Overview overview7.5 Introduction Modelling parallel systems - PowerPoint PPT Presentation

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda 25 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : 26 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 27 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 simulation for ( T 1 , T 2 ) ( T 1 , T 2 ) ( T 1 , T 2 ): 28 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 simulation for ( T 1 , T 2 ) ( T 1 , T 2 ) ( T 1 , T 2 ): � � � ( pay , pay ) , ( pay , pay ) , ( pay , pay ) , ( paid 1 , select ) , ( paid 1 , select ) , ( paid 1 , select ) , ( paid 2 , select ) , ( paid 2 , select ) , ( paid 2 , select ) , � � � ( coke , coke ) , ( coke , coke ) , ( coke , coke ) , ( soda , soda ) ( soda , soda ) ( soda , soda ) 29 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , but T 2 �� T 1 T 2 �� T 1 T 2 �� T 1 simulation for ( T 1 , T 2 ) ( T 1 , T 2 ) ( T 1 , T 2 ): � � � ( pay , pay ) , ( pay , pay ) , ( pay , pay ) , ( paid 1 , select ) , ( paid 1 , select ) , ( paid 1 , select ) , ( paid 2 , select ) , ( paid 2 , select ) , ( paid 2 , select ) , � � � ( coke , coke ) , ( coke , coke ) , ( coke , coke ) , ( soda , soda ) ( soda , soda ) ( soda , soda ) 30 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , but T 2 �� T 1 T 2 �� T 1 T 2 �� T 1 for AP = { pay , drink } : AP = { pay , drink } : AP = { pay , drink } : 31 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , but T 2 �� T 1 T 2 �� T 1 T 2 �� T 1 for AP = { pay , drink } : AP = { pay , drink } : AP = { pay , drink } : 32 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , but T 2 �� T 1 T 2 �� T 1 T 2 �� T 1 for AP = { pay , drink } : AP = { pay , drink } : AP = { pay , drink } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , and T 2 � T 1 T 2 � T 1 T 2 � T 1 33 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , but T 2 �� T 1 T 2 �� T 1 T 2 �� T 1 for AP = { pay , drink } : AP = { pay , drink } : AP = { pay , drink } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , and T 2 � T 1 T 2 � T 1 T 2 � T 1 ( T 1 , T 2 ) simulation for ( T 1 , T 2 ) ( T 1 , T 2 ): as before 34 / 336

Two beverage machines bseqor5.1-8 T 1 T 1 T 1 T 2 T 2 T 2 pay pay paid 1 paid 2 select coke soda coke soda for AP = { pay , coke , soda } AP = { pay , coke , soda } AP = { pay , coke , soda } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , but T 2 �� T 1 T 2 �� T 1 T 2 �� T 1 for AP = { pay , drink } : AP = { pay , drink } : AP = { pay , drink } : T 1 � T 2 T 1 � T 2 T 1 � T 2 , and T 2 � T 1 T 2 � T 1 T 2 � T 1 simulation for ( T 2 , T 1 ) ( T 2 , T 1 ) ( T 2 , T 1 ): � � � ( pay , pay ) , ( select , paid 1 ) , ( select , paid 2 ) , ( pay , pay ) , ( select , paid 1 ) , ( select , paid 2 ) , ( pay , pay ) , ( select , paid 1 ) , ( select , paid 2 ) , � � � ( coke , coke ) , ( soda , soda ) ( coke , coke ) , ( soda , soda ) ( coke , coke ) , ( soda , soda ) 35 / 336

Simulation condition bseqor5.1-9 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 ↓ ↓ ↓ ↓ ↓ can be completed to ↓ ↓ ↓ ↓ s ′ s ′ s ′ s ′ s ′ s ′ s ′ R – s ′ s ′ R – R 1 1 2 1 1 1 1 2 2 36 / 336

R Path fragment lifting for simulation R R bseqor5.1-9 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 ↓ ↓ ↓ s 1 , 1 s 1 , 1 s 1 , 1 ↓ ↓ ↓ s 1 , 2 s 1 , 2 s 1 , 2 ↓ ↓ ↓ s 1 , 3 s 1 , 3 s 1 , 3 ↓ ↓ ↓ . . . . . . . . . ↓ ↓ ↓ s 1 , n s 1 , n s 1 , n 37 / 336

R Path fragment lifting for simulation R R bseqor5.1-9 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 ↓ ↓ ↓ s 1 , 1 s 1 , 1 s 1 , 1 ↓ ↓ ↓ s 1 , 2 s 1 , 2 s 1 , 2 ↓ ↓ ↓ can be completed to s 1 , 3 s 1 , 3 s 1 , 3 ↓ ↓ ↓ . . . . . . . . . ↓ ↓ ↓ s 1 , n s 1 , n s 1 , n 38 / 336

R Path fragment lifting for simulation R R bseqor5.1-9 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ R s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 – R R – s 2 , 1 s 2 , 1 s 2 , 1 ↓ ↓ ↓ ↓ ↓ ↓ s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 ↓ ↓ ↓ ↓ ↓ ↓ can be completed to s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 ↓ ↓ ↓ ↓ ↓ ↓ . . . . . . . . . . . . . . . . . . ↓ ↓ ↓ ↓ ↓ ↓ s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n 39 / 336

