Homomorphisms and First-Order Logic

Transcription

1 Homomorphisms and First-Order Logic (Revision of August 16, 2007 not for distribution) Benjamin Rossman MIT Computer Science and Artificial Intelligence Laboratory 32 Vassar St., Cambridge, MA Abstract We prove that the homomorphism preservation theorem (h.p.t.), a classical result of mathematical logic, holds when restricted to finite structures. That is, a first-order formula is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existential-positive formula. This result, which contrasts with the known failure of other classical preservation theorems on finite structures, answers a longstanding question in finite model theory. The relevance of this result, however, extends beyond logic to areas of computer science, including constraint satisfaction problems and database theory; the database connection arises from a correspondence between existential-positive formulas and unions of conjunctive queries (also known as select-project-join-union queries). A second result of this article strengthens the classical h.p.t. by showing that a firstorder formula is preserved under homomorphisms on all structures if, and only if, it is equivalent to an existential-positive formula of equal quantifier-rank. Unlike traditional proofs of the classical h.p.t., the proof of this stronger equirank theorem is compactnessfree and constructive. While these results are logical in nature, the technical development of the article takes place almost entirely within a combinatorial framework. The concept of tree-depth, a graph parameter related to tree-width, plays an important role in our analysis (as a combinatorial counterpart to quantifier-rank). We introduce new notions of n-homomorphism and n-core, which approximate the familiar concepts of homomorphism and core up to tree-depth n. The key technical lemmas take a pair of n-homomorphically equivalent [finite] relational structures and construct corresponding [finite] co-retracts which satisfy a certain back-andforth property. Supported by an MIT Akamai Presidential Fellowship and a National Defense Science and Engineering Graduate Fellowship. This article was partially written during an internship at IBM s Almaden Research Center. 1

2 Contents 1 Introduction Preservation theorems in classical and finite model theory Main results Combinatorial perspective Prior and related work Outline of the article Preliminaries Basic definitions Structures and homomorphisms over a set X Gaifman graphs and tree-depth Retractions and cores Back-and-forth equivalence Primitive-positive and existential-positive sentences 15 4 Bounded tree-depth n-homomorphism n-cores Logical characterization of n-homomorphism Freeness Extension cores Equirank homomorphism preservation theorem The n-extension property Extendable co-retracts Finite homomorphism preservation theorem 34 7 Nonelementary length blow-up 45 8 Extensions and open questions Extensions of our results Open questions References 52 2

3 1 Introduction 1.1 Preservation theorems in classical and finite model theory Classical model theory is a branch of mathematical logic that studies properties of abstract mathematical structures expressible in first-order logic [20]. One early and fundamental class of results are the classical preservation theorems, which connect syntactic and semantic properties of first-order formulas. Three representative preservation theorems (the ones most relevant to this article) are the Loś-Tarski theorem, Lyndon s theorem and the homomorphism preservation theorem (or h.p.t., for short). These closely related theorems, all dating from the 1950s, provide semantic characterizations of the syntactic classes of existential formulas, positive formulas, and existential-positive formulas. (We do not assume the reader has any background in logic. Definitions relating to theorems stated in this introduction are concisely presented in 2.1.) Theorem 1.1 ( Loś-Tarski Theorem). A first-order formula is preserved under embeddings on all structures if, and only if, it is logically equivalent to an existential formula. 1 Theorem 1.2 (Lyndon s Theorem). A first-order formula is preserved under surjective homomorphisms on all structures if, and only if, it is logically equivalent to a positive formula. Theorem 1.3 (Homomorphism Preservation Theorem). A first-order formula is preserved under homomorphisms on all structures if, and only if, it is logically equivalent to an existentialpositive formula. In these theorems and throughout this article, the phrase all structures is inclusive of both finite and infinite structures. As we will see, this is an important qualification. Finite model theory is the study of first-order logic (and its various extensions) on finite structures [11, 23]. Computer science provides many primary motivations for studying logic on finite structures (cf. the book [13] which discusses a range of applications of finite model theory to computer science). Database theory is one area in particular where finite model theory has made an impact [1]. Ever since Codd s seminal article [9], finite relational structures have been a popular model for databases. Many questions arising in the database context have been fruitfully studied in the framework of finite model theory; even the main question addressed in this article, the status of the h.p.t. on finite structures, has a natural formulation in the language of database theory. Complexity theory is another area where logic has made a contribution; a body of work in descriptive complexity has characterized various complexity classes by means of different extensions of first-order logic on finite structures [21]. Finite model theory has also found applications in the study of constraint satisfaction problems [22]. On the face of it, finite model theory appears to be a subfield of classical model theory; after all, finite structures are a subclass of all structures. In reality, there is a large gulf between the finite and classical worlds. From the standpoint of classical model theory, the intrinsically interesting structures are infinite. Many classical theorems and key techniques break down when restricted to finite structures. The compactness theorem (one of the most important tools in classical model theory) is perhaps the most conspicuous failure on finite 1 An embedding of A into B is an isomorphism from A onto an induced substructure of B. The Loś-Tarski Theorem is sometimes stated in its dual form: a first-order formula is preserved under induced substructures if, and only if, it is logically equivalent to a universal formula. 3

