Sequential composition of schema mappings

A method for generating a schema mapping. A provided mapping M12 relates schema S1 to schema S2. A provided mapping M23 relates schema S2 to schema S3. A mapping M13 is generated from schema S1 to schema S3 as a composition of mappings M12 and M23. Mappings M12, M23, and M13 are each expressed in terms of at least one second-order nested tuple-generating dependency (SO nested tgd). Mapping M13 does not expressly recite any element of schema S2. At least one schema of the schemas S1 and S2 may comprise at least one complex type expression nested inside another complex type expression. Mapping M13 may define the composition of the mappings M12 and M23 with respect to a relationship semantics or a transformation semantics.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates to a method and system for generating a schema mapping that composes two given schema mappings.

2. Related Art

Compositional mappings between schemas may be expressed as constraints in logic-based languages. However, there is no known compositional mapping that applies to Extensible Markup Language (XML) schemas. Accordingly, there is a need for a compositional mapping that applies to XML schemas.

SUMMARY OF THE INVENTION

The present invention provides a method for generating a schema mapping, said method comprising:

providing a mapping M12from a schema S1to a schema S2, said mapping M12relating the schema S2to the schema S1, said schema S1and schema S2each comprising one or more elements, said mapping M12being expressed in terms of at least one second-order nested tuple-generating dependency (SO nested tgd);

providing a mapping M23from the schema S2to a schema S3, said mapping M23relating the schema S3to the schema S2, said schema S3comprising one or more elements, said mapping M23being expressed in terms of at least one SO nested tgd; and

generating a mapping M13from the schema S1to the schema S3, said mapping M13relating the schema S3to the schema S1, said mapping M13being expressed in terms of at least one SO nested tgd that does not expressly recite any element of the schema S2, said generating the mapping M13comprising generating the mapping M13as a composition of the mappings M12and M23, wherein at least one schema of the schemas S1and S2comprises at least one complex type expression nested inside another complex type expression.

The present invention provides a method for generating a schema mapping, said method comprising:

providing a mapping M12from a schema S1to a schema S2, said mapping M12relating the schema S2to the schema S1, said schema S1and schema S2each comprising one or more elements, said mapping M12being expressed in terms of at least one second-order nested tuple-generating dependency (SO nested tgd);

providing a mapping M23from the schema S2to a schema S3, said mapping M23relating the schema S3to the schema S2, said schema S3comprising one or more elements, said mapping M23being expressed in terms of at least one SO nested tgd; and

generating a mapping M13from the schema S1to the schema S3, said mapping M13relating the schema S3to the schema S1, said mapping M13being expressed in terms of at least one SO nested tgd that does not expressly recite any element of the schema S2, said generating the mapping M13comprising generating the mapping M13as a composition of the mappings M12and M23, wherein the mapping M13defines the composition of the mappings M12and M23with respect to a transformation semantics.

The present invention advantageously provides a compositional mapping that applies to XML schemas.

DETAILED DESCRIPTION OF THE INVENTION

A schema is a set of rules describing how stored information is organized. The stored information may be in the form of data structures (e.g., databases, tables, files, etc.). A schema mapping from a first schema to a second schema may define a composition of two sequential mappings with respect to a relationship semantics or to a transformation semantics. A schema mapping pertaining to a relationship semantics describes relationships between the first schema and the second schema. A schema mapping pertaining to a transformation semantics describes a transformation of data from the first schema to the second schema.

The present invention addresses the problem of sequential composition of two schema mappings. Given a mapping M12describing a relationship between a schema S1and a schema S2, and a mapping M23describing a relationship between schema S2and schema S3, the present invention derives a mapping M13that describes the implied relationship between schema S1and S3, without referring to schema S2in the mapping M13. This problem is of importance in areas such as information integration and schema evolution, where tools are needed to manipulate and reason about mappings between relational database schemas and/or mapping between Extensible Markup Language (XML) schemas. Mappings between schemas may be expressed as constraints in a logic-based language; these constraints specify how data stored under one schema relate to data stored under another schema.

The present invention describes a system that implements two possible semantics for the result of composing two sequential mappings that are both useful in practice. These two semantics are a relationship semantics and a transformation semantics. The relationship semantics is more general than the transformation semantics and handles composition of mappings when mappings are used to describe arbitrary relationships between schemas. The transformation semantics applies only when mappings are used to describe transformations for exchanging data from a source schema to a target schema. The present invention provides algorithms for composing sequential schema mappings under both the relationship semantics and the transformation semantics. In effect, this disclosure claims two different systems for sequential composition of schema mappings: one supporting the relationship semantics, and one supporting the transformation semantics.

The remainder of the Detailed Description of the Invention is divided into the following sections:

3. Algorithm 1: Composition with Respect to Relationship Semantics

4. Algorithm 2: Composition with Respect to Transformation Semantics

5. Computer System

Applications of the present invention comprise schema evolution, optimization of data transformation flows, and mapping deployment.

Schema mappings are abstractions or models for the more concrete scripts or queries that can be used to transform data from one schema to another. In other words, given a schema mapping M12from S1to S2, a human or a software tool can generate a script or query or program to transform a database or a document instance that conforms to schema S1into a database or document instance that conforms to schema S2. However, when schema evolution occurs, such as when schema S1changes to a new schema S1′ or schema S2changes to a new schema S2′, the earlier scripts or queries or programs from S1to S2are no longer applicable and new scripts or queries ones are generated.

The present invention can be used in solving the above problem by employing any one of the two mapping composition algorithms claimed.FIGS. 1 and 2depict diagrammatically how mapping composition can be applied for target evolution and source evolution, respectively, in accordance with embodiments of the present invention.FIG. 1shows the case of target schema evolution (the target schema S2changes to S2′), whileFIG. 2shows the case of source schema evolution (the source schema S1changes to S1′).

