Pebble weighted automata and transitive closure logics B. Bollig, - - PowerPoint PPT Presentation

Pebble weighted automata and transitive closure logics

• B. Bollig, P. Gastin, B. Monmege, M. Zeitoun

LSV, ENS Cachan, CNRS, INRIA.

DOTS meeting March 18, 2010

Motivations

Analysis of quantitative systems

◮ Probabilistic Systems ◮ Minimization of costs ◮ Maximization of rewards ◮ . . .

Motivations

Analysis of quantitative systems

◮ Probabilistic Systems ◮ Minimization of costs ◮ Maximization of rewards ◮ . . .

Models

◮ Probabilistic automata (generative, reactive) ◮ Transition systems with costs or rewards ◮ . . .

Special instances of weighted automata.

Motivations

Sequential setting: automata on (finite) words.

◮ Weighted automata: quantitative extension of classical automata.

◮ Classical: decide whether a given word is accepted or not, ◮ Weighted: compute a value in a semiring from input word.

◮ Weighted logics: a formula evaluated on a word produces a value.

◮ How often does a Boolean property hold? ◮ Is the number of nodes selected by a request at least 10?

◮ In this talk, we focus on expressiveness. Boolean setting:

Automata = FO+TC = MSO = EMSO = . . .

Weighted automata

◮ Transitions carry weights from a semiring K = (K, +, ×, 0, 1).

p q ka

Example: Semirings

◮ B = ({0, 1}, ∨, ∧, 0, 1)

Boolean

◮ P = (R+, +, ×, 0, 1)

Probabilistic

◮ T = (N ∪ {∞}, min, +, ∞, 0)

Tropical

◮ N = (N, +, ×, 0, 1)

Natural

Weighted automata

◮ Transitions carry weights from a semiring K = (K, +, ×, 0, 1).

p q ka

Example: Semirings

◮ B = ({0, 1}, ∨, ∧, 0, 1)

Boolean

◮ P = (R+, +, ×, 0, 1)

Probabilistic

◮ T = (N ∪ {∞}, min, +, ∞, 0)

Tropical

◮ N = (N, +, ×, 0, 1)

Natural

◮ Weight of a run: product of all transition weights in the semiring.

weight(p0

k1a1

− − − → p1

k2a2

− − − → · · ·

knan

− − − → pn) = k1k2 · · · kn

◮ Weight of a word: sum of all weights of accepted runs on this word.

A(w) =

• ρ run of A on w

weight(ρ)

Weighted automata

◮ Transitions carry weights from a semiring K = (K, +, ×, 0, 1).

p q ka

Example: Semirings

◮ B = ({0, 1}, ∨, ∧, 0, 1)

Boolean existence of accepting run.

◮ P = (R+, +, ×, 0, 1)

Probabilistic probability of acceptance

◮ T = (N ∪ {∞}, min, +, ∞, 0)

Tropical minimal cost

◮ N = (N, +, ×, 0, 1)

Natural

◮ Weight of a run: product of all transition weights in the semiring.

weight(p0

k1a1

− − − → p1

k2a2

− − − → · · ·

knan

− − − → pn) = k1k2 · · · kn

◮ Weight of a word: sum of all weights of accepted runs on this word.

A(w) =

• ρ run of A on w

weight(ρ)

Examples of weighted automata

◮ Alphabet Σ, on (N, +, ×, 0, 1).

1 1 2Σ A(u) = 2|u|. 1 1 1Σ 1Σ 1a −1b A(u) = |u|a − |u|b.

Weighted automata cannot compute large functions

Lemma

A = (Q, µ) weighted automaton on N. There exists M such that A(u) = O(M|u|).

◮ There are O(|Q||u|) runs on u, ◮ Each of which of weight exponential in |u| = n: k1 · · · kn ≤ (max ki)n.

Pebble weighted automata

◮ Automaton with 2-way mechanism and pebbles {1, . . . , r}. ◮ Read-only head can go either direction on ⊲u⊳ (where ⊲, ⊳ /

∈ Σ).

◮ The automaton can ←, →, drop or lift a pebble with a stack policy.

Only the more recently dropped pebble may be lifted.

◮ Applicable transitions depend on current (state,letter,pebbles).

(p, ka, Pebbles, D, q), where D ∈ {←, →, lift, drop}.

◮ Note. For Boolean word automata, this does not add expressive power.

Pebble weighted automata

◮ Automaton with 2-way mechanism and pebbles {1, . . . , r}. ◮ Read-only head can go either direction on ⊲u⊳ (where ⊲, ⊳ /

∈ Σ).

◮ The automaton can ←, →, drop or lift a pebble with a stack policy.

Only the more recently dropped pebble may be lifted.

◮ Applicable transitions depend on current (state,letter,pebbles).

(p, ka, Pebbles, D, q), where D ∈ {←, →, lift, drop}.

◮ Note. For Boolean word automata, this does not add expressive power.

1 1 2Σ

Pebble weighted automata

◮ Automaton with 2-way mechanism and pebbles {1, . . . , r}. ◮ Read-only head can go either direction on ⊲u⊳ (where ⊲, ⊳ /

∈ Σ).

◮ The automaton can ←, →, drop or lift a pebble with a stack policy.

Only the more recently dropped pebble may be lifted.

◮ Applicable transitions depend on current (state,letter,pebbles).

(p, ka, Pebbles, D, q), where D ∈ {←, →, lift, drop}.

◮ Note. For Boolean word automata, this does not add expressive power.

2Σ, ∗, → Σ, ∗, ← Σ, ∅, ← ⊲, ∗, → Σ, ∗, drop ⊲, ∗, → ⊳, ∗, ← Σ, •, lift Σ, ∗, →

◮ Computes 2|u|2: pebbles add expressive power.

Weighted MSO

Short history

Introduced by Droste & Gastin (ICALP’05) Aim: Logical characterization of weighted automata. Generalization of Elgot’s and Büchi’s theorems.

Weighted MSO

Short history

Introduced by Droste & Gastin (ICALP’05) Aim: Logical characterization of weighted automata. Generalization of Elgot’s and Büchi’s theorems. Extended to

◮ Trees

Droste & Vogler

◮ Infinite words

Droste & Kuske, Droste & Rahonis

◮ Pictures

Fischtner

◮ Traces

Meinecke

◮ Distributed systems

Bollig & Meinecke

◮ . . .

Weighted MSO

Definition: Syntax of MSO

ϕ ::= k | Pa(x) | x ≤ y | x ∈ X | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃Xϕ | ∀Xϕ where k ∈ K, a ∈ Σ, x, y are first-order variables, X is a set variable.

Definition: Semantics

◮ A formula ϕ without free variables defines a mapping ϕ : Σ+ → K. ◮ First order variables are interpreted as positions in the word. ◮ Pa(x) means “position x carries an a”. ◮ x ≤ y means “position x is before position y”.

Weighted MSO

Definition: Syntax of MSO

ϕ ::= k | Pa(x) | x ≤ y | x ∈ X | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃Xϕ | ∀Xϕ where k ∈ K, a ∈ Σ, x, y are first-order variables, X is a set variable.

Definition: Semantics

◮ A formula ϕ without free variables defines a mapping ϕ : Σ+ → K. ◮ First order variables are interpreted as positions in the word. ◮ Pa(x) means “position x carries an a”. ◮ x ≤ y means “position x is before position y”. ◮ ϕ1 ∨ ϕ2 = ϕ1 + ϕ2 and ϕ1 ∧ ϕ2 = ϕ1 × ϕ2. ◮ ∃xϕ interpreted as a sum over all positions. ◮ ∀xϕ interpreted as a product over all positions.

MSO: examples

◮ ∃xPa(x) = |u|a

recognizable

◮ ∃xPa(x) ∨ ∃x[−1 ∧ Pb(x)] = |u|a − |u|b

recognizable

◮ ∀y2(u) = 2|u|.

recognizable

MSO: examples

◮ ∃xPa(x) = |u|a

recognizable

◮ ∃xPa(x) ∨ ∃x[−1 ∧ Pb(x)] = |u|a − |u|b

recognizable

◮ ∀y2(u) = 2|u|.

recognizable

◮ ∀x∀y2(u) = 2|u|2.

not recognizable

◮ =

⇒ Recognizable are not stable under universal quantification.

MSO: examples

◮ ∃xPa(x) = |u|a

recognizable

◮ ∃xPa(x) ∨ ∃x[−1 ∧ Pb(x)] = |u|a − |u|b

recognizable

◮ ∀y2(u) = 2|u|.

recognizable

◮ ∀x∀y2(u) = 2|u|2.

not recognizable

◮ =

⇒ Recognizable are not stable under universal quantification. [DG’05] defined RMSO, a fragment of MSO. Theorem (Droste & Gastin’05) Weighted automata = RMSO (effective translation).

