equivalences on labelled transition systems ed brinksma course 2004

Equivalences on Labelled Transition Systems

Ed BrinksmaCourse 2004

© Ed Brinksma/Jan Tretmans

Observable Behaviour

a aa

a

a?

?

?

“ Some transition systems are more equal than others “


a

b

a

b

aa

b

b

a

Observable Behaviour

?

?

?

?

?


Isomorphismp q

"p and q are exactly the same modulo state names"

Isomorphism:

aa

a

bijection : Sp Sq :

s1, s2 Sp, L {} :

s1 s2 (s1) (s2)

(s0p) = s0q

a

s0

s1

a

v

u


a

b

a

b

aa

b

b

a

Isomorphism


Comparing Transition Systems

S1 S2

environmente

environmente

? ?

S1 S2 e E . obs ( e, S1 ) = obs (e, S2 )


Trace Equivalence

S1 S2

environment environment

s1 tr s2 traces ( s1 ) = traces ( s2 )

traces (s) = { L* | s }


cb

a aa

cb

cb

a

tr

Trace Equivalence

traces = { ,a,a b,a c }


Completed Trace Equivalence

p tr q traces(p) = traces (q)

p ctr q Ctraces(p) = Ctraces (q)

and traces(p) = traces (q)

Reachable states: s after = { s’ | s

s’ }

traces (s) = { L* | s }Traces:

s after refuses A s' s after : s' refuses ARefusals:

as refuses A a A : s Refusal:

Ctraces (s) = { L* | s after refuses L }

Completed traces:

Completed

trace equivalence:

Trace equivalence:



a aa

a

a

tr

ctr

tr

ctr

tr

ctr

p q

p after a refuses L q after a refuses L

q after refuses Lp after refuses L


a

b

a

b

aa

b

b

a

tr

ctr tr

ctr

tr

ctr

tr

ctr



cb

a aa

cb

cb

a

?

(Completed) Trace Equivalence : Others ?


Equivalences on Transition Systems

isomorphism

bisimulation( weak )

failure trace= refusal

failures= testing

completedtrace

trace

weak

strong


Bisimulation

p b q

"p and q simulate each other and go to states from where they can simulate each other again"

Bisimulation:

a

a

Sp Sq : s0p, s0q and

s1, s2 , L* :

whenever sp s'p then sq s'q and s'p, s'q

whenever sq s'q then sp s'p and s'p, s'q

b


Bisimulation

aaa

a

a

b b b


a

b

a

b

aa

b

b

a

Bisimulation

b

b

b

bb


cb

a aa

cb

cb

a

b

Bisimulation


Comparing Systems :Testing Equivalence

S1 S2


LTS(L) Ctraces (e||s)

S1 te S2 e E . obs ( e, S1 ) = obs (e, S2 )

? ?


Testing Equivalence

S1 S2


S1 te S2 e LTS(L) . L* .

e||S1 after refuses L e||S2 after refuses L


Testing Equivalence

a aa

a

a

te te te

p q

obs(e,p) = { a

}

aEnvironment e :

obs(e,q) = { a , }


S1 S2environment

a

b

environment

a

bcb

a aa

cb

teaa b aa b

S1 after a refuses {b} S2 after a refuses {b}

Testing Equivalence

a b aa a b


Testing Equivalence

P te q FP(p) = FP(q)

Failure equivalence = testing equivalence :

aa

cb

FP :

a , {c}

a b , L

,

Not FP :

a , {b,c}

a a ,

, L

Failure pairs of p :

FP (p) = { , A | A L, traces(p), p afer refuses A }


cb

a aa

cb

tr

b

ca

a

b

ca

a

ctr

p q

traces (p) = traces (p) = { , a, a b, a c }

Ctraces (p) = Ctraces (p) = { a b, a c }

p after a b refuses L q after a b refuses L

q after a refuses {a,c}p after a refuses {a,c}

p after a refuses L q after a refuses L

Testing Equivalence

te


Testing Equivalence

cb

a aa

cb

cb

a

te

te

te


Testing Equivalence

tea

tea

coincoin

bangcoffee

coffeebang

p

coffee

tea

coincoin

bangtea

coffeebang

q

p te

qBut: if you want coffee you will eventually always succeed in q but not p !?


Refusal Equivalence

tea

tea

coin

coin

bangcoffee

coffeebang

p

coffee

tea

coin

coin

bangtea

coffeebang

q

coffee

coffee

coin

bang

Test t : only possible

if everything else is impossible

coin bang coffee obs ( q || t )

coin bang coffee obs ( p || t )

p rf

q


Comparing Systems : Refusal Equivalence

S1 S2


LTS( L{} ) Ctraces (e||s)

S1 rf S2 e E . obs ( e, S1 ) = obs (e, S2 )

? ?


Refusal Equivalence

S1 S2


S1 rf S2 e LTS(L{}) . L* .

e||S1 after refuses L e||S2 after refuses L


Refusal Equivalence

P rf q Ftraces(p) = Ftraces

(q)

Failure trace equivalence = refusal equivalence :

( L ( L ) )* : s

Failure trace :

Failure traces of p : Ftraces (p) = { ( L ( L ) )* | p }

Failure A : A { } : s A

s s

aa

cb

Ftraces :

{b,c} a {a,c} b L

a {c} b {a} {b} {c}

a {b} {b} c

Not Ftraces :

{a,b,c} a {a,c} b L

a {c} c L

a a


Refusal Equivalence

tea

tea

coin

coin

bangcoffee

coffeebang

p

coffee

tea

coin

coin

bangtea

coffeebang

q

p rf

qFtrace of p :

coin {coffee} bang {coffee} teaNot an Ftrace of p :

coin {coffee} bang coffee

An Ftrace of q :

coin {coffee} bang coffee

Not an Ftrace of q :

coin {coffee} bang {coffee} tea


Equivalences on Transition Systems

isomorphism

bisimulation( weak )

failure trace= refusal

failures= testing

completedtrace

trace

weak

strong

observing sequences of actions and their end

observing sequences of actions

test an LTS with another LTS

test an LTS with another LTS, and try again (continue) after failure

test an LTS with another LTS, and undo, copy, repeat as often as you like

now you need to observe 's ……


Equivalences : Examples

a aa

a

a

sb

b

tr

te

sb

b

tr

te

sb

b

tr

te


cb

a aa

cb

cb

a



a

b

a

b

aa

b

b

a



bb

dc

a

Equivalence of Transition Systems

aa

bb

c ddc

b

a


d

a

b

c

p

dc d

a

bbb

c

s

aa

bb

dc

r

bb

dc

a

q




tea

tea

coincoin

bangcoffee

coffeebang

p

coffee

tea

coincoin

bangtea

coffeebang

q

equivalences on labelled transition systems ed brinksma course 2004

Documents

s q s q

s p s q

s p s p

ready s

ctraces s

s refusal

s q bb slide

testing equivalence