1. Preorders on Labelled Transition Systems
Ed Brinksma
Course 2004

2.  Formal Testing i imp s i passes T imp S i specification test
generation
test suite T 
imp
implementation i
test
execution
pass / fail

3. Implementation Relation
i imp s : implementation i implements specification s
imp reflexive ? s imp s
imp symmetric ? i imp s  s imp i
imp transitive ? i imp s, s imp t  i imp t
imp anti-symmetric? i imp s, s imp i  i = s
imp linear ? i imp s or s imp i
imp congruent ? i imp s  f( i ) imp f( s )
Yes No
Prefer
equivalence : reflexive, symmetric, transitive
preorder : reflexive, transitive
partial order : anti-symmetric preorder
linear/total order : linear partial order

4. Preorders on Transition Systems
i  s   e  E . obs ( e, i )  obs (e, s )
implementation i
specification s
i  s   e  E . obs ( e, i ) = obs (e, s ) 
environment e
   ? ? ?

5. Preorders on Transition Systems
implementation i
specification s
environment e
Suppose an environment interacts with the black box implementation i and with the specification s :
i correctly implements s iff
all observation of i can be related to observations of s

6. Preorders on Transition Systems
For almost any equivalence  you can define a corresponding preorder  such that p  q  p  q and q  p :
S1 h S2 homomorphism
S1 b S2 various simulation relations
S1 tr S2 trace preorder
S1 te S2 testing preorder
S1 ready S2 ready preorder
S1 Q S2 queue preorder
S1 ft S2 failure (trace) preorder
……… ………

7. Equivalences on Transition Systems
now you need to observe 's ……
isomorphism
test an LTS with another LTS, and undo, copy, repeat as often as you like
bisimulation ( weak )
weak
strong
test an LTS with another LTS, and try again (continue) after failure
failure trace = refusal
test an LTS with another LTS
failures = testing
observing sequences of actions and their end
completed trace
observing sequences of actions
trace

8. Preorders on Transition Systems
now you need to observe 's ……
isomorphism
test an LTS with another LTS, and undo, copy, repeat as often as you like
bisimulation ( weak )
weak
strong
test an LTS with another LTS, and try again (continue) after failure
failure trace = refusal
test an LTS with another LTS
failures = testing
observing sequences of actions and their end
completed trace
observing sequences of actions
trace

9. tr Trace Preorder i tr s  traces ( i )  traces ( s )
implementation i
specification s
environment e
i tr s  traces ( i )  traces ( s )
traces (s) = {   L* | s  }
Traces:

10. Trace Preorder tr tr tr tr tr tr coffee coffee coffee i tr s =
dub
tr
coffee
dub
tea
tr
tr
tr
dub
coffee
tea
tr
tr
i tr s =
traces(i)  traces(s)

11. Trace Preorder tr tr tr coffee coffee i tr s = coffee
dub
coffee
dub
tea
tr
dub
coffee
tea
i tr s =
traces(i)  traces(s)

12. te Testing Preorder i te s   e  E . obs ( e, i )  obs (e, s )
implementation i
specification s
environment e
i te s   e  E . obs ( e, i )  obs (e, s )
 
LTS(L) Ctraces (e||s)

13. te Testing Preorder e||i after  refuses L  e||s after  refuses L
implementation i
specification s
environment e
i te s   e  LTS(L) .    L* .
e||i after  refuses L  e||s after  refuses L
 FP (i)  FP (s)
FP (p) = {  , A  | A  L,  traces(p), p afer  refuses A }

14. Testing Preorder te te p q c b a a c b p after a b refuses L
q after a b refuses L
p after a refuses {b}
q after a refuses {b}
p after a refuses {c}
q after a refuses {c}
p after a refuses {b,c}
q after a refuses {b,c}

15. Testing Preorder te te p q a a c b b p after a refuses {a}
q after a refuses {a}
p after a refuses {b}
q after a refuses {b}
p after a refuses {c}
q after a refuses {c}
p after a refuses {b,c}
q after a refuses {b,c}
p after a refuses {a,b,c}
q after a refuses {a,b,c}

16. Testing Preorder te te te te te te  c b a a c b c b a i te s =
FP(i)  FP(s)

17. Testing preorder te te te te te te p q   a a a a
Environment e :
obs(e,p) = { a  }
obs(e,q) = { a ,  }

18. Testing Preorder a
? te a b
? te a b
? te a  b
? te b  a
? te

19. Testing Preorder p te q p te q ? q te p ? p q tea coin bang coffee

20. rf Refusal Preorder i rf s   e  E . obs ( e, i )  obs (e, s )
implementation i
specification s
environment e
i rf s   e  E . obs ( e, i )  obs (e, s )
 
LTS(L{}) Ctraces (e||i)

21. rf Refusal Preorder e||i after  refuses L  e||s after  refuses L
implementation i
specification s
environment e
i rf s   e  LTS(L{}) .    L* .
e||i after  refuses L  e||s after  refuses L
 Ftraces( i )  Ftraces ( s )

22. Refusal Preorder    A  {  } : s    ( L ( L ) )* : s
Failure A : 
   A  {  } : s A
s s 
  ( L ( L ) )* : s 
Failure trace  :
Failure traces of p :
Ftraces (p) = {   ( L ( L ) )* | p } 
p rf q  Ftraces(p)  Ftraces (q)
Failure trace preorder
= refusal preorder : a c b
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 

23. Refusal Preorder q rf p p rf q p q Ftrace of p :
tea
coin
bang
coffee p
coffee
tea
coin
bang q
q rf p
p rf q
Ftrace of p :
coin {coffee} bang {coffee} tea
Not an Ftrace of q :
coin {coffee} bang {coffee} tea
Not an Ftrace of p :
coin {coffee} bang coffee
An Ftrace of q :
coin {coffee} bang coffee

24. Preorders on Transition Systems
failure trace = refusal preorder
test an LTS with another LTS, and try again (continue) after failure
weak
strong
failures = testing preorder
test an LTS with another LTS
trace preorder
observing sequences of actions