4 structures. Other classical theorems are either meaningless or irrelevant in the finite setting (e.g., the Lowenheim-Skolem theorem). Others still are curiously inverted: whereas the set of valid first-order formulas is r.e. (recursively enumerable) [Gödel s completeness theorem] but not co-r.e. [Church s theorem], the set of first-order formulas that are valid on finite structures is co-r.e. but not r.e. [Trakhtenbrot s theorem]. A handful of classical results and techniques survive transition to the finite setting (e.g., Ehrenfeucht-Fraïssé games), but not enough to bridge the gulf between the finite and classical worlds. And, of course, a great many results are native to finite model theory, having no classical counterpart. The project to classify the status of classical theorems when restricted to finite structures has been an active line of research since the 1980s [16]. Results which hold true are typically those whose proofs in the classical setting work just as well when one considers only finite structures. On the other hand, the failure of compactness theorem presages the collapse of its many corollaries, among them the classical preservation theorems (whose original proofs rely on compactness arguments) [30]. In a survey article [29] on classical and finite model theory from 2002, Eric Rosen write that there seems to be no example of a theorem that remains true when relativized to finite structures but for which there are entirely different proofs for the two cases. It would be interesting to find a theorem proved using the compactness theorem that can be established using a new method over finite structures. It is known that many of the candidates for such a theorem, such as preservation and interpolation theorems, fail in the finite. Indeed, both the Loś-Tarski theorem and Lyndon s theorem are known to fail in the finite. Theorem 1.4 ( Loś-Tarski fails). There exists a first-order formula that is preserved under embeddings on finite structures, but is not equivalent in the finite to an existential formula. Theorem 1.5 (Lyndon fails). There exists a first-order formula that is preserved under surjective homomorphisms on finite structures, but is not equivalent in the finite to a positive formula. Theorem 1.4 was proved by Tait [33] in 1959 and rediscovered by Gurevich and Shelah [4, 15] in Theorem 1.5 is the work of Ajtai and Gurevich [2] from In neither case is the failure on finite structures as obvious as the blatant failure of the compactness theorem. The counterexample in Theorem 1.5 in particular is highly nontrivial. (A somewhat simplified counterexample was later obtained by Stolboushkin [32].) However, the status of the h.p.t. on finite structures remained an open problem, despite a number of partial solutions [3, 5, 14, 28] and an incorrect (and quickly retracted) claim in [17]. 1.2 Main results Our main result shows that classical h.p.t. holds on finite structures. This is somewhat surprising since the h.p.t. seems to live at the intersection of the Loś-Tarski and Lyndon preservation theorems, which both fail on finite structures. A secondary main result (which we prove along the way) improves the classical h.p.t. We call this result the equirank h.p.t., because of a clause concerning quantifier-rank by which it strengthens the classical h.p.t. In stating our main results, below, we switch from talking about formulas to sentences (i.e., formulas without free variables). This is merely a matter of convenience; both theorems remain valid when stated more generally for formulas instead of sentences (see discussion in 8.1). 4

5 Equirank Hom. Preservation Theorem (Theorem 5.12) A first-order sentence is preserved under homomorphisms on all structures if, and only if, it is equivalent to an existential-positive sentence of equal quantifier-rank. Finite Hom. Preservation Theorem (Theorem 6.16) A first-order sentence of quantifier-rank n is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existential-positive sentence of quantifier-rank ρ(n) for some computable function ρ : N N. We actually obtain slightly sharper results, stated as interpolation theorems (Theorems 5.11 and 6.15). Recapping, the equirank h.p.t. tells us that every homomorphism-preserved first-order sentence Φ is equivalent to an existential-positive sentence Ψ of the same (or lesser) quantifier-rank. Said another way, there is no blow-up in quantifier-rank from Φ to Ψ. Moreover, we can effectively compute Ψ. But what about the length of (the shortest equivalent) Ψ? No algorithm which computes Ψ is of any practical use if Ψ is vastly longer than Φ. Unfortunately, this turns out to be the case. An previously unpublished result of Gurevich and Shelah (which we include here as Theorem 7.1) says that, despite the non-increase in quantifier-rank, there is a possibly nonelementary blow-up in the length of Ψ. The same counterexample of Gurevich and Shelah also implies that the best quantifier-rank bound ρ(n) we manage to achieve in the finite h.p.t. is a nonelementary function. (In 8.2 we conjecture that the true minimal bound is ρ(n) = n, that is, that the equirank h.p.t. holds on finite structures.) 1.3 Combinatorial perspective Our main results have purely combinatorial counterparts. In fact, once we establish a correspondence between primitive-positive sentences and finite cores (or existential-positive sentences and finite antichains in the homomorphism lattice), the entire technical development of the article takes place within a combinatorial framework. From the combinatorial perspective, we are interested in the following equivalences on relational structures. (The second and third equivalences have logical as well as combinatorial formulations.) homomorphic equivalence (A B) there exist homomorphisms A B and B A n-homomorphic equivalence (A n B) combinatorial: C A if, and only if, C B for every structure C of tree-depth n logical: A and B satisfy the same existential-positive sentences of quantifier-rank n n-back-and-forth equivalence (A n B) combinatorial: there exists an n-back-and-forth system of partial isomorphisms on A, B logical: A and B satisfy the same first-order sentences of quantifier-rank n 5

