Decidability of Cylindric Set Algebras of Dimension Two and First-Order Logic with Two Variables

The aim of this paper is to give a new proof for the decidability and nite model property of rst-order logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse 2). The new proof also shows the known results that the universal theory of Pse 2 is decidable and that every nite Pse 2 can be represented on a nite base. Since the class Cs 2 of cylindric set algebras of dimension 2 forms a reduct of Pse 2 , these results extend to Cs 2 as well. We hasten to remark that the results proved here are not new, and indeed there are several rather diierent proofs available (references below). We felt justiied publishing this new proof, since we believe it is simpler than the proofs known, and accessible to both algebraists and logicians. The proof uses only very elementary ideas from universal algebra and model theory and one heavy combinatorial theorem, due to Herwig (Theorem 8). After the proof, we will indicate how to obtain the decidability result using the more elementary Fra ss e's Theorem. The mosaic method used here was originally devised to show decidability of the equational theories of the classes Crs (!) of relativised cylindric