1. Finding Optimal Probabilistic Generators for XML Collections Serge Abiteboul, Yael Amsterdamer, Daniel Deutch, Tova Milo, Pierre Senellart

2. Adding probabilities to an XML Schema Given a collection of XML documents, we sometimes have a schema the documents conform to. – E.g., DTD or XSD – Restricts the structure, mostly parent-child node relations (using regular expressions) The schema may be very general (e.g., xhtml, RSS) We want to add probabilities that reflect the likelihood of different parts of the schema – We will use the probabilities to turn the schema into a probabilistic generative model for XML documents – In particular, we want them to maximize the likelihood of a given XML document or document collection - 2 - Motivation Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

3. One Application: XML Auto- Completion [SIGMOD 2012] Based on previous document versions / corpus of example documents – Suggest nodes / sub-trees / node values to the user For example: Challenges: – Allow editing in every part of the document – What kind of completion to suggest? – Finding the top-k best completions - 3 - Motivation Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer XML for Beginners M. Jones H. Q. David L. Martin S. Smith Advanced XML M. Jones J. E. Peterson G. L. Williams

4. Many Other Usages for a Probabilistic Schema Testing – e.g., generating many XML messages to simulate network load and test system performance. Explaining – e.g., a probabilistic schema for DBLP may show which types of publications are rarely used, which kinds of attributes are not filled for BibTex, etc. Schema Evaluation – how well a given schema describes a given corpus. … - 4 - Motivation Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

5. Our solution - An Outline - 5 -Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer Preliminaries – Tree Automata Generators for Schemas without Constraints Restart Generators Continuation-Test Generators Leaf Values Adding Constraints

6. Schema as a Deterministic Tree Automaton - 6 - Preliminaries Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer q0q0 q1q1 q2q2 b ac $ An XML document is modeled as an ordered tree. Document d 0 : Schema validation: the children of an a-labeled node are accepted by DFA A a Automaton A r : (L( A r ) = a*bc*$) Validation is performed for the children of every inner node. abcd 532 $ r abc

7. Using the Schema as a Generator Recall that we want to turn the schema from an acceptor into a probabilistic generative model. Straightforward nondeterministic generator: repeatedly choose an accepting run for a node's automaton, and generate children accordingly. Adding probabilities: we consider two problem settings 1.Generating documents that are accepted by the schema, while maximizing the likelihood of a corpus. 2.Additionally, imposing integrity constraints on the documents (e.g., key constraints) - 7 - Preliminaries Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

8. Probabilistic Generator Each transition is assigned a probability We assume independent choices, (a Markovian process) thus the document probability is the product. In this case, Pr( d )=p a ∙p a ∙p b ∙p $ The schema and generator ignore leaf values (for now!) - 8 - Without Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer b a c $ papa pcpc pbpb p$p$ q0q0 q1q1 q2q2 $ r aab

9. Formal Problem Definition Given a corpus D of documents, and a deterministic schema S that accepts every document in D We want to find an optimal generator based on S : – Find probabilities for the transitions of S that maximize the probability of generating D, – i.e., the maximum likelihood estimator (MLE). - 9 - Without Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

10. A Learning Algorithm - 10 - Without Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer b ac $ $ The frequency of using each transition during the corpus verification process is recorded. (q 0, a) (q 0, b) (q 1, c) (q 1, $) 1 1 1 1 q0q0 q1q1 q2q2 r abc

11. An Algorithm for Probabilities Learning (Cont.) This is repeated for every node in every corpus document. We set the probability of each transition to be its relative frequency. - 11 - Without Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer (q 0, a)1 (q 0, b)1 (q 1, c)1 (q 1, $)1 /2 Theorem: This efficient algorithm learns the MLE probabilities – finds an optimal probabilistic generator

12. An Additional Result Theorem: generation terminates with probability 1. – Guaranteed only because of the choice of probabilities according to the corpus. - 12 - Without Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

13. Integrity Constraints We want to support integrity constraints, which are used in XML schema languages. Key Constraint: the leaves of a-labeled leaves have unique values (unary key) Inclusion Constraint: the values of a-labeled leaves are contained in those of b-labeled leaves Domain Constraint: the values of a-labeled leaves belong to some (finite or infinite) domain Different types are considered in the literature [Fan & Libkin 2001; David Libkin & Tan 2011] - 13 - Adding Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

14. New Problem We want to find optimal generators for XML schemas with constraints. Valid generator output: an XML document, which 1.is a accepted by the schema, and 2.there exists a valid leaf value assignment – which does not violate the constraints – Example: each of a, b, c is unique, and contained the others - 14 - Adding Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer $ r aab c r ab b c … b

15. Restart Generators A simple idea: – Use a probabilistic generator to generate a document – Check if it has a value assignment valid w.r.t. the constraints – If not, 'restart' and try again until a valid document is generated Problem definition -- same as in the case without constraints (but now the schema includes constraints) Proposition: Given a document with no values, checking for the existence of a valid value assignment is in PTIME – Proof: By translating the constraints to bounds on the number of unique values for each leaf label Bad news: number of restarts can be unboundedly large in an optimal generator - 15 - Adding Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

