Nested Words Weighted Automata and Logics Weighted Automata and Logics for Infinite Nested Words Manfred Droste and Stefan D¨ uck Leipzig University, Germany 8th LATA, March 2014 Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Nested Words (Alur and Madhusudan JACM 2009) Usage Representation of data with linear and hierarchical structure (e.g. structured programs, XML documents) , ˆ � � � � Σ = a , b Σ = a , � a , a � , b , � b , b � word over Σ : a b a b a b a Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Nested Words (Alur and Madhusudan JACM 2009) Usage Representation of data with linear and hierarchical structure (e.g. structured programs, XML documents) , ˆ � � � � Σ = a , b Σ = a , � a , a � , b , � b , b � word over Σ : a b a b a b a nested word : a b a b a b a Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Nested Words (Alur and Madhusudan JACM 2009) Usage Representation of data with linear and hierarchical structure (e.g. structured programs, XML documents) , ˆ � � � � Σ = a , b Σ = a , � a , a � , b , � b , b � word over Σ : a b a b a b a nested word : a b a b a b a call internal return Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Nested Words (Alur and Madhusudan JACM 2009) Usage Representation of data with linear and hierarchical structure (e.g. structured programs, XML documents) , ˆ � � � � Σ = a , b Σ = a , � a , a � , b , � b , b � word over Σ : a b a b a b a nested word : a b a b a b a call internal return representation over ˆ Σ : � a � b a � b a � � b a � Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Nested Words (Alur and Madhusudan JACM 2009) Usage Representation of data with linear and hierarchical structure (e.g. structured programs, XML documents) , ˆ � � � � Σ = a , b Σ = a , � a , a � , b , � b , b � word over Σ : a b a b a b a nested word : a b a b a b a call internal return representation over ˆ Σ : � a � b a � b a � � b a � with nesting relation ν : ( w , ν ) ∈ NW (Σ) = ( abababa , { (1 , 5) , (2 , 3) , (6 , 7) } ) Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Nested Word Automata NWA (Alur and Madhusudan) A = ( Q , q 0 , ( δ call , δ int , δ ret ) , Q f ) δ call , δ int : Q × Σ → Q δ ret : Q × Q × Σ → Q Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Nested Word Automata NWA (Alur and Madhusudan) A = ( Q , q 0 , ( δ call , δ int , δ ret ) , Q f ) δ call , δ int : Q × Σ → Q δ ret : Q × Q × Σ → Q � a b � / q 1 � a b � / q 1 b � / q 0 q 0 q 1 q 2 q 3 b � / q 0 ⇒ L ( A ) = { ( � a ) k ( b � ) k | k ≥ 0 } Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics MSO Logic for Nested Words Definition ( MSO ( NW (Σ))) β ::= Lab a ( x ) | call( x ) | ret( x ) | ν ( x , y ) | x ≤ y | x ∈ X | ¬ β | β ∧ β | ∀ x .β | ∀ X .β with a ∈ Σ, x , y , X ∈ V , V finite set of FO- and SO-variables Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics MSO Logic for Nested Words Definition ( MSO ( NW (Σ))) β ::= Lab a ( x ) | call( x ) | ret( x ) | ν ( x , y ) | x ≤ y | x ∈ X | ¬ β | β ∧ β | ∀ x .β | ∀ X .β with a ∈ Σ, x , y , X ∈ V , V finite set of FO- and SO-variables Theorem (Alur and Madhusudan) L language of nested [ ω -] words over Σ , TFAE: (1) L = L ( A ) for an [sM] NWA A (2) L = L ( β ) for an MSO ( NW (Σ)) -sentence β Our goal: Quantitative version of this result Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Valuation Monoids (Droste and Meinecke 2012) Definition ω -valuation monoid ( D , + , Val ω , 0): complete monoid ( D , + , 0) (infinite sums defined) ω -valuation function Val ω : D ω → D Val ω (( d i ) i ∈ N ) = 0 if d i = 0 for an i ∈ N Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Valuation Monoids (Droste and Meinecke 2012) Definition ω -valuation monoid ( D , + , Val ω , 0): complete monoid ( D , + , 0) (infinite sums defined) ω -valuation function Val ω : D ω → D Val ω (( d i ) i ∈ N ) = 0 if d i = 0 for an i ∈ N Examples 1 totally complete semirings ( K , + , · , 0 , 1) 2 Chatterjee, Doyen, Henzinger 2008: ( R ∪ {−∞ , ∞} , sup , lim avg , −∞ ) , n 1 � lim avg(( d i ) i ∈ N ) := lim inf d i n n →∞ i =1 Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Weighted Stair Muller NWA wsMNWA A = ( Q , I , ( δ call , δ int , δ ret ) , F ), F ⊆ 2 Q δ call , δ int : Q × Σ × Q → D δ ret : Q × Q × Σ × Q → D Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Weighted Stair Muller NWA wsMNWA A = ( Q , I , ( δ call , δ int , δ ret ) , F ), F ⊆ 2 Q � δ call , δ int : Q × Σ × Q → D run r � wt A ( r , nw , i ) δ ret : Q × Q × Σ × Q → D Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Weighted Stair Muller NWA wsMNWA A = ( Q , I , ( δ call , δ int , δ ret ) , F ), F ⊆ 2 Q � δ call , δ int : Q × Σ × Q → D run r � wt A ( r , nw , i ) δ ret : Q × Q × Σ × Q → D A over D = (¯ R , sup , lim avg , −∞ ) with F = {{ q 0 }} Σ(1) Σ(0) , � Σ(0) , Σ � / q 1 (0) � Σ(1) q 0 q 1 Σ � / q 0 (1) Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Weighted Stair Muller NWA wsMNWA A = ( Q , I , ( δ call , δ int , δ ret ) , F ), F ⊆ 2 Q � δ call , δ int : Q × Σ × Q → D run r � wt A ( r , nw , i ) δ ret : Q × Q × Σ × Q → D A over D = (¯ R , sup , lim avg , −∞ ) with F = {{ q 0 }} �A� ( nw ) Σ(1) Σ(0) , � Σ(0) , Σ � / q 1 (0) � Val ω (( wt A ( r , nw , i )) i ∈ N ) := r acc � Σ(1) q 0 q 1 Σ � / q 0 (1) Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Weighted Stair Muller NWA wsMNWA A = ( Q , I , ( δ call , δ int , δ ret ) , F ), F ⊆ 2 Q � δ call , δ int : Q × Σ × Q → D run r � wt A ( r , nw , i ) δ ret : Q × Q × Σ × Q → D A over D = (¯ R , sup , lim avg , −∞ ) with F = {{ q 0 }} �A� ( nw ) Σ(1) Σ(0) , � Σ(0) , Σ � / q 1 (0) � Val ω (( wt A ( r , nw , i )) i ∈ N ) := r acc � Σ(1) q 0 q 1 = lim avg(( wt A ( r , nw , i )) i ∈ N ) = ’ratio’ of top-level positions Σ � / q 0 (1) of nw Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Weighted MSO Logic – based on Droste and Gastin 2006 Definition ( MSO ( D , NW (Σ))) ϕ ::= d | β | ϕ ∨ ϕ | ϕ ∧ ϕ | ∀ x .ϕ | ∃ x .ϕ | ∃ X .ϕ with d ∈ D , β ∈ MSO ( NW (Σ)) ( boolean ), x , y , X ∈ V as before Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Weighted MSO Logic – based on Droste and Gastin 2006 Definition ( MSO ( D , NW (Σ))) ϕ ::= d | β | ϕ ∨ ϕ | ϕ ∧ ϕ | ∀ x .ϕ | ∃ x .ϕ | ∃ X .ϕ with d ∈ D , β ∈ MSO ( NW (Σ)) ( boolean ), x , y , X ∈ V as before Definition ( ω -pv-monoid – Droste and Meinecke) product ω -valuation monoid ( D , + , Val ω , ⋄ , 0 , 1) ω -valuation monoid ( D , + , Val ω , 0) 1 ∈ D , Val ω (1 ω ) = 1 ⋄ : D 2 → D , 0 ⋄ d = d ⋄ 0 = 0 and 1 ⋄ d = d ⋄ 1 = d ∀ d ∈ D Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Semantics (1) given ϕ ∈ MSO ( D , NW (Σ)) nw ∈ NW ω (Σ) σ assignment of free variables of ϕ to positions / set of positions Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Semantics (1) given ϕ ∈ MSO ( D , NW (Σ)) nw ∈ NW ω (Σ) σ assignment of free variables of ϕ to positions / set of positions define � ϕ � ( nw , σ ) ∈ D inductively as follows Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
Nested Words Weighted Automata and Logics Semantics (2) � β � ( nw , σ ) := ✶ L ( β ) ( nw , σ ) � d � ( nw , σ ) := d for all d ∈ D Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words
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