Optimal Probabilistic Generators for XML Collections

Serge Abiteboul, Yael Amsterdamer, Daniel Deutch, Tova Milo, Pierre Senellart

[ICDT 2012]

Adding probabilities to an XML Schema

• XML schemas are useful for describing the structures of XML documents.

– E.g., DTD or XSD

• Schemas 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

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Motivation

Optimal Probabilistic Generators for XML Collections

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 every part of the document

– What kind of completion to suggest?

– Finding the top-k best completions

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Motivation

Optimal Probabilistic Generators for XML Collections

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<title>XML for Beginners</title>

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<title>Advanced XML</title>

<author>M. Jones</author>

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Many Other Usages for a Probabilistic Schema

...

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Motivation

Optimal Probabilistic Generators for XML Collections

• 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.

✗

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Our solution - An Outline

- 5 -Optimal Probabilistic Generators for XML Collections

Preliminaries – Tree Automata

Generators for Schemas without Constraints

Restart Generators

Continuation-Test Generators

Leaf Values

Adding Constraints

Schema as a Deterministic Tree Automaton

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Preliminaries

Optimal Probabilistic Generators for XML Collections

q0 q1 q2b

a c

$

An XML document is modeled as an ordered tree.

Document d0:

Schema validation: the children of an a-labeled node are accepted by DFA Aa

Automaton Ar: (L(Ar) = a*bc*$)

Validation is performed for the children of every inner node.

abcd abcd532

$

r

a b c

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)

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Preliminaries

Optimal Probabilistic Generators for XML Collections

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)=pa p∙ a p∙ b p∙ $

• The schema and generator ignore leaf values (for now!)- 8 -

Without Constraints

Optimal Probabilistic Generators for XML Collections

ba c

$pa pc

pb p$

q0

q1

q2 $

r

a a b

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).

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Without Constraints

Optimal Probabilistic Generators for XML Collections

A Learning Algorithm

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Without Constraints

Optimal Probabilistic Generators for XML Collections

b

a c

$

$

The frequency of using each transition during the corpus verification process is recorded.

(q0, a)

(q0, b)

(q1, c)

(q1, $)

1111

q0 q1 q2

r

a b c

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.

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Without Constraints

Optimal Probabilistic Generators for XML Collections

(q0, a) 1(q0, b) 1(q1, c) 1(q1, $) 1

/2/2

/2/2Theorem: This efficient algorithm

learns the MLE probabilities – finds an optimal probabilistic generator

Termination

• Theorem: generation terminates with probability 1.

– Guaranteed only because of the choice of probabilities according to the corpus.

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Without Constraints

Optimal Probabilistic Generators for XML Collections

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

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Adding Constraints

Optimal Probabilistic Generators for XML Collections

New Problem• We want to find optimal generators for XML schemas with

constraints.

• Valid generator output: an XML document, which1. is a accepted by the schema, and

2. there exists a valid leaf value assignment – which does not violate the constraints

– Example: a, b, c are unique and contain each other

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Adding Constraints

Optimal Probabilistic Generators for XML Collections

$

r

a a bc

r

a b

b

c

…

b

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

• 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

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Adding Constraints

Optimal Probabilistic Generators for XML Collections

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 d0 the constraints:

– c is included in a– c is unique

• The generation process:

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Adding Constraints

Optimal Probabilistic Generators for XML Collections

ba c

$$

pa pc

pb p$

q0

q1

q2

r

a b c

Pr(d) = pa p∙ b p∙ c∙1

Perform a continuation-test before taking the

transition

Implies |c|≤|a|

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

Optimal Probabilistic Generators for XML Collections

(q0, a) 1(q0, b) 1(q1, c) 1(q1, $) 0

/2/2/1/1

(q1, $) was chosen only when (q1, c) was not available.

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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 polynomial when using a continuation-test oracle.

– 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?

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Adding Constraints

Optimal Probabilistic Generators for XML Collections

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

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Leaf Values

Optimal Probabilistic Generators for XML Collections

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

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Leaf Values

Optimal Probabilistic Generators for XML Collections

$

r

a b c

abcdabcd efg

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]

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Summary

Optimal Probabilistic Generators for XML Collections

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– More Efficient combinations of restart and continuation-test generators

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Summary

Optimal Probabilistic Generators for XML Collections

Thank You!Thank You!

Q&A