Limits of Schema Mappings

Summary of Results

Our contributions are both conceptual and technical. At the conceptual level, we develop a framework for studying sequences of schema mappings by first introducing a natural notion of distance on the powerset \(\mathcal {P}(\text {Inst}(\mathbf {T}))\) of the set Inst(T) of finite instances over a schema T. Intuitively, this notion of distance expresses how “close” two sets of finite T-instances are as regards the certain answers of conjunctive queries on these sets. The pair \((\mathcal {P}(\text {Inst}(\mathbf {T})),dist)\) is a pseudometric space, which means that the distance function d i s t(⋅,⋅) is symmetric and obeys the triangle inequality, but different sets of finite target instances may have distance zero; however, two such sets have distance zero if and only if they are C Q-equivalent, i.e., every conjunctive query has the same certain answers on these two sets. Thus, we will also work with the metric space obtained by considering the C Q-equivalence classes of members of \(\mathcal {P}(\text {Inst}(\mathbf {T}))\), and will use the same notation for it.

Sequences of functions from some set to a metric space occupy a central place in the study of metric spaces (see, e.g., [18]). In particular, there are natural notions of a pointwise limit and of a uniform limit of a sequence (f _n ) _{n ≥ 1} of functions from some set to a metric space; moreover, there are companion notions of a pointwise Cauchy and of a uniformly Cauchy sequence of such functions. We now describe briefly how these notions can be applied to sequences of schema mappings. In its most general formulation, a schema mapping \(\mathcal {M}\) over a source schema S and a target schema T is a set of pairs (I, J), where I is a finite S-instance and J is a finite T-instance. It follows that a schema mapping \(\mathcal {M}\) can be also be viewed as a function f from the set Inst(S) of all finite S-instances to the powerset \(\mathcal {P}(\text {Inst}(\mathbf {T}))\) of the set of all finite T-instances, where \(f(I) =\{J: (I,J) \in \mathcal {M}\}\). This way, a sequence \((\mathcal {M}_{n})_{n\geq 1}\) of schema mappings over a source schema S and a target schema T can be viewed as a sequence of functions from Inst(S) to the (pseudo)metric space \((\mathcal {P}(\text {Inst}(\mathbf {T})),dist)\).

After the conceptual framework has been laid out, we study in depth the limiting behavior of sequences of GAV mappings and the convergence of sequences of LAV mappings. We establish a number of technical results that reveal rather dramatic and perhaps unanticipated differences between GAV schema mappings and LAV schema mappings.

For sequences of GAV mappings, we point out that every uniformly Cauchy sequence of GAV mappings is eventually constant, hence it has a GAV mapping as uniform limit. We also show that every pointwise Cauchy sequence of GAV mappings has a pointwise limit, but it need not have a uniform limit; moreover, there are pointwise Cauchy sequences of GAV mappings such that no GAV mapping is their pointwise limit. This raises the question as to when a sequence of GAV mappings has a GAV mapping as a pointwise limit. We prove that a sequence of GAV mappings has a GAV mapping as a pointwise limit if and only if it has a pointwise limit that allows for C Q-rewriting¹.

For sequences of LAV mappings, we show that the notions of uniform limit and pointwise limit coincide; moreover, the same holds true for the notions of uniformly Cauchy and pointwise Cauchy sequences. However, there are uniformly Cauchy sequences of LAV mappings that have no uniform limit. We also establish that a uniformly Cauchy sequence of LAV mappings has a LAV mapping as a uniform limit if and only if it has a uniform limit that admits universal solutions. The aforementioned results lift to sequences of premise-bounded sequences of GLAV mappings, i.e., sequences of GLAV mappings for which there is a k ≥ 1 such that, for every mapping in the sequence, the left-hand side of every GLAV constraint has at most k source atoms (LAV mappings have k = 1).

In terms of techniques, we use systematically the structural characterizations of schema-mapping languages established in [19], thus creating a link with a different line of research.

The metric space \((\mathcal {P}(\text {Inst}(\mathbf {T})),dist)\) is incomplete, i.e., there are Cauchy sequences of elements of \(\mathcal {P}(\text {Inst}(\mathbf {T}))\) that have no limit in \(\mathcal {P}(\text {Inst}(\mathbf {T}))\). It is well known that every incomplete metric space (X, d) has a completion, which means that it can be embedded into a complete metric space (X ^∗, d ^∗) so that X is a dense subset of X ^∗. Moreover, pointwise (respectively, uniformly) Cauchy sequences of functions on X have pointwise (respectively, uniform) limits that take values in X ^∗. The construction of X ^∗ from X involves equivalence classes of Cauchy sequences of elements of X, thus, in general, the members of X ^∗ do not have a concrete representation. In the last part of the paper, we show that the members of \(\mathcal {P}(\text {Inst}(\mathbf {T}))^{*}\) can be represented by suitably constructed infinite T-instances. As a consequence of this, the pointwise (respectively, uniform) limits of Cauchy sequences of schema mappings can be represented by generalized schema mappings, i.e., schema mappings that allow for infinite target instances as solutions to finite source instances.

Schemas, Instances, and Conjunctive Queries

A schema R is a finite sequence 〈R ₁,…, R _k 〉 of relation symbols, where each R _i has a fixed arity. An instance I over R, or an R -instance, is a sequence (R1I,…, R k I), where each \({R^{I}_{i}}\) is a finite relation of the same arity as R _i . We will often use R _i to denote both the relation symbol and the relation \({R^{I}_{i}}\) that interprets it. The active domain a d o m(I) of an instance I is the set of all values occurring in the relations of I. A fact of an instance I (over R) is an expression \({R_{i}^{I}}(t_{1}, \ldots , t_{m})\) (or simply R _i (t ₁,…, t _m )), where R _i is a relation symbol of R and \((t_{1},\ldots ,t_{m}) \in {R_{i}^{I}}\).

A conjunctive query is a first-order formula of the form ∃z 𝜃(x, z), where 𝜃(x, z) is a conjunction of atomic formulas R _i (v ₁,..., v _m ) and each v _j is one of the variables in x and z. A boolean conjunctive query is a conjunctive query with no free variables. We write C Q for the class of all conjunctive queries over some schema. For every n ≥ 1, we let C Q _n denote the class of all conjunctive queries with at most n variables. We also let C Q ₀ denote the singleton consisting of a trivially true query. If I is an instance and q is a conjunctive query, then we write q(I) for the result of evaluating q on I; in particular, for boolean conjunctive queries q we have that q(I) = t r u e if and only if I satisfies q.

Schema Mappings, Universal Solutions, Certain Answers

Motivated by the terminology in data exchange [6], we typically work with two schemas, a source schema S and a target schema T with no relation symbols in common. We refer to S-instances as source instances, and to T-instances as target instances. We assume that the values occurring in the active domains of instances come from two fixed countably infinite disjoint sets, the set Const of all constants and the set Null of (labeled) nulls. We also assume that the active domains of source instances consist entirely of constants; the active domains of target instances may contain both constants and nulls.

In its most general form, a schema mapping \(\mathcal {M}\) between a source schema S and a target schema T is a set of pairs (I, J), where I is source instance and J a target instance. To avoid anomalies that arise from such a relaxed notion, we will assume that a schema mapping \(\mathcal {M}\) must also possess a mild closure property, namely, that \(\mathcal {M}\) is closed under isomorphisms that rename nulls by other nulls. This is a natural “genericity” condition that is akin to the condition that database queries are closed under arbitrary isomorphisms. The precise definitions are as follows.

