Example number of -symbols. length of the largest sequence of - - PowerPoint PPT Presentation

Finite Automata (FA) and Monadic Second Order logic (MSO). FA: executable model with good (decidable) properties. MSO (over words): very expressive and yet simple logic. Both equally expressive over words and trees (Büchi). Qualitative properties over words. Quantitative properties are also important (today).

Example

number of

• symbols.

length of the largest sequence of

• symbols.

How can we extend finite automata or MSO to define these properties (or functions)?

Weighted automata

General automata framework to define quantitative properties over words. (Boolean) automata, Probabilistic automata, Distance automata, Multiplicity automata, etc... Extension of finite automata with weights from a fix semiring.

Semiring (reminder)

Definition

A (commutative) semiring is an algebraic structure S = (S, ⊕, ⊙, 0, 1) where: (S, ⊕, 0) and (S, ⊙, 1) are commutative monoids, multiplication distributes over addition, and 0 ⊙ s = s ⊙ 0 = 0 for each s ∈ S.

Example

Natural numbers: (N, +, ⋅, 0, 1). Boolean: ({true, false}, ∨, ∧, false, true). Min-plus: (N∞, min, +, ∞, 0). Max-plus: (N−∞, max, +, −∞, 0).

Weighted automata (definition)

Fix a semiring S and a finite alphabet Γ.

Definition

A weighted automata over S and Γ is a tuple A = (Γ, S, Q, E, I, F): E ∶ Q × Γ × Q → S is the transition relation (p a/s

• → q), and

I, F ∶ Q → S is the initial and final function.

Semantics

A run ρ of A over a1 . . . an ∈ Γ∗ is: ρ = q0

a1/s1

• → q1

a2/s2

• → ⋯ an/sn
• → qn

The weight of run ρ of A: weight(ρ) = I(q0) ⊙

n

⊙

i=1

si ⊙ F(qn) A defines the function ⟦A⟧ ∶ Γ∗ → S: ⟦A⟧(w) = ⊕

ρ∈RunA(w)

weight(ρ)

Weighted automata (examples)

Over (N,+,⋅, 0, 1)

f(w) = 3 ⋅ ∣w∣a + 4 ⋅ ∣w∣b a, b/1 a, b/1 a/3 b/4

Over (N∞, min,+,∞, 0)

f(w) = min{∣w∣a, ∣w∣b} a/1 b/0 a/0 b/1

Over (N−∞, max,+,−∞, 0)

f(w) = maximum length of all infix sequences of b’s a, b/0 b/1 a, b/0 b/1 a/0

What is a good logic to define quantitative properties?

Weighted MSO (Droste & Gastin 2005) Disadvantages: Semantical definition of valid formulas. Inherits the undecidability results of weighted automata. We want a quantitative logic that:

• 1. has a simple and purely syntactical definition,
• 2. as expressive as weighted automata, and
• 3. with good decidability properties.

We propose: Quantitative Monadic Second Order Logic (QMSO)

• 1. General framework for adding quantitative properties to any boolean logic.
• 2. Subfragments of QMSO capture different subclasses of WA.
• 3. Subfragments of QMSO with good decidability properties.

More results in the paper: Evalution of QMSO with respect to counting complexity classes.

Quantitative Monadic Second-Order Logic

Cristian Riveros University of Oxford Stephan Kreutzer Technische Universität Berlin LICS 2013

QMSO and WA QMSO and subclasses of WA Beyond WA Conclusions

Outline

Quantitative Monadic Second Order Logic (QMSO)

For each w ∈ Γ∗, we represent w ∶= ({1, . . . , ∣w∣}, ≤, {Pa}a∈Γ).

Syntax of QMSO[S,Γ]

ϕ ∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ ϕ ∨ ϕ ∣ ¬ϕ ∣ ∃x. ϕ ∣ ∃X. ϕ θ ∶= ϕ ∣ s ∈ S ∣ θ ⊕ θ ∣ θ ⊙ θ ∣ Σx. θ ∣ Πx. θ ∣ ΣX. θ

Semantic of QMSO[S,Γ]

⟦ϕ⟧(w, σ) ∶= { 1 if (w, σ) ⊧ ϕ

• therwise

⟦s⟧(w, σ) ∶= s ⟦θ1 ⊕ θ2⟧(w, σ) ∶= ⟦θ1⟧(w, σ) ⊕ ⟦θ2⟧(w, σ) ⟦Πx. θ(x)⟧(w, σ) ∶= ⊙

i ∈ dom(w)

⟦θ(x)⟧(w, σ[x → i]) ⟦ΣX. θ(X)⟧(w, σ) ∶= ⊕

I ⊆ dom(w)

⟦θ(X)⟧(w, σ[X → I])

Quantitative Monadic Second Order Logic (QMSO)

The syntax of QMSO[S, Γ] depends on the semiring.

