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On the Complexity of Horn and Krom Fragments of Second-Order Boolean Logic

Authors Miika Hannula , Juha Kontinen , Martin Lück, Jonni Virtema

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Author Details

Miika Hannula
  • Department of Mathematics and Statistics, University of Helsinki, Finland
Juha Kontinen
  • Department of Mathematics and Statistics, University of Helsinki, Finland
Martin Lück
  • Institut für Theoretische Informatik, Leibniz Universität Hannover, Germany
Jonni Virtema
  • Faculty of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan


We wish to thank the anonymous referees for their very careful and thorough reviews.

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Miika Hannula, Juha Kontinen, Martin Lück, and Jonni Virtema. On the Complexity of Horn and Krom Fragments of Second-Order Boolean Logic. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 27:1-27:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Second-order Boolean logic is a generalization of QBF, whose constant alternation fragments are known to be complete for the levels of the exponential time hierarchy. We consider two types of restriction of this logic: 1) restrictions to term constructions, 2) restrictions to the form of the Boolean matrix. Of the first sort, we consider two kinds of restrictions: firstly, disallowing nested use of proper function variables, and secondly stipulating that each function variable must appear with a fixed sequence of arguments. Of the second sort, we consider Horn, Krom, and core fragments of the Boolean matrix. We classify the complexity of logics obtained by combining these two types of restrictions. We show that, in most cases, logics with k alternating blocks of function quantifiers are complete for the kth or (k-1)th level of the exponential time hierarchy. Furthermore, we establish NL-completeness for the Krom and core fragments, when k = 1 and both restrictions of the first sort are in effect.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • quantified Boolean formulae
  • computational complexity
  • second-order logic
  • Horn and Krom fragment


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