Simulations of Weighted Tree Automata Zoltn sik 1 and Andreas Maletti - PowerPoint PPT Presentation

Simulations of Weighted Tree Automata Zoltán Ésik 1 and Andreas Maletti 2 1 University of Szeged, Szeged, Hungary 2 Universitat Rovira i Virgili, Tarragona, Spain andreas.maletti@urv.cat Winnipeg — August 12, 2010 Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 1 ·

Motivation Simulation of weighted string automata Theorem (B ÉAL , L OMBARDY , S AKAROVITCH 2005 & 2006) For all equivalent weighted string automata over . . . there exists a chain of simulations connecting them. a field the integers (more generally, an E UCLIDIAN domain) the natural numbers the B OOLEAN semiring (functional transducers) Consequence Equivalence = Chain of Simulations Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 2 ·

Motivation Simulation of weighted string automata Theorem (B ÉAL , L OMBARDY , S AKAROVITCH 2005 & 2006) For all equivalent weighted string automata over . . . there exists a chain of simulations connecting them. a field the integers (more generally, an E UCLIDIAN domain) the natural numbers the B OOLEAN semiring (functional transducers) Consequence Equivalence = Chain of Simulations Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 2 ·

Motivation Minimization of weighted tree automata In fields [S EIDL 1990, B OZAPALIDIS 1991] A B equivalent minimizes to minimizes to C automata collapsed by equivalence relation the canonical homomorphism is a simulation Consequence Equivalence = Chain of Simulations Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 3 ·

Motivation Minimization of weighted tree automata In fields [S EIDL 1990, B OZAPALIDIS 1991] A B equivalent minimizes to minimizes to C automata collapsed by equivalence relation the canonical homomorphism is a simulation Consequence Equivalence = Chain of Simulations Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 3 ·

Weighted Tree Automaton Contents Motivation 1 Weighted Tree Automaton 2 Simulation 3 Simulation vs. Equivalence 4 Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 4 ·

Weighted Tree Automaton Semiring Definition A commutative semiring is an algebraic structure ( K , + , · , 0 , 1 ) with commutative monoids ( K , + , 0 ) and ( K , · , 1 ) a · ( b + c ) = ( a · b ) + ( a · c ) a · 0 = 0 Example natural numbers tropical semiring ( N ∪ {∞} , min , + , ∞ , 0 ) B OOLEAN semiring ( { 0 , 1 } , max , min , 0 , 1 ) any commutative ring Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 5 ·

Weighted Tree Automaton Semiring Definition A commutative semiring is an algebraic structure ( K , + , · , 0 , 1 ) with commutative monoids ( K , + , 0 ) and ( K , · , 1 ) a · ( b + c ) = ( a · b ) + ( a · c ) a · 0 = 0 Example natural numbers tropical semiring ( N ∪ {∞} , min , + , ∞ , 0 ) B OOLEAN semiring ( { 0 , 1 } , max , min , 0 , 1 ) any commutative ring Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 5 ·

Weighted Tree Automaton Weighted tree automaton Definition (B ERSTEL , R EUTENAUER 1982) A weighted tree automaton (wta) is a tuple A = ( Q , Σ , K , I , R ) with rules of the form σ c q → q 1 q k . . . where q , q 1 , . . . , q k ∈ Q are states c ∈ K is a weight (taken from a semiring) σ ∈ Σ k is an input symbol Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 6 ·

Weighted Tree Automaton Run S NP VP JJ NNS VBP ADVP RB Colorless ideas sleep furiously Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 7 ·

Weighted Tree Automaton Run S q NP q ′ VP q 1 JJ q ′ NNS q ′′ VBP q ′ ADVP q 2 1 RB q 2 Colorless w ideas w sleep w furiously w Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 7 ·

Weighted Tree Automaton Run S . 4 q NP . 2 VP . 4 q ′ q 1 JJ . 3 NNS . 3 VBP . 2 ADVP . 3 q ′ q ′′ q ′ q 2 1 RB . 2 Colorless . 1 ideas . 1 sleep . 1 q 2 w w w furiously . 1 w Definition The weight of a run is obtained by multiplying the weights in it. Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 7 ·

Weighted Tree Automaton Run S . 4 q NP . 2 VP . 4 q ′ q 1 JJ . 3 NNS . 3 VBP . 2 ADVP . 3 q ′ q ′′ q ′ q 2 1 RB . 2 Colorless . 1 ideas . 1 sleep . 1 q 2 w w w furiously . 1 w Weight of the run 0 . 4 · 0 . 2 · 0 . 3 · 0 . 1 · 0 . 3 · 0 . 1 · 0 . 4 · 0 . 2 · 0 . 1 · 0 . 3 · 0 . 2 · 0 . 1 Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 7 ·

Weighted Tree Automaton Semantics Definition The weight of an input tree t is weight ( t ) = � I ( root ( r )) · weight ( r ) r run on t Definition Two wta are equivalent if they assign the same weights to all trees. Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 8 ·

Weighted Tree Automaton Semantics Definition The weight of an input tree t is weight ( t ) = � I ( root ( r )) · weight ( r ) r run on t Definition Two wta are equivalent if they assign the same weights to all trees. Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 8 ·

