Causality Abstractions in Non-Deterministic Automata Networks Loc - PowerPoint PPT Presentation

Causality Abstractions in Non-Deterministic Automata Networks Loïc Paulevé LRI, CNRS / Université Paris-Sud, France loic.pauleve@lri.fr http://loicpauleve.name Joint work with O. Roux , M. Magnin , M. Folschette (IRCCyN), G. Andrieux (IRISA) November 5, 2014 - Nice, France

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Causality ( event A and event B ) or event C cause event D event A or event C necessary for event D event B or event C necessary for event D event A and event B sufficient for event D event A and event C sufficient for event D Overview (1) • Events are automaton state changes • We focus on local causality, i.e. with very limited scope • ⇒ formal reasoning on local causalities to capture global dynamics. Loïc Paulevé 2/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Causality ( event A and event B ) or event C cause event D event A or event C necessary for event D event B or event C necessary for event D event A and event B sufficient for event D event A and event C sufficient for event D Overview (1) • Events are automaton state changes • We focus on local causality, i.e. with very limited scope • ⇒ formal reasoning on local causalities to capture global dynamics. Loïc Paulevé 2/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Causality ( event A and event B ) or event C cause event D event A or event C necessary for event D event B or event C necessary for event D event A and event B sufficient for event D event A and event C sufficient for event D Overview (1) • Events are automaton state changes • We focus on local causality, i.e. with very limited scope • ⇒ formal reasoning on local causalities to capture global dynamics. Loïc Paulevé 2/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Non-Deterministic Finite Automata Networks a b 3 3 ℓ 4 ℓ 3 ℓ 1 ℓ 1 ℓ 5 2 2 c d ℓ 2 ℓ 2 ℓ 6 1 2 1 2 ℓ 3 ℓ 4 ℓ 5 ℓ 6 1 1 ℓ • transition ℓ pre-condition: • ℓ = { a i | a i − → a j } : s → s ′ ∆ ⇔ ∃ ℓ ∈ L : ∀ a i ∈ • ℓ, s ( a ) = a i ∧ ∀ a j ∈ ℓ • , s ′ ( a ) = a j ∧∀ b ∈ Σ , LS ( b ) ∩ • ℓ = ∅ ⇒ s ( b ) = s ′ ( b ) . • (or 1-safe Petri nets with mutually exclusive places) Loïc Paulevé 3/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Non-Deterministic Finite Automata Networks a b 3 3 ℓ 4 ℓ 3 ℓ 1 ℓ 1 ℓ 5 2 2 c d ℓ 2 ℓ 2 ℓ 6 1 2 1 2 ℓ 3 ℓ 4 ℓ 5 ℓ 6 1 1 ℓ • transition ℓ pre-condition: • ℓ = { a i | a i − → a j } : s → s ′ ∆ ⇔ ∃ ℓ ∈ L : ∀ a i ∈ • ℓ, s ( a ) = a i ∧ ∀ a j ∈ ℓ • , s ′ ( a ) = a j ∧∀ b ∈ Σ , LS ( b ) ∩ • ℓ = ∅ ⇒ s ( b ) = s ′ ( b ) . • (or 1-safe Petri nets with mutually exclusive places) Loïc Paulevé 3/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Non-Deterministic Finite Automata Networks a b 3 3 ℓ 4 ℓ 3 ℓ 1 ℓ 1 ℓ 5 2 2 c d ℓ 2 ℓ 2 ℓ 6 1 2 1 2 ℓ 3 ℓ 4 ℓ 5 ℓ 6 1 1 ℓ • transition ℓ pre-condition: • ℓ = { a i | a i − → a j } : s → s ′ ∆ ⇔ ∃ ℓ ∈ L : ∀ a i ∈ • ℓ, s ( a ) = a i ∧ ∀ a j ∈ ℓ • , s ′ ( a ) = a j ∧∀ b ∈ Σ , LS ( b ) ∩ • ℓ = ∅ ⇒ s ( b ) = s ′ ( b ) . • (or 1-safe Petri nets with mutually exclusive places) Loïc Paulevé 3/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Non-Deterministic Finite Automata Networks a b 3 3 ℓ 4 ℓ 3 ℓ 1 ℓ 1 ℓ 5 2 2 c d ℓ 2 ℓ 2 ℓ 6 1 2 1 2 ℓ 3 ℓ 4 ℓ 5 ℓ 6 1 1 ℓ • transition ℓ pre-condition: • ℓ = { a i | a i − → a j } : s → s ′ ∆ ⇔ ∃ ℓ ∈ L : ∀ a i ∈ • ℓ, s ( a ) = a i ∧ ∀ a j ∈ ℓ • , s ′ ( a ) = a j ∧∀ b ∈ Σ , LS ( b ) ∩ • ℓ = ∅ ⇒ s ( b ) = s ′ ( b ) . • (or 1-safe Petri nets with mutually exclusive places) Loïc Paulevé 3/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Non-Deterministic Finite Automata Networks a b 3 3 ℓ 4 ℓ 3 ℓ 1 ℓ 1 ℓ 5 2 2 c d ℓ 2 ℓ 2 ℓ 6 1 2 1 2 ℓ 3 ℓ 4 ℓ 5 ℓ 6 1 1 ℓ • transition ℓ pre-condition: • ℓ = { a i | a i − → a j } : s → s ′ ∆ ⇔ ∃ ℓ ∈ L : ∀ a i ∈ • ℓ, s ( a ) = a i ∧ ∀ a j ∈ ℓ • , s ′ ( a ) = a j ∧∀ b ∈ Σ , LS ( b ) ∩ • ℓ = ∅ ⇒ s ( b ) = s ′ ( b ) . • (or 1-safe Petri nets with mutually exclusive places) Loïc Paulevé 3/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Non-Deterministic Finite Automata Networks a b 3 3 ℓ 4 ℓ 3 ℓ 1 ℓ 1 ℓ 5 2 2 c d ℓ 2 ℓ 2 ℓ 6 1 2 1 2 ℓ 3 ℓ 4 ℓ 5 ℓ 6 1 1 ℓ • transition ℓ pre-condition: • ℓ = { a i | a i − → a j } : s → s ′ ∆ ⇔ ∃ ℓ ∈ L : ∀ a i ∈ • ℓ, s ( a ) = a i ∧ ∀ a j ∈ ℓ • , s ′ ( a ) = a j ∧∀ b ∈ Σ , LS ( b ) ∩ • ℓ = ∅ ⇒ s ( b ) = s ′ ( b ) . • (or 1-safe Petri nets with mutually exclusive places) Loïc Paulevé 3/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Indeterministic Finite Automata Networks Comments • Transition-centered specification • Can model indeterministic discrete function: � 1 if x [ b ] ≥ 1 ∨ x [ c ] ≥ 1 f a ( x ) = 0 if x [ b ] = 0 ∨ x [ c ] = 0 • Can model any discrete network async/sync update. Specializations (sub-classes) Asynchronous Automata Network • Only one automaton is updated at each transition ℓ ( ∀ ℓ, # { a j | a i − → a j , j � = i } = 1) AAN + binary pre-condition (a.k.a. Process Hitting) • Asynchronous Automata Network • + any transition concerns at most two automata ( ∀ ℓ, # • ℓ ≤ 2). Loïc Paulevé 4/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Interaction Networks with Automata Networks a + + 0 1 c b a 