Definition 1

Proposition 1

Assume that \(\mathcal {M}\) is a schema mapping, I is a source instance, and J is a universal solution for I w.r.t. \(\mathcal {M}\).If q is a conjunctive query, then \(cert(q,I,\mathcal {M}) = q(J)_{\downarrow }\) , where q(J) _↓ is the set of all null-free tuples in q(J).

Proof 1

First, assume that \(\textbf {t} \in cert(q,I,\mathcal {M})\). Then, as discussed earlier, t must be a null-free tuple. Since J is a solution for I w.r.t. \(\mathcal {M}\), we have that t ∈ q(J), hence we have that t ∈ q(J) ↓. Next, assume that t is a null-free tuple in q(J). If J ^′ is an arbitrary solution for I w.r.t. \(\mathcal {M}\), then, since J is a universal solution for I w.r.t. I, there is a homomorphism h from J to J ^′. Since conjunctive queries are preserved under homomorphisms, it follows that h(t) = t ∈ q(J ^′). Thus, \(\textbf {t} \in cert(q,I,\mathcal {M})\). □

Structural Properties of Schema Mappings

We now present a number of structural properties that a schema mapping may or may not possess. These properties were investigated in their own right in [19], where they were used to obtain characterizations of schema-mapping languages that will be of great interest to us in this paper.

Schema Mapping Languages

A GLAV (global-and-local-as-view) constraint is a first-order formula of the form ∀x(φ(x) →∃y ψ(x, y)), where φ(x) is a conjunction of atoms over the source schema S, each variable in x occurs in at least one atom in φ(x), and ψ(x, y) is a conjunction of atoms over the target schema T with variables in x and y. We refer to φ(x) as the left-hand side, or premise, and ∃y ψ(x, y) as the right-hand side, or conclusion of the constraint. Another name for GLAV constraints is source-to-target tuple-generating dependencies or, in short, s-t tgds.

A LAV (local-as-view) constraint is a GLAV constraint whose left-hand side is a single atom over the source, while a GAV (global-as-view) constraint is a GLAV constraint whose right-hand side contains no existential quantifiers and consists of a single atom over the target. For example, ∀x, y(E(x, y) →∃z(F(x, z) ∧ F(z, y))) is a LAV constraint, and ∀x, y, z(E(x, z) ∧ E(z, y) → F(x, y)) is a GAV constraint.

A GLAV (global-and-local-as-view) mapping is a schema mapping \(\mathcal {M}=(\mathbf {S}, \mathbf {T}, {\Sigma })\) such that Σ is a finite set of GLAV constraints. The notions of a LAV mapping and of a GAV mapping are defined analogously.

Every GLAV mapping \(\mathcal {M}\) admits universal solutions [6]; furthermore, given a source instance I, a canonical universal solution \(chase(I,\mathcal {M})\) can be produced via the oblivious chase procedure as follows: whenever the antecedent of an s-t tgd in \(\mathcal {M}\) becomes true, fresh null values are introduced and facts involving these nulls are added to \(chase(I,\mathcal {M})\), so that the conclusion of the s-t tgd becomes true. Every GLAV mapping is also known to allow for C Q-rewriting and to be n-modular, for some n ≥ 1. Moreover, every LAV mapping is closed under unions, while every GAV mapping is closed under target intersections.

Every SO tgd allows for C Q-rewriting and admits universal solutions; however, an SO tgd may not be closed under target homomorphisms and there may not exist any n ≥ 1 such that the SO tgd is n-modular (see [8, 19]).

Pseudometric Spaces and Metric Spaces

A pseudometric space is a pair (X, d), where X is a set and d is a function from X × X to the set R ⁺ of non-negative real numbers with the following properties: (i) d(x, x) = 0, for every x in X; (ii) d(x, y) = d(y, x), for every x and y in X; (iii) d(x, y) ≤ d(x, z) + d(y, z), for every x, y, z in X (triangle inequality). A metric space is a pseudometric space (X, d) such that if d(x, y) = 0, then x = y. It is easy to see that if (X, d) is a pseudometric space, then the relation R _d = {(x, y) ∈ X × X∣d(x, y) = 0} is an equivalence relation on X. From this, it follows that every pseudometric space (X, d) gives rise to a metric space \((\widehat {X},\widehat {d})\), where \(\widehat {X}\) is the set of equivalence classes of elements of X modulo the equivalence relation R _d and \(\widehat {d}([x],[y]) = d(x,y)\).

Let (X, d) be a pseudometric space. A sequence of elements x ₁, x ₂,… of X converges to an element x of X, denoted by \(\lim \limits _{n\to \infty } x_{n} = x\), if for every 𝜖 > 0, there is an integer n ₀ such that d(x _n , x) < 𝜖, for every n ≥ n ₀. We say that x is a limit of this sequence. The limit is unique if (X, d) is a metric space. A sequence x ₁, x ₂,… of elements of X is Cauchy if for every 𝜖 > 0, there is an integer n ₀ such that \(d(x_{n},x_{n^{\prime }}) < \epsilon \), for every n, n ^′≥ n ₀.

Using the triangle inequality, it is easy to see that if a sequence of elements in a (pseudo)metric space has a limit, then the sequence is Cauchy. The converse, however, does not hold for arbitrary (pseudo)metric spaces. A (pseudo)metric space (X, d) is complete if every Cauchy sequence of elements of X has a limit in X; otherwise, it is incomplete.

It is well known that every incomplete (pseudo)metric space (X, d) can be embedded into a complete (pseudo)metric space (X ^∗, d ^∗), called the completion of (X, d), in such a way that X is a dense subset of X ^∗, i.e., every member of X ^∗ is the limit of a sequence of members of X. The members of X ^∗ are equivalence classes of Cauchy sequences of X, where two Cauchy sequences x ₁, x ₂,... and y ₁, y ₂,… of elements of X are equivalent if \(\lim \limits _{n\to \infty } d(x_{n},y_{n}) = 0\), while the distance function d ^∗ is defined as \(d^{*}([x_{1},x_{2},\ldots ],[y_{1},y_{2},\ldots ]) = \lim \limits _{n\to \infty } d(x_{n},y_{n})\). The proof of correctness of this construction can be found in [18] or any other book on metric spaces.

As a concrete example, the metric space of the real numbers is the completion of the metric space of the rational numbers (both with the standard distance).

Definition 2

Definition 3

Definition 4

Let T be a schema. If J is a T-instance, then we write v(J) to denote the member of \(\mathcal {P}(\text {Inst}(\mathbf {T}))\) consisting of all isomorphic copies of J via isomorphisms that rename nulls. In other words, v(J) consists of all T-instances J ^′ such that J ^′ is isomorphic to J via an isomorphism h that maps each constant to itself and maps each null to a (possibly different) null.

Lemma 1

Proof 2

For the first two parts of the lemma, let J be a T-instance whose active domain consists entirely of nulls. For every non-boolean query q in C Q _k , we have that c e r t(q, v(J)) = ∅, because v(J) contains instances with disjoint active domains. For every boolean query q, we have c e r t(q, v(J)) = q(J) for the following reason: first, J is a member of v(J), so if c e r t(q, v(J)) = t r u e, then q(J) = t r u e as well; second, since every member of v(J) is isomorphic to J and since boolean conjunctive queries are preserved under isomorphisms, we have that if q(J) = t r u e, then c e r t(q, v(J)) = t r u e.

