Recursive Views and Global Views

Zachary G. IvesUniversity of Pennsylvania

CIS 550 – Database & Information Systems

November 9, 2004

Some slide content courtesy of Susan Davidson, Dan Suciu, & Raghu Ramakrishnan

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Where We Are…

We’ve seen how views are useful both within a data model, and as a way of going from one model to another

You read the Shanmugasundaram paper on relational XML conversion There have been many follow-up pieces of work There have been attempts to build “native XML” databases

instead Now we’re going to talk about another important way

views can be used to fix a limitation of the XML relational mappings We’ll also talk about how certain classes of views can be

manipulated and reasoned about in interesting ways Then we’ll consider the use of views in integrating data

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An Important Set of Questions

Views are incredibly powerful formalisms for describing how data relates: fn: rel … rel rel

Can I define a view recursively? Why might this be useful in the XML construction

case? When should the recursion stop? Suppose we have two views, v1 and v2

How do I know whether they represent the same data?

If v1 is materialized, can we use it to compute v2? This is fundamental to query optimization and data

integration, as we’ll see later

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Reasoning about Queries and Views

SQL or XQuery are a bit too complex to reason about directly Some aspects of it make reasoning about SQL

queries undecidable

We need an elegant way of describing views (let’s assume a relational model for now) Should be declarative Should be less complex than SQL Doesn’t need to support all of SQL –

aggregation, for instance, may be more than we need

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Let’s Go Back a Few Weeks…Domain Relational Calculus

Queries have form:

{<x1,x2, …, xn>| p }

Predicate: boolean expression over x1,x2, …, xn We have the following operations:

<xi,xj,…> R xi op xj xi op const const op xi

xi. p xj. p pq, pq p, pqwhere op is , , , , , and

xi,xj,… are domain variables; p,q are predicates Recall that this captures the same

expressiveness as the relational algebra

domain variables predicate

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A Similar Logic-Based Language:Datalog

Borrows the flavor of the relational calculus but is a “real” query language Based on the Prolog logic-programming language A “datalog program” will be a series of if-then rules

(Horn rules) that define relations from predicates

Rules are generally of the form:Rout(T1) R1(T2), R2(T3), …, c(T2 [ … Tn)

where Rout is the relation representing the query result, Ri are predicates representing relations, c is an expression using arithmetic/boolean predicates

over vars, and Ti are tuples of variables

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Datalog Terminology

An example datalog rule:idb(x,y) r1(x,z), r2(z,y), z < 10

Irrelevant variables can be replaced by _ (anonymous var)

Extensional relations or database schemas (edbs) are relations only occurring in rules’ bodies – these are base relations with “ground facts”

Intensional relations (idbs) appear in the heads – these are basically views

Distinguished variables are the ones output in the head

Ground facts only have constants, e.g., r1(“abc”, 123)

head subgoals

body

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Datalog in Action

As in DRC, the output (head) consists of a tuple for each possible assignment of variables that satisfies the predicate We typically avoid “8” in Datalog queries:

variables in the body are existential, ranging over all possible values

Multiple rules with the same relation in the head represent a union

We often try to avoid disjunction (“Ç”) within rules Let’s see some examples of datalog queries

(which consist of 1 or more rules): Given Professor(fid, name), Teaches(fid, serno, sem),

Courses(serno, cid, desc), Student(sid, name) Return course names other than CIS 550 Return the names of the teachers of CIS 550 Return the names of all people (professors or students)

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Datalog is Relationally Complete

We can map RA Datalog: Selection p: p becomes a datalog subgoal

Projection A: we drop projected-out variables from head Cross-product r s: q(A,B,C,D) r(A,B),s(C,D) Join r ⋈ s: q(A,B,C,D) r(A,B),s(C,D), condition Union r U s: q(A,B) r(A,B) ; q(C, D) :- s(C,D) Difference r – s: q(A,B) r(A,B), : s(A,B)

(If you think about it, DRC Datalog is even easier)

Great… But then why do we care about Datalog?