InFIG. 1for target schema evolution, the first step after the mapping M12has been previously specified is to generate a schema mapping M22′from S2to S2′. The second step is to apply mapping composition for M12and M22′to obtain a new schema mapping M12′. This new schema mapping M12′can then be used to guide the generation of the new scripts, queries or programs that transform data from S1to the new schema S2′.

InFIG. 2for source schema evolution, the first step after the mapping M12has been previously specified is to generate a schema mapping M1′1from S1′ to S1. The second step is to apply mapping composition for M1′1and M12to obtain a new schema mapping M1′2. This new schema mapping M1′2can then be used to guide the generation of the new scripts, queries or programs that transform data from the new schema S1′ to S2.

Both source schema evolution and target schema evolution may be implemented with respect to a transformation semantics.

1.2 Optimization of Data Transformation Flows

A data transformation flow is a sequence or, more generally, a directed graph, where the nodes of the graph are schemas and the edges are transformations or mappings between the schemas. Data transformation flows are the common abstraction for ETL (extract-transform-load) commercial systems. The methods of the current invention can be used to compose two schema mappings that appear sequentially in the data transformation flow and, thus, eliminate intermediate stages in the graph, with obvious performance benefits.

Furthermore, the repeated application of composition can yield an end-to-end schema mapping that eliminates all the intermediate nodes in the graph. This results in a high-level view of the entire data transformation flow; this high-level view, in turn, may allow for subsequent be re-optimization of the data flow graph into a different, but equivalent, data flow graph, with better performance.

Schema mappings may be designed at a logical level. More concretely, a schema mapping MSTmay be designed between two logical schemas S and T. At deployment time however, such mapping needs to be deployed (usually by a different person) between two physical schemas S′ and T′ that are not exactly the same as the logical schemas. Once mappings between the physical and the logical schemas are obtained (e.g., mappings MS′Sfrom S′ to S, and MTT′from T to T′), the methods of the current invention can be applied to generate a physical schema mapping MS′T′from S′ to T′ by composing MS′Swith the “logical” mapping MSTand then with MTT′.

2.1 Schemas and Instances

The schemas S1, S2, and S3can be any schema expressed in a language of nested elements of different types that include atomic types, record types, choice types and set types respectively corresponding to atomic elements, record elements, choice elements, and set elements. This language is called herein the nested relational schema language and can encode relational database schemas as well as XML and hierarchical schemas.

Atomic types (or primitive types) expressions are the usual data types: String, Int, Float, etc. Record types are used to encapsulate together groups of elements which in turn can be atomic types or complex types expressions. A complex type expression is a non-atomic type (e.g., a set, record, or choice type) expression. For example, RCD [ssn: Int, name: String] represents a record type whose components are ssn (social security number) and name. Another example is RCD [ssn: Int, person: RCD [name: String, address: String]], which denotes a record with two components: ssn, of integer type, and person, of record type (with two components: name and address, of string type). Set types are used to represent collections of elements or records. For example, SET of RCD [ssn: Int, person: RCD [name: String, address: String]] will represent a collection of records, where each record will denote a person. Choice types are used to represent elements that can be one of multiple components. For example, CHOICE [name: String, full_name: RCD [firstName: string, lastName: String]] will represent elements that can include either a name component (of type String), or of a full_name component (which itself consists of firstName and lastName).

A schema is a collection of roots, that is, names with associated nested types. The following are examples of nested relational schemas S1, S2, and S3:

In this example, the schemas represent information about students (student id or sid, name), courses they take, and the results (grade, evaluation_file) that are assigned to students for each course. The first schema S1contains two different roots, src1and src2, which are complementary sources of information about students. The second schema S2is a reorganization of the first schema that merges the information under one root, tgt. The third schema S3has one root new_tgt and can be thought of an evolution of the second schema S2, where individual courses are no longer of interest but the results still are.

Given a schema, any element of set type that is not nested inside other set type (directly or indirectly) is referred to as a top-level set-type element. For example, in the schema S2above, “students” and “evaluations” are top-level set-type elements, while “courses” is not (since it is nested inside the set type associated with “students”).

Given a schema S with roots r1of type T1, . . . , rnof type Tn, an instance I over S is a collection of values v1, . . . , vnfor the roots r1, . . . , rnof S, such that for each k from 1 to n, vkis of type Tk. For example,FIG. 3illustrates two instances over the schema S2shown supra, in accordance with embodiments of the present invention. InFIG. 3, the instances I and I′ appear on the left and right, respectively.

Schema mappings are used to specify relationships between schemas. A schema mapping between schema S1and schema S2specifies how a database or document instance conforming to S1relates to a database or document instance conforming to S2. Additionally, a schema mapping can be seen as a specification of how a database or document instance conforming to schema S1can be transformed into a database or document instance conforming to schema S2.

Schema mappings are expressed in a constraint language called second-order nested tuple-generating dependencies (or SO nested tgds). A SO nested tgd may be characterized by at least one schema of the schemas S1and S2comprising at least one complex type expression nested inside another complex type expression.

Each SO nested tgd includes one or more formulas. Each formula includes clauses such as: a for clause, an exists clause, a where clause, and a with clause. The for clause identifies source tuples to which the formula applies. The exists clause identifies tuples that must exist in the target. The where clause describes constraints on the tuples of source and/or target. The with clause describes how values in fields of source and target tuples are matched.

The following is an example of a schema mapping, M12, between the schemas S1and S2that were illustrated before. M12includes two formulas, m1and m2, each formula being an SO nested tgd.

