On Second-Order Monadic Groupoidal Quantifiers

Juha Kontinen, Heribert Vollmer

Abstract: We study logics defined in terms of so-called second-order monadic groupoidal quantifiers. These are generalized quantifiers defined by groupoid word-problems or equivalently by context-free languages. We show that, over strings with built-in arithmetic, the extension of monadic second-order logic by all second-order monadic groupoidal quantifiers collapses to its fragment \({{\rm mon-}Q^1_{{\rm Grp}}}{\text{\rm FO}}\). We also show a variant of this collapse which holds without built-in arithmetic. Finally, we relate these results to an open question regarding the expressive power of finite leaf automata with context-free leaf languages.