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 finite model property of first-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 (Pse2). The new proof also shows the known results that the universal theory of Pse, is decidable and that every finite Pse2 can be represented on a finite base. Since the class Cs, of cylindric set algebras of dimension 2 forms a reduct of Pse,. these results extend to Cs2 as well. We hasten to remark that the results proved here are not new, and indeed there are several rather different proofs available (references below). We felt justified 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 Fraisse's Theorem. The mosaic method used here was originally devised to show decidability of the equational theories of the classes Crs, (a < co) of relativised cylindric set algebras [13]. As shown here, the same method works with unrelativised algebras in dimension two, so we obtain a unified proof-method for both Crs, and Cs2. The original method uses an infinite step-by-step argument to construct a model from a finite set of mosaics. This has the drawback that the method shows decidability without being able to say anything about the finite model property. The idea of combining the mosaic approach with Herwig's Theorem originates with Ian Hodkinson, in an attempt to remove any step-by-step argument from these type of decidability proofs [10]. In [1], Herwig's Theorem is used directly without the intermediate step of mosaics to show the finite base property (every finite algebra can be represented as an algebra with finite base) for a wide range of classes of relativised algebras. Implicitly the mosaic idea was already used by G6del in 1933 to show that every satisfiable prenex V2 ' -sentence can be satisfied on a finite model [4]. A step-by-step argument instead of Gddel's model construction readily shows decidability of the satisfaction problem of these type of sentences. The construction Received August 29, 1996; revised February 16, 1998. Supported by EPSRC grant No. GR/K54946. Supported by EPSRC grant Nos. GR/K54946 and GR/L82441 and OTKA grant Nos. F17452 and T16448. ( 1999. Association for Symbolic Logic 0022-4812/99/6404-001 4/$2.00