Weighted Tree Automata II. A Kleene theorem for wta over M-monoids - PowerPoint PPT Presentation

1 Weighted Tree Automata II. – A Kleene theorem for wta over M-monoids Zolt´ an F¨ ul¨ op University of Szeged Department of Foundations of Computer Science fulop@inf.u-szeged.hu November 10, 2009

2 Contents • Multioperator monoids (M-monoids) • Uniform tree valuations • Wta over M-monoids, recognizable uniform tree valuations • Rational operations, rational uniform tree valuations • Kleene theorem for recognizable uniform tree valuations • Kleene theorem for (non commutative) semirings • Kleene theorem for commutative semirings is a corollary • References

3 Multioperator monoid A multioperator monoid (for short: M-monoid) ( A, ⊕ , 0 , Ω) consists of - a commutative monoid ( A, ⊕ , 0) and - an Ω -algebra ( A, Ω) - with id A ∈ Ω (1) and 0 m ∈ Ω ( m ) for m ≥ 0 . A is distributive if n n M M ω A ( b 1 , . . . , b i − 1 , a j , b i +1 , . . . , b m ) = ω A ( b 1 , . . . , b i − 1 , a j , b i +1 , . . . , b m ) j =1 j =1 holds for every m, n ≥ 0 , ω ∈ Ω ( m ) , b 1 , . . . , b m ∈ A , 1 ≤ i ≤ m , and a 1 , . . . , a n ∈ A . In particular, ω A ( . . . , 0 , . . . , ) = 0 .

4 Operations on Ops( A ) Ops( A ) ( Ops k ( A ) ) is the set of operations ( k -ary operations) on A . Let ( A, ⊕ , 0 , Ω) be an M-monoid and k ≥ 0 . • Let ω 1 , ω 2 ∈ Ops k ( A ) . The sum of ω 1 and ω 2 is the k -ary operation a ∈ A k , by ( ω 1 ⊕ ω 2 )( � ω 1 ⊕ ω 2 that is defined, for every � a ) ⊕ ω 2 ( � a ) = ω 1 ( � a ) . • Let ω ∈ Ops k ( A ) and ω j ∈ Ops l j ( A ) with l j ≥ 0 for every 1 ≤ j ≤ k . The composition of ω with ( ω 1 , . . . , ω k ) is the ( l 1 + · · · + l k ) -ary operation ω ( ω 1 , . . . , ω k ) that is defined by ` ´ ω ( ω 1 , . . . , ω k ) ( � a k ) = ω ( ω 1 ( � a 1 ) , . . . , ω k ( � a k )) a 1 , . . . , � a j ∈ A l j with 1 ≤ j ≤ k . for every � (Ops k ( A ) , ⊕ , 0 k ) is a commutative monoid for every k ≥ 0 , for k = 0 is isomorphic to the monoid ( A, ⊕ , 0 ) . Sum is left- and right- distributive, and composition is associative.

5 Uniform tree valuations | t | Z is the number of occurrences of variables of Z in t Uvals(Σ , Z, A ) is the class of mappings S : T Σ ( Z ) → Ops( A ) such that the arity of ( S, t ) is | t | Z . Such mappings are called uniform tree valuations over Σ , Z and A . • Hence Uvals(Σ , ∅ , A ) = A � � T Σ � � . 0 , t ) = 0 | t | Z for every t ∈ T Σ ( Z ) . • ( e • The sum of S 1 , S 2 ∈ Uvals(Σ , Z, A ) is the uniform tree valuation S 1 ⊕ u S 2 defined by ( S 1 ⊕ u S 2 , t ) = ( S 1 , t ) ⊕ ( S 2 , t ) for every t ∈ T Σ ( Z ) . • (Uvals(Σ , Z, A ) , ⊕ u , e 0 ) is a commutative monoid; for Z = ∅ it is nothing � , ⊕ , e but ( A � � T Σ � 0 ) . • For S ∈ Uvals(Σ , Z, A ) we write S = L u t ∈ T Σ ( Z ) ( S, t ) .t .

6 Weighted tree automata (wta) over M-monoids Syntax A system M = ( Q, Σ , Z, A, F, µ, ν ) (over Σ , Z and A ) - Q, Σ , Z as before, - ( A, ⊕ , 0 , Ω) is an M-monoid, - F : Q → Ω (1) is the root weight, - µ = ( µ m | m ≥ 0) is the family of transition mappings with µ m : Q m × Σ ( m ) × Q → Ω ( m ) , - ν : Z × Q → Ω (1) , the variable assignment. Such a wta recognizes a uniform tree valuation , i.e., a mapping S M : T Σ ( Z ) → Ops( A ) in Uvals(Σ , Z, A ) . In case Z = ∅ it recognizes a tree series in A � � T Σ � � .