6 (On finite structures, the sequence of equivalences 0, 1, 2,... approximates, while 0, 1, 2,... converges to isomorphism. On infinite structures, things are slightly more subtle. The intersection of equivalences n N ( n) is coarser than in general, while n N ( n), called elementary equivalence in model theory, is weaker than isomorphism on infinite structures.) The key combinatorial lemmas behind our main theorems can now be stated as follows. Equirank Theorem (combinatorial essence) For all structures A and B such that A n B, there exist A and B such that A A n B B. Finite Theorem (combinatorial essence) For all finite structures A and B such that A ρ(n) B, there exist finite A and B such that A A n B B. 1.4 Prior and related work Prior to this article, the h.p.t. was known to hold on finite structure in various special cases. Rosen [28] proved that a first-order formula in the prefix class is preserved under homomorphisms on finite structures if, and only if, it is equivalent to an existential-positive formula. Similarly, Grädel and Rosen [14] proved that the h.p.t. hold on finite structures for first-order formulas that involve only two distinct variables. Ajtai and Gurevich [3] showed that if a class of finite structures is definable both by a firstorder sentence and by Datalog expression, then it is also definable by an existential-positive sentence. Although the technique of Ajtai and Gurevich is completely different from the method of this article, this result follows as a special case of our finite h.p.t. (Theorem 6.16), since all Datalog expressions are preserved under homomorphisms. Another result that now follows Theorem 6.16, but was originally proved by a different technique, comes from the field of constraint satisfaction problems. For a finite structure B, let CSP(B) denote the class of finite structures A which admit a homomorphism to B. The membership problem for the class CSP(B) is referred to as the constraint satisfaction problem with fixed template structure B. Atserias [5] proved that if CSP(B) is definable by a first-order sentence, then the complement CSP(B) (i.e., the class of finite structures not homomorphic to B) is definable by an existential-positive sentence. Finally, we mention two results which relate to the theme of this article, but are not directly implied by either of our main theorems. Feder and Vardi [12] proved various homomorphism preservation theorems on finite structures for some non-first-order logics including L ω, SNP and 2SNP. In a different vein, Atserias, Dawar and Kolaitis [6] showed that the classical (firstorder) h.p.t. holds on several restricted classes of finite structures. Specifically, they proved that if K is any class of finite structures either of bounded degree, or whose cores exclude a minor, then the classical h.p.t. holds when restricted to K (i.e., a first-order formula is preserved under homomorphisms on K if, and only if, it is logically equivalent to an existential-positive formula on K). Building on the work of [3], the basic technique of Atserias, Dawar and Kolaitis is rather different from the methods of this article (see Remark 3.10). 6

7 1.5 Outline of the article We review the basics of structures, homomorphisms and first-order logic in 2, with emphasis on the notions of tree-depth and cores. The fundamentals of primitive-positive and existentialpositive sentences are treated in 3. We establish an equivalence between existential-positive sentences and finite antichains in the homomorphism lattice, essentially due to Chandra and Merlin [8]. The main novelty in this section is an emphasis on the correspondence between quantifier-rank and tree-depth. After this section, we dispense with logic and proceed in a combinatorial vein until 7 (except for brief interludes). In 4 we introduce and study new notions of n-homomorphism, n-cores and n-freeness, which approximate the familiar concepts of homomorphism, cores and separation up to treedepth n. (A equivalent logical theory of n-existential-positive types was developed in [31], a preliminary version of this article. The improved combinatorial presentation of the present article was inspired by the book [19].) The equirank homomorphism preservation theorem is proved in 5. The main tool is a technical notion of n-extendability (called n-existential-positive saturation in [31]), which resembles similar extension properties in model theory. While the equirank h.p.t. concerns infinite structures (and is thus perhaps more relevant to model theory than computer science), the proof serves as a warm-up for the more complicated proof yet to come. In 6, we finally prove the finite homomorphism preservation theorem. At the heart of the proof is an intricate back-and-forth argument. We return to logic in 7, proving a previously unpublished theorem of Gurevich and Shelah (announced in [17]) that the shortest existential-positive sentence equivalent to a given homomorphism-preserved first-order sentence has a potentially nonelementary blow-up in length. This result neatly complements our equirank h.p.t., which implies that there is no blow-up in quantifier-rank. We conclude in 8 by mentioning a few extensions of our results and raising a few intriguing questions. 2 Preliminaries We begin in 2.1 with the basic definitions of structures, homomorphisms and first-order logic required for the statement of our main results (and all other theorems stated in 1). Following these basic definitions, in 2.2 we look at the category of structures and homomorphisms over a set X. In 2.3, we define the Gaifman graph G(A) of a structure A; this graph contains the information about connectivity among elements and subsets of A. Also in 2.3, we introduce the key concept of tree-depth td X (A) of a finite structure A over a subset X A. Retractions and cores are defined in 2.4, as is the canonical core Core X (A) of a finite structure A over a subset X A. In 2.5 we look at n-back-and-forth equivalence of structures (corresponding to indistinguishability by first-order sentences of quantifier-rank n). We state a lemma characterizing n-equivalence in terms of an n-back-and-forth system of partial isomorphisms. Most lemmas in this section are stated without proof and should be considered easy exercises. References are given where results are not folklore. 7

8 2.1 Basic definitions Structures A (relational) structure is an object A = A, R1 A,..., RA m where A is a (finite or infinite) set, called the universe of A, and each Ri A is a relation on A, that is, a subset of A k i for some positive integer k i. The sequence of relation symbols R 1,..., R m together the corresponding arities k 1,..., k m comprise the vocabulary of A. The relation Ri A is called the interpretation of relation symbol R i in A. We will always consider structures with the same vocabulary, which we denote by σ. We emphasize that σ consists of finitely many relation symbols; in other words, σ is a finite relational vocabulary. 2 We consistently employ boldface letters A, B, C,... for structures and italic letters A, B, C,... for the corresponding universes. A structure B is a substructure of A (symbolically: B A) if B A and R B R A for every relation symbol R in σ. It is an induced substructure if R B is the restriction of R A to A k for every R in σ of arity k. For a subset X A, A X denotes the (unique) induced substructure of A with universe X. Homomorphisms Let A = A, R1 A,..., RA m and B = B, R1 B,..., RB m be structures (in the same vocabulary). A homomorphism from A to B is a function h : A B such that h(ri A) RB i (i.e., if (a 1,..., a ki ) Ri A then (h(a 1 ),..., h(a ki )) Ri B ) for each relation symbol R i. First-order logic following rules: Formulas of first-order logic (in the vocabulary σ) are built up using the x = y, Rx 1... x k atomic formulas φ, φ ψ, φ ψ negation, conjunction and disjunction x φ, x ψ existential and universal quantification where x, y and x 1,..., x k are variables, R is a k-ary relation symbol in σ, and φ and ψ are formulas. We assume familiarity with the concept of free and bound variables, as well as the semantics of first-order logic (i.e., when a formula is true in (satisfied by) a structure for a given assignment of free variables). A formula is often written out followed by an ordered list of its free variables in the style of φ(x 1,..., x k ). For a structure A and tuple a A k, we write A = φ( a) if the formula φ( x) is true in A with variables x 1,..., x k taking values a 1,..., a k. Formulas with no free variables are called sentences. A few special classes of formulas are defined below. Primitive-positive formulas are built out of atomic formulas using only conjunction and existential quantification. (In the database context, these formulas define conjunctive queries.) Existential-positive formulas are built out of atomic formulas using conjunction and existential quantification, as well as disjunction. (The database analogue are unions of conjunctive queries, also known as select-project-join-union queries.) Positive formulas are simply first-order formulas without negations. 2 In general, vocabularies may involve infinitely many relation symbols, as well as constant symbols and function symbols. In 8.1, we consider how our results extend to structures in more general vocabularies. 8

