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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

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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

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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) Σ =

  • a , b
  • , ˆ

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

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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) Σ =

  • a , b
  • , ˆ

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

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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) Σ =

  • a , b
  • , ˆ

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

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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) Σ =

  • a , b
  • , ˆ

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

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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) Σ =

  • a , b
  • , ˆ

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

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Nested Words Weighted Automata and Logics

Nested Word Automata

NWA (Alur and Madhusudan) A = (Q, q0, (δcall, δint, δret), Qf) δcall, δint : Q × Σ → Q δret : Q × Q × Σ → Q

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Nested Word Automata

NWA (Alur and Madhusudan) A = (Q, q0, (δcall, δint, δret), Qf) δcall, δint : Q × Σ → Q δret : Q × Q × Σ → Q q0 q1 q2 q3 a a b/q1 b/q1 b/q0 b/q0 ⇒ L(A) = { (a)k (b)k | k ≥ 0}

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

MSO Logic for Nested Words

Definition (MSO(NW (Σ))) β ::= Laba(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

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Nested Words Weighted Automata and Logics

MSO Logic for Nested Words

Definition (MSO(NW (Σ))) β ::= Laba(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

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Nested Words Weighted Automata and Logics

Valuation Monoids (Droste and Meinecke 2012)

Definition ω-valuation monoid (D, +, Valω, 0): complete monoid (D, +, 0) (infinite sums defined) ω-valuation function Valω : Dω → D Valω((di)i∈N) = 0 if di = 0 for an i ∈ N

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Valuation Monoids (Droste and Meinecke 2012)

Definition ω-valuation monoid (D, +, Valω, 0): complete monoid (D, +, 0) (infinite sums defined) ω-valuation function Valω : Dω → D Valω((di)i∈N) = 0 if di = 0 for an i ∈ N Examples

1 totally complete semirings (K, +, ·, 0, 1) 2 Chatterjee, Doyen, Henzinger 2008:

(R ∪ {−∞, ∞}, sup, lim avg, −∞), lim avg((di)i∈N) := lim inf

n→∞

1 n

n

  • i=1

di

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Weighted Stair Muller NWA

wsMNWA A = (Q, I, (δcall, δint, δret), F), F ⊆ 2Q δcall, δint : Q × Σ × Q → D δret : Q × Q × Σ × Q → D

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Weighted Stair Muller NWA

wsMNWA A = (Q, I, (δcall, δint, δret), F), F ⊆ 2Q δcall, δint : Q × Σ × Q → D δret : Q × Q × Σ × Q → D

  • run r

wtA(r, nw, i)

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Weighted Stair Muller NWA

wsMNWA A = (Q, I, (δcall, δint, δret), F), F ⊆ 2Q δcall, δint : Q × Σ × Q → D δret : Q × Q × Σ × Q → D

  • run r

wtA(r, nw, i) A over D = (¯ R, sup, lim avg, −∞) with F = {{q0}} q0 q1 Σ(1) Σ(0), Σ(0), Σ/q1(0) Σ(1) Σ/q0(1)

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Weighted Stair Muller NWA

wsMNWA A = (Q, I, (δcall, δint, δret), F), F ⊆ 2Q δcall, δint : Q × Σ × Q → D δret : Q × Q × Σ × Q → D

  • run r

wtA(r, nw, i) A over D = (¯ R, sup, lim avg, −∞) with F = {{q0}} q0 q1 Σ(1) Σ(0), Σ(0), Σ/q1(0) Σ(1) Σ/q0(1) A(nw) :=

  • r acc

Valω((wtA(r, nw, i))i∈N)

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Weighted Stair Muller NWA

wsMNWA A = (Q, I, (δcall, δint, δret), F), F ⊆ 2Q δcall, δint : Q × Σ × Q → D δret : Q × Q × Σ × Q → D

  • run r

wtA(r, nw, i) A over D = (¯ R, sup, lim avg, −∞) with F = {{q0}} q0 q1 Σ(1) Σ(0), Σ(0), Σ/q1(0) Σ(1) Σ/q0(1) A(nw) :=

  • r acc

Valω((wtA(r, nw, i))i∈N) = lim avg((wtA(r, nw, i))i∈N) = ’ratio’ of top-level positions

  • f nw

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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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

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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, +, Valω, ⋄, 0, 1) ω-valuation monoid (D, +, Valω, 0) 1 ∈ D, Valω(1ω) = 1 ⋄ : D2 → 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

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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

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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

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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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Nested Words Weighted Automata and Logics

Semantics (2)

β (nw, σ) := ✶L(β)(nw, σ) d (nw, σ) := d for all d ∈ D ϕ ∨ ψ (nw, σ) := ϕ (nw, σ) + ψ (nw, σ) ϕ ∧ ψ (nw, σ) := ϕ (nw, σ) ⋄ ψ (nw, σ)

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Semantics (2)

β (nw, σ) := ✶L(β)(nw, σ) d (nw, σ) := d for all d ∈ D ϕ ∨ ψ (nw, σ) := ϕ (nw, σ) + ψ (nw, σ) ϕ ∧ ψ (nw, σ) := ϕ (nw, σ) ⋄ ψ (nw, σ) ∃x.ϕ (nw, σ) :=