For the third part of the lemma, let J and J ^′ be T-instances whose active domains consist entirely of nulls and let k be a positive integer. If \(v(J) \equiv _{\mathsf {CQ}_{k}} v(J^{\prime })\), then J and J ^′ must satisfy the same boolean conjunctive queries in C Q _k because J ∈ v(J) and J ^′∈ v(J ^′). For the converse, assume that J and J ^′ satisfy the same boolean conjunctive queries in C Q _k . We have to show that c e r t(q, v(J)) = c e r t(q, v(J ^′)), for every conjunctive query q in C Q _k . If q is a non-boolean conjunctive query in C Q _k , then, by the first part of the lemma, we have that c e r t(q, v(J)) = ∅ = c e r t(q, v(J ^′)). If q is a boolean query in C Q _k , then, by the second part of the lemma and the hypothesis about J and J ^′, we have that c e r t(q, v(J)) = q(J) = q(J ^′) = c e r t(q, v(J ^′)). □

Example 1

Let T be a schema consisting of a single binary relation E and let C _m be the undirected cycle of length m, m ≥ 1, where the vertices of the cycle are pairwise distinct labeled nulls. Consider the sequence (v(C _2n+1)) _{n ≥ 1} arising from the cycles of odd size. Then, for every m ≥ 1, we have that \(\lim \limits _{n\to \infty }v(C_{2n+1}) = v(C_{2m})\). In particular, \(\lim \limits _{n\to \infty }v(C_{2n+1}) = v(C_{2})\).

We first show that v(C _2m ) ≡ _{C Q} v(C ₂), for every m ≥ 1. By Lemma 1, it suffices to show that C _2m and C ₂ satisfy the same boolean conjunctive queries. This is true because C _2m and C ₂ are homomorphically equivalent (and boolean conjunctive queries are preserved under homomorphisms). Indeed, there is a homomorphism from C ₂ to C _2m because C ₂ is a subgraph of C _2m , and there is a homomorphism from C _2m to C ₂ because C _2m is 2-colorable.

We will show that \(\lim \limits _{n\to \infty }v(C_{2n+1}) = v(C_{2})\) by showing that for every k, there exists n ₀ such that for all n ≥ n ₀, we have that \(v(C_{2n+1}) \equiv _{\mathsf {CQ}_{k}} v(C_{2})\). For this, we take n ₀ = k and show that if n ≥ k, then \(v(C_{2n+1}) \equiv _{\mathsf {CQ}_{k}} v(C_{2})\). By the third part of Lemma 1, it suffices to show if q is a boolean conjunctive query in C Q _k , then q(C _2n+1) = q(C ₂). Since C ₂ is a subgraph of C _2n+1, we have that if q(C ₂) = t r u e, then also q(C _2n+1) = t r u e. Assume that q(C _2n+1) = t r u e. Since q ∈C Q _k , there is a subgraph H of C _2n+1 with at most k distinct nodes such that q(H) = t r u e. Since 2n + 1 > n ≥ k, we have that H is a proper subgraph of C _2n+1. Consequently, H is 2-colorable and so there is a homomorphism from H to C ₂, which, in turn, implies that q(C ₂) = t r u e.

Proposition 2

Let T be a schema consisting of a single binary relation E and let K _n be the clique of size n, for n ≥ 1, where the vertices are pairwise distinct labeled nulls. The sequence (v(K _n )) _{n ≥ 1} is Cauchy, but has no limit in \(\mathcal {P}(\text {Inst}(\mathbf {T}))\).

Proof 3

The sequence (v(K _n )) _{n ≥ 1} is Cauchy, because if m ≥ n, then v(K _m ) and v(K _n ) satisfy the same conjunctive queries in C Q _n . To show this, by the third part of Lemma 1, it suffices to show that if m ≥ n, then K _m and K _n satisfy the same boolean conjunctive queries in C Q _n . Let q be a boolean conjunctive query in C Q _n . Since K _n is a subgraph of K _m , if q(K _n ) = t r u e, then q(K _m ) = t r u e. Conversely, if q(K _m ) = t r u e, then there is a subgraph H of K _m with at most n distinct nodes such that q(H) = t r u e. But then H is also a subgraph of K _n , hence q(K _m ) = t r u e.

It remains to show that the sequence (v(K _n )) _{n ≥ 1} has no limit in \(\mathcal {P}(\text {Inst}(\mathbf {T}))\). Assume to the contrary that there does exist a set \(\mathcal {J}\) of finite instances over T such that \(\lim \limits _{n\to \infty } v(K_{n})= \mathcal {J}\). We distinguish three cases.

First, if \(\mathcal {J}=\emptyset \), then \(cert(q,\mathcal {J})= {\mathit {true}}\), for every conjunctive query q. In particular, this holds for the query q = ∃x E(x, x), which asserts the existence of a self-loop. In contrast, for this conjunctive query, we have that c e r t(q, v(K _n )) = f a l s e, for every n ≥ 1, since K _n ∈ v(K _n ) and none of the graphs K _n , n ≥ 1 contains a self-loop.

Second, if \(\mathcal {J} \neq \emptyset \) and if every member J of \(\mathcal {J}\) contains a self-loop, then we again consider the query q = ∃x E(x, x). We thus have \(cert(q, \mathcal {J}) = {\mathit {true}}\), whereas c e r t(q, v(K _n )) = f a l s e, for every n ≥ 1.

Definition 5

Proposition 3

Consider a sequence \((\mathcal {M}_{n})_{n\geq 1}\) of schema mappings and a schema mapping \(\mathcal {M}\).Then \(\lim \limits _{n\to \infty }^{u} \mathcal {M}_{n} = \mathcal {M}\) if and only if for every integer k ≥ 1, there is an integer n ₀ ≥ 1such that for all integers n ≥ n ₀ , we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}_{k}} \mathcal {M}\).

Proof 4

The result follows by unfolding and comparing the definitions. Specifically, \(\lim \limits _{n\to \infty }^{u} \mathcal {M}_{n} = \mathcal {M}\) means that for every 𝜖 > 0, there is an integer n ₀ ≥ 1 such that for every integer n ≥ n ₀ and for every source instance I we have that \(dist(\text {Sol}(I,\mathcal {M}_{n}), \text {Sol}(I,\mathcal {M}))< \epsilon \). In turn, this means that for every integer k ≥ 1, there is an integer n ₀ ≥ 1 such that for every integer n ≥ n ₀ and for every source instance I we have that \(\text {Sol}(I,M_{n}) \equiv _{\mathsf {CQ}_{k}} \text {Sol}(I,\mathcal {M})\). Thus, for every integer k ≥ 1, there is an integer n ₀ ≥ 1 such that for every integer n ≥ n ₀, we have that \(\mathcal {M}_{n}\equiv _{\mathsf {CQ}_{k}}\mathcal {M}\). □

Theorem 1

(implicit in[5]) Every SO tgd is a uniform limit of a sequence of GLAV mappings.