16. Continuation-test Generators Never make choices that lead to a 'dead end', thus always generate a valid document. We use a binary test to check if a choice has a continuation. Example: add to the schema of d 0 the constraints: – c is included in a – c is unique The generation process: - 16 - Adding Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer b a c $ $ papa pcpc pbpb p$p$ q0q0 q1q1 q2q2 r abc Pr( d ) = p a ∙p b ∙p c ∙1 Perform a continuation-test before taking the transition Implies |c|≤|a|

17. Learning Algorithm for Continuation-test Generators The probabilities are again relative frequencies, but – only in cases where there was an alternative choice. The learned generator will generate as many c-s as a-s Adding Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer (q 0, a)1 (q 0, b)1 (q 1, c)1 (q 1, $)0 /2 /1 (q 1, $) was chosen only when (q 1, c) was not available. - 17 -

18. Results for Continuation- test Generators Theorem: The algorithm learns an optimal continuation-test generator, for automata with binary choices. – Extensions to non-binary are discussed in the paper Theorem: Continuation-test is NP-Complete – But only in the size of the schema; it is polynomial in the document size – Both generation and finding the optimal generator are exponential in the schema size unless P=NP. – Based on schema satisfiability test [David et al. 2011] Theorem: probability of termination for a continuation-test generator may be arbitrarily small! – Proof – by construction of a simple, non-recursive schema – Can be handled by adding a constraint on the document size. – Sub-classes of schemas that guarantee termination? - 18 - Adding Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

19. Adding Values to the Structure So far our generators were used only for the document structure Leaf values may also have a distribution according to which they can be generated – The distribution may be learned from the same document collection We will focus on the interesting case – generating leaf values for a schema with constraints - 19 - Leaf Values Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

20. Suggested Algorithm We start with a valid document skeleton Order labels by inclusion constraints (e.g., c, b, a) Choose a leaf from the 'smallest' (most included) label, and including leaves Draw a value (from the domain) according to a given distribution. Use PTIME test to verify validity, if not revert the step Improvements presented in the paper - 20 - Leaf Values Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer $ r abc abcd efg

21. Possible improvement to the basic algorithm Annotate the leaves with 'old' or 'new' For 'old' a-labeled leaves choose values already chosen for some a-labeled leaf For 'new' choose a value unused by a-labeled leaves yet Annotations can be learned from the corpus, and generated: – Offline – after the document generation, using a PTIME validity test – Online – during document generation, using a continuation test. – Both methods are incomparable in terms of quality - 21 - Leaf Values Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer newoldnew $ r aab

22. Related Work Schema Satisfiability tests [Fan & Libkin 2001; David, Libkin & Tan 2011] Probabilistic XML and Probabilistic Schemas [e.g., Benedikt, Kharlamov, Olteanu & Senellart 2010] Probabilistic XML generation [e.g., Antonopoulos, Geerts, Martens & Neven 2011] Schema Inference [e.g., Bex, Gelade, Neven & Vansummeren 2008] AXML [Abiteboul, Benjelloun & Milo 2008] PCFGs [e.g., Chi & Geman 1998] - 22 - Summary Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

23. Conclusion A model for a probabilistic XML generators Unconstrained case – Generation and learning optimal generators can be done efficiently – Termination is guaranteed Constrained case – Restart generator # of restarts is unbounded – Continuation-test generators Generation and learning optimal generators are expensive Termination is not guaranteed Leaf Value generation In the talk labels and states are coupled (as in a DTD), but all the results hold when they are uncoupled. Future work – Efficient combinations of restart and continuation-test generators – Experimental study - 23 - Summary Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

24. Thank You! Q&A

25. Using a Tree Automaton for Schema Verification Preliminaries Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer q0q0 q1q1 q2q2 b ac $ r abc $ An XML is modeled as an ordered tree. Document d 0 : The children of a-labeled node are accepted by automaton A a Automaton A r : This is done for every inner node in a fixed order (BF-LTR)

26. Sentence Generation Example Input: a simple paragraph in an XML format Sam is a student. She goes to school on Weekdays. Marley thinks Sam is nice. Input: a (manually created) schema Output: randomly generated paragraphs a student is nice. a student thinks Sam thinks Sam thinks a student is nice. Sam thinks Sam is nice. Sam thinks She is nice. She is Sam. Sam is Marley. Marley thinks a student goes to school on weekdays. Sam goes to school on weekdays. Sam is nice. Marley thinks Sam is nice. Marley is Sam. Challenges: – Can constraints be useful here? – Creating an elaborate schema (classical NLP problem) - 18 - Implementation Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer

27. An Algorithm for Probabilities Learning - 27 - Without Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer b ac $ $ The frequency each transition is chosen during the corpus verification process is recorded. (q 0, a) (q 0, b) (q 1, c) (q 1, $) 1 1 1 1 q0q0 q1q1 q2q2 r abc

28. Example for Unboundedly Many Restarts Consider the following schema and corpus The schema allows 0 or 1 a -labeled leaves. We want to choose α that maximizes the likelihood of d The probability of d is the probability of generating it on the 1 st attempt + the probability of restarting once and generating d on the 2 nd attempt, and so on – a geometric series. Monotonically increases as α approaches 1. But so is also the probability of restarting… - 28 - Adding Constraints Finding Optimal Probabilistic Generators for XML Collections – Yael Amsterdamer r a d $ a $ α 1-α q1q1 q2q2 S a is unique and taken from {0}