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A Query We Can’tAnswer in RA/TRC/DRC…

Recall our example of a binary relation for graphs or trees (similar to an XML Edge relation):

edge(from, to)

If we want to know what nodes are reachable:

reachable(F, T, 1) :- edge(F, T) distance 1reachable(F, T, 2) :- edge(F, X), edge(X, T) dist. 2reachable(F, T, 3) :- edge(F, X), dist2(X, T) dist. 3

But how about all reachable paths? (Note this was easy in XPath over an XML representation -- //edge)

(another way of writing )

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Recursive Datalog Queries

Define a recursive query in datalog:reachable(F, T, 1) :- edge(F, T) distance 1reachable(F, T, D + 1) :- edge(F, X),

reachable(X, T, D) distance >1

What does this mean, exactly, in terms of logic? There are actually three different (equivalent)

definitions of semantics All make a “closed-world” assumption: facts should

exist only if they can be proven true from the input – i.e., assume the DB contains all of the truths out there!

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Fixpoint Semantics

One of the three Datalog models is based on a notion of fixpoint: We start with an instance of data, then derive

all immediate consequences We repeat as long as we derive new facts

In the RA, this requires a while loop! However, that is too powerful and needs to be

restricted Special case: “inflationary semantics”

(which terminates in time polynomial in the size of the database!)

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Our Query in RA + while(inflationary semantics, no negation)

Datalog:reachable(F, T, 1) :- edge(F, T)reachable(F, T, D+1) :- edge(F, X), reachable(X, T, D)

RA procedure with while:reachable += edge ⋈ literal1

while change {reachable += F, T, D(T ! X(edge) ⋈ F ! X,D ! D0(reachable) ⋈ add1)

}

Note literal1(F,1) and add1(D0,D) are actually arithmetic and literal functions modeled here as relations.

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Negation in Datalog

Datalog allows for negation in rules It’s essential for capturing RA set difference-

style ops:Professor(, name), : Student(, name)

But negation can be tricky… … You may recall that in the DRC, we had a

notion of “unsafe” queries, and they return here…

Single(X) Person(X), : Married(X,Y)

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Safe Rules/Queries

Range restriction, which requires that every variable: Occurs at least once in a positive relational predicate in

the body, Or it’s constrained to equal a finite set of values by

arithmetic predicatesUnsafe:q(X) r(Y)q(X) : r(X,X)q(X) r(X) Ç t(Y)

Safe:q(X) r(X,Y)q(X) X = 5 q(X) : r(X,X), s(X)q(X) r(X) Ç (t(Y),u(X,Y))

For recursion, use stratified semantics: Allow negation only over edb predicates Then recursively compute values for the idb

predicates that depend on the edb’s (layered like strata)

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Conjunctive Queries

A single Datalog rule with no “Ç,” “:,” “8” can express select, project, and join – a conjunctive query

Conjunctive queries are possible to reason about statically (Note that we can write CQ’s in other languages, e.g., SQL!)

We know how to “minimize” conjunctive queriesAn important simplification that can’t be done for general SQL

We can test whether one conjunctive query’s answers always contain another conjunctive query’s answers (for ANY instance)

Why might this be useful?

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Example of Containment

Suppose we have two queries:

q1(S,C) :- Student(S, N), Takes(S, C), Course(C, X), inCSE(C),

Course(C, “DB & Info Systems”)

q2(S,C) :- Student(S, N), Takes(S, C), Course(C, X)

Intuitively, q1 must contain the same or fewer answers vs. q2: It has all of the same conditions, except one extra conjunction

(i.e., it’s more restricted) There’s no union or any other way it can add more data

We can say that q2 contains q1 because this holds for any instance of our DB {Student, Takes, Course}

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Wrapping up Datalog…

We’ve seen a new language, Datalog It’s basically a glorified DRC with a special feature,

recursion It’s much cleaner than SQL for reasoning about … But negation (as in the DRC) poses some

challenges

We’ve seen that a particular kind of query, the conjunctive query, is written naturally in Datalog Conjunctive queries are possible to reason about We can minimize them, or check containment Conjunctive queries are very commonly used in our

next problem, data integration

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The Data Integration Problem We’ve seen that even with normalization and the

same needs, different people will arrive at different schemas

In fact, most people also have different needs! Often people build databases in isolation, then want

to share their data Different systems within an enterprise Different information brokers on the Web Scientific collaborators Researchers who want to publish their data for others to

use This is the goal of data integration: tie together

different sources, controlled by many people, under a common schema Typically it’s based on conjunctive queries, as with Datalog