R Path fragment lifting for simulation R R bseqor5.1-9 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ R s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 – R R – s 2 , 1 s 2 , 1 s 2 , 1 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ R s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 – R R – s 2 , 2 s 2 , 2 s 2 , 2 ↓ ↓ ↓ ↓ ↓ ↓ can be completed to s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 ↓ ↓ ↓ ↓ ↓ ↓ . . . . . . . . . . . . . . . . . . ↓ ↓ ↓ ↓ ↓ ↓ s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n 40 / 336

R Path fragment lifting for simulation R R bseqor5.1-9 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ R s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 – R R – s 2 , 1 s 2 , 1 s 2 , 1 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ R s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 – R R – s 2 , 2 s 2 , 2 s 2 , 2 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ can be completed to – R R R – s 2 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 2 , 3 s 2 , 3 ↓ ↓ ↓ ↓ ↓ ↓ . . . . . . . . . . . . . . . . . . ↓ ↓ ↓ ↓ ↓ ↓ s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n 41 / 336

R Path fragment lifting for simulation R R bseqor5.1-9 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 – R R R – s 2 s 2 s 2 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ R s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 – R R – s 2 , 1 s 2 , 1 s 2 , 1 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ R s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 – R R – s 2 , 2 s 2 , 2 s 2 , 2 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ can be completed to – R R R – s 2 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 2 , 3 s 2 , 3 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ s 1 , n s 1 , n – R R R – s 2 , n s 2 , n s 1 , n s 1 , n s 1 , n s 1 , n s 2 , n 42 / 336

Correct or wrong? bseqor5.1-12 � � � 43 / 336

Correct or wrong? bseqor5.1-12 s 1 s 2 s 1 s 1 s 2 s 2 � � � s ′ s ′ s ′ s ′ s ′ s ′ 1 1 1 2 2 2 correct. 44 / 336

Correct or wrong? bseqor5.1-12 s 1 s 2 s 1 s 1 s 2 s 2 � � � s ′ s ′ s ′ s ′ s ′ s ′ 1 1 1 2 2 2 � ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ � � � � � correct. simulation: 2 ) 2 ) 2 ) 45 / 336

Correct or wrong? bseqor5.1-12 s 1 s 2 s 1 s 1 s 2 s 2 � � � s ′ s ′ s ′ s ′ s ′ s ′ 1 1 1 2 2 2 � ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ � � � � � correct. simulation: 2 ) 2 ) 2 ) � � � 46 / 336

Correct or wrong? bseqor5.1-12 s 1 s 2 s 1 s 1 s 2 s 2 � � � s ′ s ′ s ′ s ′ s ′ s ′ 1 1 1 2 2 2 � ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ � � � � � correct. simulation: 2 ) 2 ) 2 ) s 2 s 2 s 2 s 1 s 1 s 1 � � � s ′ s ′ s ′ s ′ s ′ s ′ 2 2 2 1 1 1 wrong. there is no path fragment in T 2 T 2 T 2 corresponding to the path fragment s 1 s ′ s 1 s ′ s 1 s ′ 1 s ′ 1 s ′ 1 s ′ 1 1 1 47 / 336

Correct or wrong? bseqor5.1-13 � � � 48 / 336

Correct or wrong? bseqor5.1-13 s 2 s 2 s 2 s 1 s 1 s 1 s ′ s ′ s ′ � 2 � � 2 2 s ′ s ′ s ′ s ′′ s ′′ s ′′ 1 1 1 2 2 2 ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ 2 ) , ( s ′ 2 ) , ( s ′ 2 ) , ( s ′ 1 , s ′′ 1 , s ′′ 1 , s ′′ � � � � � � correct. simulation: 2 ) 2 ) 2 ) 49 / 336

Correct or wrong? bseqor5.1-13 s 2 s 2 s 2 s 1 s 1 s 1 s ′ s ′ s ′ � 2 � � 2 2 s ′ s ′ s ′ s ′′ s ′′ s ′′ 1 1 1 2 2 2 ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ 2 ) , ( s ′ 2 ) , ( s ′ 2 ) , ( s ′ 1 , s ′′ 1 , s ′′ 1 , s ′′ � � � � � � correct. simulation: 2 ) 2 ) 2 ) � � � 50 / 336

Correct or wrong? bseqor5.1-13 s 2 s 2 s 2 s 1 s 1 s 1 s ′ s ′ s ′ � 2 � � 2 2 s ′ s ′ s ′ s ′′ s ′′ s ′′ 1 1 1 2 2 2 ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ 2 ) , ( s ′ 2 ) , ( s ′ 2 ) , ( s ′ 1 , s ′′ 1 , s ′′ 1 , s ′′ � � � � � � correct. simulation: 2 ) 2 ) 2 ) s ′ s ′ s ′ t ′ t ′ t ′ � � � 1 1 1 2 2 2 s ′ s ′ s ′ 2 2 2 wrong. s ′ s ′ s ′ 1 �� s ′ 1 �� s ′ 1 �� s ′ 2 and s ′ s ′ s ′ 1 �� t ′ 1 �� t ′ 1 �� t ′ 2 2 2 2 2 51 / 336

Simulation preorder ... bseqor5.1-29 • as a relation that compares two transition systems 52 / 336

