Associativity of Tree-Based Formalisms Yael Sygal and Shuly Wintner - - PowerPoint PPT Presentation

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Associativity of Tree-Based Formalisms

Yael Sygal and Shuly Wintner Department of Computer Science University of Haifa Haifa, Israel XMG Workshop June 21-22, 2007 Nancy, France

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Tree-Based Formalisms

Two kinds of formalisms: Formalisms which use grammatical objects (e.g., trees, d-trees

• r graphs) as basic units: D-Tree Grammars (Rambow,

Vijay-Shanker, and Weir, 1995) PUG (Kahane, 2006) etc. Formalisms which use grammatical descriptions (e.g., formulas describing structures such as trees or d-trees) as basic units: Interaction Grammars (Perrier, 2000), Tree Description Grammars (Kallmeyer, 2001), XMG (Crabb´ e, 2005; Duchier, Le Roux, and Parmentier, 2004) etc.

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

PUG: The Tree Combination Operation

Polarized unification grammar (PUG): A grammar is a set of structures which can combine Assume these structures are trees Tree combination: given two trees, nodes from the two combined trees are identified where:

When two nodes are identified, they must belong to different trees When two nodes are identified, all their ancestors must identify as well At least two nodes (each from a different tree) must be identified

These conditions guarantee that the resulting graph is indeed a tree All the results can be trivially extended to general graphs

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Overview

Tree combination is a non-associative operation The consequence is overgeneration Introducing polarities:

Kahane (2006) conjectures that combination of tree-based grammar fragments with polarities is associative Existing polarity systems do not render the combination

• peration associative

There is no other non-trivial polarity system for which grammar combination is associative

The solution (inspired by Cohen-Sygal and Wintner (2006)) – forest combination Forest combination and XMG

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Tree Combination is Non-Associative

Theorem Tree combination is a non-associative operation

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Proof

T1 T2 T3 T4

b b b b b b b b b b b b

T4 ∈ T1 + (T2 + T3) T4 ∈ (T1 + T2) + T3

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Introducing Polarities

A system of polarities is a pair (P, ·) where P is a set (of polarities) and ‘·’ is an associative and commutative product

• ver P

Each node is associated with a polarity Nodes can only be identified if their polarities are unifiable; the resulting node has the unified polarity A non-empty, strict subset of polarities, the neutral polarities, determines which of the resulting trees are valid: A polarized tree is saturated if all its polarities are neutral Polarized formalisms include PUG, Interactive grammar and XMG colors

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

PUG Grammar Combination

Kahane and Lareau (2005) uses the following system of polarities: · ⊥ The neutral polarities are black and gray

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

PUG Grammar Combination

Kahane (2006) extends this system by adding two more polarities, plus and minus: · − + − + − + − − − ⊥ ⊥ + + + ⊥ ⊥ ⊥ ⊥ ⊥

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

PUG Grammar Combination

T7 T8 T9 T7 + T8 (T7 + T8) + T9 T8 + T9 T7 + (T8 + T9)

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

XMG Color Table

XMG uses colors to sanction tree node identification The color combination table is: · ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ where ⊥ represents the impossibility to combine

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

XMG Color Table

T1 T2 T3 T1 + T2 (T1 + T2) + T3 T2 + T3 T1 + (T2 + T3) No No Solution Solution

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

XMG Color Table

T4 T5 T6 T4 + T5 (T4 + T5) + T6 T5 + T6 T4 + (T5 + T6)

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

General polarity systems

Some existing polarity-based formalisms are non-associative This is not accidental; the only polarity scheme that induces associative tree combination is trivial Some notation:

if (P, ·) is a system of polarities and a, b ∈ P, we use the shorthand notation ab instead of a · b ab↓ means that the combination of a and b is defined ab↑ means that a and b cannot combine

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

General Polarity Systems

Definition A system of polarities (P, ·) is trivial if for all a, b ∈ P, ab↑. Theorem Let (P, ·) be a non-trivial system of polarities. Then polarized tree combination based on (P, ·) is not associative.

