Reflections on Finite Model Theory Phokion G. Kolaitis∗ IBM Almaden Research Center San Jose, CA 95120, USA [email protected] Abstract model theory, to examine some of the obstacles that were encountered, and to discuss some open problems that have Advances in finite model theory have appeared in LICS stubbornly resisted solution. proceedings since the very beginning of the LICS Sympo- sium. The goal of this paper is to reflect on finite model 2 Early Beginnings theory by highlighting some of its successes, examining ob- stacles that were encountered, and discussing some open In the first half of the 20th Century, finite models were problems that have stubbornly resisted solution. used as a tool in the study of Hilbert’s Entscheidungsprob- lem, also known as the classical decision problem, which is the satisfiability problem for first-order logic: given a 1 Introduction first-order sentence, does it have a model? Indeed, even before this problem was shown to be undecidable by Turing During the past thirty years, finite model theory has devel- and Church, logicians had identified decidable fragments of oped from a collection of sporadic, but influential, early re- first-order logic, such as the Bernays-Schonfinkel¨ Class of sults to a mature research area characterized by technical all ∃∗∀∗ sentences and the Ackermann Class of all ∃∗∀∃∗ depth and mathematical sophistication. In this period, fi- sentences. The decidability of these two classes was estab- nite model theory has been explored not only for its con- lished by proving that the finite model property holds for nections to other areas of computer science (most notably, them: if a sentence in these classes has a model, then it has computational complexity and database theory), but also in a finite model (see [14, Chapter 6] for a modern exposition its own right as a distinct area of logic in computer science. of these results). After hard toil over many years, it turned Since the very first LICS Symposium in 1986, LICS has out that these two are the only quantifier prefix classes of been a natural home for communicating state-of-the-art ad- first-order logic (with equality) over relational vocabular- vances in finite model theory. Moreover, at least five times ies for which the satisfiability problem is decidable [14]. A since its inception in 1995, the Kleene Award for Best Stu- third important class is the Godel¨ Class of all ∃∗∀∀∃∗ sen- dent Paper has been given for work in finite model theory tences. The equality-free fragment of this class has the finite [7, 11, 66, 69, 70]. model property, hence it is decidable [35]; in contrast, the The invitation to give a talk at LICS 2007 presents an full Godel¨ Class (with equality) is undecidable [36]. opportunity to reflect on the development and the state of fi- Trakhtenbrot’s Theorem [75] is generally regarded as the nite model theory today. This paper is not a comprehensive first important result in finite model theory during the sec- survey of finite model theory. To begin with, space limita- ond half of the 20th Century. Here, finite models are the ob- tions in a conference proceedings make this an impossible ject of study, as this result is about finitely valid first-order task. More importantly, there is no real need for such a sur- sentences, i.e., first-order sentences true on all finite models. vey, given that, by now, there are two books on the subject [26, 62], a monograph on descriptive complexity [48], and Theorem 1 Let σ be a relational vocabulary containing a a new book with comprehensive overviews of the main top- non-unary relation symbol. The set of all finitely valid first- ics in finite model theory and its applications [37]. At the order sentences over σ is not recursively enumerable. same time, this paper is not a “personal perspective” [31] Trakhtenbrot’s Theorem says that there is no effective on the development of finite model theory either. Instead, axiomatization of the set of all finitely valid first-order sen- it is an attempt to highlight some of the successes of finite tences. It contrasts sharply with Godel’s¨ Completeness The- ∗On leave from UC Santa Cruz. orem about the set of all valid first-order sentences, and it can be construed as an “anti-completeness” theorem for the of all finite ordered graphs, the class of all planar graphs, set of all finitely valid first-order sentences. the class of graphs of treewidth bounded by some fixed con- The development of finite model theory was also influ- stant, and the class of all finite strings. enced by the quest to resolve certain problems that were After a number of pioneering results obtained in the late articulated in the 1950s. A set S of