The meaning of the above formulas for m1and m2is as follows.FIG. 4also shows a pictorial view of how the above formulas for m1and m2relate elements in schema S1and elements in schema S2, in accordance with embodiments of the present invention.

The formula m1is a constraint asserting, via the exists clause, that for each tuple s that appears in the set “students” under src1(of S1), there exists a tuple s′ in the set “students” under tgt (of S2), a tuple c′ in the set “courses” of s′, and a tuple e′ in the set “evaluations” under tgt.

Moreover, the where clause of m1specifies that the tuples c′ and e′ must not be arbitrary but they are constrained so that they have the same “eval_key” field.

Furthermore, the tuples s′, c′, and e′ are also constrained by the with clause of m1: the “sid” and “name” fields of s′ must respectively equal (i.e., have the same value of) the “sid” and “name” fields of s, the “course” field of c′ must equal the “course” field of s, and the “grade” field of e′ must equal the “grade” field of s. The value of “evaluation_file” of e′ is left unspecified by m1, due to the fact that src1does not contain any element that corresponds to an evaluation file.

In general, the with clause of a formula such as m1will contain a sequence of equalities relating elements in a source schema such as S1with elements (not necessarily with the same name) in a target schema such as S2. In the case of m1, these equalities are shown pictorially inFIG. 4as the set of arrows grouped under the name “m1”.

The formula m2is a similar constraint asserting, via the exists clause, that for each tuple s that appears in the set “students” under src2(of S1) and for each tuple c that appears in the set “courses” under src2, where s and c satisfy the condition that they have the same “course_code” value (as stated in the first where clause of m2), there must exist a tuple s′ in the set “students” under tgt (of S2), a tuple c′ in the set “courses” of s′, and a tuple e′ in the set “evaluations” under tgt.

As in m1, the where clause of m2specifies that the tuples c′ and e′ must not be arbitrary but they are constrained so that they have the same “eval_key” field. Furthermore, the tuples s′, c′, and e′ are also constrained by the with clause of m2: the “sid” and “name” fields of s′ must equal the “sid” and “name” fields of s, the “course” field of c′ must equal the “course” field of c, and the “evaluation_file” field of e′ must equal the “evaluation_file” field of c. The “grade” of e′ is left unspecified by m2, due to the fact that src2does not contain any element that corresponds to a grade.

As a notational matter, clauses appearing in mapping such as M12are underlined (e.g., for clause, exists clause, where clause, with clause) and may alternatively be denoted in upper case letters (e.g., FOR clause, EXISTS clause, WHERE clause, WITH clause, respectively).

2.3 Mapping Language: General Syntax of SO Nested tgds

In general, given a schema S1(the source schema) and a schema S2(the target schema), a nested tgd (m) is a formula of the form:

2) e1s, . . . eks, e1t, . . . , ektare expressions, where in general expressions are defined by the following grammar: e::=x|r|e.A (i.e., an expression can be a variable, a schema root, or a record component of another expression). In the above mapping, e1s, . . . eksare source expressions, that is, they are required to use only variables from the for clause (i.e., x1, . . . , xn) and schema roots from the source schema S1. Moreover, e1t, . . . ektare target expressions, that is, they are required to use only variables from the exists clause (i.e., y1, . . . , ym) and schema roots from the target schema S2. Furthermore, all the expressions that appear in the with clause must be of atomic type.

3) g1s, . . . gns, g1t, . . . , gmtare generators, where in general a generator is defined by the following grammar: g::=e1|case e2of A (where e1is an expression of set type and e2is an expression of a choice type that must include the choice of an A component). In the above mapping, g1s, . . . gnsare source generators, that is, they are required to use only variables from the for clause (i.e., x1, . . . , xn) and schema roots from the source schema S1. Moreover, g1t, . . . gktare target generators, that is, they are required to use only variables from the exists clause (i.e., y1, . . . , ym) and schema roots from the target schema S2. Furthermore, for every i from 1 to n, the ith source generator in the for clause can only use variables x1, . . . , xi-1. Similarly, for every j from 1 to m, the jth target generator in the exists clause can only use variables y1, . . . , yj-1.

4) B1(x1, . . . , xn) and B2(y1, . . . , ym) are predicates of the form (e1=e1′) and . . . and (e1=e1′) where e1, e1′, . . . , e1, e1′ are expressions of atomic type. In the case of B1(x1, . . . , xn), which is called a source predicate, these expressions can only be source expressions, while in the case of B2(y1, . . . , ym), which is called a target predicate, these expressions can only be target expressions.

The earlier formulas m1and m2are examples of nested tgds over the schema S1and schema S2. Second-order nested tgds (or, SO nested tgds, in short) are defined as an extension of nested tgds in the following way.

Each source expression that can appear in B1and each source expression eisthat can appear in the with clause can now be a function term t, defined by the grammar t::=e|F(t) where e is an expression as before (over the source variables x1, . . . , xn) while F is a function name (out of a possibly infinite set of function names that are available). It is possible that one function name is shared among multiple nested SO tgds.

The function F is sometimes called a Skolem function and a corresponding function term F(t) is sometimes called a Skolem term. One reason for this terminology is that the subsequent Algorithm 1, described infra, includes a step (step31) that generates such function terms (Skolem terms) based on a procedure called Skolemization.

As an example,FIG. 5depicts schemas S and T to illustrate a Skolem function, in accordance with embodiments of the present invention. The following set MSTis a schema mapping comprising SO nested tgds over the schemas S and T that both use the Skolem function F:

The source schema S includes a Takes table storing student names and courses the students take. The schema T includes two separate tables: a Student table, storing student ids and student names, and an Enrolls table relating student ids and courses. The mapping MSTsplits a tuple (name, course) into two tuples of Student and Enrolls, one containing the name the other one containing the course. At the same time, a Skolem function F is used to assign an “unknown” student id for each student name. The Skolem function F is used consistently (i.e., having the name as parameter) in both formulas to express the fact that a given student name is assigned the same student id in both tables.

