Hyper-minimization for deterministic weighted tree automata Andreas - PowerPoint PPT Presentation

Hyper-minimization for deterministic weighted tree automata Andreas Maletti and Daniel Quernheim Institute of Computer Science, Universität Leipzig, Germany maletti@informatik.uni-leipzig.de May 29, 2014

Overview Weighted Tree Language ◮ Assigns weight (e.g. a probability) to each tree ◮ Weight drawn from commutative semiring; e.g. ( Q , + , · , 0 , 1 ) A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 2

Overview Weighted Tree Language ◮ Assigns weight (e.g. a probability) to each tree ◮ Weight drawn from commutative semiring; e.g. ( Q , + , · , 0 , 1 ) Weighted Tree Automaton ◮ Finitely represents weighted tree language ◮ Defines the recognizable weighted tree languages A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 2

Overview Weighted Tree Language ◮ Assigns weight (e.g. a probability) to each tree ◮ Weight drawn from commutative semiring; e.g. ( Q , + , · , 0 , 1 ) Weighted Tree Automaton ◮ Finitely represents weighted tree language ◮ Defines the recognizable weighted tree languages Application ◮ Re-ranker for parse trees ◮ Representation of parses large models A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 2

Basics A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 3

Semiring Definition A commutative semiring is an algebraic structure A = ( A , + , · , 0 , 1 ) ◮ ( A , + , 0 ) commutative monoid ◮ ( A , · , 1 ) commutative monoid ◮ · distributes over + a · ( a 1 + a 2 ) = ( a · a 1 ) + ( a · a 2 ) ◮ 0 · a = 0 for all a ∈ A Examples: ( N , + , · , 0 , 1 ) and ( Q , + , · , 0 , 1 ) A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 4

Semiring Definition A commutative semiring is an algebraic structure A = ( A , + , · , 0 , 1 ) ◮ ( A , + , 0 ) commutative monoid ◮ ( A , · , 1 ) commutative monoid ◮ · distributes over + a · ( a 1 + a 2 ) = ( a · a 1 ) + ( a · a 2 ) ◮ 0 · a = 0 for all a ∈ A Examples: ( N , + , · , 0 , 1 ) and ( Q , + , · , 0 , 1 ) Definition A commutative semifield is a commutative semiring A = ( A , + , · , 0 , 1 ) ◮ for all a ∈ A \ { 0 } there exists a − 1 ∈ A with a · a − 1 = 1 Example: ( Q , + , · , 0 , 1 ) A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 4

Syntax Definition Weighted tree automaton (WTA) is tuple ( Q , Σ , A , F , µ ) where ◮ finite set Q states ◮ ranked alphabet Σ input symbols ◮ commutative semiring A = ( A , + , · , 0 , 1 ) weight structure ◮ F ⊆ Q final states A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 5

Syntax Definition Weighted tree automaton (WTA) is tuple ( Q , Σ , A , F , µ ) where ◮ finite set Q states ◮ ranked alphabet Σ input symbols ◮ commutative semiring A = ( A , + , · , 0 , 1 ) weight structure ◮ F ⊆ Q final states ◮ µ = ( µ k ) k ∈ N with µ k : Σ k → A Q × Q k weighted transitions A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 5

Syntax Definition Weighted tree automaton (WTA) is tuple ( Q , Σ , A , F , µ ) where ◮ finite set Q states ◮ ranked alphabet Σ input symbols ◮ commutative semiring A = ( A , + , · , 0 , 1 ) weight structure ◮ F ⊆ Q final states ◮ µ = ( µ k ) k ∈ N with µ k : Σ k → A Q × Q k weighted transitions Sample Transition with weight µ k ( σ ) q , q 1 ··· q k q σ q 1 . . . q k A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 5

Syntax — Illustration Sample Automaton 3 6 f/ 0 . 3 f/ 0 . 5 1 2 4 5 a/ 1 b/ 1 a/ 1 a/ 1 A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 6

Semantics Definition Let t ∈ T Σ ( Q ) and W = pos ( t ) . ◮ Run on t : map r : W → Q with r ( w ) = t ( w ) if t ( w ) ∈ Q A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 7

Semantics Definition Let t ∈ T Σ ( Q ) and W = pos ( t ) . ◮ Run on t : map r : W → Q with r ( w ) = t ( w ) if t ( w ) ∈ Q ◮ Weight of r � wt ( r ) = µ k ( t ( w )) r ( w ) , r ( w 1 ) ··· r ( wk ) w ∈ W t ( w ) ∈ Σ A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 7

