1 Semantics Q S EMANTICS (Q1,’07) Week 6 Jacob Andersen PhD student

2 Semantics Q Week 6 - Outline A Tale of Two Coca-Cola Dispensers About Equivalences Strong + Weak Bisimulation Bisimulation Games CCS Tools

3 Semantics Q CCS Syntax CCS Syntax: “0”// inaction “ .P ”// action prefix,   Act “ P+P ”// non-deterministic choice “ P|P ”// parallel composition “ P\L ”// restriction (private name), L ⊆ L “ P[f] ”// action relabelling “ K ”// process variable, K  K P ::= 0 | .P | P+P | P|P | P\L | P[f] | K K 1 = P 1, K 2 = P 2, … def  a: f(a) = f(a)  f(  ) =  f:Act  Act … where Note: restrictions on f

4 Semantics Q SOS for CCS Structural Operational Semantics: Q: why  (tau) in communication “ P|Q ” (instead of propagating a or a ) ?  ~ “the unobservable hand-shake” [ RES ] [ REL ] [ COM 1 ] [ COM 2 ] [ COM 3 ] [ ACT ] [ SUM j ][ CON ]

5 Semantics Q A T ALE OF TWO C OCA -C OLA D ISPENSERS

6 Semantics Q Once upon a time... Dispenser: Dispenser’: ’ ’ Would you consider them equal ‘=’ ? coin. (coke + sprite) coin.coke + coin.sprite Would you consider them equivalent ‘  ’ ? What does it at all mean for them to be equivalent ‘  ’ ?!?

7 Semantics Q Equal vs. Equivalent Equal (concrete identity): 3 = 3 Equivalent (abstract): 3   0x03 16  \003 8   three 3  3   let n=2 in n*(n-1)+(n-2)  i i i=0 2 more abstract

8 Semantics Q Trace Equivalence Definition: Trace Equivalence: Two processes P and Q are trace equivalent “  tr ” iff: »They can produce the same traces: » Example: Traces( ) = { , coin, coin;coke, coin;sprite } Traces( ) = { , coin, coin;coke, coin;sprite } coin. (coke + sprite) coin.coke + coin.sprite ’ ’ ’ ’  tr Hence: Traces(P) = {  Act* |  Q : P  * Q } 

9 Semantics Q Contextual Composition…? Recall: "Coke-only-drinker": Contextual composition: coin. (coke + sprite) coin.coke + coin.sprite ’ ’ ’ ’  tr What the.. !? coin. coke. drink The coke drinker is certainly able to to distinguish the two dispensers !! Idea (can we…?): put the two dispensers in a context where they can be differentiated ! problematic equality(!) Trace equivalence cannot distinguish the two dispensers.

10 Semantics Q A BOUT E QUIVALENCES

11 Semantics Q Purpose of equivalences Recall examples from the homepage: Specification and (model of) program equivalent ? Two (models of) programs equivalent ?

12 Semantics Q CCS: “Single-Language Formalism” CCS is a so-called “Single-lang. formalism”; i.e. one may specify both: »implementation( ) »and specification( ) We would like to check via some (reasonable) equivalence, R, that: “The implementation has the intended behavior”: » R the spec. and impl. are “equivalent” IMPL = def... SPEC = def... IMPL SPEC

13 Semantics Q Equivalence “wish list” …so we would like: reflexitivity : »SYS R SYS (same behavior as itself) ! transitivity (for stepwise modelling/refinement) !!! : »S 0 R S 1 R... R S n R IMPL => S 0 R IMPL symmetry (just a nice property to have): »S R S’  S’ R S

14 Semantics Q Def: Equivalence Relation Let R be a binary relation over set A: R  A  A R is an equivalence relation iff: Reflexive: » Symmetric: » Transitive: »  x  A: x R x  x,y  A: x R y  y R x  x,y,z  A: x R y  y R z  x R z Q: is trace equivalence “  tr ” an equivalence relation ? Yes

15 Semantics Q Equivalence “wish list” (cont’d)… Furthermore, we would like these properties: P+Q R Q+P // ‘ + ’ commutative (P+Q)+R R P+(Q+R) // ‘ + ’ associative P|Q R Q|P // ‘ | ’ commutative (P|Q)|R R P|(Q|R) // ‘ | ’ associative 0+P R P // ‘ 0 ’ is neutral wrt. ‘+’ 0|P R P // ‘ 0 ’ is neutral wrt. ‘ | ’... Trace equivalence ‘  tr ’ ? Yes

16 Semantics Q …and (congruence wrt. CCS) Definition: “ R ” congruence (wrt. CCS): P R Q  C[P] R C[Q], for all contexts C[] »“relation is preserved under contextual substitution” A context = a process with a gap: Examples: P R Q  P+R R Q+R P R Q  P|S R Q|S P R Q  a.P R a.Q P R Q  ((a.P|R)+S)\x R ((a.Q|R)+S)\x C : .[] | []+P | P+[] | []|P | P|[] | [][f] | []\a []+R []|S a.[] Congruence ‘  tr ’ ?