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Proof

Two possible cases:

1 There exist a ∈ P such that aa↓: Take the non-polarized

example and attach all nodes the polarity ‘a’

2 There exist a, b ∈ P such that a = b and ab↓, aa↑ and bb↑ Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Proof

Consider the following trees: T1 T2 T3 b a a a b a Of all the trees in (T1 + T2) + T3 and T1 + (T2 + T3), focus

• n trees of this structure:

b b b b

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Proof

Then: (T1 + T2) + T3 T1 + (T2 + T3) a No Solution ab ab a

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Non-Associativity Analysis

T1 T2 T3 T4

b b b b b b b b b b b b

T4 ∈ T1 + (T2 + T3) , T4 ∈ (T1 + T2) + T3 In T4, T1 and T2 are substructures separated by T3

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Non-Associativity Analysis

T7 T8 T9 T7 + T8 (T7 + T8) + T9 T8 + T9 T7 + (T8 + T9)

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Non-Associativity Analysis

When two trees are combined, at least two nodes (each from a different tree) must identify Hence, the two trees must be connected in the resulting tree Other combination orders that allow two trees to be separated (by other trees) can yield results which cannot be obtained when the two trees are first combined together

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

From Trees to Forests

The solution: Move to the powerset domain Enables the operator to ‘remember’ all the possibilities After the combination, an extra stage is added in which the

• riginal entities are restored

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Forest Combination

The basic units become forests rather than trees Forest combination is defined in much the same way as above:

Two forests are combined by identifying some of their nodes When two nodes are identified then they must belong to the two different forests When two nodes are identified then all their ancestors must be identified as well Two forests can be combined even if none of their nodes are identified

Polarities can be added as before

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

XMG Forest Combination

F1 F2 F3 F1 + F2 F2 + F3

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

XMG Forest Combination

F1 + (F2 + F3) = (F1 + F2) + F3

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

XMG Forest Combination – Resolution

F1 + (F2 + F3) = (F1 + F2) + F3

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Forest Combination

Theorem (Polarized) forest combination is an associative operation

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Forest Combination and XMG

XMG:

Syntax: grammatical descriptions (tree description language) Semantics: grammatical objects (trees)

Forest combination amounts in the semantic level to the conjunction operation in the syntactic level

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Conclusion

Tree-based grammar combination is a non-associative

• peration

The consequence is overgeneration Adding polarities:

Existing polarity systems do not render the combination

• peration associative

There is no other non-trivial polarity system for which grammar combination is associative

The solution: move from trees to the power domain of forests Different formalisms use different grammatical objects as basic units. Here, we focus on trees but all the results can be trivially extended to general graphs

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

Acknowledgments

Funding: The Israel Science Foundation (grant no. 136/01) Discussion: Yannick Parmentier

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms

Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References

References

Cohen-Sygal, Yael and Shuly Wintner. 2006. Partially specified signatures: A vehicle for grammar modularity. In Proceedings of COLING-ACL 2006, pages 145–152, Sydney, Australia, July. Crabb´ e, Benoit. 2005. Grammatical development with XMG. In Proceedings of the 5th International Conference on Logical Aspects of Computational Linguistics (LACL), Bordeaux, France, April. Duchier, Denys, Joseph Le Roux, and Yannick Parmentier. 2004. The metagrammar compiler: An NLP application with a multi-paradigm architecture. In Proceedings of the Second International Mozart/Oz Conference (MOZ 2004), Charleroi, Belgium, October. Kahane, Sylvain. 2006. Polarized unification grammars. In Proceedings of COLING-ACL 2006, pages 137–144, Sydney, Australia, July. Kahane, Sylvain and Francois Lareau. 2005. Meaning-text unification grammar: modularity and polarization. In Proceedings of the 2nd International Conference on Meaning-Text Theory, pages 197–206, Moscow. Kallmeyer, Laura. 2001. Local tree description grammars. Grammars, 4(2):85–137. Perrier, Guy. 2000. Interaction grammars. In Proceedings of COLING 2000, pages 600–606. Rambow, Owen, K. Vijay-Shanker, and David Weir. 1995. D-tree grammars. In Proceedings of ACL 95, pages 151–158.

Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms