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 ) . ka p q 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 ) . ka p q 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. k 1 a 1 k 2 a 2 k n a n weight ( p 0 − − − → p 1 − − − → · · · − − − → p n ) = k 1 k 2 · · · k n ◮ Weight of a word: sum of all weights of accepted runs on this word. � � A � ( w ) = weight ( ρ ) ρ run of A on w

Weighted automata ◮ Transitions carry weights from a semiring K = ( K , + , × , 0 , 1 ) . ka p q 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. k 1 a 1 k 2 a 2 k n a n weight ( p 0 − − − → p 1 − − − → · · · − − − → p n ) = k 1 k 2 · · · k n ◮ Weight of a word: sum of all weights of accepted runs on this word. � � A � ( w ) = weight ( ρ ) ρ run of A on w

Examples of weighted automata ◮ Alphabet Σ , on ( N , + , × , 0 , 1 ) . 2 Σ 1 1 � A � ( u ) = 2 | u | . 1 Σ 1 Σ 1 a 1 1 � A � ( u ) = | u | a − | u | b . − 1 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 : k 1 · · · k n ≤ ( max k i ) 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. 2 Σ 1 1

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. Σ , ∗ , ← 2 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 | P a ( x ) | x ≤ y | x ∈ X | ¬ ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ | ∃ x ϕ | ∀ x ϕ | ∃ X ϕ | ∀ X ϕ 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. ◮ P a ( x ) means “position x carries an a ”. ◮ x ≤ y means “position x is before position y ”.

Weighted MSO Definition: Syntax of MSO ϕ ::= k | P a ( x ) | x ≤ y | x ∈ X | ¬ ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ | ∃ x ϕ | ∀ x ϕ | ∃ X ϕ | ∀ X ϕ 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. ◮ P a ( x ) means “position x carries an a ”. ◮ x ≤ y means “position x is before position y ”. ◮ � ϕ 1 ∨ ϕ 2 � = � ϕ 1 � + � ϕ 2 � and � ϕ 1 ∧ ϕ 2 � = � ϕ 1 � × � ϕ 2 � . ◮ ∃ x ϕ interpreted as a sum over all positions. ◮ ∀ x ϕ interpreted as a product over all positions.

MSO: examples ◮ � ∃ xP a ( x ) � = | u | a recognizable ◮ � ∃ xP a ( x ) ∨ ∃ x [ − 1 ∧ P b ( x )] � = | u | a − | u | b recognizable ◮ � ∀ y 2 � ( u ) = 2 | u | . recognizable

MSO: examples ◮ � ∃ xP a ( x ) � = | u | a recognizable ◮ � ∃ xP a ( x ) ∨ ∃ x [ − 1 ∧ P b ( x )] � = | u | a − | u | b recognizable ◮ � ∀ y 2 � ( u ) = 2 | u | . recognizable ◮ � ∀ x ∀ y 2 � ( u ) = 2 | u | 2 . not recognizable ◮ = ⇒ Recognizable are not stable under universal quantification.

MSO: examples ◮ � ∃ xP a ( x ) � = | u | a recognizable ◮ � ∃ xP a ( x ) ∨ ∃ x [ − 1 ∧ P b ( x )] � = | u | a − | u | b recognizable ◮ � ∀ y 2 � ( u ) = 2 | u | . recognizable ◮ � ∀ x ∀ y 2 � ( 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. Σ , ∗ , ← Σ , ∅ , ← ⊲ , ∗ , → Σ , ∗ , drop ⊲ , ∗ , → ⊳ , ∗ , ← Σ , • , lift A 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. Σ , ∗ , ← Σ , ∅ , ← ⊲ , ∗ , → Σ , ∗ , drop ⊲ , ∗ , → ⊳ , ∗ , ← Σ , • , lift A Σ , ∗ , → For ∃ : nondeterministcally drop pebble 1. Σ , ∗ , → Σ , ∗ , ← ⊲ , ∗ , → Σ , ∗ , drop ⊲ , ∗ , → A

Transitive closure logics ◮ ϕ with at least two first order free variables. ϕ 1 ( x , y ) = ( x ≤ y ) ∧ ϕ � �� ϕ n ( x , y ) = ∃ z 0 · · · ∃ z n �� x = z 0 < z 1 < · · · < z n = y ∧ 1 ≤ ℓ ≤ n ϕ ( z ℓ − 1 , z ℓ ) . ◮ The transitive closure operator is defined by TC < � ϕ n . xy ϕ = n ≥ 1 z 1 z 2 z 3 y x ϕ ϕ ϕ ϕ