Formal Testing with Labelled Transition Systems

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Formal Testing with Labelled Transition Systems

• Ed Brinksma
• Course 2004

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Goal Formal Testing with Transition Systems
Test hypothesis ?IUT?IMPS . ?iIUT ?IOTS .
?t?TTS . exec(t,IUT) obs(t,iIUT)
Proof soundness and exhaustivess ?i?IOTS . (
?t?der(s) . ?t(obs(t,i)) pass ) ? i ioco s
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Labelled Transition Systems
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Labelled Transition Systems
L dub,kwart, coffee,tea,soup
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Labelled Transition Systems
Sequences of observable actions
s
traces(s) ?, dub, dub coffee, dub tea
Reachable states
s after dub S1, S2
s after dub tea S4
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Labelled Transition Systems

• Labelled Transition System ? S, L, T, s0 ?

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Representation of LTS
S

• Explicit ? S0,S1,S2,S3, dub,coffee,tea,
  (S0,dub,S1), (S1,coffee,S2), (S1,tea,S3) ,
  S0 ?
• Transition tree / graph
• Language / behaviour expression S dub (
  coffee stop ? tea stop )

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Labelled Transition Systems
stop
Pwhere P a P
a stop b stop
a b stop
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Labelled Transition Systems
Q where Q a ( b stop Q )
a b stop a c stop
a stop ? c stop
a stop b stop
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Language of Behaviour Expressions

• Process Algebra algebra of processes with
  behaviour operators
• Operational semantics in labelled transition
  systems
• Concise, structured, compositional
  representationof labelled transition systems
• Behaviour operators

• inaction stop
• action prefix a B ? B
• choice B1 B2
• hiding hide G in B

• Parallelism B1 G B2 B1 B2 B2
  B2
• process def. p where p Bp

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Language of Behaviour Expressions

• inaction stop

• action prefix a B

• choice B1 B2

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Language of Behaviour Expressions

• hiding hide a in B

• process definition pwhere p Bp

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Language of Behaviour Expressions

• parallel composition B1 a B2

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Language of Behaviour Expressions

• parallel composition B1 a,b B2

a,b

• synchronization B1 B2 def B1 L
  B2

• interleaving B1 B2 def 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

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Example Parallel Composition
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Example Behaviour Expression

• Simple buffer(domain(x)1 states)

in ? x
out ! x
Buf in,out in ? x out ! x
Buf in,out
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Example Behaviour Expression

• Double buffer((domain(x)1)2 states)

(x,?)
in ? x
out ! x
(x,x)
(?,?)
?
(?,x)
in ? x
out ! x
DBuf in,out hide mid in Buf in,mid
mid Buf mid,out
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Example Behaviour Expression

• Six-Double buffers((domain(x)1)12 states)

SixDBuf in,out DBuf DBuf
DBuf DBuf DBuf DBuf