9 Existential formulas are formulas in which every existential quantifier ( ) falls inside the scope of an even number of negations, while every universal quantifier ( ) falls inside the scope of an odd number of negations. Equivalently, a formula is existential if no universal quantifiers remain after all negations are pushed down to the level of atomic formulas via rules φ φ and (φ ψ) φ ψ, etc. The quantifier-rank of a formula φ is the maximal nesting depth of quantifiers in φ. This is potentially less than the overall number of quantifiers in φ, its quantifier-count. For instance, the formula ( x ( y Rxy) ( z Rzx)) ( x Rxx) has quantifier-rank 2 and quantifier-count 4. For formulas φ(x 1,..., x k ) and ψ(x 1,..., x k ) with the same free variables, φ entails ψ [in the finite] if A = φ( a) implies A = ψ( a) for every [finite] structure A and tuple a A k. Formulas which entail each other [in the finite] are (logically) equivalent [in the finite]. A formula φ(x 1,..., x k ) is preserved under homomorphisms [on finite structures] provided that for all [finite] structures A and B, if A = φ(a 1,..., a k ) and h : A B is a homomorphism, then B = φ(h(a 1 ),..., h(a k )). By now we have given all definitions relating to the statements of our main theorems and all other background theorems stated in 1. The remaining definitions and lemmas in this section pertain to concepts needed for the proofs of our main theorems. Remark 2.1. It should be obvious that we do insist that formulas be expressed in prenex form, that is, with all quantifiers up front followed by a propositional (quantifier-free) formula. This is an important qualification, as our definition of quantifier-rank is sensitive to the fact that quantifiers may be interlaced with conjunctions and disjunction; the standard procedure for transforming a first-order formula into an equivalent prenex formula potentially increases quantifier-rank. Note that quantifier-rank and quantifier-count coincide for prenex formulas, but not for formulas in general. Remark 2.2. Logically equivalent formulas are clearly also logically equivalent in the finite. Similarly, a formula which is preserved under homomorphisms is also preservation under homomorphisms on finite structures. In neither case, however, is the converse true. (One can cook up counterexamples by tinkering with axioms of infinity, that is, first-order sentences which are satisfied by some infinite structure but not by any finite structure.) 2.2 Structures and homomorphisms over a set X Let X be an arbitrary set. We call a structure A whose universe includes X (i.e., X A) a structure over X. For structures A and B over X, we call a homomorphism from A to B which fixes X pointwise a homomorphism over X. We write A X B if there exists a homomorphism from A to B over X. We say A and B are homomorphically equivalent over X, and we write A X B, if A X B and B X A. We say A and B are isomorphic over X, and we write A = X B, if there exist homomorphisms f : A X B and g : B X A such that g f = id A and f g = id B ; in this case, we say f and g are isomorphisms over X. The coproduct of A and B in the category of structures and homomorphisms over X is called the X-sum of A and B and denoted by A X B. Formally, this is a structure with universe X (A \ X) (B \ X) whose relations are inherited from A and B via the natural inclusion maps A X (A\X) (B\X) B. We view X as an associative and commutative 9

