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- W2039822084 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 n th vectorization of R, corresponds to the Δ 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." @default.
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- W2039822084 date "2011-01-01" @default.
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- W2039822084 title "Extensions of MSO and the monadic counting hierarchy" @default.
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- W2039822084 doi "https://doi.org/10.1016/j.ic.2010.09.002" @default.
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