1. f a ( x ) = x [ b ] ∧ x [ c ] transitions: c b a 0 → a 1 : b 1 ∧ c 1 1 1 a 1 → a 0 : b 0 ∨ c 0 2. Non-deterministic f a 0 0 transitions: a 0 → a 1 : b 1 ∨ c 1 a 1 → a 0 : b 0 ∨ c 0 Loïc Paulevé 5/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Interaction Networks with Automata Networks a + + 0 1 c b a ℓ 1 1. f a ( x ) = x [ b ] ∧ x [ c ] transitions: c b a 0 → a 1 : b 1 ∧ c 1 ℓ 1 ℓ 1 1 1 a 1 → a 0 : b 0 ∨ c 0 2. Non-deterministic f a 0 0 transitions: a 0 → a 1 : b 1 ∨ c 1 a 1 → a 0 : b 0 ∨ c 0 Loïc Paulevé 5/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Interaction Networks with Automata Networks a + + 0 1 c b a ℓ 2 ℓ 1 1. f a ( x ) = x [ b ] ∧ x [ c ] transitions: c b a 0 → a 1 : b 1 ∧ c 1 ℓ 1 ℓ 1 1 1 a 1 → a 0 : b 0 ∨ c 0 2. Non-deterministic f a ℓ 2 0 0 transitions: a 0 → a 1 : b 1 ∨ c 1 a 1 → a 0 : b 0 ∨ c 0 Loïc Paulevé 5/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Interaction Networks with Automata Networks a + + ℓ 3 0 1 c b a ℓ 2 1. f a ( x ) = x [ b ] ∧ x [ c ] ℓ 1 transitions: c b a 0 → a 1 : b 1 ∧ c 1 ℓ 1 ℓ 1 a 1 → a 0 : b 0 ∨ c 0 1 1 2. Non-deterministic f a ℓ 2 ℓ 3 0 0 transitions: a 0 → a 1 : b 1 ∨ c 1 a 1 → a 0 : b 0 ∨ c 0 Loïc Paulevé 5/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Interaction Networks with Automata Networks a + + ℓ 3 0 1 c b a ℓ 2 1. f a ( x ) = x [ b ] ∧ x [ c ] ℓ 1 transitions: c b a 0 → a 1 : b 1 ∧ c 1 ℓ 1 ℓ 1 a 1 → a 0 : b 0 ∨ c 0 1 1 2. Non-deterministic f a ℓ 2 ℓ 3 0 0 transitions: a 0 → a 1 : b 1 ∨ c 1 a 1 → a 0 : b 0 ∨ c 0 Loïc Paulevé 5/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Interaction Networks with Automata Networks a + + ℓ 3 0 1 c b a ℓ 2 1. f a ( x ) = x [ b ] ∧ x [ c ] ℓ 1 transitions: ℓ 4 c b a 0 → a 1 : b 1 ∧ c 1 ℓ 1 ℓ 4 a 1 → a 0 : b 0 ∨ c 0 1 1 2. Non-deterministic f a ℓ 2 ℓ 3 0 0 transitions: a 0 → a 1 : b 1 ∨ c 1 a 1 → a 0 : b 0 ∨ c 0 Loïc Paulevé 5/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Interaction Networks with Automata Networks a + + ℓ 3 0 1 c b a ℓ 2 1. f a ( x ) = x [ b ] ∧ x [ c ] ℓ 1 transitions: ℓ 4 c b a 0 → a 1 : b 1 ∧ c 1 ℓ 1 ℓ 4 a 1 → a 0 : b 0 ∨ c 0 1 1 2. Non-deterministic f a ℓ 2 ℓ 3 0 0 transitions: a 0 → a 1 : b 1 ∨ c 1 a 1 → a 0 : b 0 ∨ c 0 Loïc Paulevé 5/34

Causality Abstractions in Non-Deterministic Automata Networks: Introduction Interaction Networks with Automata Networks a + + ℓ 3 0 1 c b a ℓ 2 1. f a ( x ) = x [ b ] ∧ x [ c ] ℓ 1 transitions: ℓ 4 c b a 0 → a 1 : b 1 ∧ c 1 ℓ 1 ℓ 4 a 1 → a 0 : b 0 ∨ c 0 1 1 2. Non-deterministic f a ℓ 2 ℓ 3 0 0 transitions: a 0 → a 1 : b 1 ∨ c 1 a 1 → a 0 : b 0 ∨ c 0 Loïc Paulevé 5/34