Theorem 2

Proof 5

We consider GAV mappings over a source schema S and a target schema T. Let r denote the maximum arity of the relation symbols in T. For showing the first claim, assume that \((\mathcal {M}_{n})_{n\geq 1}\) is a pointwise Cauchy sequence of schema mappings and let I be a source instance. For each n ≥ 1, consider the universal solution \(chase(I,\mathcal {M}_{n})\) for I w.r.t. \(\mathcal {M}_{n}\) obtained by using the oblivious chase procedure. Since each \(\mathcal {M}_{n}\) is a GAV schema mapping, we have that \(chase(I,\mathcal {M}_{n})\) contains constants from the active domain of I and no nulls. We claim that there exists some n ₀ such that for all n ≥ n ₀, we have that \(chase(I,\mathcal {M}_{n}) = chase(I, \mathcal {M}_{n_{0}})\). In other words, we claim that the sequence \((chase(I,\mathcal {M}_{n}))_{n\geq 1}\) is eventually constant (does not oscillate). Since every instance in the sequence \((chase(I,\mathcal {M}_{n}))_{n\geq 1}\) has no nulls, it can be identified by evaluating on that instance the atomic queries R(x ₁,…, x _k ), where R ranges over the relation symbols of T and k (with k ≤ r) denotes the arity of R. The assumption that the sequence \((chase(I,\mathcal {M}_{n}))_{n \geq 1}\) is pointwise Cauchy implies that there exists a positive integer n ₀ (that depends on I and r) such that for every integer n ≥ n ₀ and every conjunctive query q ∈C Q _r , we have that \(cert(q,I,\mathcal {M}_{n}) = cert(q,I,\mathcal {M})\). This implies that \(q(chase(I,\mathcal {M}_{n})) = q(chase(I,\mathcal {M}_{n_{0}}))\) and, consequently, for every n ≥ n ₀, we have that \(chase(I,\mathcal {M}_{n})) = chase(I,\mathcal {M}_{n_{0}})\).

We have Just shown that if \((\mathcal {M}_{n})_{n\geq 1}\) is a pointwise Cauchy sequence of GAV mappings, then for every I, there exists a positive integer m _I such that \(chase(I,\mathcal {M}_{m_{I}}) = chase(I,\mathcal {M}_{n})\), for all n ≥ m _I . It follows that the schema mapping \(\mathcal {M} = \{(I, chase(I,\mathcal {M}_{m_{I}}))\mid I \text { is a source instance}\}\) is a pointwise limit of the sequence \((\mathcal {M}_{n})_{n\geq 1}\). Note that \(\mathcal {M}\) is indeed a schema mapping because \(chase(I,\mathcal {M}_{m_{I}})\) contains no nulls.

For showing the second claim, assume that \((\mathcal {M}_{n})_{n\geq 1}\) is a uniformly Cauchy sequence of GAV mappings. We claim that \((\mathcal {M}_{n})_{n\geq 1}\) is eventually constant, i.e., there is some n ₀ such that for all n ≥ n ₀, \(\mathcal {M}_{n}\equiv _{\mathsf {CQ}} \mathcal {M}_{n_{0}}\) holds. For this, we repeat the previous argument, but also note that, since the sequence \((\mathcal {M}_{n})_{n\geq 1}\) is uniformly Cauchy, there exists a positive integer n ₀ that depends only on r such that for every source instance I, for every integer n ≥ n ₀ and every conjunctive query q ∈C Q _r , we have that \(cert(q,I,\mathcal {M}_{n}) = cert(q,I,\mathcal {M})\). This implies that for every source instance I and every n ≥ n ₀, we have that \(q(chase(I,\mathcal {M}_{n})) = q(chase(I,\mathcal {M}_{n_{0}}))\); consequently, for every source instance I and every n ≥ n ₀, we have that \(chase(I,\mathcal {M}_{n})) = chase(I,\mathcal {M}_{n_{0}})\). □

Proposition 4

There exists a sequence of GAV mappings that has a GAV mapping as a pointwise limit, but has no uniform limit.

Proof 6

For every n ≥ 2, let \(q_{n} = \bigwedge _{1 \leq i < j \leq n}(E(x_{i},x_{j}) \wedge E(x_{j},x_{i})).\) Intuitively, if E is interpreted as edge relation, then q _n yields a non-empty answer over any graph that contains a self-loop or a clique of size n. Let S be a source schema consisting of a binary relation symbol E and a unary relation symbol P, let T be a target schema consisting of a unary relation symbol P ^′. Let \((\mathcal {M}_{n})_{n\geq 1}\) be the sequence of GAV mappings, where \(\mathcal {M}_{n}\) is specified by the constraint ∀x∀x ₁,…, x _n+1(P(x) ∧ q _n+1 → P ^′(x)). Intuitively, \(\mathcal {M}_{n}\) is a “copy” schema mapping, but the copying action is triggered only if the source instance contains a self-loop or a clique of size n + 1. We will show that the GAV schema mapping \(\mathcal {M} = \{\forall x\forall y(P(x) \wedge E(y,y) \rightarrow P^{\prime }(x))\}\) is a pointwise limit of \((\mathcal {M}_{n})_{n\geq 1}\), but that this pointwise limit is not a uniform limit of \((\mathcal {M}_{n})_{n\geq 1}\) and thus no uniform limit of \((\mathcal {M}_{n})_{n\geq 1}\) exists.

Next, we show that \((\mathcal {M}_{n})_{n\geq 1}\) has no uniform limit. Towards a contradiction, suppose that such a uniform limit exists. Every uniform limit is also a pointwise limit; moreover, pointwise and uniform limits are unique up to C Q-equivalence. Hence, since the schema mapping \(\mathcal {M}\) defined above is a pointwise limit of \((\mathcal {M}_{n})_{n\geq 1}\), it follows that \(\mathcal {M}\) is also a uniform limit of \((\mathcal {M}_{n})_{n\geq 1}\). Let m = 1. Then there exists an n ₀ such that for all n ≥ n ₀ we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}_{1}} \mathcal {M}\). Take n = n ₀. Let I be the source instance K _n+1 ∪{P(c)} and let q be the target conjunctive query ∃x P ^′(x). We now claim that \(cert(q,I,\mathcal {M}_{n}) \neq cert(q,I,\mathcal {M})\), which contradicts the previously derived fact that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}_{1}} \mathcal {M}\). Indeed, since I contains a clique of size n + 1, we have P(c) is a universal solution for I w.r.t. \(\mathcal {M}_{n}\), hence \(cert(q,I,\mathcal {M}_{n}) = \mathit {true}\). However, since I contains no self-loop, we have that ∅ is a universal solution for I w.r.t. \(\mathcal {M}\), hence \(cert(q,I,\mathcal {M}) = \mathit {false}\). □

Proposition 5

There is a pointwise Cauchy sequence of GAV schema mappings such that no SO tgd is a pointwise limit of that sequence.

Proof 7

Theorem 3

Proof 8

Definition 6

Let \((\mathcal {M}_{n})_{n\geq 1}\) be a sequence of schema mappings. We say that \((\mathcal {M}_{n})_{n\geq 1}\) allows for C Q -rewriting if for every target conjunctive query q, there is a union q ^′ of source conjunctive queries having the following property: for every source instance I, there is a positive integer n _I such that \(cert(q,I,\mathcal {M}_{n}) = q^{\prime }(I)\), for every n ≥ n _I .

Corollary 1

Corollary 2

Definition 7

Let \(\mathcal {M}\) be a GLAV mapping and k a positive integer. We call \(\mathcal {M}\) a k-premise-bounded GLAV mapping if the premise of every constraint in \(\mathcal {M}\) has at most k atoms.

Let \((\mathcal {M}_{n})_{n\geq 1}\) be a sequence of GLAV mappings. We say that \((\mathcal {M}_{n})_{n\geq 1}\) is premise-bounded if there exists an integer k such that every element \(\mathcal {M}_{n}\) of \((\mathcal {M}_{n})_{n\geq 1}\) is k-premise bounded.

Theorem 4

Proof 9

We prove the first part and then use it to prove the second part.