Simulation Matrix representation Definition matrix presentation of a wta ( Q , Σ , I , R ) σ c R k ( σ ) q 1 ··· q k , q = c ⇐ ⇒ ∈ R q → q 1 q k . . . Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 9 ·

Simulation Main definition Definition (B LOOM , É SIK 1993) A wta ( Q , Σ , I , R ) simulates a wta ( P , Σ , J , S ) if there exists a matrix X ∈ K Q × P such that I = XJ � I ( q ) = X q , p · J ( p ) p ∈ P R k ( σ ) X = X k , ⊗ S k ( σ ) � � R k ( σ ) q 1 ··· q k , q · X q , p = X q 1 , p 1 · . . . · X q k , p k · S k ( σ ) p 1 ··· p k , p q ∈ Q p 1 ··· p k ∈ P k Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 10 ·

Simulation Main definition Definition (B LOOM , É SIK 1993) A wta ( Q , Σ , I , R ) simulates a wta ( P , Σ , J , S ) if there exists a matrix X ∈ K Q × P such that I = XJ � I ( q ) = X q , p · J ( p ) p ∈ P R k ( σ ) X = X k , ⊗ S k ( σ ) � � R k ( σ ) q 1 ··· q k , q · X q , p = X q 1 , p 1 · . . . · X q k , p k · S k ( σ ) p 1 ··· p k , p q ∈ Q p 1 ··· p k ∈ P k Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 10 ·

Simulation Main definition Definition (B LOOM , É SIK 1993) A wta ( Q , Σ , I , R ) simulates a wta ( P , Σ , J , S ) if there exists a matrix X ∈ K Q × P such that I = XJ � I ( q ) = X q , p · J ( p ) p ∈ P R k ( σ ) X = X k , ⊗ S k ( σ ) � � R k ( σ ) q 1 ··· q k , q · X q , p = X q 1 , p 1 · . . . · X q k , p k · S k ( σ ) p 1 ··· p k , p q ∈ Q p 1 ··· p k ∈ P k Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 10 ·

Simulation Main definition Definition (B LOOM , É SIK 1993) A wta ( Q , Σ , I , R ) simulates a wta ( P , Σ , J , S ) if there exists a matrix X ∈ K Q × P such that I = XJ � I ( q ) = X q , p · J ( p ) p ∈ P R k ( σ ) X = X k , ⊗ S k ( σ ) � � R k ( σ ) q 1 ··· q k , q · X q , p = X q 1 , p 1 · . . . · X q k , p k · S k ( σ ) p 1 ··· p k , p q ∈ Q p 1 ··· p k ∈ P k Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 10 ·

Simulation Main definition Definition (B LOOM , É SIK 1993) A wta ( Q , Σ , I , R ) simulates a wta ( P , Σ , J , S ) if there exists a matrix X ∈ K Q × P such that R k ( σ ) I X k , ⊗ X J S k ( σ ) Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 10 ·

Simulation Relation to simple simulations Definition A matrix X ∈ K Q × P is relational if X ∈ { 0 , 1 } Q × P functional if X is relational and induces a mapping surjective, injective, . . . Definition (H ÖGBERG , ∼ , M AY 2007) wta A forward simulates wta B if A simulates B with a functional transfer matrix. wta A backward simulates wta B if B simulates A with transfer matrix X such that X T is functional. Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 11 ·

Simulation Relation to simple simulations Definition A matrix X ∈ K Q × P is relational if X ∈ { 0 , 1 } Q × P functional if X is relational and induces a mapping surjective, injective, . . . Definition (H ÖGBERG , ∼ , M AY 2007) wta A forward simulates wta B if A simulates B with a functional transfer matrix. wta A backward simulates wta B if B simulates A with transfer matrix X such that X T is functional. Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 11 ·

Simulation Relation to simple simulations (cont’d) Definition (B ÉAL , L OMBARDY , S AKAROVITCH 2005) The semiring K is equisubtractive if for every a 1 , a 2 , b 1 , b 2 ∈ K with a 1 + b 1 = a 2 + b 2 there exist c 1 , c 2 , d 1 , d 2 ∈ K such that a 1 = c 1 + d 1 and b 1 = c 2 + d 2 a 2 = c 1 + c 2 and b 2 = d 1 + d 2 a 1 a 2 b 1 b 2 Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 12 ·

Simulation Relation to simple simulations (cont’d) Definition (B ÉAL , L OMBARDY , S AKAROVITCH 2005) The semiring K is equisubtractive if for every a 1 , a 2 , b 1 , b 2 ∈ K with a 1 + b 1 = a 2 + b 2 there exist c 1 , c 2 , d 1 , d 2 ∈ K such that a 1 = c 1 + d 1 and b 1 = c 2 + d 2 a 2 = c 1 + c 2 and b 2 = d 1 + d 2 c 1 a 1 a 2 c 2 d 1 c 2 d 1 b 1 b 2 d 2 Simulations of Weighted Tree Automata Z. Ésik and A. Maletti 12 ·