1. Datalog, View Unfolding, Recursion and Foundations of Integration
Zachary G. Ives
University of Pennsylvania
CIS 550 – Database & Information Systems
December 4, 2018
Some slide content courtesy of Susan Davidson, AnHai Doan, Dan Suciu, & Raghu Ramakrishnan

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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)

8. 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?

9. View Unfolding
We can define views (idbs) using one or more datalog rules:
dbirProfs(p,subj) :- Teaches(fid,cid), COURSE(cid,subj,serno,sem), subj=“DB”
dbirProfs(p,subj) :- Teaches(fid,cid), COURSE(cid,subj,serno,sem), subj=“IR”
Now if we query the relation, we can compose or unfold the query + view:
dbProfs(p,nam) :- dbirProfs(p,“DB”), PROFESSOR(p,nam)

10. 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 )

11. 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.

12. 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!)

13. 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.

14. 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)

15. 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)

16. 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?

17. 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}

18. 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

19. A 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

20. 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

21. Typical Data Integration Components
Query
Results
Data Integration System / Mediator
Mediated Schema
Source
Catalog
Mappings
in Catalog
Wrapper
Wrapper
Wrapper
Source
Relations

22. Typical Data Integration Architecture
Query
Source
Descrs.
Reformulator
Source
Catalog
Query over sources
Query Processor
Results
Queries +
bindings
Data in mediated format
Wrapper
Wrapper
Wrapper

23. 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

24. Different Aspects to Mapping
Schema matching / ontology alignment
How do we find correspondences between attributes?
Entity matching / deduplication / record linking / etc.
How do we know when two records refer to the same thing?
Mapping definition
How do we specify the constraints or transformations that let us reason about when to create an entry in one schema, given an entry in another schema?
Let’s see one influential approach to schema matching…

25. The Basic Design Pattern (Est. by a System Called LSD)
Suppose user wants to integrate 100 data sources
User:
manually creates mappings for a few sources, say 3
shows LSD these mappings
System learns from the mappings
“Multi-strategy” learning incorporates many types of info in a general way
Knowledge of constraints further helps
System proposes mappings for remaining 97 sources
Points:
1) Introduce our approach.
2) We do not manually map the schemas of all sources to mediated schema.
The goal is to manually mark up only a few sources, and be able to learn from the marked up sources to successfully propose mappings for subsequent sources.
3) Once the markup is done, there are many different types of information to learn from.

26. Example Mediated schema Learned hypotheses Schema of realestate.com
address price agent-phone description
location listed-price phone comments
Learned hypotheses
Schema of realestate.com
If “phone” occurs in the name => agent-phone
location
Miami, FL
Boston, MA
...
listed-price
$250,000
$110,000
...
phone
(305)
(617)
...
comments
Fantastic house
Great location
...
realestate.com
If “fantastic” & “great” occur frequently in data values =>
description
Points:
1) Introduce our approach.
2) We do not manually map the schemas of all sources to mediated schema.
The goal is to manually mark up only a few sources, and be able to learn from the marked up sources to successfully propose mappings for subsequent sources.
3) Once the markup is done, there are many different types of information to learn from.
homes.com
price
$550,000
$320,000
...
contact-phone
(278)
(617)
...
extra-info
Beautiful yard
Great beach
...

27. Multi-Strategy Learning
Use a set of base learners
Each exploits well certain types of information:
Name learner looks at words in the attribute names
Naïve Bayes learner looks at patterns in the data values
Etc.
Match schema elements of a new source
Apply the base learners
Each returns a score
For different attributes one learner is more useful than another
Combine their predictions using a meta-learner
Meta-learner
Uses training sources to measure base learner accuracy
Weighs each learner based on its accuracy
Points:
1) Introduce the multi-strategy learning approach
Note: In the next several slides we are going to describe the multi-strategy learning approach in more details.

28. Training the Learners Mediated schema Schema of realestate.com
address price agent-phone description
location listed-price phone comments
Schema of realestate.com
Name Learner
(location, address)
(listed-price, price)
(phone, agent-phone)
(comments, description)
...
<location> Miami, FL </>
<listed-price> $250,000</>
<phone> (305) </>
<comments> Fantastic house </>
realestate.com
Points:
1) Introduce our approach.
2) We do not manually map the schemas of all sources to mediated schema.
The goal is to manually mark up only a few sources, and be able to learn from the marked up sources to successfully propose mappings for subsequent sources.
3) Once the markup is done, there are many different types of information to learn from.
<location> Boston, MA </>
<listed-price> $110,000</>
<phone> (617) </>
<comments> Great location </>
Naive Bayes Learner
(“Miami, FL”, address)
(“$ 250,000”, price)
(“(305) ”, agent-phone)
(“Fantastic house”, description)
...

29. Applying the Learners Schema of homes.com Mediated schema
area day-phone extra-info
address price agent-phone description
Name Learner
Naive Bayes
<area>Seattle, WA</>
<area>Kent, WA</>
<area>Austin, TX</>
Meta-Learner
(address,0.8), (description,0.2)
(address,0.6), (description,0.4)
(address,0.7), (description,0.3)
Name Learner
Naive Bayes
Meta-Learner
(address,0.7), (description,0.3)
<day-phone>(278) </>
<day-phone>(617) </>
<day-phone>(512) </>
Points:
1) Explain how the example learners are applied to match “area” with “address”.
2) Note that in general an arbitrary number of learners can be plugged into the matching process. If we have a new learner that has been trained, we can simply plug it in. This show how easy it is to add a new learner in our approach.
(agent-phone,0.9), (description,0.1)
<extra-info>Beautiful yard</>
<extra-info>Great beach</>
<extra-info>Close to Seattle</>
(address,0.6), (description,0.4)

30. Two Phases Training Phase Matching Phase Mediated schema
Source schemas
Domain
Constraints
Data listings
Training data
for base learners
User Feedback
Constraint Handler L1 L2 Lk
Mapping Combination

31. Mappings between Schemas
Schema matchers provide attribute correspondences, but not complete mappings
Mappings generally are posed as views: define relations in one schema (typically either the mediated schema or the source 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

32. A Few Mapping Examples T1 T2
Movie(Title, Year, Director, Editor, Star1, Star2)
PieceOfArt(ID, Artist, Subject, Title, TypeOfArt)
MotionPicture(ID, Title, Year) Participant(ID, Name, Role)
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
CustID
CustName
1234
Smith, J. T2
PennID
EmpName
46732
John Smith
Need a concordance table from CustIDs to PennIDs