Transcript Automated Model Based Testing From Theory via Tools to

Preorders on Labelled Transition
Systems
Ed Brinksma
Course 2004
Formal Testing
S
imp
test
generation

i imp s
specification
i passes T
test suite T
implementatio
i
n
test
execution
© Ed Brinksma/Jan Tretmans
pass / fail
Implementation Relation
i imp s : implementation i implements specification s
 imp reflexive ?
s imp s
Yes
 imp symmetric ?
i imp s  s imp i
No
 imp transitive ?
i imp s, s imp t  i imp t
Prefer
 imp anti-symmetric?
i imp s, s imp i  i = s
No
 imp linear ?
i imp s or s imp i
No
 imp congruent ?
i imp s  f( i ) imp f( s )
Prefer
equivalence :
preorder :
partial order :
linear/total order :
© Ed Brinksma/Jan Tretmans
reflexive, symmetric, transitive
reflexive, transitive
anti-symmetric preorder
linear partial order
Preorders on Transition Systems

implementation
i
specification
s
environment
e
ii  ss


environment
e
 ee 
 EE .. obs
obs (( e,
e, ii )) 
= obs (e, s )


?
© Ed Brinksma/Jan Tretmans

?

?
Preorders
on Transition Systems
implementation
i

environment
e
specification
s
environment
e
Suppose an environment interacts with the
implementation i and with the specification s :
black
i correctly implements s iff
all observation of i can be related to observations of s
© Ed Brinksma/Jan Tretmans
box
Preorders on
Transition Systems
For almost any equivalence

you can define a corresponding preorder
such that
S1
S1
S1
S1
S1
S1
S1
p
 h S2
 b S2
tr S2
te S2
ready S2
Q S2
ft S2
………
………
© Ed Brinksma/Jan Tretmans

q

p

q and q

 p:
homomorphism
various simulation relations
trace preorder
testing preorder
ready preorder
queue preorder
failure (trace) preorder
………
………
Equivalences on
Transition Systems
strong
weak
isomorphism
now you need to observe 's ……
bisimulation
( weak )
test an LTS with another LTS, and
undo, copy, repeat as often as you like
failure trace
= refusal
test an LTS with another LTS, and
try again (continue) after failure
failures
= testing
test an LTS with another LTS
completed
trace
observing sequences of actions and
their end
trace
© Ed Brinksma/Jan Tretmans
observing sequences of actions
Preorders on
Transition Systems
strong
weak
isomorphism
now you need to observe 's ……
bisimulation
( weak )
test an LTS with another LTS, and
undo, copy, repeat as often as you like
failure trace
= refusal
test an LTS with another LTS, and
try again (continue) after failure
failures
= testing
test an LTS with another LTS
completed
trace
observing sequences of actions and
their end
trace
© Ed Brinksma/Jan Tretmans
observing sequences of actions
Trace Preorder
implementation
i
tr
specification
s
environment
e
i tr s
Traces:
© Ed Brinksma/Jan Tretmans

environment
e
traces ( i )

traces ( s )
traces (s) = {   L* | s 
}
Trace Preorder
tr
dub
dub
tr
tea
coffee
tr
tr
i tr s =
traces(i)  traces(s)
© Ed Brinksma/Jan Tretmans
tr
dub
tea
dub
coffee
tr
coffee
Trace Preorder
dub
tr
dub
traces(i)  traces(s)
© Ed Brinksma/Jan Tretmans
tea
dub
tea
tr
coffee
i tr s =
tr
dub
coffee
coffee
Testing Preorder
implementation
i
te
specification
s
environment
e
i
te
s

environment
e
 e  E . obs ( e, i )  obs (e, s )


LTS(L) Ctraces (e||s)
© Ed Brinksma/Jan Tretmans
Testing Preorder
implementation
i
te
environment
e
i
te
specification
s
environment
e
 e  LTS(L) .    L* .
e||i after  refuses L  e||s after  refuses L
s


FP (i)  FP (s)
FP (p) = {  , A 
© Ed Brinksma/Jan Tretmans
| A  L,  traces(p),
p afer  refuses A }
Testing Preorder
p
q
a
b
a
c
te
te
b
a
c
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}
© Ed Brinksma/Jan Tretmans
Testing Preorder
p
q
a
b
a
te
te
b
a
c
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}
© Ed Brinksma/Jan Tretmans
Testing Preorder
te
a
b
a
a
te
c
b
a
te
te
i te s =
FP(i)  FP(s)
© Ed Brinksma/Jan Tretmans

b
te
te

c
c
Testing preorder
p
a
te
te
te

a
q
te
te
te
Environment e :
obs(e,p) = { a  }
© Ed Brinksma/Jan Tretmans
a
a
obs(e,q) = { a ,  }

Testing Preorder
?
te
a
te
a
?
a
b
b
?
te
a
?
© Ed Brinksma/Jan Tretmans
te
a

b
?
te
a

b
Testing Preorder
p
coin
coffee
bang
coffee
q
coin
coin
tea
coffee
bang
tea
bang
coin
tea
bang
tea
p te q
coffee
p te q ?
q te p ?
© Ed Brinksma/Jan Tretmans
Refusal Preorder
implementation
i
rf
specification
s
environment
e
i
rf
s

environment
e
 e  E . obs ( e, i )  obs (e, s )

LTS(L{})
© Ed Brinksma/Jan Tretmans

Ctraces (e||i)
Refusal Preorder
implementation
i
environment
e
i
rf
s

rf
specification
s
environment
e
 e  LTS(L{}) .    L* .
e||i after  refuses L  e||s after  refuses L

© Ed Brinksma/Jan Tretmans
Ftraces( i )  Ftraces ( s )
Refusal Preorder
Failure A :
s
Failure trace  :
A
s

  ( L ( L ) )* :
  A{}:
s

Failure traces of p : Ftraces (p) = {   ( L ( L ) )* | p
Failure trace preorder
= refusal preorder :
a
b
a
c
© Ed Brinksma/Jan Tretmans
p rf q


s

Ftraces(p)  Ftraces (q)
Ftraces :
Not Ftraces :
{b,c} a {a,c} b L
{a,b,c} a {a,c} b L
a {c} b {a} {b} {c}
a {c} c L
 a {b} {b} c
a a 
}
Refusal Preorder
p
coin
coffee
bang
coffee
q
coin
coin
tea
bang
tea
q rf p
Ftrace of p :
coin {coffee} bang {coffee} tea
Not an Ftrace of p :
coin {coffee} bang coffee
© Ed Brinksma/Jan Tretmans
coffee
bang
tea
coin
tea
bang
coffee
p rf q
Not an Ftrace of q :
coin {coffee} bang {coffee} tea
An Ftrace of q :
coin {coffee} bang coffee
Preorders on
Transition Systems
strong
failure trace
= refusal
preorder
test an LTS with another LTS, and
try again (continue) after failure
failures
= testing
preorder
test an LTS with another LTS
trace
preorder
observing sequences of actions
weak
© Ed Brinksma/Jan Tretmans