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Building a Data Integration System

Create a middleware “mediator” or “data integration system” over the sources Can be warehoused (a data warehouse) or virtual Presents a uniform query interface and schema Abstracts away multitude of sources; consults them for

relevant data Unifies different source data formats (and possibly schemas) Sources are generally autonomous, not designed to be

integrated Sources may be local DBs or remote web sources/services Sources may require certain input to return output (e.g.,

web forms): “binding patterns” describe these

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Data Integration System / Mediator

Typical Data Integration Components

Mediated Schema

Wrapper Wrapper Wrapper

SourceRelations

Mappingsin Catalog

SourceCatalog

Query Results

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Typical Data Integration Architecture

Reformulator

QueryProcessor

SourceCatalog

Wrapper Wrapper Wrapper

Query

Query over sources

SourceDescrs.

Queries +bindings Data in mediated format

Results

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Challenges of Mapping Schemas

In a perfect world, it would be easy to match up items from one schema with another Every table would have a similar table in the other schema Every attribute would have an identical attribute in the other

schema Every value would clearly map to a value in the other schema

Real world: as with human languages, things don’t map clearly! May have different numbers of tables – different

decompositions Metadata in one relation may be data in another Values may not exactly correspond It may be unclear whether a value is the same

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A Few Simple Examples

Movie(Title, Year, Director, Editor, Star1, Star2)

Movie(Title, Year, Director, Editor, Star1, Star2)

PieceOfArt(ID, Artist, Subject, Title, TypeOfArt)

MotionPicture(ID, Title, Year)Participant(ID, Name, Role)CustI

DCustName

1234 Ives, Z.

PennID

EmpName

46732 Zachary Ives

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How Do We Relate Schemas?

General approach is to use a view to define relations in one schema, given data in the other schema This allows us to “restructure” or “recompose +

decompose” our data in a new way

We can also define mappings between values in a view We use an intermediate table defining

correspondences – a “concordance table” It can be filled in using some type of code, and

corrected by hand

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Mapping Our Examples

Movie(Title, Year, Director, Editor, Star1, Star2)

Movie(Title, Year, Director, Editor, Star1, Star2)

PieceOfArt(ID, Artist, Subject, Title, TypeOfArt)

MotionPicture(ID, Title, Year)Participant(ID, Name, Role)

CustID

CustName

1234 Ives, Z.

PennID

EmpName

46732 Zachary Ives

PieceOfArt(I, A, S, T, “Movie”) :- Movie(T, Y, A, _, S1, S2),ID = T || Y, S = S1 || S2

Movie(T, Y, D, E, S1, S2) :- MotionPicture(I, T, Y), Participant(I, D, “Dir”), Participant(I, E, “Editor”), Participant(I, S1, “Star1”), Participant(I, S2, “Star2”)

T1 T2

???

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Two Important Approaches

TSIMMIS [Garcia-Molina+97] – Stanford Focus: semistructured data (OEM), OQL-based language

(Lorel) Creates a mediated schema as a view over the sources Spawned a UCSD project called MIX, which led to a company

now owned by BEA Systems Other important systems of this vein: Kleisli/K2 @ Penn

Information Manifold [Levy+96] – AT&T Research Focus: local-as-view mappings, relational model Sources defined as views over mediated schema

Requires a special Spawned Tukwila at Washington, and eventually a company as

well Led to peer-to-peer integration approaches (Piazza, etc.)

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The Focus of these Systems

Focus: Web-based queryable sources CGI forms, online databases, maybe a few RDBMSs Each needs to be mapped into the system – not as

easy as web search – but the benefits are significant vs. query engines

A few parenthetical notes: Part of a slew of works on wrappers, source profiling,

etc. The creation of mappings can be partly automated –

systems such as LSD, Cupid, Clio, … do this Today most people look at integrating large

enterprises (that’s where the $$$ is!) – Nimble, BEA, IBM

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TSIMMIS

“The Stanford-IBM Manager of Multiple Information Sources” … or, a Yiddish stew

An instance of a “global-as-view” mediation system

One of the first systems to support semi-structured data, which predated XML by several years

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Semi-structured Data: OEM

Observation: given a particular schema, its attributes may be unavailable from certain sources – inherent irregularity

Proposal: Object Exchange Model, OEMOID: <label, type, value>

… How does it relate to XML? … What problems does OEM solve, and

not solve, in a heterogeneous system?

