XML Views & Reasoning about Views

Zachary G. IvesUniversity of Pennsylvania

CIS 550 – Database & Information Systems

November 4, 2004

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

2

A View as a Translation betweenXML and Relations

You’ll be reviewing the most-cited paper in this area (Shanmugasundaram et al), and there are many more (Fernandez et al., …)

Techniques already making it into commercial systems XPERANTO at IBM Research will appear in DB2

v8 or 9 SQL Server has some XML support; Oracle is

also doing some XML … Now you’ll know how it works!

3

Issues in Mapping Relational XML

We know the following: XML is a tree XML is SEMI-structured

There’s some structured “stuff” There is some unstructured “stuff”

Issues relate to describing XML structure, particularly parent/child in a relational encoding Relations are flat Tuples can be “connected” via

foreign-key/primary-key links

4

The Simplest Way to Encode a Tree

Suppose we had:<tree id=“0”> <content id=“1”> <sub-content>XYZ </sub-content> <i-content>14 </i-content> </content></tree>

If we have no IDs, we CREATE values…

BinaryLikeEdge(key, label, type, value, parent)

key

label type

value

parent

0 tree ref - -

1 content ref - 0

2 sub-content

ref - 1

3 i-content ref - 1

4 - str XYZ 2

5 - int 14 3What are shortcomings here?

5

Florescu/Kossmann Improved Edge Approach

Consider order, typing; separate the values Edge(parent, ordinal,

label, flag, target)

Vint(vid, value)

Vstring(vid, value)

parent

ord

label flag

target

- 1 tree ref 0

0 1 content ref 1

1 1 sub-content

str v2

1 1 i-content int v3

vid

value

v3 14

vid

value

v2 XYZ

6

How Do You Compute the XML?

Assume we know the structure of the XML tree (we’ll see how to avoid this later)

We can compute an “XML-like” SQL relation using “outer unions” – we first this technique in XPERANTO Idea: if we take two non-union-compatible

expressions, pad each with NULLs, we can UNION them together

Let’s see how this works…

7

A Relation that Mirrors theXML Hierarchy

Output relation would look like:

rLabel

rid

rOrd

clabel

cid

cOrd

sLabel sid

sOrd

str int

tree 0 1 - - - - - - - -

- 0 1 content

1 1 - - - - -

- 0 1 - 1 1 sub-content

2 1 - -

- 0 1 - 1 1 - 2 1 XYZ

-

- 0 1 - 1 2 i-content 3 1 - -

- 0 1 - 1 2 - 3 1 - 14

8

A Relation that Mirrors theXML Hierarchy

Output relation would look like:

rLabel

rid

rOrd

clabel

cid

cOrd

sLabel sid

sOrd

str int

tree 0 1 - - - - - - - -

- 0 1 content

1 1 - - - - -

- 0 1 - 1 1 sub-content

2 1 - -

- 0 1 - 1 1 - 2 1 XYZ

-

- 0 1 - 1 2 i-content 3 1 - -

- 0 1 - 1 2 - 3 1 - 14

9

A Relation that Mirrors theXML Hierarchy

Output relation would look like:

rLabel

rid

rOrd

clabel

cid

cOrd

sLabel sid

sOrd

str int

tree 0 1 - - - - - - - -

- 0 1 content

1 1 - - - - -

- 0 1 - 1 1 sub-content

2 1 - -

- 0 1 - 1 1 - 2 1 XYZ

-

- 0 1 - 1 2 i-content 3 1 - -

- 0 1 - 1 2 - 3 1 - 14

Colors are representative of separate SQL queries…

10

SQL for Outputting XML

For each sub-portion we preserve the keys (target, ord) of the ancestors

Root:select E.label AS rLabel, E.target AS rid, E.ord AS rOrd, null AS cLabel, null AS cid, null AS cOrd, null AS subOrd, null AS sid, null AS str, null AS intfrom Edge Ewhere parent IS NULL

First-level children:select null AS rLabel, E.target AS rid, E.ord AS rOrd, E1.label AS cLabel, E1.target AS cid, E1.ord AS cOrd, null AS …from Edge E, Edge E1where E.parent IS NULL AND E.target = E1.parent

11

The Rest of the Queries

Grandchild:select null as rLabel, E.target AS rid, E.ord AS rOrd, null AS cLabel, E1.target AS cid, E1.ord AS cOrd, E2.label as sLabel, E2.target as sid, E2.ord AS sOrd, null as …from Edge E, Edge E1, Edge E2where E.parent IS NULL AND E.target = E1.parent AND E1.target = E2.parent

Strings:select null as rLabel, E.target AS rid, E.ord AS rOrd, null AS cLabel, E1.target AS cid, E1.ord AS cOrd, null as sLabel, E2.target as sid, E2.ord AS sOrd, Vi.val AS str, null as intfrom Edge E, Edge E1, Edge E2, Vint Vi where E.parent IS NULL AND E.target = E1.parent AND E1.target = E2.parent AND Vi.vid = E2.target

How would we do integers?

