On the Rst-order Preex Hierarchy

We investigate the expressive power of fragments of rst-order logic that are deened in terms of preexes. The main result establishes a strict hierarchy among these fragments over the signature consisting of a single binary relation. It implies that for each preex p, there is a sentence ' p in prenex normal form with preex p, over a single binary relation, such that for all sentences in prenex normal form, if is equivalent to ' p , then p can be embedded in the preex of. This strengthens a theorem of Walkoe. In this paper we address the following question. Given two rst-order preexes p and q, is there a sentence with preex p that is not equivalent to any sentence with preex q? Walkoe 4] proved that if p and q are diierent preexes of length n, then there is such a sentence, containing a single n-ary relation symbol. Keisler and Walkoe 3] then strengthened this result by showing that it also holds over the class of nite structures. Our main theorem improves on Walkoe's result. It implies that for each preex p, there is a sentence ' p in prenex normal form with preex p, over a single binary relation, such that for all sentences in prenex normal form, if is equivalent to ' p , then p can be embedded in the preex of. (We leave its precise statement to Section 1.) This also resolves a conjecture of Grr adel and McColm 1], explained below. 1 Background and statement of the main theorem 1.1 Terminology and deenitions We adopt the following terminology and conventions. We will consider (fragments of) rst-order logic (FO) and innnitary logic (L 1!), which allows innnitary conjunctions and disjunctions. Throughout we assume that formulas are in negation normal form, that is, negation symbols only bind atomic formulas. Signatures are always purely relational and nite. A graph is a structure with signature fEg, E binary; it is simple if it is undirected and loop-free. In a (directed) graph, we say that vertex a is an E-predecessor of vertex b, if Eab ^ :Eba. Throughout, we assume that any class of structures we are talking about is closed under isomorphism.