2.4 Two Semantics of Mapping Composition

There are two semantics that can be associated with schema mappings and with their composition. The first semantics, the relationship semantics, is more general, while the second semantics, the transformation semantics is more suitable for certain specialized tasks, such as schema evolution, optimization of data transformation flows, and also generates simpler formulas.

Schema mappings can be viewed as describing relationships between instances over two schemas. Under this view, the formulas that appear in a schema mapping are considered as inter-schema constraints. More concretely, given a schema mapping M12between a schema S1and a schema S2, one can define the set
Rel(M12)={(I1,I2)|I1is an instance over S1, I2is an instance over S2, and (I1, I2) satisfies M12}  (1)

This set Rel (M12), called the binary relation of M12or the relationship induced by M12, contains all the “valid” pairs of instances (I1, I2), where “valid” means pairs of instances (I1, I2) that satisfy all the constraints that appear in M12. Given two schema mappings M12, from schema S1to schema S2, and M23from schema S2to another schema S3, the composition Rel (M12)·Rel (M23) of their induced relationships is defined as the composition of the two binary relations Rel (M12) and Rel (M23):
Rel(M12)·Rel(M23)={(I1,I3)|I1is an instance over S1, I3is an instance over S3, and there is an instance I2over S2such that (I1,I2) satisfies M12and (I2,I3) satisfies M23}  (2)
Then, by definition, a schema mapping M13defines the composition of schema mappings M12and M23, under the relationship semantics, if the following equation is satisfied:
Rel(M13)=Rel(M12)·Rel(M23)  (3)

The above definition is for schemas that are non-nested and for schema mappings that are between non-nested schemas. (A non-nested schema is a schema as defined earlier, with the restriction that there are no set-type elements that are nested within other set-type elements. Schemas of relational databases are good examples of non-nested schemas.)

In general, for schemas that are nested (that is, can have set-type elements that are nested within other set-type elements) the above definition is slightly modified by requiring that all instances (e.g., I1, I2, I3) that appear in equations (1) and (2) to be in a normal form that is called partitioned normal form (or PNF).

An instance is said to be in the partitioned normal form (PNF) if there cannot be two records in the same set (where the set can occur in any place in the instance) such that the two records have the same atomic sub-components, component-wise, but different set-valued sub-components.

For example, the instance I on the left inFIG. 3is not in PNF, because there are three records that have the same atomic components (001 and Mary for the “sid” and “name” components, respectively) but different sets of courses. In contrast, the instance I′ on the right inFIG. 3is in PNF (intuitively, all the courses for 001 and Mary have been merged under a single set.)

PNF is a goodness criterion for instances; this criterion requires a natural form of data merging (grouping) to be satisfied by instances. As an important special case, all non-nested instances are automatically in PNF.

A schema mapping can be viewed as describing a process (data movement) in which a target instance is materialized, given a source instance I and the schema mapping. More concretely, a schema mapping M12from schema S1to schema S2defines a function that, given an instance I1over S1, computes an instance I2over S2:
I2=M12(I1)  (4)

The formal definition of the function is as follows. First, all the formulas that appear in M12are Skolemized, by applying the procedure described in detail infra for step41of Algorithm 2 (seeFIG. 12described infra). Let the set of resulting Skolemized formulas from step31of Algorithm 1 be M′12. For the earlier example M12={m1, m2}, the Skolemized set M′12includes the formulas m1′ and m2′ shown in the description of step41of Algorithm 2.

In the second step, the chase with SO nested tgds is used to construct a target instance I2′ based on I1, using the mapping M′12. At the beginning of the chase, the target instance I2′ is empty. Then, for each SO nested tgd m′ in M′12, and for each binding of the for clause of m′ to tuples in the source instance I1such that the first where clause is satisfied, the chase adds tuples to the target instance I2′ such that the exists clause of m′, its associated where clause and the with clause are satisfied.

As an example,FIG. 6depicts a mapping that illustrates the chase process for constructing a target instance, in accordance with embodiments of the present invention. In particular,FIG. 6depicts a source instance I1over the schema S1illustrated earlier. The source instance I1contains two “student” tuples under the root “src1”: [001, Mary, CS120, A] for Mary and [005, John, CS500, B] for John. For simplicity,FIG. 6does not show the labels that are associated with the preceding values; i.e., the labels “sid”, “name”, “course”, and “grade”. These labels are shown instead in the schema S1, which is illustrated right near the instance. Furthermore, the instance I1contains two more tuples about Mary in the “students” set under the root “src2” (i.e., the tuples [001, Mary, K7] and [001, Mary, K4]), and two more tuples implicitly about Mary in the “courses” set under the root “src2” (i.e., the tuples [K7, CS120, file01] and [K4, CS200, file07]).

For this example, the chase works as follows. The for clause of m1′ can be instantiated to the first tuple [001, Mary, CS120, A] in the “students” set of “src1”. Then, to satisfy the exists clause, its associated where clause as well as the with clause of m1′, three tuples are added. First, a tuple [001, Mary, s1] is added to the “students” set of the root “tgt”. The value 001 is the “sid” component of this new tuple, Mary is the “name” component, and s1denotes the set value (initially empty) for the “courses” component. Then, a second tuple [CS120, E1(001, Mary, CS120, A)] is added to the set s1of the previous tuple. Here, E1(001, Mary, CS120, A) is a ground Skolem term obtained by applying the Skolem function E1to the concrete values (001, Mary, CS120, A). This Skolem function and its arguments are specified by the formula m1′ which is the result of the Skolemization step mentioned earlier. Finally, a third tuple [E1(001, Mary, CS120, A), A, F(001, Mary, CS120, A)] is added to the “evaluations” set under the root “tgt”. Here, E1(001, Mary, CS120, A) is the same ground Skolem term created before, while F(001, Mary, CS120, A) is a new ground Skolem term obtained by applying the Skolem function F to the concrete values (001, Mary, CS120, A). Again, this Skolem function and its arguments are specified by the formula m1′ which is the result of the Skolemization step mentioned earlier.

