Expressivity within second-order transitive-closure logic

TL;DR

This paper investigates the expressive power of second-order transitive-closure logic and its fragments, revealing collapses and equivalences with other logical systems, and clarifying their capabilities in capturing complexity classes.

Contribution

It proves that the existential fragment SO(2TC)(E) collapses to existential first-order logic and shows that CMSO(TC) and MSO(TC) have the same expressive power.

Findings

SO(2TC)(E) collapses to existential first-order logic

CMSO(TC) and MSO(TC) are expressively equivalent

MSO(TC) strictly subsumes order-invariant MSO on unary vocabularies

Abstract

Second-order transitive-closure logic, SO(TC), is an expressive declarative language that captures the complexity class PSPACE. Already its monadic fragment, MSO(TC), allows the expression of various NP-hard and even PSPACE-hard problems in a natural and elegant manner. As SO(TC) offers an attractive framework for expressing properties in terms of declaratively specified computations, it is interesting to understand the expressivity of different features of the language. This paper focuses on the fragment MSO(TC), as well on the purely existential fragment SO(2TC)(E); in 2TC, the TC operator binds only tuples of relation variables. We establish that, with respect to expressive power, SO(2TC)(E) collapses to existential first-order logic. In addition we study the relationship of MSO(TC) to an extension of MSO(TC) with counting features (CMSO(TC)) as well as to order-invariant MSO. We show that the expressive powers of CMSO(TC) and MSO(TC) coincide. Moreover we establish that, over unary vocabularies, MSO(TC) strictly subsumes order-invariant MSO.