Post on 19-Jan-2018
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“B is a method for specifying, designing, and coding software systems.”J.R. Abrial, The B-Book, Cambridge University Press
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B4free
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Exercise 1.7A car park has 640 parking spaces. Give an abstract machine which specifies a system to control cars entering the car park. It should keep track of the cars currently in the car park, and should provide 3 operations:– Enter, which recorders the entry of a new car.
This should occur only when the car park is not full;
– Leave, which records the exit of a car from the car park;
– Query, which outputs the number of cars currently in the car park.
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MACHINE CarParkVARIABLES contentsINVARIANT contents : NAT & contents <= 640INITIALIZATION contents := 0OPERATIONSenter =
PRE contents < 640THEN contents := contents + 1END;
leave =PRE contents > 0THEN contents := contents – 1END;
nn <-- query =PRE trueTHEN nn := contentsEND
END
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43houseset, magazine := {}, {}
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t parentancestor
parentyzparentzx
PERSONzzyxparentancestor
,,.,
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Substitutions
xEP /
Expression E is substituted for a free variable x byreplacing all occurrences of x by E.Read as P with E for x.
GOaliceCHESSalice
xaliceGOxCHESSx
/
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zyxxyzyxyx /
2)(.
/2)(.
oldlimitpagePERSONpp
limitoldlimitlimitpagePERSONpp
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3.
/3./3.
nmmmlimitnlimitmmm
limitnlimitnnn
Renaming bound variables to avoid variable capture
limitmmm
nlimitmmmnlimitnnn
./50.
/50.
If the variable being substituted does not occur free anywhere inthe predicate then it is left unchanged.
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nxxxGFEP ,...,/,..., 21
zyxyx
yxzyxyxyx
,/,
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Self test
servennextnnservenn
serveservenextnnservenn
/4.)2(
/14.)1(
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The set of all possiblestates a machine canbe in.
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See Page26 of theB-method.
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P is a predicate which describes a set of states that may be reachedafter the performance of statement S.P is referred to as the post condition of S.
The notation [S]P denotes a predicate which is true ofany initial state from which is guaranteed to achieve P.
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See Page27 of theB-method.
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[hh := min(houseset)](!hh.(hh:houseset=> hh < 163))
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[a(4) := 7](a : NAT1 >+> NAT)
= (a <+ {(4,7)}) : NAT1 >+> NAT
= ({4} <<| a) : NAT1 >+> NAT &7 /: ran({4} <<| a)
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Other Constructs
[IF E THEN S ELSE T END]P =(E & [S]P) or (not(E) & [T]P)
[IF E THEN S ELSE T END]P =(E => [S]P) or (not(E) => [T]P)
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[IF x<5 THEN x:=x+4 ELSE x:=x-3 END] (x<7)
= (x<5 & [x:=x+4](x<7)) or((not(x<5)) & [x:=x-3](x<7))
= (x<5 & (x+4<7)) or ((x>=5) & (x-3<7))
= (x<5 & x<3) or ((x>=5) & (x<10))
= (x<3) or ((x>=5) & (x<10))
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MACHINE Info(ITEM, sample, num)CONSTRAINTS sample:ITEM &
num : NAT &num > card(ITEM)
CONSTANTS storagePROPERTIES storage : NAT1 &
storage <= num
VARIABLES current, next, previous
INVARIANT current <: ITEM &next : ITEM &previous : ITEM &next /= previous
Exercise 5.1 Page 67 “the b-method”What are the proof obligations associated with the constraints below?Are they consistent?
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# ITEM, sample, num .(ITEM /= {} &sample : ITEM &num : NAT &num > card(ITEM))
Proof obligation associated with the constraints:
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(ITEM /={} & sample:ITEM & num:NAT & num > card(ITEM))
=>
# storage . (storage : NAT1 & storage <= num)
Proof obligation: It must be possible to find appropriateSETS and CONSTANTS.
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( ITEM /={} & sample:ITEM & num:NAT &num>card(ITEM) &
storage : NAT1 & storage <= num )=># current, next, previous . (
current <: ITEM &next : ITEM &previous : ITEM &next /= previous
)
Proof obligation: When all the parameters are set it must bepossible for the machine to have variables that satisfy theinvariant.
What ifITEM={a}?
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END S THEN P PRE operationeach for
][ 5.][ 4.. 3.., 2.
. 1.
ISPICBITCBIvCBBkStC
Cp
Summary of Proof Obligations:
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• Self tests (from “the b-method”)– Exercise 5.2 page 68– Exercise 6.3 Page 89
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Completing the Laws of [S]P
nnnn xxEEPPEExx
yxFEPPFEyx
xEPPEx
,...,/,...,,...,:,...,
,/,,:,
/:
1111
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PP skip
PTEPSE
PTSE
ENDELSETHENIF
PTEPSE
PTSE
ENDELSETHENIF
?ENDTHENIF PSE
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PVeEeEeEPTeE
PTeEPTeE
P
VTe
TeTe
E
n
nnnn ...
...
ENDELSE
THENOR...OR
THENORTHENEITHER
OFCASE
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aEiaPPEia /,:)(
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PTQx
PTQx
.
ENDTHENWHEREANY
Non-determinism:
PTQxx
PTQxx
n
n
.,...,ENDTHENWHERE,...,ANY
1
1
xzPSzzPSx /.:
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PSExx
PSExxLET
.
ENDINBE
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PTPSPTS ENDORCHOICE
PSPSPS
PSSS
n
n
...END...ORORCHOICE
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• Sequences