Pebble weighted automata are stable under FO

Lemma

Pebble weighted automata are stable under FO constructs. Proof idea for ∀/∃: add first pebble interpreted as free variable. For ∀: drop pebble 1 successively on each position. A

Pebble weighted automata are stable under FO

Lemma

Pebble weighted automata are stable under FO constructs. Proof idea for ∀/∃: add first pebble interpreted as free variable. For ∀: drop pebble 1 successively on each position. A Σ, ∗, ← Σ, ∅, ← ⊲, ∗, → Σ, ∗, drop ⊲, ∗, → ⊳, ∗, ← Σ, •, lift Σ, ∗, →

Pebble weighted automata are stable under FO

Lemma

Pebble weighted automata are stable under FO constructs. Proof idea for ∀/∃: add first pebble interpreted as free variable. For ∀: drop pebble 1 successively on each position. A Σ, ∗, ← Σ, ∅, ← ⊲, ∗, → Σ, ∗, drop ⊲, ∗, → ⊳, ∗, ← Σ, •, lift Σ, ∗, → For ∃: nondeterministcally drop pebble 1. A Σ, ∗, → Σ, ∗, ← ⊲, ∗, → Σ, ∗, drop ⊲, ∗, →

Transitive closure logics

◮ ϕ with at least two first order free variables.

ϕ1(x, y) = (x ≤ y) ∧ ϕ ϕn(x, y) = ∃z0 · · · ∃zn

• x = z0 < z1 < · · · < zn = y ∧
• 1≤ℓ≤n ϕ(zℓ−1, zℓ)
• .

◮ The transitive closure operator is defined by

TC<

xyϕ =

• n≥1

ϕn. x y z1 z2 z3 ϕ ϕ ϕ ϕ

Transitive closure logics

◮ ϕ with at least two first order free variables.

ϕ1(x, y) = (x ≤ y) ∧ ϕ ϕn(x, y) = ∃z0 · · · ∃zn

• x = z0 < z1 < · · · < zn = y ∧
• 1≤ℓ≤n ϕ(zℓ−1, zℓ)
• .

◮ The transitive closure operator is defined by

TC<

xyϕ =

• n≥1

ϕn. x y z1 z2 z3 ϕ ϕ ϕ ϕ

◮ ϕ(x, y)

def

= (y = x + 1) ∧ ∀z 2. Then [TC<

xyϕ](u, 1, |u|) = (2|u|)|u|−1.

TC<

xy simulates first-order universal quantification.

Transitive closure logics

◮ ϕ with at least two first order free variables.

ϕ1(x, y) = (x ≤ y) ∧ ϕ ϕn(x, y) = ∃z0 · · · ∃zn

• x = z0 < z1 < · · · < zn = y ∧
• 1≤ℓ≤n ϕ(zℓ−1, zℓ)
• .

◮ The transitive closure operator is defined by

TC<

xyϕ =

• n≥1

ϕn. x y z1 z2 z3 ϕ ϕ ϕ ϕ

◮ ϕ(x, y)

def

= (y = x + 1) ∧ ∀z 2. Then [TC<

xyϕ](u, 1, |u|) = (2|u|)|u|−1.

TC<

xy simulates first-order universal quantification. ◮ ψ = TC< x,y(2 ∧ y = x + 2).

ψ(u, 1, |u|) =

• 2n

if |u| = 2n + 1 with n ≥ 1

• therwise.

Expressiveness

Theorem (Bollig, Gastin, Monmege, Z.)

◮ FO + TC< with bounded steps = weighted pebble automata. ◮ TC<(ϕ) where ϕ Boolean captures weighted automata. ◮ Proof resembles the translation 2-way → 1-way automata. ◮ Uses an intermediate automaton model: nested automata. ◮ We do not know if allowing unbounded steps goes beyond pebble automata. ◮ Allowing forward + backward unbounded steps goes beyond pebble

automata.

Some short-term questions

• 1. Algorithms. (SAT is decidable for positive semiring.)
• 2. Relax bounded assumption.
• 3. Weak pebbles vs. strong pebbles.
• 4. How do these results extend to trees?
• 5. XPath-like logics?