7 Wta over M-monoids Semantics M = ( Q, Σ , Z, A, F, µ, ν ) a wta over the M-monoid A and t ∈ T Σ ( Z ) - a run of M on t is a mapping r : pos( t ) → Q - the set of runs of M on t is R M ( t ) - for w ∈ pos( t ) , the weight wt( t, r, w ) of w in t under r • if t ( w ) = z for some z ∈ Z , then wt( t, r, w ) = ν ( z, r ( w )) • otherwise (if t ( w ) = σ for some σ ∈ Σ ( k ) , k ≥ 0 ) wt( t, r, w ) = µ k ( r ( w 1) , . . . , r ( wk ) , t ( w ) , r ( w ))(wt( t, r, w 1) , . . . , wt( t, r, wk )) • the weight of r is wt( t, r ) = wt( t, r, ε ) . The uniform tree valuation S M : T Σ ( Z ) → A recognized by M is defined by M S M ( t ) = F ( r ( ε ))(wt( t, r )) . r ∈ R M ( t )

8 An example of a wta over M-monoids The tree series height : T Σ → N can be recognized by M = ( Q, Σ , A, F, µ ) , where • Q = { q } , • A = ( N , − , − , Ω) with { 1 + max { n 1 , . . . , n k } | k ≥ 0 } ⊆ Ω , • F ( q ) = id N , and • µ 0 ( α, q ) = 0 and for every k ≥ 1 and σ ∈ Σ ( k ) , let µ k ( q . . . q, σ, q ) = 1 + max { n 1 , . . . , n k } . Then S M = height .

9 Rational operations on Uvals(Σ , Z, A ) 1. The sum ⊕ u : ( S 1 ⊕ u S 2 , t ) = ( S 1 , t ) ⊕ ( S 2 , t ) . 2. The top-concatenation : for k ≥ 0 , σ ∈ Σ ( k ) , ω ∈ Ω ( k ) , and S 1 , . . . , S k ∈ Uvals(Σ , Z, A ) , we define M u top σ,ω ( S 1 , . . . , S k ) = ω (( S 1 , t 1 ) , . . . , ( S k , t k )) .σ ( t 1 , . . . , t k ) . t 1 ,...,t k ∈ T Σ ( Z ) 3. The z -concatenation : for every z ∈ Z and S, S ′ ∈ Uvals(Σ , Z, A ) , we define “ ” M u S · z S ′ = ( S, s ) ◦ s,z (( S ′ , t 1 ) , . . . , ( S ′ , t l )) .s [ z ← ( t 1 , . . . , t l )] . s ∈ T Σ ( Z ) , l = | s | z t 1 ,...,t l ∈ T Σ ( Z )

10 Rational operations on Uvals(Σ , Z, A ) 4. The z - K LEENE -star : for every z ∈ Z and S ∈ Uvals(Σ , Z, A ) we define: (i) S 0 z = e 0 ; and z ) ⊕ u id A .z . (ii) S n +1 = ( S · z S n z Then, the z -K LEENE star S ∗ z of S is defined as follows: If S is z -proper, i.e., ( S, z ) = 0 , then ( S ∗ z , t ) = ( S height( t )+1 , t ) z for every t ∈ T Σ ( Z ) , otherwise S ∗ z = e 0 .

11 Rational expressions (over Σ , Z and A ) RatExp(Σ , Z, A ) (over Σ , Z , and A ) is the smallest set R satisfying Conditions (i)–(v). For every ratexp η ∈ RatExp(Σ , Z, A ) we define its semantics [ [ η ] ] ∈ Uvals(Σ , Z, A ) simultaneously. (i) For every z ∈ Z and ω ∈ Ω (1) we have ω.z ∈ R and [ [ ω.z ] ] = ω.z . (ii) For every k ≥ 0 , σ ∈ Σ ( k ) , ω ∈ Ω ( k ) , and rational expressions η 1 , . . . , η k ∈ R we have top σ,ω ( η 1 , . . . , η k ) ∈ R and [ [top σ,ω ( η 1 , . . . , η k )] ] = top σ,ω ([ [ η 1 ] ] , . . . , [ [ η k ] ]) . ] ⊕ u [ (iii) For every η 1 , η 2 ∈ R we have η 1 + η 2 ∈ R and [ [ η 1 + η 2 ] ] = [ [ η 1 ] [ η 2 ] ] . (iv) For every η 1 , η 2 ∈ R and z ∈ Z we have η 1 · z η 2 ∈ R and [ [ η 1 · z η 2 ] ] = [ [ η 1 ] ] · z [ [ η 2 ] ] . (v) For every η ∈ R and z ∈ Z we have η ∗ [ η ∗ ] ∗ z ∈ R and [ z ] ] = [ [ η ] z .