Syntax of QMSO[(N,+,⋅, 0, 1),Γ]

ϕ ∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ ϕ ∨ ϕ ∣ ¬ϕ ∣ ∃x. ϕ ∣ ∃X. ϕ θ ∶= ϕ ∣ s ∈ N ∣ θ + θ ∣ θ ⋅ θ ∣ Σx. θ ∣ Πx. θ ∣ ΣX. θ

Syntax of QMSO[(N∞, min,+,∞, 0),Γ]

ϕ ∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ ϕ ∨ ϕ ∣ ¬ϕ ∣ ∃x. ϕ ∣ ∃X. ϕ θ ∶= ϕ ∣ s ∈ N∞ ∣ min{θ, θ} ∣ θ + θ ∣ Min x. θ ∣ Σx. θ ∣ Min X. θ

Syntax of QMSO[(N−∞, max,+,−∞, 0),Γ]

ϕ ∶= Pa(x) ∣ x ≤ y ∣ x ∈ X ∣ ϕ ∨ ϕ ∣ ¬ϕ ∣ ∃x. ϕ ∣ ∃X. ϕ θ ∶= ϕ ∣ s ∈ N−∞ ∣ max{θ, θ} ∣ θ + θ ∣ Max x. θ ∣ Σx. θ ∣ Max X. θ

Examples of QMSO formulas

Over (N,+,⋅, 0, 1)

f(w) = 3 ⋅ ∣w∣a + 4 ⋅ ∣w∣b Σx. ( 3 ⋅ Pa(x) + 4 ⋅ Pb(x) )

Over (N∞, min,+,∞, 0)

f(w) = min{∣w∣a, ∣w∣b} min { Σx. Pa(x) ↦ 1 , Σx. Pb(x) ↦ 1 } where Pa(x) ↦ 1 ∶= min{ Pa(x) + 1, ¬Pa(x) }.

Over (N−∞, max,+,−∞, 0)

f(w) = maximum length of all infix sequences of b’s Max x. (Σy. intervalb(x, y) ↦ 1) where intervalb(x, y) ∶= x ≤ y ∧ ∀z. (x ≤ z ∧ z ≤ y) → Pb(z).

Subfragments of QMSO

• 1. QMSO(Op) restricted to operators Op ⊆ {⊕, ⊙, Σx, Πx, ΣX }.

⊕ = semiring addition ⊙ = semiring multiplication Σx = first-order addition Πx = first-order multiplication ΣX = second-order addition

Example

Full QMSO ∶= QMSO(ΣX , Πx, Σx, ⊕, ⊙)

Subfragments of QMSO

• 1. QMSO(Op) restricted to operators Op ⊆ {⊕, ⊙, Σx, Πx, ΣX }.
• 2. Alternation and Nesting of semiring quantifiers.

Example

QMSO(ΣX ΣxΠx, ⊕, ⊙): ΣX. (Σy. Πz. ϕ(X, z)) ⊕ (Πz1. Πz2. θ(X, z1, z2)) QMSO(ΣxΠ1

x, ⊕, ⊙):

Σx. (Σy. Πz. ϕ(x, y, z)) ⊙ (Πz. θ(x, z)) QMSO(Πn

x, ⊕, ⊙), n ∈ N:

Πx1. ⋯Πxn. θ(x1, . . . , xn)

QMSO and weighted automata

QMSO is too expressive to capture weighted automata!

Over (N,+,⋅, 0, 1)

⟦Πx. Πy. 2⟧(w) = 2∣w∣2. For every weighted automata A over (N, +, ⋅, 0, 1): ⟦A⟧(w) ∈ 2O(∣w∣)

QMSO and weighted automata

QMSO is too expressive to capture weighted automata!

Definition

Quantitative Iteration Logic (QIL) ∶= QMSO(ΣX,xΠ1

x, ⊕, ⊙).

Theorem

A function f ∶ Γ∗ → S is definable by a weighted automaton over S and Γ if, and only if, f is definable by a formula in QIL[S, Γ]. Weighted Automata ≡ QIL .

Undecidable properties of QIL

Quantitative generalization of classical decision problems: Equivalence: ⟦θ1⟧(w) = ⟦θ2⟧(w) for all w ∈ Γ∗, Containment: ⟦θ1⟧(w) ≤ ⟦θ2⟧(w) for all w ∈ Γ∗.

Proposition

The following problems are undecidable:

• 1. Containment of formulas in QMSO(ΣxΠ1

x, ⊕, ⊙) over (N, +, ⋅, 0, 1).

• 2. Equivalence and containment of formulas in QMSO(ΣxΠ1

x, ⊕, ⊙)

• ver (N∞, min, +, ∞, 0).

QMSO and WA QMSO and subclasses of WA Beyond WA Conclusions

Outline

Different fragments of QMSO captures different subclasses of WA

Classes of Weighted Automata (WA) depending on the ambiguity: Deterministic WA (DWA). Unambiguous WA (unamb- WA): ∣ RunA(w)∣ ≤ 1 for all w ∈ Σ∗ Finite Ambiguous WA (fin- WA): ∣ RunA(w)∣ < k for all w ∈ Σ∗ Polynomially Ambiguous WA (poly- WA): ∣ RunA(w)∣ ∈ O(∣w∣k) DWA ⊊ unamb- WA ⊊ fin- WA ⊊ poly- WA ⊊ WA

Unambiguous and finitely ambiguous weighted automata are captured by QMSO

Subfragment QMSO(Op, ⊕b): ⊕-operator is restricted to a “base” level.