Simulation preorder ... bseqor5.1-29 • as a relation that compares two transition systems T 1 T 1 T 1 T 2 T 2 T 2 53 / 336

Simulation preorder ... bseqor5.1-29 • as a relation that compares two transition systems • as a relation on the states of one transition system 54 / 336

Simulation preorder ... bseqor5.1-29 • as a relation that compares two transition systems • as a relation on the states of one transition system T T T s 1 s 1 s 1 s 2 s 2 s 2 iff ? s 1 � T s 2 s 1 � T s 2 s 1 � T s 2 55 / 336

Simulation preorder ... bseqor5.1-29 • as a relation that compares two transition systems • as a relation on the states of one transition system T s 1 T s 2 T s 1 T s 1 T s 2 T s 2 T T T s 1 s 1 s 1 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 2 s 2 s 1 � T s 2 T s 1 � T s 2 s 1 � T s 2 s 1 � T s 2 iff T s 1 � T s 2 T s 1 � T s 2 56 / 336

Simulation preorder ... bseqor5.1-29 • as a relation that compares two transition systems • as a relation on the states of one transition system T s 1 T s 2 T s 1 T s 1 T s 2 T s 2 T T T s 1 s 1 s 1 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 2 s 2 s 1 � T s 2 T s 1 � T s 2 s 1 � T s 2 s 1 � T s 2 iff T s 1 � T s 2 T s 1 � T s 2 iff there exists a simulation R R R T ( s 1 , s 2 ) ∈ R for T T with ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R 57 / 336

Simulation preorder for a single TS bseqor5.1-30 Let T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) be a transition system. The simulation preorder � T � T � T is the coarsest relation on S S such that for all states s 1 S s 1 s 1 , s 2 ∈ S s 2 ∈ S s 2 ∈ S with s 1 � T s 1 � T s 1 � T s 2 s 2 : s 2 58 / 336

Simulation preorder for a single TS bseqor5.1-30 Let T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) be a transition system. The simulation preorder � T � T � T is the coarsest relation on S S S such that for all states s 1 s 1 s 1 , s 2 ∈ S s 2 ∈ S s 2 ∈ S with s 1 � T s 1 � T s 1 � T s 2 s 2 : s 2 L ( s 1 ) = L ( s 2 ) (1) L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) (2) . . . . . . . . . 59 / 336

Simulation preorder for a single TS bseqor5.1-30 Let T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) be a transition system. The simulation preorder � T � T � T is the coarsest relation on S S S such that for all states s 1 s 1 s 1 , s 2 ∈ S s 2 ∈ S s 2 ∈ S with s 1 � T s 1 � T s 1 � T s 2 : s 2 s 2 L ( s 1 ) = L ( s 2 ) (1) L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) (2) each transition of s 1 s 1 s 1 can be mimicked by a transition of s 2 s 2 s 2 60 / 336

Simulation preorder for a single TS bseqor5.1-30 Let T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) be a transition system. The simulation preorder � T � T � T is the coarsest relation on S S S such that for all states s 1 s 1 s 1 , s 2 ∈ S s 2 ∈ S s 2 ∈ S with s 1 � T s 1 � T s 1 � T s 2 s 2 : s 2 L ( s 1 ) = L ( s 2 ) (1) L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) (2) each transition of s 1 s 1 s 1 can be mimicked by a transition of s 2 s 2 s 2 � T � T s 1 s 1 s 1 � T � T s 2 s 2 s 2 s 1 s 1 s 1 � T � T s 2 s 2 s 2 can be          � � � � � � � � � completed to s ′ s ′ s ′ s ′ s ′ s ′ s ′ s ′ s ′ � T � T � T 1 1 2 1 1 1 1 2 2 61 / 336

Simulation preorder for a single TS bseqor5.1-30 Let T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) T = ( S , Act , → , . . . ) be a transition system. The simulation preorder � T � T � T is the coarsest relation on S S such that for all states s 1 S s 1 s 1 , s 2 ∈ S s 2 ∈ S s 2 ∈ S with s 1 � T s 1 � T s 1 � T s 2 : s 2 s 2 L ( s 1 ) = L ( s 2 ) (1) L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) (2) each transition of s 1 s 1 s 1 can be mimicked by a transition of s 2 s 2 s 2 � T � T � T is a preorder, i.e., transitive and reflexive. 62 / 336

Simulation for a TS bseqor5.1-10a Let T T T be a transition system with state space S S S . A simulation for T T T is a binary relation R ⊆ S × S R ⊆ S × S R ⊆ S × S s.t. 63 / 336

Simulation for a TS bseqor5.1-10a Let T T T be a transition system with state space S S S . A simulation for T T T is a binary relation R ⊆ S × S R ⊆ S × S s.t. R ⊆ S × S (1) if ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R then L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) (2) . . . . . . . . . 64 / 336

Simulation for a TS bseqor5.1-10a Let T T T be a transition system with state space S S S . A simulation for T T T is a binary relation R ⊆ S × S R ⊆ S × S R ⊆ S × S s.t. (1) if ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R then L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) ( s 1 , s 2 ) ∈ R (2) for all ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R : ∀ s ′ ∀ s ′ ∀ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 2 ∈ Post ( s 2 ) s.t. ( s ′ ( s ′ ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ 2 ∈ Post ( s 2 ) 2 ∈ Post ( s 2 ) 2 ) ∈ R 2 ) ∈ R 2 ) ∈ R 65 / 336