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OEM Example

Show this XML fragment in OEM:

<book> <author>Bernstein</author> <author>Newcomer</author> <title>Principles of TP</title></book>

<book> <author>Chamberlin</author> <title>DB2 UDB</title></book>

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Queries in TSIMMIS

Specified in OQL-style language called Lorel OQL was an object-oriented query language Lorel is, in many ways, a predecessor to XQuery

Based on path expressions over OEM structures:select bookwhere book.author = “DB2 UDB” and book.title = “Chamberlin”

This is basically like XQuery, which we’ll use in place of Lorel and the MSL template language. Previous query restated =

for $b in document(“my-source”)/bookwhere $b/title/text = “DB2 UDB” and $b/author/text() = “Chamberlin”return $b

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Query Answering in TSIMMIS

Basically, it’s view unfolding, i.e., composing a query with a view The query is the one being asked The views are the MSL templates for the

wrappers Some of the views may actually require

parameters, e.g., an author name, before they’ll return answers Common for web forms (see Amazon, Google, …) XQuery functions (XQuery’s version of views) support

parameters as well, so we’ll see these in action

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A Wrapper Definition in MSL

Wrappers have templates and binding patterns ($X) in MSL:B :- B: <book {<author $X>}> // $$ = “select * from book where author=“ $X //

This reformats a SQL query over Book(author, year, title)

In XQuery, this might look like:define function GetBook($X AS xsd:string) as book {

for $x in sql(“select * from book where author=‘” +

$x +”’”)return <book>$x<author>$x</author></book>

} The union of GetBook’s results, plus many others,is the view AllData()

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How to Answer the Query

Given our query:for $b in AllData()/bookwhere $b/title/text() = “DB2 UDB” and $b/author/text() = “Chamberlin”return $b

We want to find all wrapper definitions that: Either contain output enough information that

we can evaluate all of our conditions over the output

Or have already tested the conditions for us!define function AllData($x AS xsd:string) as element* {

return GetBooks($x), …

}

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Query Composition with Views

We find all views that define book with author and title, and we compose the query with each:define function GetBook($x AS xsd:string) as book {

for $b in sql(“select * from book where author=‘” + $x

+”’”)return <book>$b<author>$x</author></book>

}for $b in AllData()/book

where $b/title/text() = “DB2 UDB” and $b/author/text() = “Chamberlin”return $b

We need a value for $x!

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Matching View Output to Our Query’s Conditions

Determine that $b/book/author/text() $x by matching the pattern on the function’s output:define function GetBook($x AS xsd:string) as book {

for $b in sql(“select * from book where author=‘” +

$x +”’”)return <book>$b<author>$x</author></book>

}

where $x = “Chamberlin”for $b in GetBook($x)/bookwhere $b/title/text() = “DB2 UDB” return $b

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The Final Step: Unfolding

where $x = “Chamberlin”for $b in { for $b in

sql(“select * from book where author=‘” + $x +”’”)return <book>$b<author>$x</author></book> }/bookwhere $b/title/text() = “DB2 UDB” return $b

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What Is the Answer?

Given schema book(author, year, title) and datalog rules defining an instance:

book(“Chamberlin”, “1992”, “DB2 UDB”)book(“Chamberlin”, “1995”, “DB2/CS”)

What do we get for our query answer?

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TSIMMIS

Early adopter of semistructured data Can support irregular structure and missing

attributes Can support data from many different sources Doesn’t fully solve heterogeneity problem,

though!

Simple algorithms for view unfolding Easily can be composed in a hierarchy of

mediators

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Limitations of TSIMMIS’ Approach

Some data sources may contain data with certain ranges or properties “Books by Aho”, “Students at UPenn”, … How do we express these? (Important for

performance!)

Mediated schema is basically the union of the various MSL templates – as they change, so may the mediated schema

Next time we’ll see the opposite approach – and some very cool logical inference!