The above described process is repeated for all the tuples in the source instance I1and for all the formulas in M12′ (e.g., for m2′ in addition to m1′). At the end, each distinct ground function term is replaced by a unique value all throughout I2. For example, the two occurrences of E1(001, Mary, CS120, A) are replaced by a value E1that is generated so that it is different from every other value. A different ground function term, such as F(001, Mary, CS120, A), is replaced by a different value (e.g., F1). The instance I2′ depicted inFIG. 6at the right of schema S2is the result of applying the chase with the SO nested tgds in M12′.

After the chase finishes, the resulting target instance I2′ is further transformed (PNF-ized) into an instance I2that is in the partitioned normal form (PNF) described earlier. The PNF-ization identifies all records that appear in the same set (where the set can occur in any place in the instance) such that the records have the same atomic sub-components, component-wise, but different set-valued sub-components. For all such records, the set-valued components are unioned together. The process continues recursively until no such records can be found.

For the previous example, the instance I2′ inFIG. 6is not in PNF. It contains three tuples that have the same atomic components (001, Mary) but different sets: s1, s3and s4. To PNF-ize the instance, the three tuples are merged into one tuple, whose set is the union of s1, s3and s4.FIG. 7illustrates PNF-izing the instance I2′ into the final instance I2, in accordance with embodiments of the present invention. The instance I2is M12(I1).

Then, by definition, a schema mapping M13defines the composition of schema mappings M12and M23, with respect to the transformation semantics, if the following equation is satisfied:
M13(I1)=M23(M12(I1)), for every instance I1over S1(5)

With respect to schema evolution (see Section 1.1), it is noted that for target schema evolution the mapping M12is provided or specified before the mapping M23is provided or specified, whereas for source schema evolution the mapping M12is provided or specified after the mapping M23is provided or specified.

3. Algorithm 1: Composition with Respect to Relationship Semantics

The input to Algorithm 1 is as follows. Schema mappings M12and M23are inputs expressed as constraints in the language of second-order nested tuple-generating dependencies (or SO nested tgds). The schema mapping M12is a set of SO nested tgds that relate a schema S1and a schema S2. Similarly, schema mapping M23is a set of SO nested tgds that relate a schema S2to a schema S3.

FIG. 8illustrates a sequence of schema mappings M12and M23to illustrate an example of input to the Algorithm 1, in accordance with embodiments of the present invention. M12is a mapping from S1to S2, and M23is a mapping from S2to S3, where S1, S2, and S3are the three schemas illustrated earlier. M12is the set {m1, m2} of formulas described earlier, and M23comprises the following formula m3:

The output to Algorithm 1 is as follows. Schema mapping M13, comprising nested tgds that relate schema S1and schema S3, is an output representing the composition of M12and M23under the relationship semantics.

FIGS. 9A-9C(collectively, “FIG.9”) are flow charts depicting steps31-34of Algorithm 1 for determining a composition of schema mappings with respect to relationship semantics, in accordance with embodiments of the present invention. In the discussion ofFIG. 9, the term “target element” refers to an element of schema S2.

In step31, each SO nested tgd m of M12is Skolemized by assigning a Skolem function term to each target atomic element whose value is not determined by formula m. A target atomic element X is said to be not determined by formula m if X is an atomic component of a tuple of S2that is asserted in the exists clause of formula m but X does not appear in any equality with a source atomic element in the with clause of formula m.

For example, for the formula m1of the earlier schema mapping M12, the target atomic element e′.evaluation_file, which is a component of the tuple e′ asserted in the exists clause of m1, is not determined by m1, since e′.evaluation_file does not appear in any equality in the with clause of m1. As additional examples, for the same formula m1, the target atomic elements c′.eval_key and e′.eval_key are components of the tuples c′ and e′ in the exists clause of m1, but are not determined by m1(i.e., there is an equality that relates c′.eval_key and e′.eval_key in the where clause of m1, but not in the with clause of m1).

For each formula m, Skolemization adds an equality in the with clause of m for each target atomic element that is not determined by m. This additional equality equates the target atomic element with a Skolem term that is constructed by creating a new function symbol and applying it to a list of arguments consisting of all the source atomic elements that appear in the with clause of m. The resulting formula is another SO nested tgd m′. If two target atomic elements are constrained to be equal by the where clause of m, then the same Skolem term (i.e., the same function symbol) will be used in m′.

As an example of Skolemization, the formulas m1and m2of the earlier schema mapping M12are respectively transformed into the following formulas m1′ and m2′ (which are also SO nested tgds). The added equalities in the with clauses m1′ and m2′ are shown in italics.

Note that for both m1′ and m2′, since the eval_key component of c′ is constrained to be equal to the eval_key of e′ (in the original formulas), the same Skolem term is being used for the two components (E1(s.sid, s.name, s.course, s.grade) in m1′, and E2(s.sid, s.name, c.course, c.evaluation_file) in m2′). On the other hand, the evaluation_file component of e′ and the grade component of e′ are not equal to anything in the where clause of m1′ and m2′, respectively; therefore evaluation_file and grade are used in unique Skolem terms F(s.sid, s.name, s.course, s.grade) and G(s.sid, s.name, c.course, c.evaluation_file), respectively.

