Finite variable logics in descriptive complexity theory

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area. For each k 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider infinitary variants and extensions by so-called counting quantifiers. Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time (PTIME) or polynomial space (PSPACE) can be characterized as collections of all classes of (ordered) finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifierdepth. Although these early results from descriptive complexity theory have been put in much more elegant forms later using so-called fixed-point logics, the importance of the number of variables as a complexity measure remained. In 1990, Kolaitis and Vardi [51] proved a 0-1 law for the infinitary finite variable logics. As a corollary, they re-proved a result of Blass, Gurevich, and Kozen [7] that there is a 0-1 law for least-fixed point logic. The fact that makes Kolaitis’ and Vardi’s paper so remarkable is that it uses finite variable logics as a technical tool to obtain results concerning fixed-point logics, which are central in descriptive complexity. Once they had realized this possibility, Kolaitis, Vardi, and others (see, for example, [1, 2, 17, 52, 54, 63]) developed finite variable logics into the technical framework of a central part of finite model theory. Notably, Abiteboul and Vianu [2] introduced a machine model reflecting precisely the expressive power of these logics and used it to prove a fundamental result which states that the complexity theoretic question of whether PTIME equals PSPACE is equivalent to the purely logical question of whether least fixed-point logic has the same expressive power as partial fixed-point logic. In his PhD thesis, Dawar [12] gives a coherent picture of the theory at this stage, putting some emphasis on model theoretic aspects. Immerman [45] proposed to extend the finite-variable logics by counting quantifiers such as “there exist at least m elements x”. Counting is of a very low computational complexity. The logics Lk can only count up to k, though. So it seems reasonable to extend them by counting quantifiers. Cai, Fürer, and Immerman [9] related the counting