Simulation for a TS bseqor5.1-10a Let T T T be a transition system with state space S S S . A simulation for T T T is a binary relation R ⊆ S × S R ⊆ S × S R ⊆ S × S s.t. (1) if ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R then L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) ( s 1 , s 2 ) ∈ R (2) for all ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R : ∀ s ′ ∀ s ′ ∀ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 2 ∈ Post ( s 2 ) s.t. ( s ′ ( s ′ ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ 2 ∈ Post ( s 2 ) 2 ∈ Post ( s 2 ) 2 ) ∈ R 2 ) ∈ R 2 ) ∈ R – R R – s 2 R – R R R – s 2 s 1 s 1 s 1 s 2 s 2 s 1 s 1 s 1 s 2 s 2 can be          � � � � � � � � � completed to s ′ s ′ s ′ s ′ s ′ s ′ R – s ′ s ′ s ′ – R R 1 1 1 1 1 1 2 2 2 66 / 336

� T Simulation preorder � T � T bseqor5.1-10a Let T T T be a transition system with state space S S S . A simulation for T T T is a binary relation R ⊆ S × S R ⊆ S × S R ⊆ S × S s.t. (1) if ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R then L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) L ( s 1 ) = L ( s 2 ) ( s 1 , s 2 ) ∈ R (2) for all ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R : ∀ s ′ ∀ s ′ ∀ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 1 ∈ Post ( s 1 ) ∃ s ′ 2 ∈ Post ( s 2 ) s.t. ( s ′ ( s ′ ( s ′ 1 , s ′ 1 , s ′ 1 , s ′ 2 ∈ Post ( s 2 ) 2 ∈ Post ( s 2 ) 2 ) ∈ R 2 ) ∈ R 2 ) ∈ R � T simulation preorder � T � T : s 1 � T s 2 R T s 1 � T s 2 s 1 � T s 2 iff there exists a simulation R R for T T ( s 1 , s 2 ) ∈ R s.t. ( s 1 , s 2 ) ∈ R ( s 1 , s 2 ) ∈ R 67 / 336

� T Path fragment lifting for � T � T bseqor5.1-23 � T � T � T s 1 s 1 s 1 s 2 s 2 s 2    � � � s 1 , 1 s 1 , 1 s 1 , 1    � � � s 1 , 2 s 1 , 2 s 1 , 2    � � � s 1 , 3 s 1 , 3 s 1 , 3    � � � . . . . . . . . .    � � � s 1 , n s 1 , n s 1 , n 68 / 336

� T Path fragment lifting for � T � T bseqor5.1-23 � T � T � T s 1 s 1 s 1 s 2 s 2 s 2    � � � s 1 , 1 s 1 , 1 s 1 , 1    � � � s 1 , 2 s 1 , 2 s 1 , 2    can be completed to � � � s 1 , 3 s 1 , 3 s 1 , 3    � � � . . . . . . . . .    � � � s 1 , n s 1 , n s 1 , n 69 / 336

� T Path fragment lifting for � T � T bseqor5.1-23 � T � T � T s 1 s 1 s 1 s 2 s 2 s 2 s 1 s 1 s 1 � T � T � T s 2 s 2 s 2          � � � � � � � � � s 1 , 1 � T s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 � T � T s 2 , 1 s 2 , 1 s 2 , 1       � � � � � � s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2       can be completed to � � � � � � s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3       � � � � � � . . . . . . . . . . . . . . . . . .       � � � � � � s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n 70 / 336

� T Path fragment lifting for � T � T bseqor5.1-23 � T � T � T s 1 s 1 s 1 s 2 s 2 s 2 s 1 s 1 s 1 � T � T � T s 2 s 2 s 2          � � � � � � � � � s 1 , 1 � T s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 � T � T s 2 , 1 s 2 , 1 s 2 , 1          � � � � � � � � � s 1 , 2 � T s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 � T � T s 2 , 2 s 2 , 2 s 2 , 2       can be completed to � � � � � � s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3       � � � � � � . . . . . . . . . . . . . . . . . .       � � � � � � s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n s 1 , n 71 / 336

� T Path fragment lifting for � T � T bseqor5.1-23 � T � T � T s 1 s 1 s 1 s 2 s 2 s 2 s 1 s 1 s 1 � T � T � T s 2 s 2 s 2          � � � � � � � � � s 1 , 1 � T s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 s 1 , 1 � T � T s 2 , 1 s 2 , 1 s 2 , 1          � � � � � � � � � s 1 , 2 � T s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 s 1 , 2 � T � T s 2 , 2 s 2 , 2 s 2 , 2          can be completed to � � � � � � � � � s 1 , 3 � T s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 s 1 , 3 � T � T s 2 , 3 s 2 , 3 s 2 , 3          � � � � � � � � � . . . . . . . . . . . . . . . . . . . . . . . . . . .          � � � � � � � � � s 1 , n s 1 , n s 1 , n � T � T � T s 1 , n s 1 , n s 1 , n s 2 , n s 2 , n s 2 , n 72 / 336