In step32for each target element E in schema S2that is of set type, a rule REis computed such that REdescribes how data instances for the element E can be created based on the Skolemized SO nested tgds relevant for E obtained in step31. Each such rule REincludes a union of a plurality of query terms that creates data instances for the element E based on all the skolemized SO nested tgds that are relevant for E (i.e., each Skolemized SO nested tgd recites E in its exists clause).

For example, the element “students” in the earlier schema S2is of a set type. Moreover, among the two Skolemized SO nested tgds, m1′ and m2′, that are generated in step31, both m1′ and m2′ are relevant for “students” since “students” is recited in the exists clause of both m1′ and m2′. Hence a rule that includes a union of two query terms is generated for “students”:

Each query term joined by the union operator ∪ in Rstudentsincludes a for clause that is the same as the for clause in the corresponding SO nested tgd. Furthermore, if the SO nested tgd includes a where clause following immediately the for clause, then this where clause is also included in the query term. In addition, each query term has a return clause that specifies how to construct the atomic components of a student record, based on the with clause of the corresponding SO nested tgd. For example, in the first query term, the sid component is to be constructed by taking the value of s.sid, where s represent a student record in the source. The correct expressions (e.g., s.sid) are decided based on the with clause of the corresponding Skolemized SO nested tgd, which specifies what the values of the target atomic components should be.

In addition, in the return clause, each component that is of set type is constructed by invoking a rule that is similar to the above rule Rstudentsbut is parameterized by the values of the atomic components of the record. For example, the “courses” component is constructed by invoking a rule Rcourseswith parameters s.sid and s.name. Such invocation constructs one set of course records for each different combination of student id and student name.

The rule for “courses”, given the above two Skolemized SO nested tgds, is shown below. Again, as before, both m1′ and m2′ are relevant (their exists clauses both contain “courses”). Therefore, two query terms appear in the union. Furthermore, each query term has a filtering condition in its where clause so that it generates course records only for students whose sid and name values match the values of the parameters (11and 12).

In general, such a parameterized rule is constructed for every element of set type that is nested inside another set type. Rules for top-level set-type elements do not need to be parameterized. As another example of such top-level rule, the rule for “evaluations” is listed below.

In step33, a composition-with-rules algorithm is executed to replace in mapping M23all references to schema S2with references to schema S1. Let a rule set R12be the set of all the rules that result after applying step32. The composition-with-rules algorithm includes the following steps33A,33B, and33C.

In step33A, a mapping holder R is initialized to be M23. For the example under discussion, R includes only one formula, m3, shown earlier. Generally, R includes all formulas m comprised by mapping M23. The formulas in the mapping holder R resulting from step33A will be transformed by steps33B and33C to a form that has eliminated all references to S2to become the output mapping M13of Algorithm 1.

In step33B, for each SO nested tgd m in R, each top-level set-type element that is mentioned in formula m is replaced by the corresponding rule (for that element) in the rule set R12. For the example under discussion, the formula m3is transformed into the following formula:

where the notation <body of Rstudents> represents a short-hand for the union of query terms that is on the right-hand side of the equal symbol in the above definition of Rstudents. The notation <body of Revaluations> is a similar short-hand for the case of Revaluations.

In step33C, each formula in R that results after step33B is rewritten by using at least one rewrite rule of four rewrite rules. These rewrite rules, called de-nesting, union separation, record projection and case reduction, are illustrated inFIGS. 10 and 11, in accordance with embodiments of the present invention.FIG. 10illustrates de-nesting, andFIG. 11illustrates union separation, record projection, and case reduction.

For a given formula m in R, the de-nesting rule removes inner nested expressions such as the inner nested expression {for (y1in Y1) . . . (ykin Yk) where B return r} inFIG. 10via replacement of each inner nested expression by the notation e[g→r] in a relevant clause(s) of formula m to simulate the functionality of each replaced inner nested expression and via insertion of the generators (y1in Y1) . . . (ykin Yk) in the outer for clause. The de-nesting rule assumes that all the variables in the inner for_where_return_expression are different from (i.e., do not conflict with) the outer variables. This will be accomplished by renaming all the inner variables before applying every de-nesting step.

The union separation rule separates the N expressions (N at least 2) joined by the union operator (∪) in a given formula m in R into N formulas. InFIG. 11, for example, the formula comprising the two expressions X1and X2joined as X1∪X2is separated into the two formulas shown, namely a first formula comprising X1and a second formula comprising X2.

The case reduction rule has two cases. First, whenever a formula m includes an appearance of <Li=ei>. Lithis appearance of <Li=ei>. Liis replaced by ei. However, if the formula m includes an appearance of <Li=ei>. Lj, where Liand Ljare different, then the formula is abandoned. The reason for this abandoning of the formula is that the formula can never be satisfied, since the choice expression <Li=ei> includes a component (element) called Libut the larger expression tries to obtain a component called Lj. Since the formula cannot be satisfied, there is no need to include the formula in the final result of composition. Here, the notation ⊥ is used to denote formally an abandoned formula (or, formula that cannot be satisfied).

The rewriting process is as follows. While there is some formula m′ in R for which some rewrite rule applies to it (or to some subexpression of it), the method of the present invention applies the rewrite rule to m′, adds the resulting formulas (if not equal to ⊥) to R and removes m′ from R.