10 operation, and we view A and B as substructures of A X B, even if this is technically a lie. 3 When considering the X-sum A X A of a single structure A, we refer the lefthand and righthand copies of A in A X A. The coproduct of an indexed family (A i ) i I of structures over X is denoted by i I X A i. For a nonempty set A of structures over X, we abbreviate the awkward-looking A AX A by instead writing X A. The product of A and B in the category of structures over X and homomorphisms over X is denoted by A X B. Formally, this is a structure with universe A B, in which the set X = {(x, x) : x X} (the diagonal over X ) is identified with X itself; more formally, A X B has universe X ((A B) \ X ). Relations in A X B are defined by R A XB = {((a 1, b 1 ),..., (a k, b k )) (A B) k : (a 1,..., a k ) R A and (b 1,..., b k ) R B }. In the special case when X =, we sometimes write A B and A B and A B instead of A B and A B and A B. In particular, A B is the familiar disjoint union of structures. Note that any structure over X is also a structure over W for every W X. Parts (1) and (2) of next lemma extends this observation. Lemma 2.3. Let A, B, C be structures over a set X and let W X. 1. A X B = A W B 2. (A W B) W (A X B) 3. (A X C and B X C) (A X B) X C 4. (C X A and C X B) C X (A X B) Proof. Statement (1) is obvious, since any homomorphism over X is also a homomorphism over W. For statement (2), we define a function h : W (A\W ) (B \W ) X (A\X) (B \X) by patching the map W X (i.e., the identity map on W ) together with natural embeddings A \ W = = (X \ W ) (A \ X) X (A \ X) and B \ W (X \ W ) (B \ X) X (B \ X). It is easy to see that h is a homomorphism from A W B to A X B over W. Statements (3) and (4) are standard facts about the coproduct and product. There are many similar rules that one might point out, such as (A X B and B Y C) = A X Y C. We have stated in Lemma 2.3 only what we will need later on. We will occasionally generalize the notation X in order to assert the existence of homomorphisms satisfying certain additional constraints. For structures A and B and tuples a A k and b B k, we write (A, a) X (B, b) if there exists a homomorphism h : A X B such that h(a i ) = b i for every i {1,..., k}. For a one-to-one partial function π from A to B, we write A π B if there exists a homomorphism from A to B which extends π, and we write A π B if A π B and B π 1 A. 3 The situation with the set-theoretic disjoint union is analogous. For sets P and Q, the disjoint union P Q is formally the set P {{P, {P, q}} : q Q} (according to one common definition). While is neither truly commutative nor associative, and while Q is not necessarily a subset of P Q, with a modicum of care one may harmlessly pretend that is both commutative and associate and that Q is an actual subset of P Q (identified with {{P, {P, q}} : q Q}). 10

11 2.3 Gaifman graphs and tree-depth By default, graphs are simple (i.e., undirected and without self-loops) though possibly infinite. We allow graphs to have zero vertices, calling the unique graph with no vertices the null graph. A finite rooted forest is a finite disjoint union of finite rooted trees. For vertices v and w in a finite rooted forest F, we write v w if v is the parent of w. The height of F is number of vertices in the longest path (via ) from a root to any leaf. The closure F of F is a graph having the same vertex set as F, in which vertices v and w are adjacent if, and only if, one of the pairs (v, w) and (w, v) belongs to the transitive closure of (i.e., v w and v and w are ancestors in F). The tree-depth td(g) of a finite graph G is the minimal height of a finite rooted forest whose closure contains G as a subgraph. (This definition of tree-depth is from [26]. Tree-depth has several combinatorial equivalents, including minimum elimination tree height and vertex ranking number [10, 25].) An inductive form of this definition is as follows (Lemma 2.2 of [26]): 0 if G is the null graph (with no vertices), td(g) = 1 + max vertex v of G td(g \ v) max component G of G td(g ) if G is non-null and connected, if G is non-null and disconnected. Here G \ v denotes the graph obtained from G by removing vertex v and all edges incident to v. Remark 2.4. We do not define tree-depth of infinite graphs, since present purposes do not require it. We mention, however, that there is a natural definition of tree-depth for certain infinite graphs. This extended notion of tree-depth, defined by transfinite induction (writing sup in place of max above), is used to good effect in [27]. Examples 2.5. We give some examples of tree-depth of finite graphs. (a) The kite pictured below has tree-depth 3. (b) A complete graph on n vertices has tree-depth n. (c) A path on n vertices has tree-depth log 2 n + 1. (d) Tree-depth is related to tree-width by the inequality [25]: tw(g) + 1 td(g) tw(g) log 2 G. The Gaifman graph G(A) of a structure A is a graph with vertex set A in which two elements are adjacent if, and only if, they appear together in some tuple in a relation of A. Essentially, G(A) distills the structure A down to the information about connectivity (and distance) among elements and subsets of A. For a subset X A, let G(A) \ X denote the induced subgraph of 11

12 G(A) with vertex set A \ X. (Do not confuse G(A) \ X with G(A \ X), that is, the Gaifman graph of the induced substructure of A with universe A \ X. For vocabularies σ containing relation symbols of arity 3, these two graphs need not coincide.) The tree-depth td X (A) of a finite structure A over a subset X A is defined as the treedepth of the Gaifman graph of A over X. In symbols, td X (A) = td(g(a) \ X). (A similar caution applies here: do not confuse td X (A) with td(g(a \ X)).) The next lemma, which follows directly from definitions, lists some useful facts about tree-depth of finite structures. Lemma 2.6 (Facts about tree-depth). Suppose A and B be finite structures and X A B. 1. td X (A X B) = max{td X (A), td X (B)}. 2. If B A, then td X (B) td X (A). 3. td X (A) td X Y (A) + Y for all Y A. 4. If G(A) \ X is non-null and connected, then td X (A) = 1 + max y A\X td X {y}(a). We remark that Lemma 2.6(1,4), together with the proposition that td X (A) = 0 if A = X, completely axiomatizes tree-depth of finite structures. 2.4 Retractions and cores Suppose B is a substructure of A. Homomorphisms A B B are called retractions. If there exists a retraction from A to B, then we write A retr B (rather than A B B), we say that B is a retract of A and A is a co-retract of B, and we call the inclusion map B A (i.e., the identity map on B) is a co-retraction. Note that retracts are always induced substructures; in other words, A retr B implies A B = B. The next lemma states some easily proved facts about retractions. Lemma 2.7 (Properties of retractions). 1. A retr B retr C = A retr C 3. A B i i I retr A ( B i i I ) retr A 2. (A X B) retr A B X A 4. n N ( A n+1 ) retr A n = n N A n retr A 0 retr A 1 In Lemma 2.7(4), n N A n denotes the union of the chain of retracts A 0 retr A 2, that is, the structure with universe n N A n in which a relation symbol R is interpreted as the union of relations R A n. A structure A is a core over a subset X A if every homomorphism A X A is an automorphism (i.e., an isomorphism from A onto itself). The next lemma characterizes finite cores. It also associates a unique core with every finite structure. (In the graph theory context, these results go back to [18].) Lemma 2.8. Let A be a finite structure and let X A. retr 12

