Choice Construct and Lindström Logics

Abstract

A choice construct can be added to fixpoint logic to give a more expressive logic, as shown in [GH]. On the other hand, a more straightforward way of increasing the expressive power of fixpoint logic is to add generalized quantifiers, corresponding to undefinable properties, in the sense of Lindström. The paper studies the expressive power of the choice construct proposed in [GH] in its relationships to the logics defined with generalized quantifiers. We show that no extension of fixpoint logic by a set of quantifiers of bounded arity captures all properties of finite structures definable in choice fixpoint logic. Consequently, no extension of fixpoint logic with a finite set of quantifiers is more expressive than the extension of fixpoint logic with choice construct. On the other hand, we give a characterization of choice fixpoint logic by an extension of fixpoint logic with a countable set of quantifiers.