For example, for the above m′3, since <body of Rstudents> is the union of two query terms (which we can be denoted here, in short, T1and T2), it follows that the union separation rule is applicable and results in the following two formulas:for (s in T1) (c in s.courses) (e in <body of Revaluations>)where c.eval_key=e.eval_keyexists (s′ in new_tgt.students) (r′ in s′.results) (e′ in new_tgt.evaluations)where r′.eval_key=e′.eval_keywith s.sid=s′.sid and e.eval_key=e′.eval_keye.grade=e′.grade and e.evaluation_file=e′.evaluation_filefor (s in T2) (c in s.courses) (e in <body of Revaluations>)where c.eval_key=e.eval_keyexists (s′ in new_tgt.students) (r′ in s′.results) (e′ in new_tgt.evaluations)where r′.eval_key=e′.eval_keywith s.sid=s′.sid and e.eval_key=e′.eval_keye.grade=e′.grade and e.evaluation_file=e′.evaluation_file

The mapping m′3is removed and the above two formulas are added to R. The rewriting process continues by trying to rewrite the first of the two formulas. Since T1is the expression

Note that the original variable s has been replaced, in each of the two places where it occurred, by the record expression [sid=s1.sid, name=s1.name, courses=Rcourses(s1.sid, s1.name)] that is in the return clause of T1. Now the record projection rule can be applied twice and the above formula is replaced by the following:for (s1in src1.students)(c in Rcourses(s1.sid, s1.name))(e in <body of Revaluations>)where c.eval_key=e.eval_keyexists (s′ in new_tgt.students) (r′ in s′.results) (e′ in new_tgt.evaluations)where r′.eval_key=e′.eval_keywith s1.sid=s′.sid and e.eval_key=e′.eval_keye.grade=e′.grade and e.evaluation_file=e′.evaluation_file

This formula is the same as:for (s1in src1.students)(c in <body of Rcourseswhere l1is replaced by s1.sid and l2is replaced by s1.name>)(e in <body of Revaluations>)where c.eval_key=e.eval_keyexists (s′ in new_tgt.students) (r′ in s′.results) (e′ in new_tgt.evaluations)where r′.eval_key=e′.eval_keywith s1.sid=s′.sid and e.eval_key=e′.eval_keye.grade=e′.grade and e.evaluation_file=e′.evaluation_file

This process continues then by applying the union separation rule for the body of Rcourses, then de-nesting, and so on.

The algorithm terminates with a set M13of SO nested tgds that mention only schema S1and schema S3. This set M13is the schema mapping that is the composition of M12and M23, with respect to the relationship semantics. For the example, M13includes the below formulas (and more):

In consideration of the preceding discussion, step33C is implemented for each formula in R that results after step33B, by performing the process depicted in the flow chart ofFIG. 9Cfor each formula. The flow chart ofFIG. 9Ccomprises steps51-57.

Step51determines whether union separation is applicable. If step51determines that union separation is not applicable, then step53is next executed. If step51determines that union separation is applicable then the union separation is executed in step52, followed by iterative re-execution of the loop of steps51-52until union separation no longer applies in step51and the process next executes step53.

Step53determines whether a de-nesting of a complex type expression is applicable. If step53determines that said de-nesting is not applicable, then step55is next executed. If step53determines that said de-nesting is applicable then said de-nesting is executed in step54, followed by iterative re-execution of the loop of steps53-53until de-nesting no longer applies in step53and the process next executes step55. Each de-nesting de-nests an outermost complex type expression in the formula being processed.

Step55performs all case reductions and record projections that are applicable.

Step56determines whether the formula being processed comprises an expression of the form R(e1, . . . , en), wherein R(e1, . . . , en) is a rule of the ruleset R12that results from step32ofFIG. 9A. If step56determines that the formula being processed does not comprise such an expression of the form R(e1, . . . , en) then the process ends. If step56determines that the formula being processed comprises such an expression of the form R(e1, . . . , en) then step57replaces R(e1, . . . , en) by the body of R such that the parameters of R (namely, l1, . . . , ln) are respectively replaced by e1, . . . , en. Step57was illustrated supra in an example in which Rcourseswas replaced by the body of Rcourses, and the parameters l1and l2in the body of Rcourseswere replaced by s1.sid and s1.name, respectively.

After step57is executed, the process re-executes the loop51-57iteratively until there no longer remains an expression of the form R(e1, . . . , en) and the process ends.

The set of all such formulas, when taken in their entirety, describes the relationship between instances over S1and instances over S2that is implied by the relationships induced by the two given schema mappings (from S1to S2and from S2to S3, respectively), characterized by not mentioning schema S2(i.e., the elements of tgt.students, tgt.students.courses, and tgt.evaluations of schema S2are not recited in formulas m13and m′13of mapping M13).

As illustrated in the preceding examples, the particular rewrite rules invoked, and the order and number of times that said particular rewrite rules are invoked, depends on the structure of each formula m in R, such that application of the particular rewrite rules to m eliminates all references to schema S2.

4. Algorithm 2: Composition with Respect to Transformation Semantics

The input to Algorithm 1 is as follows. Schema mappings M12and M23are inputs expressed as constraints in the language of second-order nested tuple-generating dependencies (or SO nested tgds). The schema mapping M12is a set of SO nested tgds that relate a schema S1and a schema S2. Similarly, schema mapping M23is a set of SO nested tgds that relate a schema S2to a schema S3.

The output to Algorithm 1 is as follows. Schema mapping M13, consisting of a set of SO nested tgds that relate schema S1and schema S3, is an output representing the composition of M12and M23under the transformation semantics.

FIG. 12is a flow chart depicting steps41-45of Algorithm 2 for determining a composition of schema mappings with respect to transformation semantics, in accordance with embodiments of the present invention. In the discussion ofFIG. 12, the term “target element” refers to an element of schema S2.