13 1. A is a core over X if, and only if, it has no proper retract over X (i.e., A retr B = A = B or X B). 2. A has a retract which is a core over X. Moreover, if A retr B 1 and A retr B 2 such that both B 1 and B 2 are cores over X, then B 1 =X B 2. For every finite set X, we fix some set C X of finite cores over X containing exactly one representative from each = X -equivalence class of finite structures. 4 Since C X contains only finite structures, each which is unique up to isomorphism over X, it follows that that C X is a countably infinite set. We will call members of C X canonical cores over X. Corollary 2.9. For every finite structure A and X A, there exists a unique C C X such that A X C. Moreover, td X (C) td X (A) and every homomorphism h : C X A is injective and has the property that A retr h(c). We call C the (canonical) core of A over X and denoted it by Core X (A). For the special case where X =, we write C instead of C and Core(A) instead of Core (A). The homomorphism lattice The relation X of homomorphism over X clearly partially orders the set C X of canonical cores over X. (This is tantamount to the statement that X quasi-orders the X -equivalence classes of finite structures over X.) Moreover, the poset (C X, X ) is easily seen to be a lattice: Lemma 2.3(3,4), together with the observation that both A X B and A X B are finite whenever A and B are finite, implies that Core X (C 1 X C 2 ) and Core X (C 1 X C 2 ) are respectively the least upper bound and greatest lower bound of canonical cores C 1, C 2 C X. (C X, X ) is known as the homomorphism lattice. The homomorphism lattice is well-studied in combinatorics, with interesting results on a variety of phenomena such as density, gaps and maximal antichains [19]. 2.5 Back-and-forth equivalence Recall that the quantifier-count of a formula is the overall number of quantifiers it contains, while its quantifier-rank is the maximal nesting depth of quantifiers. Formally, these are defined by induction: qcount(φ) = qrank(φ) = 0 if φ is atomic, qcount( φ) = qcount(φ), qrank( φ) = qrank(φ), qcount(φ ψ) = qcount(φ ψ) = qcount(φ) + qcount(ψ), qrank(φ ψ) = qrank(φ ψ) = max{qrank(φ), qrank(ψ)}, qcount( x φ) = qcount( x φ) = 1 + qcount(φ), qrank( x φ) = qrank( x φ) = 1 + qrank(φ). Quantifier-count is obviously a lower bound on the length of a formula. 4 We can rely on the axiom of choice for this purpose. There are, however, ways to select canonical representative structures that avoid the axiom of choice. 13

14 For structures A and B and n N, we write A n B and say that A and B are n-backand-forth equivalent if they satisfy exactly the same first-order sentences of quantifier-rank n. The sequence of equivalence relations 0, 1, 2,..., each one a refinement of the previous, measures the extent to which A and B look alike from the standpoint of first-order logic. Remark Another name for n is elementary equivalent up to quantifier-rank n. (Our nonstandard choice of terminology n-back-and-forth equivalence refers to the combinatorial interpretation of n.) Structures A and B are said to be elementarily equivalent if A n B for every n N, that is, if no first-order sentence (of any quantifier-rank) distinguishes between A and B. Elementary equivalence is a highly nontrivial equivalence on infinite structures: a large part of classical model theory studies the variety of non-isomorphic structures within a given elementary equivalence class. On finite structures, the story is much simpler: elementary equivalence implies isomorphism. In fact, it is easy to show that A min( A, B )+1 B A = B for all finite structures A and B. Remark It is well-known folklore that for every n N, there are only finitely many n - equivalence classes of structures (although the exact number depends on the finite relational vocabulary σ). A related fact is that there are only finitely many first-order sentences of quantifier-rank n up to logical equivalence. There are a number of useful combinatorial characterizations of n (e.g., Ehrenfeucht-Fraïssé games). The notion of an n-back-and-forth system of partial isomorphisms, due to Fraïssé, is the most convenient for us. Recall that a partial isomorphism from A to B is a partial function π from A to B which restricts to an isomorphism from A Dom(π) to B Range(π). Definition For n N, an n-back-and-forth system on structures A and B is a sequence Π 0 Π 1 Π n of sets Π i of partial isomorphisms from A to B such that for every i < n and π Π i, (forth) a A π a Π i+1 s.t. π a extends π and a Dom(π a ), (back) b B π b Π i+1 s.t. π b extends π and b Range(π b ). Lemma Structures A and B are n-back-and-forth equivalent if, and only if, there exists an n-back-and-forth system on A and B. Proof of this fundamental lemma can be found in any model theory text [11, 20, 23]. Remark Extending n to a relation between structures with distinguished tuples (of the same arity), a simple inductive characterization emerges: (A, a) 0 (B, b) if, and only if, a b is a valid partial isomorphism from A to B. For n 1, (A, a) n (B, b) if, and only if, α A β B (A, aα) n 1 (B, bβ) and β B α A (A, aα) n 1 (B, bβ). 14

