1. XML Views & Reasoning about Views Zachary G. Ives University of Pennsylvania CIS 550 – Database & Information Systems November 4, 2004 Some slide content courtesy of Susan Davidson, Dan Suciu, & Raghu Ramakrishnan

2. 2 A View as a Translation between XML 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. 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. 4 The Simplest Way to Encode a Tree  Suppose we had: XYZ 14  If we have no IDs, we CREATE values…  BinaryLikeEdge(key, label, type, value, parent) keylabeltypevalueparent 0treeref-- 1contentref-0 2sub- content ref-1 3i-contentref-1 4-strXYZ2 5-int143 What are shortcomings here?

5. 5 Florescu/Kossmann Improved Edge Approach  Consider order, typing; separate the values  Edge(parent, ordinal, label, flag, target)  Vint(vid, value)  Vstring(vid, value) parentordlabelflagtarget -1treeref0 01contentref1 11sub-contentstrv2 11i-contentintv3 vidvalue v314 vidvalue v2XYZ

6. 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. 7 A Relation that Mirrors the XML Hierarchy  Output relation would look like: rLabelridrOrdclabelcidcOrdsLabelsidsOrdstrint tree01-------- -01content11----- -01-11sub-content21-- -01-11-21XYZ- -01-12i-content31-- -01-12-31-14

8. 8 A Relation that Mirrors the XML Hierarchy  Output relation would look like: rLabelridrOrdclabelcidcOrdsLabelsidsOrdstrint tree01-------- -01content11----- -01-11sub-content21-- -01-11-21XYZ- -01-12i-content31-- -01-12-31-14

9. 9 A Relation that Mirrors the XML Hierarchy  Output relation would look like: rLabelridrOrdclabelcidcOrdsLabelsidsOrdstrint tree01-------- -01content11----- -01-11sub-content21-- -01-11-21XYZ- -01-12i-content31-- -01-12-31-14 Colors are representative of separate SQL queries…

10. 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 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

11. 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 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?

12. 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. 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. 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 ?

15. 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. 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. 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. 18 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, id, @id) Sub-content(parentID, varchar) I-content(parentID, int)

19. 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. 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, v 1 and v 2  How do I know whether they represent the same data?  If v 1 is materialized, can we use it to compute v 2 ?  This is fundamental to query optimization and data integration, as we’ll see later

21. 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. 22 Let’s Go Back a Few Weeks… Domain Relational Calculus Queries have form: { | p } Predicate: boolean expression over x1,x2, …, xn  We have the following operations:  Rx i op x j x i op constconst op x i  x i. p  x j. p p  q, p  q  p, p  q where op is , , , , ,  and x i,x j,… are domain variables; p,q are predicates  Recall that this captures the same expressiveness as the relational algebra domain variables predicate

23. 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: R out (T 1 )  R 1 (T 2 ), R 2 (T 3 ), …, c(T 2 [ … T n ) where R out is the relation representing the query result, R i are predicates representing relations, c is an expression using arithmetic/boolean predicates over vars, and T i are tuples of variables

24. 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) headsubgoals body

25. 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. 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. 27 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) :- edge(F, T)distance 1 reachable(F, T) :- edge(F, X), edge(X, T)dist. 2 reachable(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. 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. 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. 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 += edge while change { reachable +=  F, T (  T ! X (edge) ⋈  F ! X (reachable)) }

31. 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. 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 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))

33. 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. 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. 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 “-” edges v1(x,y) :- r(x,y), : s(x,y) q(x,y) :- v1(x,y), : s(x,y) s r v1 q - + +

36. 36 Stratifying Datalog Execution foreach predicate p, set stratum(p) = 1 do 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. 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 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?

38. 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. 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. 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) StudentTakesCourseinCSE SN SC CX CDB & Info Systems S Need to get tuple in executing q2 over this database

41. 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