17 Semantics Q Congruence (cont’d) Trace equivalence, ‘  tr ’, and contexts? Recall the two Coca-cola machines: » » Now take the “Coke-only drinker”: » –Although, we have that: ’ ’  tr What the.. !? coin. (coke + sprite) coin.coke + coin.sprite coin. coke. drink ’ ’ problematic equality The coke drinker is certainly able to to distinguish the two dispensers !! Idea (can we…?): put the two dispensers in a context where they can be differentiated !

18 Semantics Q Trace Equiv. ~ DFM Acceptance Recall: a deterministic finite automaton, A: is completely identified by its set of traces: L (A) Trace equivalence ~ DFA acceptance: (without accept states - by construction) P  tr Q iff they can produce the same traces This point of view is totally justified and natural if we view our LTSs as non- deterministic devices that may generate or accept sequences of actions. However, is it still a reasonable one if we view our automata as reactive machines that interact with their environment ? -- [Aceto, Larsen, Ingólfsdóttir, p. 41]

19 Semantics Q S TRONG B ISIMULATION: ( ~ )

20 Semantics Q Def: A Strong Bisimulation Let (Proc, Act,  ) be a LTS Def: a bin. rel. R  Proc  Proc is a strong bisimulation iff whenever ( s, t )  R :  a  Act : if s  s’ then t  t’ for some t’ such that ( s’, t’ )  R if t  t’ then s  s’ for some s’ such that ( s’, t’ )  R Note: 1. Definition on LTS (not necessarily wrt. processes) 2. Definition relative to a (SOS) semantics (via LTS) a a a a a Intuition: “Only equate as consistently allowed by the semantics”

21 Semantics Q Def: Strongly Bisimilar ( ~ ) A Strong Bisimulation: Def: a bin. rel. R  Proc  Proc is a strong bisimulation iff whenever ( s, t )  R :  a  Act : if s  s’ then t  t’ for some t’ such that ( s’, t’ )  R if t  t’ then s  s’ for some s’ such that ( s’, t’ )  R The Strong Bisimilarity relation ( ~ ): Def: two (processes) s and t are strongly bisimilar ( s ~ t ) iff  strong bisimulation R : ( s, t )  R. i.e. a a a a ‘ ~ ’ :=  { R | R is a strong bisimulation }

22 Semantics Q Basic Properties of ( ~ ) Theorem: ‘ ~ ’ is an equivalence relation (exercise…) Theorem: ‘ ~ ’ is the largest strong bisimulation »i.e. for any bisimulation R we have that: R  ‘ ~ ’ Theorem: –s ~ t iff  a  Act : if s  s’ then t  t’ for some t’ such that s’ ~ t’ if t  t’ then s  s’ for some s’ such that s’ ~ t’ a a a a

23 Semantics Q How to Prove Strong Bisimilarity ? How to prove strong bisimilarity for two processes ? i.e. ?: Exhibit a (any) bisimulation R, for which: –By definition we get that: » since ‘ ~ ’ was the largest bisimulation ( s, t )  R  ‘ ~ ’ ( s, t )  R s ~ ts ~ t

24 Semantics Q Example Proof of Bisimilarity Example: Buffer (capacity 1): Buffer (capacity 2): Show that: A 0 = def in. A 1 A 1 = def out. A 0 B 0 = def in. B 1 B 1 = def in. B 2 + out. B 0 B 2 = def out. B 1 B 0 ~ A 0 |A 0 B0B0 B1B1 B2B2 A 0 |A 0 A 1 |A 0 A 0 |A 1 A 1 |A 1 R = { ( B 0, A 0 |A 0 ), ( B 1, A 1 |A 0 ), ( B 1, A 0 |A 1 ), ( B 2, A 1 |A 1 ) }

25 Semantics Q How to Prove Non-Bisimilarity ? How to prove non-bisimilarity ? i.e. ? Enumerate all binary relations: Check that none are bisimulations and contain (p,q) »However: extremely expensive O(2 |p||q| ) Use “Feynman Problem-Solving Algorithm”: »(1). Write down the problem; »(2). Think very hard; »(3). Write down the answer. Or… s ~ ts ~ t

26 Semantics Q ( ~ ) B ISIMULATION G AMES

27 Semantics Q The (Strong) Bisimulation Game Let (Proc, Act,  ) be a LTS and s, t  Proc Define 2-player game: [ attacker v defender ] The game is played in “rounds” and the configurations of the game are (Proc  Proc); »The game starts (first round) in ( s, t )  Proc  Proc Intuition (objectives): The defender wants to show that: The attacker wants to show that: a s ~ ts ~ t s ~ ts ~ t Cf. the note on SI proofs section 6 (p. 4).