positive integers is 1960s and the 1970s [28, 30, 34, 50], finite model theory said to be a spectrum if there is a first-order sentence ψ was pursued in its own right in the 1980s and beyond. It such that S = {m: ψ has a finite model with m elements}. turned out that new phenomena emerge, when one focuses Scholz [71] in 1952 and Asser [6] in 1955 posed the follow- on classes of finite structures; these phenomena gave finite ing problems about spectra. model theory its own distinct character and set it apart from other areas of mathematical logic. At the same time, finite Problem 1 The Spectrum Problem. model theory benefitted from a continuous interaction with • (Scholz) Characterize all spectra. certain areas of computer science, especially computational complexity and database theory. • (Asser) Are spectra closed under complement? In Research in finite model theory has branched into four other words, is the complement of a spectrum also a areas. The first is the study of the connections between spectrum? computational complexity and uniform definability in log- ics on finite structures, an area that is known as descriptive A problem of a different character was motivated by the complexity. The second (and closely related to the first) is preservation-under-substructures theorem of Łos-Tarski,´ the study of the expressive power of logics on finite struc- which asserts that if a first-order sentence ψ is preserved tures: what can and what cannot be expressed in various under substructures on all (finite and infinite) models, then logics on classes of finite structures? The third is the study there is a universal first-order sentence ψ∗ such that ψ is ∗ of the connections between logic and asymptotic combina- logically equivalent to ψ . In 1958, Scott and Suppes [72] torics; here the focus is on 0–1 laws and convergence laws asked whether the preservation-under-substructures theo- for the asymptotic probabilities of sentences of various log- rem holds in the finite, and conjectured that it does. ics on classes of finite structures. The final area is the study of classical model theory in the finite: do the classical re- Conjecture 1 (Scott and Suppes) If a first-order sentence ψ sults of model theory (eg., the various preservation theo- is preserved under substructures on all finite models, then rems, Craig’s Interpolation Theorem) hold in the finite? there is a universal first-order sentence ψ∗ such that ψ is In what follows, we highlight some of the achievements equivalent to ψ∗ on all finite models. in these areas, but also comment on some of the obstacles We will discuss the status of the Spectrum Problem and encountered and on certain problems that still remain open. the Scott-Suppes Conjecture in later sections. We assume that σ is a non-empty relational vocabulary; we will write F to denote the class of all finite σ-structures. 3 Main Themes in Finite Model Theory 3.1 Descriptive Complexity and Expressive Power The traditional focus of mathematical logic has been the study of logics on the class of all (finite and infinite) struc- Let C be a class of finite σ-structures and let k be a posi- tures or on a fixed infinite structure of mathematical signif- tive integer. A k-ary query on C is a mapping Q defined icance. The Completeness Theorem and the Compactness on C and such that if A is a structure in C, then Q(A) is Theorem for first-order logic are two key results in the first a k-ary relation on the universe of A that is invariant un- category. Godel’s¨ Incompleteness Theorem and Tarski’s der isomorphisms, i.e., if f : A → B is an isomorphism, Theorem about elimination of quantifiers on the reals are then Q(B) = f(Q(A)). A Boolean query on C is a map- two key results in the second category, since these are about ping from C to {0, 1} that is invariant under isomorphisms; first-order logic on the structure N = (N, +, ×) of the in- if Q(A) = 1, then we say that A satisfies Q, and write tegers and on the structure R = (R, +, ×) of the reals. A |= Q. Queries formalize and generalize the concept of In contrast, finite model theory focuses on the study of a decision problem on a class of finite structures, such as logics on classes of finite structures. In addition to first- CONNECTIVITY and 3-COLORABILITY. From a computa- order logic, various other logics have been explored in the tional standpoint, we are interested in determining the com- context of finite model theory; they include fragments of putational complexity of a given query q. From a logical second-order logic, logics with fixed-point operators, infini- standpoint, we are interested in determining whether a given tary logics, and logics with generalized quantifiers.