� T Example: simulation preorder � T � T bseqor5.1-33 s 1 s 1 s 1 s 2 s 2 s 2 { a } { a } { a } { a } { a } { a } s ′ s ′ s ′ s ′ s ′ s ′ 1 1 1 2 2 2 ∅ ∅ ∅ ∅ ∅ ∅ s 1 � T s 2 s 1 � T s 2 s 1 � T s 2 73 / 336

� T Example: simulation preorder � T � T bseqor5.1-33 s 1 s 1 s 1 s 2 s 2 s 2 { a } { a } { a } { a } { a } { a } s ′ s ′ s ′ s ′ s ′ s ′ 1 1 1 2 2 2 ∅ ∅ ∅ ∅ ∅ ∅ s 1 � T s 2 s 1 � T s 2 s 1 � T s 2 as ( s 1 , s 2 ) , ( s ′ 1 , s ′ 2 ) , ( s ′ 1 , s ′ � � � ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 2 ) , ( s ′ 2 ) , ( s ′ 1 , s ′ 1 , s ′ � � � 1 ) T 1 ) 1 ) is a simulation for T T 74 / 336

� T Example: simulation preorder � T � T bseqor5.1-33 T s 1 T s 1 T s 1 T s 2 T s 2 T s 2 s 1 s 1 s 1 s 2 s 2 s 2 { a } { a } { a } { a } { a } { a } � � � s ′ s ′ s ′ s ′ s ′ s ′ 1 1 1 2 2 2 ∅ ∅ ∅ ∅ ∅ ∅ s 1 � T s 2 s 1 � T s 2 s 1 � T s 2 as ( s 1 , s 2 ) , ( s ′ 1 , s ′ 2 ) , ( s ′ 1 , s ′ � � � ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 2 ) , ( s ′ 2 ) , ( s ′ 1 , s ′ 1 , s ′ � � � 1 ) T 1 ) 1 ) is a simulation for T T 75 / 336

� T Example: simulation preorder � T � T bseqor5.1-33 T s 1 T s 1 T s 1 T s 2 T s 2 T s 2 s 1 s 1 s 1 s 2 s 2 s 2 { a } { a } { a } { a } { a } { a } � � � s ′ s ′ s ′ s ′ s ′ s ′ 1 1 1 2 2 2 ∅ ∅ ∅ ∅ ∅ ∅ s 1 � T s 2 s 1 � T s 2 s 1 � T s 2 as ( s 1 , s 2 ) , ( s ′ 1 , s ′ 2 ) , ( s ′ 1 , s ′ � � � ( s 1 , s 2 ) , ( s ′ ( s 1 , s 2 ) , ( s ′ 1 , s ′ 1 , s ′ 2 ) , ( s ′ 2 ) , ( s ′ 1 , s ′ 1 , s ′ � � � 1 ) T 1 ) 1 ) is a simulation for T T s 1 → s ′ s 1 → s ′ s 1 → s ′ 1 → s ′ 1 → s ′ 1 → s ′ 1 → s ′ 1 → s ′ 1 → s ′ 1 → ... 1 → ... 1 → ... is simulated by s 2 → s ′ s 2 → s ′ s 2 → s ′ 2 → s ′ 2 → s ′ 2 → s ′ 1 → s ′ 1 → s ′ 1 → s ′ 1 → ... 1 → ... 1 → ... 76 / 336

Abstraction and simulation grm5.5-6 77 / 336

Abstraction and simulation grm5.5-6 transition system T T T with state space S S S 78 / 336

Abstraction and simulation grm5.5-6 transition system T T T “small” abstract S ′ state space S ′ S ′ with state space S S S 79 / 336

Abstraction and simulation grm5.5-6 abstraction function f f f s s s f ( s ) f ( s ) f ( s ) transition system T T T abstract transition system S ′ T f with state space S ′ S ′ T f with state space S S S T f 80 / 336

Abstraction and simulation grm5.5-6 abstraction function f f f s s s f ( s ) f ( s ) f ( s ) transition system T T T abstract transition system S ′ T f with state space S ′ S ′ T f with state space S S S T f lifting of transitions: → s ′ → s ′ → s ′ s − s − s − → f ( s ′ ) → f ( s ′ ) → f ( s ′ ) f ( s ) − f ( s ) − f ( s ) − 81 / 336

Abstraction and simulation grm5.5-6 abstraction function f f f s s s f ( s ) f ( s ) f ( s ) s ′ s ′ s ′ f ( s ′ ) f ( s ′ ) f ( s ′ ) transition system T T T abstract transition system S ′ T f with state space S ′ S ′ T f with state space S S S T f lifting of transitions: → s ′ → s ′ → s ′ s − s − s − → f ( s ′ ) → f ( s ′ ) → f ( s ′ ) f ( s ) − f ( s ) − f ( s ) − 82 / 336

Abstraction and simulation grm5.5-6a given: transition system T = ( S , Act , − T = ( S , Act , − T = ( S , Act , − → , S 0 , AP , L ) → , S 0 , AP , L ) → , S 0 , AP , L ) S ′ and abstraction function f : S → S ′ S ′ f : S → S ′ set S ′ f : S → S ′ L ( s ) = L ( t ) f ( s ) = f ( t ) s , t ∈ S s.t. L ( s ) = L ( t ) L ( s ) = L ( t ) if f ( s ) = f ( t ) f ( s ) = f ( t ) for all s , t ∈ S s , t ∈ S 83 / 336