Part 1. It is obvious that every uniformly Cauchy sequence of mappings is also pointwise Cauchy. We focus on the reverse direction. Let \((\mathcal {M}_{n})_{n\geq 1}\) be a pointwise Cauchy sequence of premise bounded GLAV mappings. We have to show that for every m, there is an N ₀ such that for all n, n ^′≥ N ₀, we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}_{m}} \mathcal {M}_{n^{\prime }}\).

Fix an integer m. Since \((\mathcal {M}_{n})_{n\geq 1}\) is pointwise Cauchy, for every source instance I, there is an integer n ₀(I) such that for all n, n ^′≥ n ₀(I) and for every conjunctive query q in C Q _m , we have that \(cert(q,I,\mathcal {M}_{n}) = cert(q,I,\mathcal {M}_{n^{\prime }})\). Let p be the number of relation symbols in the target schema, let r be their maximum arity, and let k be the bound on the number of atoms in the premises of the members of the sequence \((\mathcal {M}_{n})_{n\geq 1}\). We write \(\mathcal {I}\) to denote the class of all source instances with at most k ⋅ p ⋅ m ^r atoms. Clearly, up to isomorphism, there are only finitely many instances \(I \in \mathcal {I}\). Moreover, if I ^′≅I ^″, then n ₀(I ^′) = n ₀(I ^″). Consequently, the quantity \(N_{0} = \mathop {max}\{n_{0}(I) \mid I \in \mathcal {I}\}\) is a positive integer. We claim that for all n, n ^′≥ N ₀, we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}_{m}} \mathcal {M}_{n^{\prime }}\).

Let I be an arbitrary source instance and let q be an arbitrary conjunctive query in C Q _m . We have to show that \(cert(q,I, \mathcal {M}_{n}) = cert(q, I, \mathcal {M}_{n^{\prime }})\), for all n, n ^′≥ N ₀. Let a be a tuple of constants such that \(\mathbf {a} \in cert(q, I, \mathcal {M}_{n})\), hence \(\mathbf {a} \in q(chase(I, \mathcal {M}_{n}))_{\downarrow }\). Since the query q has at most m variables, it must consist of at most p ⋅ m ^r atoms. Let \(h: \mathit {atoms}(q) \to chase(I, \mathcal {M}_{n})\) be a homomorphism establishing that \(\mathbf {a} \in q(chase(I,\mathcal {M}_{n}))\). It follows that there are at most p ⋅ m ^r facts in \(chase(I,\mathcal {M}_{n})\) witnessing that \(\mathbf {a} \in q(chase(I, \mathcal {M}_{n}))_{\downarrow }\). Each of these facts must be produced in a single step while chasing the source instance I with \(\mathcal {M}_{n}\), which implies that each of these facts is produced using at most k facts from I. Let I ^∗ be the subinstance of I consisting of all the aforementioned facts of I used to produce the facts in \(chase(I,\mathcal {M}_{n})\) witnessing that \(\mathbf {a} \in q(chase(I, \mathcal {M}_{n}))_{\downarrow }\). We then have that |I ^∗|≤ k ⋅ p ⋅ m ^r and \(\mathbf {a} \in q(chase(I^{*}, \mathcal {M}_{n}))_{\downarrow }\). Since n, n ^′≥ N ₀, we have that \(q(chase(I^{*}, \mathcal {M}_{n}))_{\downarrow } = q(chase(I^{*}, \mathcal {M}^{\prime }_{n}))_{\downarrow }\), hence \(\mathbf {a} \in q(chase(I^{*}, \mathcal {M}^{\prime }_{n}))_{\downarrow }\). By the monotonicity of the chase procedure, we have that \(\mathbf {a} \in q(chase(I, \mathcal {M}^{\prime }_{n}))_{\downarrow }\). It follows that \(q(chase(I, \mathcal {M}_{n}))_{\downarrow } \subseteq q(chase(I, \mathcal {M}^{\prime }_{n}))_{\downarrow }\). A symmetric argument establishes the containment \(q(chase(I, \mathcal {M}^{\prime }_{n}))_{\downarrow } \subseteq q(chase(I, \mathcal {M}_{n}))_{\downarrow }\), hence \(q(chase(I, \mathcal {M}_{n}))_{\downarrow } = q(chase(I, \mathcal {M}^{\prime }_{n}))_{\downarrow }\), which, in turn, implies that \(cert(q,I, \mathcal {M}_{n}) = cert(q, I, \mathcal {M}_{n^{\prime }})\).

Part 2. It is obvious that if a sequence of schema mappings has a uniform limit, then it has a pointwise limit. We focus on the reverse direction. Let \((\mathcal {M}_{n})_{n\geq 1}\) be a sequence of premise bounded GLAV mappings that has a pointwise limit \(\mathcal {M}\). We claim that \(\mathcal {M}\) is also a uniform limit of \((\mathcal {M}_{n})_{n\geq 1}\).

Since \((\mathcal {M}_{n})_{n\geq 1}\) has a pointwise limit, we have that \((\mathcal {M}_{n})_{n\geq 1}\) is pointwise Cauchy. The previous part implies that \((\mathcal {M}_{n})_{n\geq 1}\) is uniformly Cauchy as well. Fix an integer m. Since \((\mathcal {M}_{n})_{n\geq 1}\) is uniformly Cauchy, there exists an n ₀ such that for all n, n ^′≥ n ₀, we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}_{m}} \mathcal {M}_{n^{\prime }}\). We claim that also \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}_{m}} \mathcal {M}\) holds, for every n ≥ n ₀. To show this, fix some n ≥ n ₀ and let I be a source instance and q a conjunctive query in C Q _m . We have to show that \(cert(q,I,\mathcal {M}_{n}) = cert(q,I,\mathcal {M})\). Since \(\mathcal {M}\) is a pointwise limit of \((\mathcal {M}_{n})_{n\geq 1}\), there is an \(n^{\prime }_{0}(I)\) such that for all \(n^{\prime } \geq n^{\prime }_{0}(I)\), we have that \(cert(q,I,\mathcal {M}_{n^{\prime }})=cert(q,I,\mathcal {M})\). Take an integer n ^′ such that \(n^{\prime } \geq \max \{n_{0}, n^{\prime }_{0}(I)\}\). Since n ^′≥ n ₀, we have that \(cert(q,I,\mathcal {M}_{n}) = cert(q,I,\mathcal {M}_{n^{\prime }})\). Since \(n^{\prime }\geq n^{\prime }_{0}(I)\), we have that \(cert(q,I,\mathcal {M}_{n^{\prime }}) = cert(q,I,\mathcal {M})\). Thus, \(cert(q,I,\mathcal {M}_{n}) = cert(q,I,\mathcal {M})\). □

Proposition 6

There exists a uniformly Cauchy sequence of LAV mappings that has no pointwise limit; in particular, it has no uniform limit either.

Proof 10

We first show that the sequence\((\mathcal {M}_{n})_{n\geq 1}\) is uniformly Cauchy. Let k ≥ 1. We claim that if we take n ₀ = k, then for every source instance I, for every n, m ≥ n ₀, and every q ∈C Q _k , we have that \(cert(q, I, \mathcal {M}_{n}) = cert(q, I, \mathcal {M}_{m})\). To see this, note that for every source instance I and for every t ≥ 1, the universal solutions of I w.r.t. \(\mathcal {M}_{t}\) have active domains consisting entirely of labeled nulls. Hence, only boolean queries may return a non-empty result. Moreover, observe that these universal solutions have no self-loops, i.e., they contain no atoms of the form F(v, v) for some labeled null v.