Semantics Definition Let t ∈ T Σ ( Q ) and W = pos ( t ) . ◮ Run on t : map r : W → Q with r ( w ) = t ( w ) if t ( w ) ∈ Q ◮ Weight of r � wt ( r ) = µ k ( t ( w )) r ( w ) , r ( w 1 ) ··· r ( wk ) w ∈ W t ( w ) ∈ Σ ◮ Recognized weighted tree language � � M � ( t ) = wt ( r ) r run on t r ( ε ) ∈ F A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 7

Semantics — Illustration Sample Automaton 3 6 f/ 0 . 3 f/ 0 . 5 1 2 4 5 a/ 1 b/ 1 a/ 1 a/ 1 Sample Runs Input tree: Runs: with weight 0 f 6 a b 1 2 A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 8

Semantics — Illustration Sample Automaton 3 6 f/ 0 . 3 f/ 0 . 5 1 2 4 5 a/ 1 b/ 1 a/ 1 a/ 1 Sample Runs Input tree: Runs: with weight f f a b a b A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 8

Semantics — Illustration Sample Automaton 3 6 f/ 0 . 3 f/ 0 . 5 1 2 4 5 a/ 1 b/ 1 a/ 1 a/ 1 Sample Runs Input tree: Runs: with weight 1 f f a b 1 b A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 8

Semantics — Illustration Sample Automaton 3 6 f/ 0 . 3 f/ 0 . 5 1 2 4 5 a/ 1 b/ 1 a/ 1 a/ 1 Sample Runs Input tree: Runs: with weight 1 f f a b 1 2 A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 8

Semantics — Illustration Sample Automaton 3 6 f/ 0 . 3 f/ 0 . 5 1 2 4 5 a/ 1 b/ 1 a/ 1 a/ 1 Sample Runs Input tree: Runs: with weight 0 . 3 f 3 a b 1 2 A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 8

Determinism Definition Deterministic WTA: for every σ ∈ Σ k and w ∈ Q k there exists exactly one q ∈ Q such that µ k ( σ ) q , w � = 0 A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 9

Determinism Definition Deterministic WTA: for every σ ∈ Σ k and w ∈ Q k there exists exactly one q ∈ Q such that µ k ( σ ) q , w � = 0 Notes ◮ Deterministic WTA does not use addition A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 9

Determinism Definition Deterministic WTA: for every σ ∈ Σ k and w ∈ Q k there exists exactly one q ∈ Q such that µ k ( σ ) q , w � = 0 Notes ◮ Deterministic WTA does not use addition ◮ Recognizable � = deterministically recognizable A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 9

Determinism Definition Deterministic WTA: for every σ ∈ Σ k and w ∈ Q k there exists exactly one q ∈ Q such that µ k ( σ ) q , w � = 0 Notes ◮ Deterministic WTA does not use addition ◮ Recognizable � = deterministically recognizable ◮ Determinization possible in locally-finite semirings [Borchardt, Vogler 2003] A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 9

Determinism Definition Deterministic WTA: for every σ ∈ Σ k and w ∈ Q k there exists exactly one q ∈ Q such that µ k ( σ ) q , w � = 0 Notes ◮ Deterministic WTA does not use addition ◮ Recognizable � = deterministically recognizable ◮ Determinization possible in locally-finite semirings [Borchardt, Vogler 2003] ◮ Partial determinization for probabilities [May, Knight 2006] ◮ Systematic presentation [Büchse, Vogler 2009] A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 9

Hyper-minimization A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 10

Assumption We assume a commutative semifield A = ( A , + , · , 0 , 1 ) A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 11

Minimization equivalent = same recognized weighted tree language Problem Given deterministic WTA, return ◮ equivalent deterministic WTA such that ◮ no equivalent deterministic WTA is smaller A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 12

Minimization equivalent = same recognized weighted tree language Problem Given deterministic WTA, return ◮ equivalent deterministic WTA ◮ minimal A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 12

Minimization equivalent = same recognized weighted tree language Problem Given deterministic WTA, return ◮ equivalent deterministic WTA ◮ minimal Theorem (M., Q. 2011) Minimization of deterministic WTA can be done in time O ( m log n ) ◮ m = size of automaton ◮ n = number of states A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 12

Minimization context = tree with exactly one occurrence of special symbol � c [ t ] = tree obtained from context c by replacing � by t Definition States p and q are equivalent if there exists a ∈ A \ { 0 } such that � M � ( c [ p ]) = a · � M � ( c [ q ]) for all contexts c ∈ C Σ A. Maletti and D. Quernheim Hyper-minimization for deterministic WTA 13