Example

(Πx. Pa(x) ⊕ Pb(x)) ⊙ (Πx. ∃z. x ≤ z ∧ Pa(z)) ∈ QMSO(Π1

x, ⊕b, ⊙)

(Πx. Pa(x) ⊕ Pb(x)) ⊕ (Πx. ∃z. x ≤ z ∧ Pa(z)) ∉ QMSO(Π1

x, ⊕b, ⊙)

Unambiguous and finitely ambiguous weighted automata are captured by QMSO

Subfragment QMSO(Op, ⊕b): ⊕-operator is restricted to a “base” level.

Theorem

unamb- WA ≡ QMSO(Π1

x, ⊕b, ⊙)

fin- WA ≡ QMSO(Π1

x, ⊕, ⊙)

Proof idea.

From QMSO(Π1

x, ⊕b, ⊙) to unamb- WA:

Exploit unambiguity to express formulas of the form Πx. ⊕

i∈I

⊙

j∈J

ϕi,j(x). From QMSO(Π1

x, ⊕, ⊙) to fin- WA:

Use disambiguation theorem presented in Klimann et all, 2004.

Polynomial ambiguous weighted automata are also captured by QMSO

Theorem

poly- WA ≡ QMSO(ΣxΠ1

x, ⊕, ⊙)

Proof idea.

From poly- WA to QMSO(ΣxΠ1

x, ⊕, ⊙):

Exploit structural properties of the components of a poly- WA.

Which fragment captures deterministic weighted automata?

The forward-iterator (⋅)→ and the backward-iterator (⋅)← ⟦θ→⟧(w, σ) =

n

⊙

i=1

⟦θ⟧(w[1..i], σ) ⟦θ←⟧(w, σ) =

n

⊙

i=1

⟦θ⟧(w[i..n], σ)

Over (N∞, min,+,∞, 0)

f(w) = number of prefixes of w that satisfy ϕ. (min{ ϕ + 1, ¬ϕ })→.

Which fragment captures deterministic weighted automata?

The forward-iterator (⋅)→ and the backward-iterator (⋅)← ⟦θ→⟧(w, σ) =

n

⊙

i=1

⟦θ⟧(w[1..i], σ) ⟦θ←⟧(w, σ) =

n

⊙

i=1

⟦θ⟧(w[i..n], σ)

Theorem

DWA ≡ QMSO(→, ⊕b, ⊙) co- DWA ≡ QMSO(←, ⊕b, ⊙)

∗ the (⋅)→- and (⋅)←-operator cannot be nested.

Connection of determinization of WA with logic.

Fragments with good decidability properties

Corollary

The following problems are decidable:

• 1. Equivalence and containment problem of formulas in QMSO(Π1

x, ⊕b, ⊙)

• ver (N, +, ⋅, 0, 1).
• 2. Equivalence and containment problem of formulas in QMSO(Π1

x, ⊕, ⊙)

• ver (N∞, min, +, ∞, 0).

QMSO(Π1

x, ⊕b, ⊙) and QMSO(Π1 x, ⊕, ⊙) are good fragments.

QMSO and WA QMSO and subclasses of WA Beyond WA Conclusions

Outline

How to go further from these (good) fragments?

• 1. Additive fragment: QMSO(Σk

xΠ1 x, ⊕, ⊙b).

Theorem

For all k ∈ N: polyk- WA ≡ QMSO(Σk

xΠ1 x, ⊕, ⊙b).

• 2. Multiplicative fragment: QMSO(Πk

x, ⊕b, ⊙).

Two-way weighted automata with nested pebbles.

Two-way weighted automata with nested pebbles

1 3 2 In the boolean case: Two-way weighted automata with nested pebbles ≡ regular languages Different subclasses of 2WA: Two-way WA with k-nested pebbles (2WA-k). Deterministic 2WA-k (2DWA-k). Unambiguous 2WA-k (unamb- 2WA-k).

Multiplicative fragment and two-way WA with nested pebbles

Theorem

The following classes of WA and subfragments of QMSO are equally expressive over Γ and S:

• 1. 2DWA-0,
• 2. unamb- 2WA-0,
• 3. unamb- WA, and
• 4. QMSO(Π1

x, ⊕b, ⊙).

Theorem

For every k ∈ N, there exists an effective translation between the following classes of WA and subfragments of QMSO over Γ and S:

• 1. 2DWA-k,
• 2. unamb- 2WA-k, and
• 3. QMSO(Πk+1

x

, ⊕b, ⊙).

QMSO and WA QMSO and subclasses of WA Beyond WA Conclusions

Outline

Future work

Logic-side: Relation between (inner) boolean logic and semiring operators. Expressibility of QMSO over more general structures. Automata-side: Decidability properties of subclasses of WA motivated by QMSO. Determinization of WA.