Monadic second-order logic

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In mathematical logic, monadic second order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets.[1] It is particularly important in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second-order formulas over certain types of graphs.

Second-order logic allows quantification over predicates. However, MSO is the fragment in which second-order quantification is limited to monadic predicates (predicates having a single argument). This is often described as quantification over "sets" because monadic predicates are equivalent in expressive power to sets (the set of elements for which the predicate is true).

Existential monadic second-order logic (EMSO) is the fragment of MSO in which all quantifiers over sets must be existential quantifiers, outside of any other part of the formula. The first-order quantifiers are not restricted. By analogy to Fagin's theorem, according to which existential (non-monadic) second-order logic captures precisely the descriptive complexity of the complexity class NP, the class of problems that may be expressed in existential monadic second-order logic has been called monadic NP. The restriction to monadic logic makes it possible to prove separations in this logic that remain unproven for non-monadic second-order logic; for instance, testing the connectivity of graphs belongs to monadic co-NP (the complement of monadic NP) but not to monadic NP itself.[2][3]


  1. ^ Courcelle, Bruno; Engelfriet, Joost (2012-01-01). Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach. Cambridge University Press. ISBN 978-0521898331. Retrieved 2016-09-15. 
  2. ^ Fagin, Ronald (1975), "Monadic generalized spectra", Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 21: 89–96, MR 0371623 .
  3. ^ Fagin, R.; Stockmeyer, L.; Vardi, M. Y. (1993), "On monadic NP vs. monadic co-NP", Proceedings of the Eighth Annual Structure in Complexity Theory Conference, Institute of Electrical and Electronics Engineers, doi:10.1109/sct.1993.336544 .