First Order Logic, Fixed Point Logic and Linear Order

The Ordered conjecture of Kolaitis and Vardi asks whether xed point logic di ers from rst order logic on every in nite class of nite ordered structures In this paper we develop the tool of bounded variable element types and illustrate its application to this and the orig inal conjectures of McColm which arose from the study of inductive de nability and in nitary logic on pro cient classes of nite structures those admitting an unbounded induction In particular for a class of nite structures we introduce a compactness notion which yields a new proof of a rami ed version of McColm s second conjecture Furthermore we show a connection between a model theoretic preservation property and the Ordered Conjecture allowing us to prove it for classes of strings colored orderings We also elaborate on complexity theoretic implica tions of this line of research