We now distinguish two cases: First, suppose that q ∈C Q _k is a boolean conjunctive query which contains a “self-loop”, i.e., an atom of the form F(z, z) for some variable z. Then we clearly have \(cert(q, I, \mathcal {M}_{n}) = \mathit {false} = cert(q, I, \mathcal {M}_{m})\). It remains to consider the case that q ∈ C Q _k is a boolean C Q containing no self-loop. Then we clearly have \(cert(q, I, \mathcal {M}_{n}) = \mathit {true} = cert(q, I, \mathcal {M}_{m})\), since we are assuming that m, n ≥ k holds.

Using an argument similar to the one in the proof of Proposition 2, we now show that the sequence \((\mathcal {M}_{n})_{n\geq 1}\) has no pointwise limit. Towards a contradiction, assume that \((\mathcal {M}_{n})_{n\geq 1}\) does have a pointwise limit \(\mathcal {M}\). Let I be a non-empty source instance. We consider three cases.

First, assume that \(\text {Sol}(I,\mathcal {M})\) is empty. Then, for every boolean conjunctive query q, it holds trivially that \(cert(q,I,\mathcal {M}) = \mathit {true}\). This is, in particular, the case for the query q = ∃z F(z, z), which asks for the existence of a self-loop. However, for this query q, we have that \(cert(q, I, \mathcal {M}_{n}) = \mathit {false}\) for every n ≥ 1.

Second, assume that \(\text {Sol}(I,\mathcal {M})\) is non-empty and that all solutions \(J \in \text {Sol}(I,\mathcal {M})\) contain a self-loop. For the query q = ∃z F(z, z) as above, we again have \(cert(q, I, \mathcal {M}) = {\mathit {true}}\), whereas \(cert(q, I, \mathcal {M}_{n}) = \mathit {false}\), for every n ≥ 1.

Proposition 7

There exists a sequence \((\mathcal {M}_{n})_{n\geq 1}\) of LAV mappings that has a uniform limit, but no uniform limit of \((\mathcal {M}_{n})_{n\geq 1}\) admits universal solutions. In particular, no SO tgd is a uniform limit of the sequence \((\mathcal {M}_{n})_{n\geq 1}\).

Proof 11

Let n ₀ = m. Since each \(\mathcal {M}_{n}\) has solutions consisting entirely of nulls, it suffices to consider boolean C Q s only. Let q be a boolean C Q with m variables and assume that \(cert(q, I, \mathcal {M}_{n})= true\), where n ≥ m. This implies that there is a homomorphism from the body of q into P _n , where P _n is the simple path with n nodes. In turn, this implies that C _k ⊧q, for every k. Thus, \(cert(q, I, \mathcal {M}) = \mathit {true}\) as well. In the other direction, assume that \(cert(q, I, \mathcal {M}) = \mathit {true}\). Note that q cannot contain a directed cycle, since no directed cycle can be mapped homomorphically in every cycle of length greater than one. Let h be a homomorphism from the body of q into C _m+1. Since q ∈C Q _m , the variables of q have at most m distinct images among the nodes of C _m+1. This means that \(\tilde C_{m+1} \models q\), where \(\tilde C_{m+1}\) is obtained from C _m+1 by removing the facts that contain at least one element that is not the image of one of the variables of q under h. Note that \(\tilde C_{m+1}\) has at least one fact less than C _m+1, and so it is a collection of simple paths of length at most m; therefore, there is a homomorphism from \(\tilde C_{m+1}\) to P _n , hence P _n ⊧q.

Part 2. For the second part of the claim and towards a contradiction, assume that \(\mathcal {M}^{\prime }\) is a uniform limit of \((\mathcal {M}_{n})_{n\geq 1}\) such that there exists a non-empty source instance I and a finite universal solution J for I w.r.t. \(\mathcal {M}^{\prime }\). Note that for every i, we have that \(cert(\exists P_{i}, I, \mathcal {M}^{\prime }) = \mathit {true}\), because \(\mathcal {M}^{\prime }\) is a (uniform and, hence also pointwise) limit of the sequence \((\mathcal {M}_{n})_{n\geq 1}\). Then we also have that J⊧∃P _i , since J is universal. Since J is finite, this is possible only if J contains a directed cycle.

We can now derive a contradiction as follows. For each positive integer l, let ∃C _l be the boolean conjunctive query asserting the existence of a cycle of length l. Then there is no n such that \(cert(\exists C_{l}, I, \mathcal {M}_{n}) = \mathit {true}\). Thus, \(cert(\exists C_{l}, I, \mathcal {M}^{\prime }) = \mathit {false}\) must hold for every l, since \(\mathcal {M}^{\prime }\) is a limit of \((\mathcal {M}_{n})_{n\geq 1}\). Hence, J cannot contain cycles.

Since every SO tgd admits universal solutions, it follows that no SO tgd is a (uniform or pointwise) limit of \((\mathcal {M}_{n})_{n\geq 1}\). □

Theorem 5

Moreover, if \((\mathcal {M}_{n})_{n\geq 1}\) is a sequence of LAV mappings, then \((\mathcal {M}_{n})_{n\geq 1}\) has a LAV mapping as a uniform limit if and only \((\mathcal {M}_{n})_{n\geq 1}\) has a uniform limit that admits universal solutions.

Lemma 2

If \(\mathcal {M}\) is the uniform limit of a sequence \((\mathcal {M}_{n})_{n\geq 1}\) of schema mappings each of which allows for C Q -rewriting, then also \(\mathcal {M}\) allows for C Q -rewriting.

Proof 12

Let q be a target conjunctive query with m variables. Since \(\mathcal {M}\) is a uniform limit of \((\mathcal {M}_{n})_{n\geq 1}\), there exists an integer n ₀ such that for every n ≥ n ₀ and every source instance I, we have that \(cert(q,I,\mathcal {M}) = cert(q,I,\mathcal {M}_{n})\). In particular, \(cert(q,I,\mathcal {M}) = cert(q,I,\mathcal {M}_{n_{0}})\). Since \(\mathcal {M}_{n_{0}}\) allows for C Q-rewriting, there is a source conjunctive query q ^′ such that \(cert(q,I,\mathcal {M}_{n_{0}}) = q^{\prime }(I)\), for every source instance I. Hence, \(cert(q,I,\mathcal {M}) = q^{\prime }(I)\) holds, for every source instance I. □

Lemma 3

Let \(\mathcal {M}\) be a uniform limit of a sequence \((\mathcal {M}_{n})_{n\geq 1}\) of LAV mappings. If \(\mathcal {M}\) admits universal solutions, then it is closed under unions.

Proof 13

The proof proceeds through several stages and involves four claims, each of which builds on preceding ones. We first state the claims without proof and then use the last claim to show the desired conclusion. After this, we complete the proof of the lemma by proving each claim.

Claim A.

The notions of u ^′-limit and uniform limit coincide. Formally, for every sequence \((\mathcal {M}_{n})_{n\geq 1}\) of schema mappings and every schema mapping \(\mathcal {M}\), we have that \(\mathcal {M}_{n} \stackrel {u}{\longrightarrow } \mathcal {M}\) if and only if \(\mathcal {M}_{n} \stackrel {u^{\prime }}{\longrightarrow } \mathcal {M}\).

Next, we use the given sequence \((\mathcal {M}_{n})_{n\geq 1}\) to construct another sequence \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\) of LAV mappings that possesses some desirable properties. To define the sequence \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\), we need another claim.

