Labeled Transition Systems 2IT70 Finite Automata and Process Theory - PowerPoint PPT Presentation

Labeled Transition Systems 2IT70 Finite Automata and Process Theory Technische Universiteit Eindhoven June 4, 2014

The lady or the tiger open open marry eat 2 IT70 (2014) Labeled Transition Systems 2 / 26

The lady or the tiger open open open marry eat marry eat 2 IT70 (2014) Labeled Transition Systems 2 / 26

The lady or the tiger open open open marry eat marry eat ↔ S right S left ≈ S right while S left / 2 IT70 (2014) Labeled Transition Systems 2 / 26

A testing machine reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open open marry eat S left open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open open marry eat S left open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open open marry eat S left open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open open marry eat S left open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open open marry eat S left open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open open marry eat S left open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open marry eat S right open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open marry eat S right open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

A testing machine open marry eat S right open eat marry reset 2 IT70 (2014) Labeled Transition Systems 3 / 26

Labeled transition system labeled transition system S = ( Q , Σ , → S , q 0 ) finite/infinite set of states Q finite/infinite set of actions Σ transition relation → S ⊆ Q × Σ τ × Q initial state q 0 → S q ′ for action α ∈ Σ τ α transitions q � 2 IT70 (2014) Labeled Transition Systems 4 / 26

Example LTS a buffer of capacity 2 out 0 in 1 0 1 ε out 0 in 1 out 1 in 0 in 0 out 1 out 1 out 0 00 11 in 1 in 0 10 01 2 IT70 (2014) Labeled Transition Systems 5 / 26

An infinite LTS a counter process up up up up q 0 q 1 q 2 q 3 q 4 down down down p 1 p 2 p 3 p 4 down down down 2 IT70 (2014) Labeled Transition Systems 6 / 26

Bisimilarity of states LTS S = ( Q , Σ , → S , q 0 ) bisimulation relation R ⊆ Q × Q : for all q , p ∈ Q and α ∈ Σ τ → S q ′ then p → S p ′ such that R ( q ′ , p ′ ) α α (i) if R ( q , p ) and q � � → S p ′ then q → S q ′ such that R ( q ′ , p ′ ) α α (ii) if R ( q , p ) and p � � states q , p ∈ Q bisimilar if R ( q , p ) for bisimulation R for S notation q ↔ p 2 IT70 (2014) Labeled Transition Systems 7 / 26

Bisimilarity of LTS LTS S 1 = ( Q 1 , Σ , → 1 , q 0 ) and LTS S 2 = ( Q 2 , Σ , → 2 , p 0 ) bisimulation relation R ⊆ Q 1 × Q 2 : for all q , p ∈ Q and α ∈ Σ τ → S q ′ then p → S p ′ such that R ( q ′ , p ′ ) α α (i) if R ( q , p ) and q � � → S p ′ then q → S q ′ such that R ( q ′ , p ′ ) α α (ii) if R ( q , p ) and p � � LTS S 1 , S 2 bisimilar if R ( q 0 , p 0 ) for bisimulation R for S 1 and S 2 notation S 1 ↔ S 2 2 IT70 (2014) Labeled Transition Systems 8 / 26

Example bisimilarity a a b b 2 IT70 (2014) Labeled Transition Systems 9 / 26

Example bisimilarity a a b b bisimilarity of states 2 IT70 (2014) Labeled Transition Systems 9 / 26

Example bisimilarity a a a b b b 2 IT70 (2014) Labeled Transition Systems 9 / 26

Example bisimilarity a a a b b b bisimilarity of LTS 2 IT70 (2014) Labeled Transition Systems 9 / 26

Clicker question L121 a a a Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 10 / 26

Clicker question L121 a a a Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 10 / 26

Clicker question L122 a b a b c c c Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 11 / 26

Clicker question L122 a b a b c c c Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 11 / 26

Clicker question L123 a a a b b Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 12 / 26

Clicker question L123 a a a ? ? b b Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 12 / 26

Clicker question L124 a b a b a a b b a b a b Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 13 / 26