In step44, each of the SO nested tgds in the mapping holder R that result after step43, with no remaining reference to schema S2, is reduced by the applying the following procedure. Each equality that appears in the first where clause of the SO nested tgd and involves Skolem terms is processed as follows. If the equality is of the form F(t1, . . . , tn)=F(t′1, . . . , t′n) (i.e., equating Skolem terms with the same Skolem function) then the equality of said form is replaced by t1=t′1and . . . and tn=t′nto construct a new SO nested tgd that replaces the old SO nested tgd. If this new SO nested tgd still contains some equality involving Skolem terms in the first where clause, then step44is applied again (recursively) to the new SO nested tgd. If the equality is of the form F(t1, . . . , tn)=G(t′1, . . . , t′n) (i.e., equating Skolem terms with different Skolem functions) or of the form e=F(t1, . . . , tn) then eliminate the current SO nested tgd from any further processing. In other words, step44is applied recursively to eliminate all Skolem term equalities in which different Skolem functions are equated or in which a Skolem term is equated to a non Skolem term.

Since none of the equalities in the first where clause contain a Skolem term, step44finishes here, for the above SO nested tgd (that is, there is no need for a recursive application of step44).

As another example, the formula m′13shown earlier is eliminated in step44, because m′13contains in its first where clause the equality E1(s2.sid, s2.name, s2.course, s2.grade)=E2(s3.sid, s3.name, s3.course, s3.grade), between two Skolem terms with different functions, E1and E2.

Step45minimizes the resulting SO nested tgds in the mapping holder R. For each SO nested tgd that results after step44, step45finds an equivalent SO nested tgd that has a minimal number of logically required variables in the for clause of each SO nested tgd. For example, the above formula can be shown to be equivalent to the following formula that uses just one variable in the for clause:for (s3in src1.students)exists (s′ in new_tgt.students) (r′ in s′.results) (e′ in new_tgt.evaluations)where r′.eval_key=e′.eval_keywith s3.sid=s′.sid and E1(s3.sid, s3.name, s3.course, s3.grade)=e′.eval_keyand s3.grade=e′.grade andF(s3.sid, s3.name, s3.course, s3.grade)=e′.evaluation_file

Intuitively, the reason why the earlier formula is equivalent to the above formula is that all that is needed from variables s1and s2in the with clause is s1.sid. But this can be replaced with s3.sid, since these two expressions are equal. Furthermore, whenever the variable s3can be instantiated to a tuple, the variables s1and s2can also be instantiated to the same tuples such that their pattern of equalities is satisfied. Hence, the formula is the same whether or not we assert s1and s2in the for clause (s3is enough).

The formal procedure for minimization of SO nested tgds is similar to the minimization of conjunctive queries in database query optimization.

The output of the algorithm is the set of all minimized SO nested tgds that result after step45. As an example, the above SO nested tgd is part of the final result of composing schema mappings M12and M23, under the transformation semantics. Another formula that is also part of this final result is:for (s in src2.students) (c in src2.courses)where s.course_code=c.course_codeexists (s′ in new_tgt.students) (r′ in s′.results) (e′ in new_tgt.evaluations)where r′.eval_key=e′.eval_keywith s.sid=s′.sid and E2(s.sid, s.name, c.course, c.evaluation_file)=e′.eval_keyand G(s.sid, s.name, c.course, c.evaluation_file)=e′.grade andc.evaluation_file=e′.evaluation_file

The set of all such formulas in M13capture all the different ways of transforming the data from schema S1to schema S3that are equivalent to all the different ways of first transforming data from schema S1to the intermediate schema S2(as dictated by M12) followed by all the different ways of transforming the resulting data from the intermediate schema S2to schema S3(as dictated by M23).

It is noted that, although Algorithm 2 is Algorithm 1 with added steps44and45, the result of Algorithm 2 is a set of formulas that is simpler (has fewer and also simpler formulas) than the result of Algorithm 1. If the main intention of schema mappings is for data transformation (and not for describing relationships between schemas) Algorithm 1 is to be preferred to Algorithm 2. However, if the primary intention of schema mappings is representing relationships between schemas, then Algorithm 1 is to be preferred, since Algorithm 2 is not guaranteed to return a schema mapping with equivalent relationship semantics.

It is noted that Algorithms 1 and 2 are each applicable to a variety of schemas, including relational database schemas, XML schemas, hierarchical schemas, etc.

5. Computer System

FIG. 13illustrates a computer system90used for generating a schema mapping, in accordance with embodiments of the present invention. The computer system90comprises a processor91, an input device92coupled to the processor91, an output device93coupled to the processor91, and memory devices94and95each coupled to the processor91. The input device92may be, inter alia, a keyboard, a mouse, etc. The output device93may be, inter alia, a printer, a plotter, a computer screen, a magnetic tape, a removable hard disk, a floppy disk, etc. The memory devices94and95may be, inter alia, a hard disk, a floppy disk, a magnetic tape, an optical storage such as a compact disc (CD) or a digital video disc (DVD), a dynamic random access memory (DRAM), a read-only memory (ROM), etc. The memory device95includes a computer code97. The computer code97includes an algorithm for generating a schema mapping. The processor91executes the computer code97. The memory device94includes input data96. The input data96includes input required by the computer code97. The output device93displays output from the computer code97. Either or both memory devices94and95(or one or more additional memory devices not shown inFIG. 13) may be used as a computer usable medium (or a computer readable medium or a program storage device) having a computer readable program code embodied therein and/or having other data stored therein, wherein the computer readable program code comprises the computer code97. Generally, a computer program product (or, alternatively, an article of manufacture) of the computer system90may comprise said computer usable medium (or said program storage device).

Thus the present invention discloses a process for deploying or integrating computing infrastructure, comprising integrating computer-readable code into the computer system90, wherein the code in combination with the computer system90is capable of performing a method for generating a schema mapping.