Claim B.

Assume that \(\mathcal {M}_{n} \stackrel {u}{\longrightarrow } \mathcal {M}\). Then, there exists a strictly increasing sequence (n _i ) _{i ≥ 1} of positive integers, such that for every ℓ ≥ 1 and for every n ≥ n _ℓ , we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\).

Claim C.

Let (n _i ) _{i ≥ 1} be the strictly increasing sequence of positive integers according to Claim B and let \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\) be the sequence of LAV mappings constructed above. Then, for every ℓ ≥ 1, the following properties hold: (i) for every n ≥ n _ℓ , we have that \(\mathcal {M}^{\prime }_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\); (ii) the conclusion of every LAV constraint in \(\mathcal {M}^{\prime }_{n_{\ell }}\) is of length at most ℓ.

We now make the following claim about the sequence \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\).

Claim D.

For every source instance I, there exists an integer n ₀ ≥ 1 such that for every I ^′⊆ I, we have that \(\text {Sol}(I^{\prime },\mathcal {M}^{\prime }_{n_{0}}) = \text {Sol}(I^{\prime },\mathcal {M})\).

Next, we use Claim D to show that \(\mathcal {M}\) is closed under unions, i.e., given \((I_{1},J_{1}) \in \mathcal {M}\) and \((I_{2},J_{2}) \in \mathcal {M}\), we must show that \((I,J) \in \mathcal {M}\) with I = I ₁ ∪ I ₂ and J = J ₁ ∪ J ₂. From Claim D, we know that there exists n ₀ such that \(\text {Sol}(I^{\prime },\mathcal {M}^{\prime }_{n_{0}}) = \text {Sol}(I^{\prime },\mathcal {M})\), for every I ^′⊆ I. In particular, I ₁, I ₂ ⊆ I. Hence, for each i ∈{1,2}, we have \(J_{i} \in \text {Sol}(I_{i}, \mathcal {M}^{\prime }_{n_{0}})\), that is, \((I_{1}, J_{1})\in \mathcal {M}^{\prime }_{n_{0}}\) and \((I_{2}, J_{2})\in \mathcal {M}^{\prime }_{n_{0}}\). Since \(\mathcal {M}^{\prime }_{n_{0}}\) is a LAV mapping, it is closed under unions. Hence, \((I,J) \in \mathcal {M}^{\prime }_{n_{0}}\), and, since \(\text {Sol}(I,\mathcal {M}^{\prime }_{n_{0}}) = \text {Sol}(I,\mathcal {M})\), we conclude that \(J \in \text {Sol}(I,\mathcal {M})\), i.e., \((I,J)\in \mathcal {M}\).

To complete the proof of the lemma, it remains to prove Claims A-D.

Claim A.

The notions of u ^′-limit and uniform limit coincide. Formally, for every sequence \((\mathcal {M}_{n})_{n\geq 1}\) of schema mappings and every schema mapping \(\mathcal {M}\), we have that \(\mathcal {M}_{n} \stackrel {u}{\longrightarrow } \mathcal {M}\) if and only if \(\mathcal {M}_{n} \stackrel {u^{\prime }}{\longrightarrow } \mathcal {M}\). (⇒) Assume \(\mathcal {M}_{n} \stackrel {u}{\longrightarrow } \mathcal {M}\). We have to show that also \(\mathcal {M}_{n} \stackrel {u^{\prime }}{\longrightarrow } \mathcal {M}\) holds. Consider an arbitrary ℓ ≥ 1 and let r be the maximal arity of the target schema of \(\mathcal {M}\). Any conjunctive query with at most ℓ atoms can have at most m = ℓ ⋅ r variables. Hence, the inclusion C Q ^′ _ℓ ⊆C Q _m holds.

We are assuming \(\mathcal {M}_{n} \stackrel {u}{\longrightarrow } \mathcal {M}\). Hence, there exists n ₀(m) such that for all n ≥ n ₀(m), we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}_{m}} \mathcal {M}\). That is, for all q ∈C Q _m and for all I, it holds that \(cert(q, I, \mathcal {M}_{n}) = cert(q, I, \mathcal {M})\). Since C Q ^′ _ℓ ⊆C Q _m , we may conclude that for all q ∈C Q ^′ _ℓ and for all I, it holds that \(cert(q, I, \mathcal {M}_{n}) = cert(q, I, \mathcal {M})\). Hence, \(\mathcal {M}_{n} \stackrel {u^{\prime }}{\longrightarrow } \mathcal {M}\) indeed holds.

(⇐) Assume \(\mathcal {M}_{n} \stackrel {u^{\prime }}{\longrightarrow } \mathcal {M}\). We have to show that also \(\mathcal {M}_{n} \stackrel {u}{\longrightarrow } \mathcal {M}\) holds. Consider an arbitrary m ≥ 1. As above, let r be the maximal arity of the target schema of \(\mathcal {M}\). Moreover, let p be the number of target relation symbols. Any conjunctive query with at most m variables can have at most ℓ = p ⋅ m ^r atoms. Hence, the inclusion C Q _m ⊆C Q ^′ _ℓ holds.

We are assuming \(\mathcal {M}_{n} \stackrel {u^{\prime }}{\longrightarrow } \mathcal {M}\). Hence, there exists n ₀(ℓ) such that for all n ≥ n ₀(ℓ), we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\). That is, for all \(q \in \mathsf {CQ}^{\prime }_{\ell }\) and for all I, it holds that \(cert(q, I, \mathcal {M}_{n}) = cert(q, I, \mathcal {M})\). Since C Q _m ⊆C Q ^′ _ℓ , we may conclude that for all q ∈C Q _m and for all I, it holds that \(cert(q, I, \mathcal {M}_{n}) = cert(q, I, \mathcal {M})\). Hence, \(\mathcal {M}_{n} \stackrel {u}{\longrightarrow } \mathcal {M}\) indeed holds.

Claim B.

Assume that \(\mathcal {M}_{n} \stackrel {u}{\longrightarrow } \mathcal {M}\). Then, there exists a strictly increasing sequence (n _i ) _{i ≥ 1} of positive integers, such that for every ℓ ≥ 1 and for every n ≥ n _ℓ , we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\). Since \(\mathcal {M}_{n} \stackrel {u^{\prime }}{\longrightarrow } \mathcal {M}\), for each ℓ ≥ 1 there exists an integer \(n^{\prime }_{\ell }\) such that for all \(n \geq n^{\prime }_{\ell } \), we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\). We may choose n _ℓ as follows to ensure strict monotonicity: \(n_{1} \text {:=}\, n^{\prime }_{1}\)

…\(n_{\ell } \text {:=}\, \max (n_{\ell -1} + 1, n^{\prime }_{\ell })\)Then the sequence (n _i ) _{i ≥ 1} is strictly increasing and for all ℓ ≥ 1 and for all n ≥ n _ℓ , we have that \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\).

Claim C.

Let (n _i ) _{i ≥ 1} be the strictly increasing sequence of positive integers according to Claim B and let \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\) be the sequence of LAV mappings constructed above. Then, for every ℓ ≥ 1, the following properties hold: (i) for every n ≥ n _ℓ , we have that \(\mathcal {M}^{\prime }_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\); (ii) the conclusion of every LAV constraint in \(\mathcal {M}^{\prime }_{n_{\ell }}\) is of length at most ℓ. Consider an arbitrary ℓ ≥ 1. By the construction of the sequence \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\), every LAV constraint in \(\mathcal {M}^{\prime }_{n}\) has a conclusion of length at most ℓ. Hence, property (ii) clearly holds.