12

Finally…

Union them all together:( select E.label as rLabel, E.target AS rid, E.ord AS rOrd, … from Edge E where parent IS NULL)UNION ( select null as rLabel, E.target AS rid, E.ord AS rOrd, E1.label AS cLabel, E1.target AS cid, E1.ord AS cOrd, null as … from Edge E, Edge E1 where E.parent IS NULL AND E.target = E1.parent) UNION ( . :) UNION ( . :)

Then another module will add the XML tags, and we’re done!

13

“Inlining” Techniques

Folks at Wisconsin noted we can exploit the “structured” aspects of semi-structured XML If we’re given a DTD, often the DTD has a lot of

required (and often singleton) child elements Book(title, author*, publisher)

Recall how normalization worked: Decompose until we have everything in a relation

determined by the keys … But don’t decompose any further than that

Shanmugasundaram et al. try not to decompose XML beyond the point of singleton children

14

Inlining Techniques

Start with DTD, build a graph representing structure

tree

content

sub-content i-content

*

* *

• The edges are annotated with ?, * indicating repetition,optionality of children

• They simplify the DTD to figure this out

@id

@id

?

15

Building Schemas

Now, they tried several alternatives that differ in how they handle elements w/multiple ancestors Can create a separate relation for each path Can create a single relation for each element Can try to inline these

For tree examples, these are basically the same Combine non-set-valued things with parent Add separate relation for set-valued child elements Create new keys as needed

name

book author

16

Schemas for Our Example

TheRoot(rootID) Content(parentID, id, @id) Sub-content(parentID, varchar) I-content(parentID, int)

If we suddenly changed DTD to <!ELEMENT content(sub-content*, i-content?) what would happen?

17

XQuery to SQL

Inlining method needs external knowledge about the schema Needs to supply the tags and info not stored in

the tables

We can actually directly translate simple XQuery into SQL over the relations – not simply reconstruct the XML

18

An Example

for $X in document(“mydoc”)/tree/contentwhere $X/sub-content = “XYZ”return $X

The steps of the path expression are generally joins … Except that some steps are eliminated by the fact

we’ve inlined subelements Let’s try it over the schema:

TheRoot(rootID)Content(parentID, id, @id)Sub-content(parentID, varchar)I-content(parentID, int)

19

XML Views of Relations

We’ve seen that views are useful things Allow us to store and refer to the results of

a query We’ve seen an example of a view that

changes from XML to relations – and we’ve even seen how such a view can be posed in XQuery and “unfolded” into SQL

20

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? 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

21

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

22

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

23

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

24

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

25

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)

26

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?

27

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

28

Recursive Datalog Queries

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

reachable(F, T) :- edge(F, X), reachable(X, T)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!

29

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!)

30

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

Datalog:reachable(F, T) :- edge(F, T)reachable(F, T) :- edge(F, X), reachable(X, T)

RA procedure with while:reachable += edgewhile change {

reachable += F, T(T ! X(edge) ⋈ F ! X(reachable))

}

31

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)

32

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))

33

Negation and Recursion

Unfortunately, the fixpoint semantics we mentioned previously for recursion “breaks” with negation… q(x) : q(x) No fixpoint!

p(x) : q(x) Multiple minimal fixpoints!q(x) : p(x)

Or the fixpoint may not “converge” (or converge to a minimal fixpoint)

This is all bad news…

34

One Way to Fix Things: Stratified Semantics

Start with semipositive datalog: negation only over edb predicates From here, we can compute a set of values for

the idb predicates that depend on the edb’s

Now we can “materialize” the results of the first set of idbs; we’ll remove their rules and treat them as edbs to compute a next “stratum”

r (1,1)r (1,2)s (1,1)v1(x,y) :- r(x,y), : s(x,y)q(x,y) :- v1(x,y), : s(x,y)

r (1,1)r (1,2)s (1,1)v1 (1,2)q(x,y) :- v1(x,y), : s(x,y)

r (1,1)r (1,2)s (1,1)v1 (1,2)q (1,2)

35

Precedence Graphs

Take a set of datalog rules, create a node for each relation (edb or idb) If r r’ then add an edge labeled “+” from r

to r’ If r : r’ then add an edge labeled “-” from r

to r’ We can stratify if there are no cycles with “-”

edgesv1(x,y) :- r(x,y), : s(x,y)q(x,y) :- v1(x,y), : s(x,y)

sr

v1 q-

+

+

36

Stratifying Datalog Execution

foreach predicate p, set stratum(p) = 1do until no change, or some stratum > # of predicates

foreach rule h b {foreach negated subgoal of b with predicate q {stratum(p) = max(stratum(p), 1+stratum(q))}foreach positive subgoal of b with predicate q {stratum(p) = max(stratum(p), stratum(q)}

}

37

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?

38

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}

39

Checking Containment via Canonical DBs

To test for q1 µ q2: Create a “canonical DB” that contains a tuple for

each subgoal in q1 Execute q2 over it If q2 returns a tuple that matches the head of q1,

then q1 µ q2

(This is an NP-complete algorithm in the size of the query. Testing for full first-order logic queries is undecidable!!!)

Let’s see this for our example…

40

Example Canonical DB

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)

Student Takes Course inCSE

S N S C C X

C DB & Info

Systems

S

Need to get tuple <S,C> in executing q2 over this database

41

Wrapping up…

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 saw an example of testing for containment Next time we’ll see some further examples