Abstraction and simulation grm5.5-6a given: transition system T = ( S , Act , − T = ( S , Act , − T = ( S , Act , − → , S 0 , AP , L ) → , S 0 , AP , L ) → , S 0 , AP , L ) S ′ and abstraction function f : S → S ′ S ′ f : S → S ′ set S ′ f : S → S ′ L ( s ) = L ( t ) f ( s ) = f ( t ) s , t ∈ S s.t. L ( s ) = L ( t ) L ( s ) = L ( t ) if f ( s ) = f ( t ) f ( s ) = f ( t ) for all s , t ∈ S s , t ∈ S goal: define abstract transition system T f T f T f S ′ s.t. T � T f S ′ with state space S ′ T � T f T � T f 84 / 336

Abstraction and simulation grm5.5-6a f : S → S ′ s.t. abstraction function f : S → S ′ f : S → S ′ L ( s ) = L ( t ) f ( s ) = f ( t ) s , t ∈ S L ( s ) = L ( t ) L ( s ) = L ( t ) if f ( s ) = f ( t ) f ( s ) = f ( t ) for all s , t ∈ S s , t ∈ S transition system T = ( S , Act , − → , S 0 , AP , L ) T = ( S , Act , − T = ( S , Act , − → , S 0 , AP , L ) → , S 0 , AP , L ) � � � � � � � � � abstract transition system T f = ( S ′ , Act ′ , − T f = ( S ′ , Act ′ , − T f = ( S ′ , Act ′ , − → f , S ′ 0 , AP , L ′ ) → f , S ′ → f , S ′ 0 , AP , L ′ ) 0 , AP , L ′ ) 85 / 336

Abstraction and simulation grm5.5-6a f : S → S ′ s.t. abstraction function f : S → S ′ f : S → S ′ L ( s ) = L ( t ) f ( s ) = f ( t ) s , t ∈ S L ( s ) = L ( t ) L ( s ) = L ( t ) if f ( s ) = f ( t ) f ( s ) = f ( t ) for all s , t ∈ S s , t ∈ S transition system T = ( S , Act , − → , S 0 , AP , L ) T = ( S , Act , − T = ( S , Act , − → , S 0 , AP , L ) → , S 0 , AP , L ) � � � � � � � � � abstract transition system T f = ( S ′ , Act ′ , − T f = ( S ′ , Act ′ , − T f = ( S ′ , Act ′ , − → f , S ′ 0 , AP , L ′ ) → f , S ′ → f , S ′ 0 , AP , L ′ ) 0 , AP , L ′ ) S ′ L ′ ( f ( s )) = L ( s ) where S ′ S ′ � � � � � � and L ′ ( f ( s )) = L ( s ) L ′ ( f ( s )) = L ( s ) 0 = f ( s 0 ) : s 0 ∈ S 0 0 = 0 = f ( s 0 ) : s 0 ∈ S 0 f ( s 0 ) : s 0 ∈ S 0 → s ′ → s ′ → s ′ s − s − s − → f f ( s ′ ) → f f ( s ′ ) → f f ( s ′ ) f ( s ) − f ( s ) − f ( s ) − 86 / 336

Abstraction and simulation grm5.5-6a f : S → S ′ s.t. abstraction function f : S → S ′ f : S → S ′ L ( s ) = L ( t ) f ( s ) = f ( t ) s , t ∈ S L ( s ) = L ( t ) L ( s ) = L ( t ) if f ( s ) = f ( t ) f ( s ) = f ( t ) for all s , t ∈ S s , t ∈ S transition system T = ( S , Act , − → , S 0 , AP , L ) T = ( S , Act , − T = ( S , Act , − → , S 0 , AP , L ) → , S 0 , AP , L ) � � � � � � � � � abstract transition system T f = ( S ′ , Act ′ , − T f = ( S ′ , Act ′ , − T f = ( S ′ , Act ′ , − → f , S ′ 0 , AP , L ′ ) → f , S ′ → f , S ′ 0 , AP , L ′ ) 0 , AP , L ′ ) Then T � T f T � T f T � T f 87 / 336

Abstraction and simulation grm5.5-6a f : S → S ′ s.t. abstraction function f : S → S ′ f : S → S ′ L ( s ) = L ( t ) f ( s ) = f ( t ) s , t ∈ S L ( s ) = L ( t ) L ( s ) = L ( t ) if f ( s ) = f ( t ) f ( s ) = f ( t ) for all s , t ∈ S s , t ∈ S transition system T = ( S , Act , − → , S 0 , AP , L ) T = ( S , Act , − T = ( S , Act , − → , S 0 , AP , L ) → , S 0 , AP , L ) � � � � � � � � � abstract transition system T f = ( S ′ , Act ′ , − T f = ( S ′ , Act ′ , − T f = ( S ′ , Act ′ , − → f , S ′ 0 , AP , L ′ ) → f , S ′ → f , S ′ 0 , AP , L ′ ) 0 , AP , L ′ ) � � � � � � R = R = R = � s , f ( s ) � : s ∈ S � s , f ( s ) � : s ∈ S � s , f ( s ) � : s ∈ S is a Then T � T f T � T f T � T f ← ← ← − − − simulation for ( T , T f ) ( T , T f ) ( T , T f ) 88 / 336