Clicker question L124 a b a b a a b b a b a b Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 13 / 26

Clicker question L124 a b a b a a a b b b a b a b Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 13 / 26

Clicker question L124 a b a b a a a b b b a b a b Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 13 / 26

Clicker question L124 a b a b a a a b b b a b a b Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 13 / 26

Clicker question L124 a b a b a a a b b b a b a b Are these two LTS bisimilar? A. Yes B. No C. Can’t tell 2 IT70 (2014) Labeled Transition Systems 13 / 26

Coloring for bisimulation coloring scheme ( c n ) ∞ n = 0 with c n ∶ Q → N satisfies c n + 1 ( q ) = c n + 1 ( p ) � ⇒ c n ( q ) = c n ( p ) 2 IT70 (2014) Labeled Transition Systems 14 / 26

Coloring for bisimulation coloring scheme ( c n ) ∞ n = 0 with c n ∶ Q → N satisfies c n + 1 ( q ) = c n + 1 ( p ) � ⇒ c n ( q ) = c n ( p ) finite LTS S = ( Q , Σ , → S , q 0 ) , coloring scheme ( c n ) ∞ n = 0 such that for all n ⩾ 0, p , q ∈ Q and α ∈ Σ τ c n + 1 ( q ) = c n + 1 ( p ) iff → S q ′ then p → S p ′ such that c n ( p ′ ) = c n ( q ′ ) α α (i) if q � � → S p ′ then q → S q ′ such that c n ( p ′ ) = c n ( q ′ ) α α (ii) if p � � 2 IT70 (2014) Labeled Transition Systems 14 / 26

Coloring for bisimulation coloring scheme ( c n ) ∞ n = 0 with c n ∶ Q → N satisfies c n + 1 ( q ) = c n + 1 ( p ) � ⇒ c n ( q ) = c n ( p ) finite LTS S = ( Q , Σ , → S , q 0 ) , coloring scheme ( c n ) ∞ n = 0 such that for all n ⩾ 0, p , q ∈ Q and α ∈ Σ τ c n + 1 ( q ) = c n + 1 ( p ) iff → S q ′ then p → S p ′ such that c n ( p ′ ) = c n ( q ′ ) α α (i) if q � � → S p ′ then q → S q ′ such that c n ( p ′ ) = c n ( q ′ ) α α (ii) if p � � define R ⊆ Q × Q by R ( q , p ) ⇐ ⇒ ∀ n ∶ c n ( q ) = c n ( p ) then R is a bisimulation relation for S 2 IT70 (2014) Labeled Transition Systems 14 / 26

Coloring an LTS 0 q 0 a a 0 0 q 1 q 2 c c b b 0 0 0 q 3 q 4 q 5 c c e d 0 0 q 6 q 7 2 IT70 (2014) Labeled Transition Systems 15 / 26

Coloring an LTS 1 q 0 a a 2 2 q 1 q 2 c c b b 3 5 4 q 3 q 4 q 5 c c e d 6 6 q 6 q 7 2 IT70 (2014) Labeled Transition Systems 15 / 26

Coloring an LTS 7 q 0 a a 8 9 q 1 q 2 c c b b 10 12 11 q 3 q 4 q 5 c c e d 6 6 q 6 q 7 2 IT70 (2014) Labeled Transition Systems 15 / 26

Dealing with silent steps s s s τ τ a a a b b t t a a b u v u v u v S 1 S 0 S 2 2 IT70 (2014) Labeled Transition Systems 16 / 26

A testing machine for τ -transitions s s τ τ a a b t t a a b u v u v reset 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a t a b u v S 1 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a t a b u v S 1 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a t a b u v S 1 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a t a b u v S 1 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a t a b u v S 1 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a b t a u v S 2 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a b t a u v S 2 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a b t a u v S 2 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a b t a u v S 2 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26

A testing machine for τ -transitions s τ a b t a u v S 2 a reset b 2 IT70 (2014) Labeled Transition Systems 17 / 26