### Small Substructures and Decidability Issues for First-Order Logic with Two Variables

E. Kieronski, M. Otto
20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05)
We study first-order logic with two variables FO 2 and establish a small substructure property. Similar to the small model property for FO 2 we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO 2 under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO 2 has the finite
more » ... property and is complete for non-deterministic exponential time, just as for plain FO 2 . With two equivalence relations, FO 2 does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO 2 is undecidable. * Full version of LICS 2005 paper [21]. -namely bisimulation games for modal logics, and k-pebble games for FO k . Correspondingly, and unlike prefix classes, these fragments enjoy natural closure properties which support some characteristic model theory. In particular, the finite variable fragments play a prominent role in finite model theory. In terms of satisfiability, FO 2 is decidable while FO 3 is undecidable. FO 2 here stands for the fragment of first-order logic with equality with only two variable symbols x and y, in finite relational vocabularies (without constants or function symbols). Without loss of generality, we also only consider vocabularies of width 2, without relation symbols of arities greater than 2. The first decidability proof for FO 2 was given by Scott [30] , via a reduction to the so-called Gödel prefix class ∃ * ∀∀∃ * , which however is only decidable in the absence of equality [10] . Full decidability for FO 2 with equality is due to Mortimer [25] . Mortimer shows FO 2 has the finite model property, and in fact every satisfiable FO 2 sentence has a model of size at most doubly exponential in the length of the sentence. This bound on the size of small models is improved to single exponential by Grädel, Kolaitis and Vardi in [12] , which leads to their result that FO 2 is decidable in NEXPTIME, and in fact NEXPTIME-complete. The study of FO 2 is also motivated by the fact that it embeds propositional modal logic K, via the standard translation. Numerous variants and extensions of modal logic find applications in various areas of computer science, including verification of software and hardware, distributed systems, knowledge representation and artificial intelligence. These applications are supported by the very good algorithmic and model-theoretic behaviour of modal logics, including their remarkably robust decidability which persists under various extensions towards greater expressiveness. Some of these extensions are equally well motivated in the context of two-variable logic. Description logics in particular naturally fit into the range between modal and twovariable logics, see [2] and in particular [6] for their connection with finite variable fragments. The extension of FO 2 by counting quantifiers [17] , for instance, is decidable by [14, 28] although it does not have the finite model property. In analogy with graded modalities, it covers certain description logics with number constraints. But also in systematic terms the question naturally arises, to which extent FO 2 shares the good algorithmic behaviour of modal logics. The picture that emerged in [15] shows that, with the notable exception of the counting extension, most extensions of FO 2 are undecidable, compare also [13] . This includes e.g. various extensions by mechanisms for fixed points or transitive closures, in analogy with the modal µ-calculus or computation tree logics. In many cases, the results of these investigations can be phrased either for satisfiability of extensions of FO 2 , or, alternatively, of FO 2 itself over restricted classes of structures. This interplay is fruitfully employed in [15, 26] . In connection with modal logics, or with applications of modal or two-variable logics in areas like knowledge representation or for description logics, a restriction of the underlying class of models is often very natural. Modal correspondence theory, for instance, associates transitivity of accessibility relations with the modal logic K4; equivalence relations with the modal logic S5. Multi-S5 systems with k equivalence relations among their accessibility relations can be used to model knowledge systems for k independent agents; linear orders as accessibility relations play an obvious role for linear temporal logics, etc. For FO 2 over such classes of structures, undecidability is established under several such constraints in [15, 13] and in particular in the presence