Existential second-order logic over strings
Existential second-order logic (ESO)and monadic second-order logic (MSO) have attracted much interest in logic and computer science. ESO isa much more expressive logic over successor structures than MSG. However, little was known about the relationship between MSO and syntactic 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(2) denote the prefix class containing all sentences of the shape There Exists RQ phi, where R is a list of predicate variables, Q is a first-order quantifier prefix from the prefix set L, and phi is quantifier-free. We show that ESO(There Exists*For All There Exists*) and ESO(There Exists*For All For All) are the maximal standard ESO-prefix classes contained in MSG, 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), ur 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.
second-order logic strings prefix classes model checking regular languages finite model theory existential fragment finite words s1s descriptive complexity np decision problem finite satisfiability