  • i∈N

( ϕ (nw, σ[x → i])) ∃X.ϕ (nw, σ) :=

  • I⊆N

( ϕ (nw, σ[X → I])) ∀x.ϕ (nw, σ) := Valω( ϕ (nw, σ[x → i]))i∈N

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Logic Fragments

Motivation: REC wMSO for finite words (Droste and Gastin)

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Logic Fragments

Motivation: REC wMSO for finite words (Droste and Gastin) Definition (almost boolean formulas) ψ ::= d | β | ϕ ∨ ϕ | ϕ ∧ ϕ with d ∈ D, β ∈ MSO(NW (Σ))

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Logic Fragments

Motivation: REC wMSO for finite words (Droste and Gastin) Definition (almost boolean formulas) ψ ::= d | β | ϕ ∨ ϕ | ϕ ∧ ϕ with d ∈ D, β ∈ MSO(NW (Σ)) Definition (ϕ – syntactically restricted) ∀-restricted: subformula ∀x.ψ ⇒ ψ almost boolean strongly-∧-restricted: subformula ψ ∧ θ ⇒ ψ and θ almost boolean or ψ or θ boolean

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

First Main Result

Theorem D regular ω-pv-monoid (constant series are recognizable), S : NW ω(Σ) → D, TFAE: (1) S = A for a wsMNWA A (2) S = ϕ for a syntactically restricted sentence ϕ

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

First Main Result

Theorem D regular ω-pv-monoid (constant series are recognizable), S : NW ω(Σ) → D, TFAE: (1) S = A for a wsMNWA A (2) S = ϕ for a syntactically restricted sentence ϕ Additional assumptions on D ⇒ less restricted formulas possible

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Logic Fragments (2)

Definition We call ϕ ∈ MSO(D, NW (Σ))

1 strongly-∧-restricted: subformula ψ ∧ θ

⇒ ψ and θ almost boolean or ψ or θ boolean

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Logic Fragments (2)

Definition We call ϕ ∈ MSO(D, NW (Σ))

1 strongly-∧-restricted: subformula ψ ∧ θ

⇒ ψ and θ almost boolean or ψ or θ boolean

2 ∧-restricted: subformula ψ ∧ θ

⇒ ψ almost boolean or θ boolean

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Logic Fragments (2)

Definition We call ϕ ∈ MSO(D, NW (Σ))

1 strongly-∧-restricted: subformula ψ ∧ θ

⇒ ψ and θ almost boolean or ψ or θ boolean

2 ∧-restricted: subformula ψ ∧ θ

⇒ ψ almost boolean or θ boolean

3 weakly-∧-restricted: subformula ψ ∧ θ

⇒ ψ almost boolean or const(ψ) and const(θ) commute (const(ϕ) := set of all elements of D occurring in ϕ)

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Second Main Result

Theorem

1

D left-distributive ϕ ∧-restricted ⇒ ∃ strongly-∧-restricted ϕ′ : ϕ′ = ϕ

2

D cc-ω-valuation semiring ϕ weakly-∧-restricted ⇒ ∃ strongly-∧-restricted ϕ′ : ϕ′ = ϕ

3 If ϕ is here ∀-restricted, we can construct ϕ′ also ∀-restricted. Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

Restating First Main Result

Theorem D ω-pv-monoid, S : NW ω(Σ) → D

1 D regular, TFAE:

(1) S = A for a wsMNWA A (2) S = ϕ for a syntactically restricted sentence ϕ

2 D left-distributive, TFAE:

(1) S = A for a wsMNWA A (2) S = ϕ for a ∀-restricted and ∧-restricted sentence ϕ

3 D cc-ω-valuation semiring, TFAE:

(1) S = A for a wsMNWA A (2) S = ϕ for a ∀-restricted and weakly-∧-restricted sentence ϕ

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Restating First Main Result

Theorem D ω-pv-monoid, S : NW ω(Σ) → D

1 D regular, TFAE:

(1) S = A for a wsMNWA A (2) S = ϕ for a syntactically restricted sentence ϕ

2 D left-distributive, TFAE:

(1) S = A for a wsMNWA A (2) S = ϕ for a ∀-restricted and ∧-restricted sentence ϕ

3 D cc-ω-valuation semiring, TFAE:

(1) S = A for a wsMNWA A (2) S = ϕ for a ∀-restricted and weakly-∧-restricted sentence ϕ

Thank you for your attention!

