1. XML Views & Reasoning about Views
Zachary G. Ives
University of Pennsylvania
CIS 550 – Database & Information Systems
December 9, 2018
Some slide content courtesy of Susan Davidson, Dan Suciu, & Raghu Ramakrishnan

2. How Do You Compute XML from a SQL Query over Relational Data?
Assume we know the structure of the XML tree and that it’s finite-depth (we’ll see how to avoid this later)
We can compute an “XML-like” SQL relation using “outer unions” – we first used this technique in XPERANTO (now IBM DB2 XTABLES)
Idea: if we take two non-union-compatible expressions, pad each with NULLs, we can UNION them together
Let’s see how this works…

3. A Relation that Mirrors the XML Hierarchy
Output relation would look like:
rLabel
rid
rOrd
clabel
cid
cOrd
sLabel
sid
sOrd
str
int
tree 1 -
content
sub-content 2
XYZ
i-content 3 14

4. A Relation that Mirrors the XML Hierarchy
Output relation would look like:
rLabel
rid
rOrd
clabel
cid
cOrd
sLabel
sid
sOrd
str
int
tree 1 -
content
sub-content 2
XYZ
i-content 3 14

5. A Relation that Mirrors the XML Hierarchy
Output relation would look like:
rLabel
rid
rOrd
clabel
cid
cOrd
sLabel
sid
sOrd
str
int
tree 1 -
content
sub-content 2
XYZ
i-content 3 14
Colors are representative of separate SQL queries…

6. 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 int from Edge E where 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 E1 where E.parent IS NULL AND E.target = E1.parent

7. The Rest of the Queries Grandchild: Strings: How would we do integers?
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 E2 where 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 int from 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?

8. 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!

9. “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

10. Inlining Techniques
Start with DTD, build a graph representing structure
tree ?
@id *
content
@id * *
sub-content
i-content
The edges are annotated with ?, * indicating repetition, optionality of children
They simplify the DTD to figure this out

11. 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
book
author
name

12. Schemas for Our Example
TheRoot(rootID)
Content(parentID,
Sub-content(parentID, varchar)
I-content(parentID, int)
If we suddenly changed DTD to <!ELEMENT content(sub-content*, i-content?) what would happen?

13. 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

14. An Example
for $X in document(“mydoc”)/tree/content where $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,
Sub-content(parentID, varchar)
I-content(parentID, int)

15. 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

16. 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

17. 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

18. 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 pq, pq p, pq
where 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

19. 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

20. 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)
body
head
subgoals

21. 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)

22. 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?

23. A Query We Can’t Answer 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 1
reachable(F, T, 2) :- edge(F, X), edge(X, T) dist. 2
reachable(F, T, 3) :- reachable(F, X, 2), edge(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 )

24. Recursive Datalog Queries
Define a recursive query in datalog:
reachable(F, T, 1) :- edge(F, T) distance 1
reachable(F, T, D + 1) :- reachable(F, X, D), edge(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!
The example not-fully-recursive query should be the following:
reachable(From, To, 1) :- edge(From, To)
reachable(From, To, 2) :- reachable(From, Midpoint, 1), edge(Midpoint, To)
reachable(From, To, 3) :- reachable(From, Midpoint, 2), edge(Midpoint, To)
...
The first rule gets all connections between points where there's a direct edge.
The second rule finds all connections from distance-1 nodes to one further node (i.e., distance-2 connections)
The third rule finds all connections between nodes separated by distance 2, followed by yet another connection of distance 1 (distance-3 connections)
etc.
We can describe it recursively as:
reachable(From, To, D+1) :- reachable(From, Midpoint, D), edge(Midpoint, To)
which will expand out to the above example.
Now this example is somewhat different from our XML example, where we're trying to find all elements at distance k from the *root* node, where k <= the depth of the tree.
Here we can create a very similar query:
elementsAt(Parent, ID, 1) :- edge(Parent, ID), Parent = NULL
elementsAt(Parent, ID, D+1) :- elementsAt(Parent, LastNode, D), edge(LastNode, ID)
The base case finds the root node. The recursive case finds the nodes connected by an edge from the root; then the nodes connected by an edge from those; etc.

25. 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!)

26. Our Query in RA + while (inflationary semantics, no negation)
Datalog:
reachable(F, T, 1) :- edge(F, T)
reachable(F, T, D+1) :- reachable(F, X, D), edge(X, T)
RA procedure with while:
reachable += edge ⋈ literal1
while change {
reachable += F, T, D(F ! X(edge) ⋈ T ! X,D ! D0(reachable) ⋈ add1) }
Note literal1(F,1) and add1(D0,D) are actually arithmetic and literal functions modeled here as relations.

27. 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)

28. 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 predicates
Unsafe: 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)

29. A Special Type of Query: 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 queries
An 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?

30. Example of Containment
Suppose we have two queries: q1(S,C) :- Student(S, N), Takes(S, C), Course(C, X), inCIS(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}

31. 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