To prove property (i), consider an arbitrary n ≥ n _ℓ . We have to show that \(\mathcal {M}^{\prime }_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\), i.e., for arbitrary source instance I and arbitrary conjunctive query q ∈C Q ^′ _ℓ , we have to show that \(cert(q, I, \mathcal {M}^{\prime }_{n}) = cert(q, I, \mathcal {M})\). By Claim B, we have \(\mathcal {M}_{n} \equiv _{\mathsf {CQ}^{\prime }_{\ell }} \mathcal {M}\). Hence, it suffices to show that \(cert(q, I, \mathcal {M}^{\prime }_{n}) = cert(q, I, \mathcal {M}_{n})\) holds. We prove the two inclusions separately.

By the construction of \(\mathcal {M}^{\prime }_{n}\), we clearly have \(chase(I,\mathcal {M}^{\prime }_{n}) \to chase(I,\mathcal {M}_{n})\). From this, it follows immediately that \(cert(q,I,\mathcal {M}^{\prime }_{n}) \subseteq cert(q,I,\mathcal {M}_{n})\).

For the reverse inclusion, consider an arbitrary tuple \(\mathbf {a} \in cert(q, I,\mathcal {M}_{n})\). Then, there exists a homomorphism \(h_{n} \colon q \to chase(I,\mathcal {M}_{n})\) with h(z) = a, where z denotes the free variables of q. Let \(h_{n}(q) = \{A_{1}, {\ldots } A_{k}\} \subseteq chase(I,\mathcal {M}_{n})\) with k ≤ ℓ. By construction, \(\mathcal {M}^{\prime }_{n}\) is obtained by restricting the conclusions of the LAV constraints \(\tau \in \mathcal {M}_{n}\) in all possible ways to at most ℓ atoms. Hence, since k ≤ ℓ, we have that also \(chase(I,\mathcal {M}^{\prime }_{n})\) contains the set {A ₁,…A _k } of atoms (up to renaming of labeled nulls). Thus, there exists a homomorphism \(h \colon \{A_{1}, {\ldots } A_{k}\} \to chase(I,\mathcal {M}^{\prime }_{n})\) and h(h _n (⋅)) is a homomorphism \(q \to chase(I,\mathcal {M}^{\prime }_{n})\) with h(z) = a. Therefore, \(\mathbf {a} \in cert(q, I,\mathcal {M}^{\prime }_{n})\) holds.

Before presenting the proof of Claim D, we need to bring the notion of fact block size into the picture; this notion was introduced in [7].

Fact Blocks. Let J be an instance. The Gaifman graph of facts G _J of J is the graph whose nodes are the facts of J and there is an edge between two facts if they have a null in common. The fact blocks (or f-blocks) of J are the sets of nodes of the connected components of G _J . The block size of an undirected graph G is the size of the maximal connected component of G _J , where the size of a component is given as the number of nodes. The fact block size (f-block size) of an instance J is the block size of the Gaifman graph of facts of J.

Claim D.

“ ⊆”: Let \(K \in \text {Sol}(I^{\prime },\mathcal {M}^{\prime }_{n_{0}})\). Since J ^′ is a universal solution for I ^′ w.r.t. \(\mathcal {M}^{\prime }_{n_{0}}\), there exists a homomorphism g ^′ : J ^′→ K. By composing g ^′ with the homomorphism h : J → J ^′, we obtain a homomorphism from J to K. By the closure under target homomorphisms, we conclude that \(K \in \text {Sol}(I^{\prime },\mathcal {M})\)

“ ⊇”: Now let \(K \in \text {Sol}(I^{\prime },\mathcal {M})\). Since J is a universal solution for I ^′ w.r.t. \(\mathcal {M}\), there exists a homomorphism g : J → K. By composing g with the homomorphism h ^′ : J ^′→ J, we obtain a homomorphism from J ^′ to K. Since LAV mapping \(\mathcal {M}^{\prime }_{n_{0}}\) is closed under target homomorphisms, we conclude that \(K \in \text {Sol}(I^{\prime },\mathcal {M}^{\prime }_{n_{0}})\).

The proof of Lemma 3 is now complete.

□

Proof 14 (Proof of Theorem 5)

The direction (1) ⇒ (2) is obvious. For the direction (2) ⇒ (1), we start with the case when \((\mathcal {M}_{n})_{n\geq 1}\) is a sequence of LAV mappings.

Theorem 3.1 in [19] asserts that if a schema mapping admits universal solutions, allows for query rewriting, and is closed under both target homomorphisms and unions, then it is logically equivalent to a LAV mapping. Consequently, we have that \(\mathcal {M}\) is logically equivalent to a LAV mapping.For the case when \((\mathcal {M}_{n})_{n\geq 1}\) is a sequence of premise-bounded GLAV mappings (but not necessarily LAV mappings), we apply yet another structural characterization of GLAV mappings from [19], namely, Theorem 3.9, which asserts that if a schema mapping allows for C Q-rewriting, admits universal solutions, is closed under target homomorphisms, and is n-modular, for some fixed n, then it is logically equivalent to a GLAV mapping.

Let k be the constant bounding the length of premises in \((\mathcal {M}_{n})_{n\geq 1}\). We proceed exactly as in the proof of Lemma 3 and construct a sequence \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\), in which the premises of tgds are the same as in tgds in \((\mathcal {M}_n)_{n\geq 1}\), hence each tgd in \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\) has at most k atoms in its premise. We proceed exactly as in the proof of Lemma 3 to establish the following analog of Claim D.Claim D (in the proof of Lemma 3) For every source instance I, there exists an integer n ₀ ≥ 1 such that for every I ^′⊆ I , we have that \(\text {Sol}(I^{\prime },\mathcal {M}^{\prime }_{n_{0}}) = \text {Sol}(I^{\prime },\mathcal {M})\).

Now, since each tgd in every element of \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\) has at most k atoms in its premise, it follows that there is a positive integer N _k so that each mapping \(\mathcal {M}^{\prime }_{n}\) in \((\mathcal {M}^{\prime }_{n})_{n\geq 1}\) is N _k -modular. It is easy to see that N _k ≤ k ⋅ r holds where r is the maximum relation arity in the source schema.

We now prove that \(\mathcal {M}\) is N _k -modular. Assume that J is not a solution for I w.r.t. to \(\mathcal {M}\). Take an integer n ₀ as in Claim D and consider \(\mathcal {M}^{\prime }_{n_{0}}\). It follows that J is not a solution for I w.r.t. \(\mathcal {M}^{\prime }_{n_{0}}\). Since \(\mathcal {M}^{\prime }_{n_{0}}\) is N _k -modular, there is a subinstance I ^′ of I such that J is not a solution for I ^′ w.r.t. \(\mathcal {M}^{\prime }_{n_{0}}\) and |d o m(I ^′)|≤ N _k . Again by Claim D, we have that J is not a solution for I ^′ w.r.t. \(\mathcal {M}\), hence M is N _k -modular.

Thus, \(\mathcal {M}\) has the following properties: it admits C Q-rewriting (since it is the uniform limit of GLAV mappings that admit C Q-rewriting), it admits universal solutions, is closed under target homomorphisms (if it is not, we take its closure before we begin the construction), and, as just shown, it is N _k -modular. Consequently, by Theorem 3.9 in [19], we have that \(\mathcal {M}\) is logically equivalent to a GLAV schema mapping, which completes the proof. □

Conjecture 1