28 Semantics Q Rules of the Bisimulation Game In round k the players change the current configuration ( s k, t k ) as follows: First, the attacker chooses: »1) one of the processes (e.g. t k ); i.e., left or right; »2) a legal action from that process: a  Act; »3) a legal transition according to the LTS: t k  t k+1 Then, the defender chooses: »-) a “counter-move” using same action, a: s k  s k+1 ( s k+1, t k+1 ) becomes the next round’s configuration… Winning: If one player (only) cannot move, the other player wins If the game is infinite (repeats configuration), the defender wins a a

29 Semantics Q Game Characterization of ( ~ ) Theorem: States (processes) s and t are not strongly bisimilar iff the attacker has a universal winning strategy States (processes) s and t are strongly bisimilar iff the defender has a universal winning strategy ( s ~ t ) basically means that: “the ‘perfect attacker’ always wins” ( s ~ t ) basically means that: “the ‘perfect defender’ always wins”

30 Semantics Q Let’s Play… Let’s play…: ~ / ~ ? show of hands… ~ ’ ’ ? coin. (coke + sprite) coin.coke + coin.sprite ’ ’ coin ’ ’ sprite coke

31 Semantics Q Another Game… Are the following two LTS(/processes) s and t strongly bisimilar: s ~ t ? There’s a universal attack strategy  hence, they are not strongly bisimilar : s ~ t

32 Semantics Q ( ~ ) C ONGRUENCE

33 Semantics Q ( ~ ) is a Congruence for CCS Theorem: Let P and Q be processes such that P ~ Q ; then: » .P ~ .Q   Act »P+R ~ Q+R  R  Proc »R+P ~ R+Q  R  Proc »P|R ~ Q|R  R  Proc »R|P ~ R|Q  R  Proc »P[f] ~ Q[f]  f : P(Act)  P(Act) relabellings »P\a ~ Q\a  a  Act \ {  } i.e. ‘ ~ ’ is a congruence for CCS

34 Semantics Q Other Properties of ( ~ ) The following properties hold  P, Q, R : P+Q ~ Q+P // ‘ + ’ commutative (P+Q)+R ~ P+(Q+R) // ‘ + ’ associative P|Q ~ Q|P // ‘ | ’ commutative (P|Q)|R ~ P|(Q|R) // ‘ | ’ associative P+0 ~ P // ‘ 0 ’ neutral wrt. ‘ + ’ P|0 ~ P // ‘ 0 ’ neutral wrt. ‘ | ’...

35 Semantics Q Summary: Strong Bisimilarity ( ~ ) Properties of ( ~ ): an equivalence relation: »reflexive, symmetric, and transitive the largest strong bisimulation: »for proving bisimilarity (exhibit a bisimulation) strong bisimulation game: »for proving non-bisimilarity (winning attack strategy) a congruence: »P ~ Q => C[P] ~ C[Q] obeys the following algebraic laws: »‘ + ’ and ‘ | ’ commutative, associative, and ‘ 0 ’ neutrality, … should we look any further ?!?

36 Semantics Q W EAK B ISIMULATION: (  )

37 Semantics Q Problems with Internal Actions: ‘  ’ Q: a.τ.0 ~ a.0 ? How would you show this? Problem ‘ ~ ’ does not: abstract away from internal actions Example: Disp = def coin. (coke.Disp + sprite.Disp) Stud = def study. coin. coke. Stud Impl = def (Disp | Stud) \ {coin,coke} Spec = def study. Spec Impl ~ Spec ?