15 3 Primitive-positive and existential-positive sentences In this section, we take a look at a well-known correspondence between primitive-positive sentences and (cores of) finite structures (originally described by Chandra and Merlin in the database context [8]). This correspondence extends to existential-positive sentences and finite antichains in the homomorphism lattice (C, ). An important feature of this correspondence, which seems to have been overlooked in the past, is a connection between quantifier-rank on one side and tree-depth on the other side. Recall that existential-positive formulas are built out of atomic formulas using only conjunction, disjunction and existential quantification. Primitive-positive formulas are precisely the existential-positive formulas containing no disjunctions. Lemma 3.1. Every existential-positive formula ψ is logically equivalent to a finite disjunction θ 1 θ m of primitive-positive formulas θ i such that qcount(ψ) max i qcount(θ i ) and qrank(ψ) max i qrank(θ i ). Proof. Repeated apply the following three syntactic transformations to subformulas of ψ, beginning with the innermost subformulas and working outward: φ 1 (φ 2 φ 3 ) (φ 1 φ 2 ) (φ 1 φ 3 ), (φ 1 φ 2 ) φ 3 (φ 1 φ 3 ) (φ 2 φ 3 ), x (φ 1 φ 2 ) ( x φ 1 ) ( x φ 2 ). These syntactic transformations have the effect of pulling disjunctions to the outermost rungs of ψ, while preserving semantics. If ψ was not primitive-positive to start with, then the resulting formula has the form ψ 1 ψ 2 where the total number of disjunctions in ψ 1 and ψ 2 is one less than the number of disjunctions in ψ. Similarly for i = 1, 2, if ψ i is not primitive-positive, then it has the form ψ i,1 ψ i,2. Proceeding to peel away disjunctions in this manner, we eventually reach a frontier of finitely many primitive-positive formulas θ 1,..., θ m, whose total disjunction is logically equivalent to ψ. Note that all three syntactic transformations preserve quantifier-rank; thus qrank(ψ) max i qrank(θ i ) (in fact, this holds with equality). Finally, each transformation transforms the lefthand formula into a disjunction of two formulas, neither of which have more quantifiers than the lefthand formulas. Therefore, we have qcount(ψ) max i qcount(θ i ). Lemma 3.2. For every primitive-positive sentence θ, there is a finite structure A θ such that B = θ A θ B for all structures B. Conversely, for every finite structure A, there is a primitive-positive sentence θ A such that B = θ A A B for all structures B. Moreover, transformations θ A θ and A θ A satisfy A θ qcount(θ), td(a θ ) qrank(θ), qcount(θ A ) = A, qrank(θ A ) = td(a). Proof. We first describe the transformation θ A θ. By renaming variables in θ, we may assume that each variable in θ gets quantified exactly once. It is easy to see that equality subformulas (of the form x = y for variables x and y) can be eliminated from θ in a manner that increases neither quantifier-count nor quantifier-rank. We may thus assume that θ contains no equality subformulas. 15

16 Suppose x 1,..., x m are the variables of θ. We create elements a 1,..., a m and declare the universe of A θ to be the set A θ = {a 1,..., a m }. Relations in A θ are defined by the rule: (a i1,..., a ik ) R A θ if, and only if, Rx i1... x ik occurs as a subformula of θ. It is easy to see that B = θ A θ B and A θ qcount(θ) (we have instead of = since by eliminating equality subformulas, we may have decreased the quantifier-count of θ). To confirm that td(a θ ) qrank(θ), consider the directed arc relation on A θ defined by a i a j if, and only if, the quantification x j lies within the scope of quantification x i in θ. For example, if θ is the sentence ( x 1 ( x 2 Rx 1 x 2 ) ( x 3 Rx 3 x 1 )) ( x 4 Rx 4 x 4 ), we have arcs x 1 x 2 and x 1 x 3. The relation determines a rooted forest F on the set A θ (namely, is the ancestor of relation on F, i.e., the transitive closure of parent of relation ). It is clear that G(A θ ) is a subgraph of the closure F. Therefore, we have td(a θ ) height(f) = qrank(θ). In the other direction, we define the transformation A θ A. This time let a 1,..., a m enumerate the elements of A, and let x 1,..., x m be a corresponding sequence of variables. By definition of tree-depth, there exists a rooted forest F of height td(a) such that G(A) is a subgraph of F. We now define primitive-positive formulas θ i for all i {1,..., m} by the following induction. For every i such that a i is a leaf in F, let θ i be the conjunction of all atomic formulas Rx j1... x jk such that (a j1,..., a jk ) R A and a j1,..., a jk are contained in the unique branch from a i to a root. For i such that a i is not a leaf, we define θ i inductively as x i ψ i where ψ i is the conjunction of formulas θ j for all j such that a i a j (i.e., a j is a child of a i ). Notice that θ i is a primitive-positive sentence (with no free variables) whenever a i is a root in F. Let θ A be the conjunction of sentences θ i for all i such that a i is a root in F. It is easy to see that A B B = θ A for all B and furthermore qcount(θ A ) = A and qrank(θ A ) = td(a). Corollary 3.3. For every primitive-positive sentence θ, there is a unique canonical core Core(θ) C such that B = θ Core(θ) B for all structures B. Moreover, it holds that Core(θ) qcount(θ) and td(core(θ)) qrank(θ). Proof. Let Core(θ) = Core(A θ ) where A θ is (as in Lemma 3.2) a finite structure such that A θ B B = θ for all structures B. To prove uniqueness, suppose that C C also satisfies B = θ Core(θ) B for all B. Since both Core(θ) = θ and C = θ (by virtue of the fact that Core(θ) Core(θ) and C C), it follows that Core(θ) C and C Core(θ). Since members of C are unique up to homomorphic equivalence, we have Core(θ) = C. We say that a first-order sentence φ is preserved under products if A B = φ whenever A = φ and B = φ. Corollary 3.4. Every primitive-positive sentence is closed under products. Proof. Let θ be a primitive-positive sentence and suppose A = θ and B = θ. Then Core(θ) A and Core(θ) B. It follows that Core(θ) A B by Lemma 2.3(4), and so we have A B = θ. Therefore, θ is closed under products. We now state a key proposition about existential-positive sentences. Proposition 3.5. Let ψ be an existential-positive sentence. There exists a unique finite set {C 1,..., C n } of canonical cores C i C such that 16

