  • Judy Goldsmith
  • Matthias Hagen
  • Martin Mundhenk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)


We investigate the complexity of finding prime implicants and minimal equivalent DNFs for Boolean formulas, and of testing equivalence and isomorphism of monotone formulas. For DNF related problems, the complexity of the monotone case strongly differs from the arbitrary case. We show that it is DP-complete to check whether a monomial is a prime implicant for an arbitrary formula, but checking prime implicants for monotone formulas is in L. We show PP-completeness of checking whether the minimum size of a DNF for a monotone formula is at most k. For k in unary, we show the complexity of the problem to drop to coNP. In [Uma01] a similar problem for arbitrary formulas was shown to be \(\Sigma^P_2\)-complete. We show that calculating the minimal DNF for a monotone formula is possible in output-polynomial time if and only if P = NP. Finally, we disprove a conjecture from [Rei03] by showing that checking whether two formulas are isomorphic has the same complexity for arbitrary formulas as for monotone formulas.


Polynomial Time Conjunctive Normal Form Boolean Formula Disjunctive Normal Form Satisfying Assignment 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Judy Goldsmith
    • 1
  • Matthias Hagen
    • 2
  • Martin Mundhenk
    • 2
  1. 1.Dept. of Computer ScienceUniversity of KentuckyLexington
  2. 2.Institut für InformatikFriedrich-Schiller-Universität JenaJena

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