Extensions of MSO and the monadic counting hierarchy

  • Authors:

  • Juha Kontinen;Hannu Niemistö

  • Affiliations:

  • Department of Mathematics and Statistics, P.O. Box 68, University of Helsinki, Helsinki FI-00014, Finland;Department of Mathematics and Statistics, P.O. Box 68, University of Helsinki, Helsinki FI-00014, Finland

  • Venue:
  • Information and Computation
  • Year:
  • 2011

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Abstract

In this paper, we study the expressive power of the extension of first-order logic by the unary second-order majority quantifier Most^1. In 1 it was shown that the extension of FO by second-order majority quantifiers of all arities describes exactly the problems in the counting hierarchy. We consider first certain sublogics of FO(Most^1) over unary vocabularies. We show that over unary vocabularies the logic MSO(R), where MSO is monadic second-order logic and R is the first-order Rescher quantifier, can be characterized by Presburger arithmetic, whereas the logic MSO(R^n)"n"@?"Z"""+, where R^n is the nth vectorization of R, corresponds to the @D"0-fragment of arithmetic. Then we show that FO(Most^1)=MSO(R^n)"n"@?"Z"""+ and that, on unary vocabularies, FO(Most^1) collapses to uniform-TC^0. Using this collapse, we show that first-order logic with the binary second-order majority quantifier is strictly more expressive than FO(Most^1) over the empty vocabulary. On the other hand, over strings, FO(Most^1) is shown to capture the linear fragment of the counting hierarchy. Finally we show that, over non-unary vocabularies, FO(Most^1) can express problems complete via first-order reductions for each level of the counting hierarchy.