17 C 1,..., C n are homomorphically incomparable (i.e., C i C j for all i j), and A = ψ n i=1 ( Ci A ) for all structures A. Moreover, it holds that qcount(ψ) max i C i and qrank(ψ) max i td(c i ). Proof. By Lemma 3.1, there exist primitive-positive sentence θ 1,..., θ M such that ψ is logically equivalent to θ 1 θ M and qcount(ψ) max i qcount(θ i ) and qrank(ψ) max i qrank(θ i ). Consider the canonical cores Core(θ 1 ),..., Core(θ M ). Without loss of generality, there exists m M such that Core(θ 1 ),..., Core(θ m ) are homomorphically incomparable, and for all j {m+1,..., M}, there exists i {1,..., m} such that Core(θ i ) Core(θ j ). For all structures A, we have M M A = ψ A = θ i i=1 i=1 ( ) Core(θ i ) A m i=1 ( ) Core(θ i ) A. We now prove uniqueness of the set {Core(θ 1 ),..., Core(θ m )}. Suppose {C 1,..., C n } is another set of homomorphically incomparable canonical cores such that A = ψ A = n ( j=1 Cj A ). Since Core(θ i ) = ψ for all i {1,..., m} and C j = ψ for all j {1,..., n}, we have i j (i) C j (i) Core(θ i ) and j i (j) Core(θ i (j)) C j. We claim that j = j (i (j)) for every j {1,..., n}. If not, then there exists j such that j = j (i (j)). But then C j (i (j)) Core(θ i (j)) C j, which contradicts the assumption that C 1,..., C n are homomorphically incomparable. By a similar argument, we have i = i (j (i)) for every i {1,..., m}. This implies that i is a bijection from {1,..., n} to {1,..., m}. It follows that C j Core(θ i (j)) C j for every j {1,..., n}; hence C j = Core(θ i (j)) since homomorphically equivalent canonical cores are identical. Therefore, sets {Core(θ 1 ),..., Core(θ m )} and {C 1,..., C n } are the same. Finally, Lemma 3.1 and Corollary 3.3 imply qcount(ψ) max i qrank(ψ) max i qcount(θ i ) max Core(θ i ), i td(core(θ i )). qrank(θ i ) max i Definition 3.6. Canonical cores C 1,..., C n C of Proposition 3.5 are called the characteristic cores of existential-positive sentence ψ. Note that an existential-positive sentence is logically equivalent to a primitive-positive sentence if, and only if, it has exactly one characteristic core. The following lemma is the converse of Proposition 3.5 and shows that the correspondence between existential-positive sentences (up to logical equivalence) and finite antichains in the homomorphism lattice (C, ) is really a bijection. Lemma 3.7. For every finite antichain {C 1,..., C n } C, there exists an existential-positive sentence ψ such that C 1,..., C n are precisely the characteristic cores of ψ and qrank(ψ) = max i (C i ). 17

18 Proof. Let ψ be the disjunction n i=1 θ C i where primitive-positive sentence θ Ci is as defined in Lemma 3.2. The next lemma gives an alternative description of the characteristic cores of an existentialpositive sentence. Lemma 3.8. A canonical core C C is a characteristic core of an existential-positive sentence ψ if, and only if, C = ψ and C A for every structure A such that A = ψ and A C. Proof. Let ψ be an existential-positive sentence. Suppose C C is a characteristic core of ψ. Clearly C = ψ. Suppose A is any structure such that A = ψ and A C. We must show that C A. Because A = ψ, there is some characteristic core D of ψ such that D A. Thus, we have D C. But now C and D are homomorphically comparable characteristic cores of ψ, so they must be the same. Therefore, we have C = D A as required. Now suppose C C is any canonical core such that C = ψ and C A for every structure A such that A = ψ and A C. We claim that C is a characteristic core of ψ. Since C = ψ, it follows that D C for some characteristic core of ψ. As D = ψ and D C, we have C D. Thus, C and D are homomorphically equivalent. But homomorphically equivalent canonical cores are equal, so C = D. Therefore, C is a characteristic core of ψ. One half of the homomorphism preservation theorem [on finite structures] follows immediately from Proposition 3.5. Corollary 3.9. Every existential-positive sentence is preserved under homomorphisms. Proof. Let θ be an existential-positive sentence. Suppose A and B are structures such that A = θ and A B. Because A = θ, we have C A for some characteristic core C of θ. But then C B (since C A and A C), and so B = θ. Thus, θ is preserved under homomorphisms. Remark Before concluding this section, we take a moment to discuss a key respect in which our approach of studying characteristic cores differs from the approach of articles [3, 6], which also deal with homomorphism-preserved first-order properties. Let K be any class of finite structures which is closed under homomorphisms. The notion of characteristic cores has a natural extension such a class K: we define characteristic cores of K as the canonical cores in the set C K which are minimal under the homomorphism order. By the results of this section, it is clear that K is definable by an existential-positive sentence if, and only if, it has finitely many characteristic cores. We now mention an alternative way to establish that K is existential-positively definable. We say a structure A K is minimal if B / K for every proper substructure B of A. In other words, minimal simply means minimal under the substructure order, rather than the homomorphism order. It is easy to see that every minimal structure of K (which, remember, is closed under homomorphisms in the finite) is a core. It is also clear that K is definable in the finite by an existential-positive sentence if, and only if, it has finitely many minimal cores up to isomorphism. Thus, instead of characteristic cores, one may instead attempt to show that K has finitely many minimal cores up to isomorphism, in order to prove that K is definable by an existential-positive sentence. Indeed, this is the approach taken by [3, 6]. When studying existential-positive definability on restricted subclasses of finite structures, this approach leads 18