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 or 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 operation 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 T 1 T 2 T 3 T 4 b b b b b b b b b b b b T 4 ∈ T 1 + ( T 2 + T 3 ) T 4 �∈ ( T 1 + T 2 ) + T 3 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 over 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 T 7 T 8 T 9 T 7 + T 8 ( T 7 + T 8 ) + T 9 T 8 + T 9 T 7 + ( T 8 + T 9 ) 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 T 1 T 2 T 3 T 1 + T 2 ( T 1 + T 2 ) + T 3 T 2 + T 3 T 1 + ( T 2 + T 3 ) 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 T 4 T 5 T 6 T 4 + T 5 ( T 4 + T 5 ) + T 6 T 5 + T 6 T 4 + ( T 5 + T 6 ) 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: T 1 T 2 T 3 b a a a b a Of all the trees in ( T 1 + T 2 ) + T 3 and T 1 + ( T 2 + T 3 ), focus on 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: ( T 1 + T 2 ) + T 3 T 1 + ( T 2 + T 3 ) 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 T 1 T 2 T 3 T 4 b b b b b b b b b b b b T 4 ∈ T 1 + ( T 2 + T 3 ) , T 4 �∈ ( T 1 + T 2 ) + T 3 In T 4 , T 1 and T 2 are substructures separated by T 3 Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms
Introduction Tree Combination Polarized Tree Combination Forest Combination Conclusion References Non-Associativity Analysis T 7 T 8 T 9 T 7 + T 8 ( T 7 + T 8 ) + T 9 T 8 + T 9 T 7 + ( T 8 + T 9 ) 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 original entities are restored Yael Sygal and Shuly Wintner Associativity of Tree-Based Formalisms
Recommend
More recommend