38 Semantics Q Can we just erase tau-actions? Consider: However, notice: i.e. we cannot simply erase tau-actions! D nasty = def coin.coke.D nasty + coin.D nasty Stud = def study. coin. coke. Stud Impl = def (Disp | Stud) \ {coin,coke} (coin.coke.D nasty +coin.D nasty | coin.coke.Stud) \ … (coke.D nasty | coke.Stud) \ …(D nasty | coke.Stud) \ …  

39 Semantics Q Define Weak Transition Relation Def: the “weak transition relation”: ‘  ’ e.g., P  Q iff P  …  P’  Q’  …  Q –Means that we can perform an action, a, by: »first, “eating” any number of tau actions: »then, performing the a action; »last, “eating” any number of tau actions: This “eating” is precisely what will give the abstraction!       a a    :   ( ‘  ’ )* o ‘  ’ o ( ‘  ’ )*, if    ( ‘  ’ )*, if  =      

40 Semantics Q Def: A Weak Bisimulation Let (Proc, Act,  ) be a LTS Def: a bin. rel. R  Proc  Proc is a weak bisimulation iff whenever ( s, t )  R :  a  Act : if s  s’ then t  t’ for some t’ such that ( s’, t’ )  R if t  t’ then s  s’ for some s’ such that ( s’, t’ )  R Note: 1. Definition on LTS (not necessarily wrt. processes) 2. Definition relative to a (SOS) semantics (via LTS) a a a a a Intuition: “Only equate as consistently allowed by the semantics; …abstracting away from tau actions”

41 Semantics Q Def: Weakly Bisimilar (  ) A Weak Bisimulation: Def: a bin. rel. R  Proc  Proc is a weak bisimulation iff whenever ( s, t )  R :  a  Act : if s  s’ then t  t’ for some t’ such that ( s’, t’ )  R if t  t’ then s  s’ for some s’ such that ( s’, t’ )  R The Weak Bisimilarity relation (  ): Def: two (processes) s and t are weakly bisimilar ( s  t ) iff  weak bisimulation R : ( s, t )  R. i.e. a a a a ‘  ’ :=  { R | R is a weak bisimulation }

42 Semantics Q Example (from earlier) Example (from earlier): 1. Does it hold? 2. How would you show this? Disp = def coin. (coke.Disp + sprite.Disp) Stud = def study. coin. coke. Stud Impl = def (Disp | Stud)\{coin,coke,sprite} Spec = def study. Spec Impl  Spec ?

43 Semantics Q (  ) B ISIMULATION G AMES

44 Semantics Q Weak Game Characterization Let (Proc, Act,  ) be a LTS and s, t  Proc The game is “similar” except: The defender may now use the ‘  ’ relation »…and thus abstract away from  -actions The attacker still uses: ‘  ’ Theorem (  ): s  t iff  winning attack strategy s  t iff  winning defensive strategy a a a

45 Semantics Q Let’s Play… Example: a. .0  a.0 ? »How would you show this? Example’: 0  .0 ? a  a.0 + .0 ? Any remarks?!? Any consequences of the above? yes no  Thus “  ” not a congruence(!); due to ( P+[] ) yes

46 Semantics Q Summary: Weak Bisimilarity (  ) Properties of (  ): an equivalence relation: »reflexive, symmetric, and transitive the largest weak bisimulation: »for proving bisimilarity (exhibit a bisimulation) weak bisimulation game: »for proving non-bisimilarity (winning attack strategy) not a congruence: »P  Q => C[P]  C[Q] obeys the following algebraic laws: »‘ + ’ and ‘ | ’ commutative, associative, and ‘ 0 ’ neutrality, … abstracts away from internal tau-actions

47 Semantics Q (  ):“Fair Abstraction from Divergence” Consider: A = def a.0 + .B B = def b.0 + .A –Note that: »A  B  a.0 + b.0 !!!..and even: Div = def .Div »0  Div !!! Intuition: “Fair Abstraction from Divergence”: »  “assumes processes (eventually) escape from loops”

48 Semantics Q CCS T OOLS

49 Semantics Q Example: A Protocol Implementation: » Specification: » Verification: » and what does that mean? P spec = def acc. del. P spec P impl = def ( Send | Med | Rec ) \ L L = {send,error,trans,ack} accept deliver P spec ~ P impl P spec  P impl ? ? senderreceiver (transport) medium

50 Semantics Q The Concurrency Workbench CWB: Recommendation: try it out (before the exam)! Command: eq(Spec,Impl); // weak bisimilarity CWB’s answer Command: strongeq(Spec,Impl); // strong bisimilarity CWB’s answer

51 Semantics Q The Bisimulation Game Game The Bisimulation Game Game ® Special Family Edition (w/ TV cables & remote control) »Beat your sister at Process Algebra simulation… “Special Family Edition!” “ The Bisimulation Game ” Only $31,95 The Bisimulation Game [ ]

52 Semantics Q "Three minutes paper" Please spend three minutes writing down the most important things that you have learned today (now). After 1 day After 1 week After 3 weeks After 2 weeks Right away

53 Semantics Q Next week: Program Equivalence, Imperative Blocks, Implementation, and other Semantic formalisms Any Questions? Thanks to Jiří Srba for inspiration to many of the slides