12 Rational tree valuations (over Σ , Z and A ) We call S ∈ Uvals(Σ , Z, A ) rational , if there exists a rational expression η ∈ RatExp(Σ , Z, A ) such that [ [ η ] ] = S . Rat(Σ , Z, A ) is the class of rational uniform tree valuations over Σ , Z and A . Then Rat(Σ , Z, A ) is the smallest class of uniform tree valuations which contains the uniform tree valuation ω.z for every z ∈ Z and ω ∈ Ω (1) and is closed under the rational operations.

13 Kleene theorem for wta over M-monoids a) Recognizable ⇒ rational: Theorem. If A is distributive, then for every wta M = ( Q, Σ , Z, A, F, µ, ν ) there exists a rational expression η ∈ RatExp(Σ , Z ∪ Q, A ) such that ] | T Σ ( Z ) . S M = [ [ η ] Hence we have Rec(Σ , Z, A ) ⊆ Rat(Σ , fin , A ) | T Σ ( Z ) , where [ Rat(Σ , fin , A ) = Rat(Σ , Z, A ) . Z finite set

14 Kleene theorem for wta over M-monoids The M-monoid ( A, ⊕ , 0 , Ω) is • sum closed , if ω 1 ⊕ ω 2 ∈ Ω ( k ) for every k ≥ 0 and ω 1 , ω 2 ∈ Ω ( k ) . • (1 , ⋆ ) -composition closed , if ω ( ω ′ ) ∈ Ω ( k ) for every k ≥ 0 , ω ∈ Ω (1) , and ω ′ ∈ Ω ( k ) . • ( ⋆, 1) -composition closed , if ω ( ω 1 , . . . , ω k ) ∈ Ω ( k ) for every k ≥ 0 , ω ∈ Ω ( k ) , and ω 1 , . . . , ω k ∈ Ω (1) . b) Rational ⇒ recognizable: Theorem. Let A be a distributive, (1 , ⋆ ) -composition closed and sum closed. Then Rec(Σ , Z, A ) contains the uniform tree valuation ω.z for every z ∈ Z and ω ∈ Ω (1) , and it is closed under the rational operations. Hence, Rat(Σ , Z, A ) ⊆ Rec(Σ , Z, A ) .

15 Kleene theorem for wta over M-monoids In case Z = ∅ : Theorem. For every (1 , ⋆ ) -composition closed and sum closed DM-monoid A , we have Rec(Σ , ∅ , A ) = Rat(Σ , fin , A ) | T Σ . Proof. We have Rec(Σ , ∅ , A ) ⊆ Rat(Σ , fin , A ) | T Σ ⊆ Rec(Σ , fin , A ) | T Σ ⊆ Rec(Σ , ∅ , A ) where the last inclusion can be seen as follows. Let S ∈ Rec(Σ , fin , A ) | T Σ . Thus, there exist a wta M = ( Q, Σ , Z, A, F, µ, ν ) such that S = S M | T Σ . It is easy to see that for the wta N = ( Q, Σ , ∅ , A, F, µ, ∅ ) we have that S N = S M | T Σ . Thus S ∈ Rec(Σ , ∅ , A ) .

16 Wta over (arbitrary) semirings M = ( Q, Σ , Z, K, F, δ, ν ) a wta, K is a semiring, t ∈ T Σ ( Z ) - a run of M on t is a mapping r : pos( t ) → Q - the set of runs of M on t is R M ( t ) - for w ∈ pos( t ) , the weight wt( t, r, w ) of w in t under r • if t ( w ) = z for some z ∈ Z , then wt( t, r, w ) = ν ( z, r ( w )) • otherwise (if t ( w ) = σ for some σ ∈ Σ ( k ) , k ≥ 0 ) wt( t, r, w ) = δ k ( r ( w 1) , . . . , r ( wk ) , t ( w ) , r ( w )) • the weight of r is wt( t, r ) = Q w ∈ pos ( t ) wt( t, r, w ) , where the order of the product is the postorder tree walk. The tree series S M : T Σ ( Z ) → K recognized by M is X S M ( t ) = wt( t, r ) · F ( r ( ε )) . r ∈ R M ( t ) The class of recognizable tree series by such wta: Rec sr (Σ , Z, K ) .