Data abstraction grm5.5-7 WHILE x > 0 x > 0 x > 0 DO x := x − 1; x := x − 1; x := x − 1; y := y +1 y := y +1 y := y +1 OD IF even ( y ) even ( y ) even ( y ) THEN return “1 1 1” ELSE return “0 0 0” FI x ∈ N x ∈ N x ∈ N y ∈ N y ∈ N y ∈ N 89 / 336

Data abstraction grm5.5-7 WHILE x > 0 x > 0 x > 0 DO x := x − 1; x := x − 1; x := x − 1; y := y +1 y := y +1 y := y +1 data OD abstr. IF even ( y ) even ( y ) even ( y ) − → − − → → THEN return “1 1 1” ELSE return “0 0 0” FI x ∈ N x ∈ N x ∈ N − − − → → → x x ∈ ∈ { gzero , zero } ∈ { gzero , zero } { gzero , zero } x y ∈ N y ∈ N y ∈ N − − − → → → y y ∈ ∈ ∈ { even , odd } { even , odd } { even , odd } y 90 / 336

Data abstraction grm5.5-7 WHILE x = gzero x = gzero DO x = gzero x := gzero x := gzero x := gzero or x := zero x := zero x := zero WHILE x > 0 x > 0 DO x > 0 x := x − 1; x := x − 1; x := x − 1; IF y = even y = even y = even THEN y := odd y := odd y := odd y := y +1 y := y +1 y := y +1 data y := even ELSE y := even y := even OD abstr. FI IF even ( y ) even ( y ) even ( y ) − → − − → → OD THEN return “1 1 1” IF y = even y = even y = even ELSE return “0 0 0” THEN return “1 1 1” FI 0 ELSE return “0 0” FI x ∈ N x ∈ N x ∈ N − − − → → → x x ∈ ∈ { gzero , zero } ∈ { gzero , zero } { gzero , zero } x y ∈ N y ∈ N y ∈ N − − − → → → y y ∈ ∈ ∈ { even , odd } { even , odd } { even , odd } y 91 / 336

Data abstraction grm5.5-7 WHILE x = gzero x = gzero x = gzero DO x := gzero x := gzero x := gzero or x := zero x := zero x := zero WHILE x > 0 x > 0 DO x > 0 x := x − 1; x := x − 1; x := x − 1; IF y = even y = even y = even THEN y := odd y := odd y := odd y := y +1 y := y +1 y := y +1 data y := even ELSE y := even y := even OD abstr. FI IF even ( y ) even ( y ) even ( y ) − → − − → → OD THEN return “1 1 1” IF y = even y = even y = even ELSE return “0 0 0” THEN return “1 1” 1 FI 0 ELSE return “0 0” FI concrete operation abstract operation � � � 92 / 336

Data abstraction grm5.5-7 WHILE x = gzero x = gzero DO x = gzero x := gzero x := gzero x := gzero or x := zero x := zero x := zero WHILE x > 0 x > 0 x > 0 DO x := x − 1; x := x − 1; x := x − 1; IF y = even y = even y = even THEN y := odd y := odd y := odd y := y +1 y := y +1 y := y +1 data y := even ELSE y := even y := even OD abstr. FI IF even ( y ) even ( y ) even ( y ) − → − − → → OD THEN return “1 1 1” IF y = even y = even y = even ELSE return “0 0 0” THEN return “1 1” 1 FI 0 ELSE return “0 0” FI concrete operation abstract operation, e.g., � � � x := x − 1 x := x − 1 x := x − 1 gzero �→ gzero or zero gzero �→ gzero or zero gzero �→ gzero or zero 93 / 336

Abstraction and simulation grm5.5-8 abstract TS simulates the concrete one 94 / 336

WHILE x = gzero x = gzero DO x = gzero x > 0 WHILE x > 0 x > 0 DO x := gzero x := gzero x := gzero or x := zero x := zero x := zero x := x − 1 x := x − 1 x := x − 1 IF y = even y = even y = even y := y +1 y := y +1 y := y +1 THEN y := odd y := odd y := odd OD ELSE y := even y := even y := even FI OD even ( y ) IF even ( y ) even ( y ) IF y = even y = even y = even THEN return 1 1 1 1 THEN return 1 1 ELSE return 0 0 0 0 ELSE return 0 0 FI 95 / 336

