Existential second-order logic over strings

Abstract

Existential second-order logic (ESO) and monadic second-order logic(MSO) have attracted much interest in logic and computer science. ESO is a much expressive logic over successor structures than MSO. However, little was known about the relationship between MSOand syntatic fragments of ESO. We shed light on this issue by completely characterizing this relationship for the prefix classes of ESO over strings, (i.e., finite successor structures). Moreover, we determine the complexity of model checking over strings, for all ESO-prefix classes. Let ESO( Q ) denote the prefix class containing all sentences of the shape ∃ R Q 4 , where R is a list of predicate variables, Q is a first-order predicate qualifier from the prefix set Q and 4 is quantifier-free. We show that ESO( ∃ * ∀∃∃∃ * ) and ESO( ∃ * ∀∀ ) are the maximal standard ESO-prefix classes contained in MSO, thus expressing only regular languages. We further prove the following dichotomy theorem: An ESO prefix-class either expresses only regular languages (and is thus in MSO), or it expresses some NP-complete languages. We also give a precise characterization of those ESO-prefix classes that are equivalent to MSO over strings, and of the ESO-prefix classes which are closed under complementation on strings.