Transcript Automated Model Based Testing From Theory via Tools to

Formal Testing with Labelled
Transition Systems
Ed Brinksma
Course 2004
Goal : Formal Testing
with Transition Systems
Test hypothesis :
s  LTS
ioco
der : LTS 
(TTS)
IUTIMPS . iIUT IOTS .
tTTS . exec(t,IUT) = obs(t,iIUT)
Ts  TTS
Proof soundness and exhaustivess:
iIOTS .
( tder(s) . t(obs(t,i)) = pass )
 i ioco s
pass
iIUT
IUT IOTS
IMPS
obs
: TTS
exec
:
TESTS
IOTS 
IMPS


(traces)
(OBS)
© Ed Brinksma/Jan Tretmans
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traces
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t:
(traces)
{fail,pass}
fail
Labelled Transition Systems
a
a
b
a
b
a
b
b
a
a
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b

a
a
c
c
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a
b
a
b
a
b
a
b
Labelled Transition Systems
s
S0
dub
S0
dub
transition
S1
kwart
S1
S2
coffee
S3
S0

S4
tea
S0
S0
dub coffee
kwart tea
kwart soup
S5
L = { dub,kwart,
coffee,tea,soup}
© Ed Brinksma/Jan Tretmans
executable
sequence
non-executable
sequence
all transition
systems over L
LTS(L)
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S3
transition
composition
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Labelled Transition Systems
Sequences of observable actions:
s
S0
dub
traces (s) = {   L* | s
dub
S1
S2
coffee
S3
tea
S4
}
traces(s) = { , dub, dub coffee, dub tea }
Reachable states:

s after  = { s’ | s
s after dub = {
S1, S2
s after dub tea = {
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
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}
S4
}
s’ }
Labelled Transition Systems
Labelled Transition System
 S, L, T, s0 
initial state
states
actions
transitions
T  S  (L{})  S
Discon
ConReq
Discon
Data
ConConf
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s0  S
Representation of LTS
S
 Explicit :
 {S0,S1,S2,S3},
S0
{dub,coffee,tea},
dub
{ (S0,dub,S1), (S1,coffee,S2), (S1,tea,S3) },
S1
coffee
S2
S0 
tea
S3
 Transition tree / graph
 Language / behaviour expression :
S := dub ; ( coffee ; stop [] tea ; stop )
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Labelled Transition Systems
a
a
b
a
stop
b
a
b
P
where P := a ; P
a ; stop [] b ; stop
a
b
a ; b ; stop
b
a
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b
a

a
a
c
c
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a
b
a
b
a
b
Labelled Transition Systems
Q
where Q := a ; ( b ; stop ||| Q )
a ; b ; stop [] a ; c ; stop
a ; stop []  ; c ; stop
a ; stop ||| b ; stop
a
b
b
a
a
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b

a
a
c
c
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a
b
a
b
a
b
a
b
Language of Behaviour Expressions
 Process Algebra :
algebra of processes with behaviour operators
 Operational semantics in labelled transition systems
 Concise, structured, compositional representation
of labelled transition systems
 Behaviour operators :
 inaction
stop
 action prefix
a ; B
B1 || B2
 ;
B2 ||| B2
 Parallelism
B
 choice
B1 [] B2
 hiding
hide G in B
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 process def.
where
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B1 |[ G ]| B2
p
p := Bp
Language of Behaviour Expressions
 inaction
stop
 action prefix
a ;B
a
B
 choice
B1 [] B2
B1
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TT 2004, LTS
B2
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B2
Language of Behaviour Expressions
 hiding
hide a in B
a
hide
B2 a
in B2
hide a
B1
in
B1
 process definition p
where p := Bp
P =
Bp
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Language of Behaviour Expressions
 parallel composition B1 |[a]| B2
a
Bx
a
a
|[a]|
=
By
Bx|[a]|By
b
b
c
|[a]|
BxTretmans
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c
=
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Bx|[a]|c;By
b;Bx|[a]|By
Language of Behaviour Expressions
 parallel composition B1 |[a,b]| B2
a
Bx
b
|[a,b]|
 synchronization
 interleaving
© Ed Brinksma/Jan Tretmans
=
By
B1 || B2
B1 ||| B2
==def
==def
B1 |[ L ]| B2
B1 |[  ]| B2
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Structural Operational Semantics
 Formal semantics of behaviour expressions
through Structural Operational Semantics (SOS)
 Example: parallel composition p |[ G ]| q
p
a
p’,
p |[ G ]| q
p
a
p |[ G ]| q
p’,
a
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q
a
a
aG
q’,
p’ |[ G ]| q’
aG
q
p’ |[ G ]| q
a
p |[ G ]| q
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q’,
a
aG
p |[ G ]| q’
Example : Parallel Composition
U
M
dub
coffee
kwart
coffee
tea
U'
M'
M || U
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dub
dub coffee
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M' || U'
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tea
Example : Behaviour Expression
 Simple buffer
(|domain(x)|+1 states)
in ? x
x
out ! x
in ? x
out ! x
Buf [in,out]
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:=
in ? x ; out ! x ; Buf [in,out]
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Example : Behaviour Expression
 Double buffer
((|domain(x)|+1)2 states)
mid ! x
in ? x
in ? x
(x,)
(,x’)
out ! x’
out ! x’

(,)
out ! x’
mid ? x’
(x,x’)
in ? x
DBuf [in,out] :=
hide mid in
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Buf [in,mid] |[mid]| Buf [mid,out]
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Example : Behaviour Expression
Six-Double buffers
((|domain(x)|+1)12 states)
out ! x
out ! x
out ! x
out ! x
out ! x
out ! x
in ? x
in ? x
in ? x
in ? x
in ? x
in ? x
SixDBuf [in,out] :=
DBuf ||| DBuf ||| DBuf ||| DBuf ||| DBuf ||| DBuf
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