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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  • App. Properties (ω-)Valuation Monoids

Definition ω-pv-monoid (D, +, Valω, ⋄, 0, 1) associative/commutative: ⋄ is associative/commutative left-+-distributive: d ⋄

i∈I di = i∈I(d ⋄ di)

(right-+-distributive, +-distributive resp.) ω-valuation semiring: +-distributive and associative left-multiplicative: d ⋄ Valω((di)i∈N) = Valω(d ⋄ d1, (di)i≥2) left-Valω-distributive: d ⋄ Valω((di)i∈N) = Valω((d ⋄ di)i∈N) left-distributive: left-+-distributive and (left-multiplicative or left-Valω-distributive) conditionally commutative: (∀j < i : di ⋄ d′

j = d′ j ⋄ di) ⇒

Valω((di)i∈N) ⋄ Valω((d′

i )i∈N) = Valω((di ⋄ d′ i )i∈N)

cc-ω-valuation semiring: semiring D conditionally commutative and left-distributive

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Nested Words Weighted Automata and Logics

  • App. Complete Monoid

Definition (complete monoid) A monoid (D, +, 0) is complete if it has infinitary sum operations

  • I : DI → D for any index set I such that
  • i∈∅ di = 0
  • i∈{k} di = dk
  • i∈{j,k} di = dj + dk for j = k
  • j∈J(

i∈Ij di) = i∈I di

if

j∈J Ij = I and Ij ∩ Ik = ∅ for j = k

Note that in every complete monoid the operation + is commutative.

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

  • App. wsMNWA - Acceptance

wsMNWA A = (Q, I, (δcall, δint, δret), F), nw = (a1a2..., ν) Definition (top-level) We call i ∈ N a top-level position if there exist no positions j, k ∈ N with j < i < k and ν(j, k). Definition (run) run r over nw: r = (q0, q1, ...) Qt

∞(r) := {q ∈ Q | qi = q for infinitely many top-level positions i}

r accepting :⇔ q0 ∈ I and Qt

∞(r) ∈ F

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

  • App. wsMNWA - Weights

Definition (weights) wtA(r, nw, i) :=    δcall(qi−1, ai, qi) , if i is a call δint(qi−1,ai, qi) , if i is an internal δret(qi−1, qj−1, ai, qi) , if ν(j, i) Definition (behaviour of A) Behaviour of the automaton A, A : NW ω(Σ) → D: A(nw) :=

  • r acc

Valω((wtA(r, nw, i))i∈N)

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Nested Words Weighted Automata and Logics

  • App. Complexity – Alur and Madhusudan 2009

Decision problems for automata Emptiness Universality Inclusion DFA Nlogspace Nlogspace Nlogspace NFA Nlogspace Pspace Pspace PDA Ptime Undecidable Undecidable NWA Ptime Ptime Ptime Nondet NWA Ptime Exptime Exptime

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

  • App. Main Theorem - Components

Lemma (Closure under conjunction) θ, ψ subformulas of a syntactically restricted formula ϕ: θ and ψ regular ⇒ θ ∧ ψ regular

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Nested Words Weighted Automata and Logics

  • App. Main Theorem - Components

Lemma (Closure under conjunction) θ, ψ subformulas of a syntactically restricted formula ϕ: θ and ψ regular ⇒ θ ∧ ψ regular Lemma (Closure under restricted ∀x) ϕ almost boolean ⇒ ∀x.ϕ regular

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

  • App. Main Theorem - Ideas

Series S regular step function :⇔ ∃ representation: S(nw) =

k

  • i=1

di✶Li(nw) Li: regular languages of nested ω-words (Li) partition of NW ω(Σ) di ∈ D S(nw) = di ⇔ nw ∈ Li

  • (regular) ω-valuation-monoid D:

S regular step function ⇒ S regular regular step functions closed under ⋄ and + (pointwise) ϕ almost boolean ⇒ ϕ regular step function

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

  • App. Definition Nesting Relation

nesting relation ν

  • f length ℓ ≥ 0: subset of {−∞, 1, ..., ℓ} × {1, ..., ℓ, ∞}
  • ver N:

subset of ({−∞} ∪ N) × (N ∪ {∞}) (i) ν(i, j) ⇒ i < j, (ii) ∀i : 1 ≤ i ≤ ℓ ⇒ |{j : ν(i, j)}| ≤ 1 ∧ |{j : ν(j, i)}| ≤ 1, (ii’) ∀i : i ∈ N ⇒ |{j : ν(i, j)}| ≤ 1 ∧ |{j : ν(j, i)}| ≤ 1 (at most one nesting edge per position (except −∞, ∞)), (iii) ν(i, j) ∧ ν(i′, j′) ∧ i < i′ ⇒ j < i′ ∨ j > j′ (nesting edges do not cross), (iv) (−∞, ∞) / ∈ ν.

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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Nested Words Weighted Automata and Logics

  • App. sMNWA Example

L := {nw ∈ NW ω(Σ) | nw has only finitely many pending calls} sMNWA with F = {{q0}, {q1}} q0 q1 Σ, Σ/q0 Σ, Σ/q1 Σ, Σ/q1 Σ, Σ/q0

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words

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  • App. Class VPL

Regular ⊂ VPL ⊂ CFL VPL = visibly pushdown languages ˆ = regular languages of nested words Closed under ∪ , ∩ and complement Substantially more expressive than regular languages Usage: Representation of data with linear and hierarchical structure (e.g. structured programs, XML documents)

Manfred Droste and Stefan D¨ uck Weighted Automata and Logics for Infinite Nested Words