ℓ 0 WHILE x = gzero x = gzero DO x = gzero ℓ 0 ℓ 0 x > 0 ℓ 0 ℓ 0 ℓ 0 WHILE x > 0 x > 0 DO x := gzero x := gzero or x := zero x := gzero x := zero x := zero ℓ 1 ℓ 1 ℓ 1 x := x − 1 ℓ 1 ℓ 1 ℓ 1 x := x − 1 x := x − 1 ℓ 2 IF y = even y = even y = even ℓ 2 ℓ 2 y := y +1 y := y +1 y := y +1 ℓ 2 ℓ 2 ℓ 2 THEN y := odd y := odd y := odd OD ELSE y := even y := even y := even FI OD even ( y ) ℓ 3 ℓ 3 ℓ 3 IF even ( y ) even ( y ) ℓ 3 IF y = even ℓ 3 y = even y = even ℓ 3 THEN return 1 1 1 ℓ 4 ℓ 4 ℓ 4 1 ℓ 4 ℓ 4 ℓ 4 THEN return 1 1 ELSE return 0 0 0 ℓ 5 ℓ 5 ℓ 5 0 ℓ 5 ℓ 5 ℓ 5 ELSE return 0 0 FI ... ... ℓ 0 ℓ 0 x =2 x =2 x =2 y =0 y =0 y =0 ℓ 0 gzero ℓ 0 gzero gzero even even even ℓ 0 ℓ 0 x =2 y =0 ℓ 1 ℓ 1 ℓ 1 x =2 x =2 y =0 y =0 ℓ 1 ℓ 1 gzero ℓ 1 gzero gzero even even even ℓ 2 x =1 x =1 x =1 y =0 y =0 y =0 ℓ 2 ℓ 2 ℓ 2 ℓ 2 gzero ℓ 2 gzero even gzero even even ℓ 2 ℓ 2 ℓ 2 zero zero zero even even even ℓ 0 ℓ 0 x =1 x =1 x =1 y =1 y =1 y =1 ℓ 0 gzero odd ℓ 0 ℓ 0 ℓ 0 gzero gzero odd odd ℓ 0 zero ℓ 0 ℓ 0 zero odd zero odd odd ... x =1 y =1 ℓ 1 ℓ 1 ℓ 1 x =1 x =1 y =1 y =1 ℓ 1 ℓ 1 gzero ℓ 1 gzero odd gzero odd odd ℓ 3 ℓ 3 ...... ℓ 3 96 / 336

ℓ 0 WHILE x = gzero x = gzero x = gzero DO ℓ 0 ℓ 0 x > 0 ℓ 0 ℓ 0 ℓ 0 WHILE x > 0 x > 0 DO x := gzero x := gzero or x := zero x := gzero x := zero x := zero ℓ 1 ℓ 1 ℓ 1 x := x − 1 ℓ 1 ℓ 1 ℓ 1 x := x − 1 x := x − 1 ℓ 2 IF y = even y = even y = even ℓ 2 ℓ 2 y := y +1 y := y +1 y := y +1 ℓ 2 ℓ 2 ℓ 2 THEN y := odd y := odd y := odd OD ELSE y := even y := even y := even FI OD even ( y ) ℓ 3 ℓ 3 ℓ 3 IF even ( y ) even ( y ) ℓ 3 IF y = even ℓ 3 y = even y = even ℓ 3 THEN return 1 1 1 ℓ 4 ℓ 4 ℓ 4 1 ℓ 4 ℓ 4 ℓ 4 THEN return 1 1 ELSE return 0 0 0 ℓ 5 ℓ 5 ℓ 5 0 ℓ 5 ℓ 5 ℓ 5 ELSE return 0 0 FI ... ... ℓ 0 ℓ 0 x =2 x =2 x =2 y =0 y =0 y =0 ℓ 0 ℓ 0 gzero gzero gzero even even even ℓ 0 ℓ 0 x =2 y =0 � ℓ 1 ℓ 1 x =2 ℓ 1 x =2 y =0 y =0 � � ℓ 1 ℓ 1 gzero ℓ 1 gzero even gzero even even ℓ 2 x =1 x =1 x =1 y =0 y =0 y =0 ℓ 2 ℓ 2 ℓ 2 ℓ 2 gzero ℓ 2 gzero even gzero even even ℓ 2 ℓ 2 ℓ 2 zero zero zero even even even ℓ 0 ℓ 0 x =1 x =1 x =1 y =1 y =1 y =1 ℓ 0 gzero odd ℓ 0 ℓ 0 ℓ 0 gzero gzero odd odd ℓ 0 zero ℓ 0 ℓ 0 zero odd zero odd odd ... x =1 y =1 ℓ 1 x =1 ℓ 1 ℓ 1 x =1 y =1 y =1 ℓ 1 ℓ 1 gzero ℓ 1 gzero odd gzero odd odd ℓ 3 ℓ 3 ...... ℓ 3 97 / 336

Simulation preorder vs. and trace inclusion bseqor5.1-25 98 / 336

Simulation preorder vs. and trace inclusion bseqor5.1-25 T 1 � T 2 T 1 � T 2 T 1 � T 2 = = = ⇒ ⇒ ⇒ Tracesfin ( T 1 ) ⊆ Tracesfin ( T 2 ) Tracesfin ( T 1 ) ⊆ Tracesfin ( T 2 ) Tracesfin ( T 1 ) ⊆ Tracesfin ( T 2 ) 99 / 336

Simulation preorder vs. and trace inclusion bseqor5.1-25 T 1 � T 2 T 1 � T 2 T 1 � T 2 = = = ⇒ ⇒ ⇒ Tracesfin ( T 1 ) ⊆ Tracesfin ( T 2 ) Tracesfin ( T 1 ) ⊆ Tracesfin ( T 2 ) Tracesfin ( T 1 ) ⊆ Tracesfin ( T 2 ) reason: path fragment lifting for � � � 100 / 336