B (10) Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 1

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1 : B (10) 2010 Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 1

2 B B Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 2 x : S S S S x := min(s) Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 3

3 allocated allocated MACHINE Allocate VARIABLES allocated INVARIANT allocated N 1 INITIALISATION allocated := Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 4 OPERATIONS choose(num) = PRE num N 1 num / allocated THEN allocated := allocated {num} ; num allocate = ANY nm WHERE nm N 1 allocated THEN num := nm allocated := allocated {nm} ; ans query(num) = PRE num N 1 THEN ans := bool(num allocated) allocate Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 5

4 REFINEMENT Allocate R REFINES Allocate VARIABLES allocated INITIALISATION allocated := Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 6 choose query OPERATIONS choose(num) =allocated := allocated {num}; num allocate = BEGIN num := min(n 1 allocated); allocated := allocated {num} ; ans query(num) = ans := bool(num allocated) allocate allocated Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 7

5 Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 8 Allocate Allocate R allocated = {6, 13} allocate Allocate 6, 13 Allocate R Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 9

6 ... Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) BOOK 2. read 3. newbook newbook Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 11

7 MACHINE Books(BOOK ) VARIABLES read INVARIANT read BOOK INITIALISATION read := OPERATIONS bk newbook = PRE read BOOK THEN ANY bk1 WHERE bk1 BOOK read THEN bk := bk1 read := read {bk1} Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 12 iseq scheme scheme ran(scheme) =BOOK read Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 13

8 Atelier B B Book REFINEMENT Books R(BOOK ) REFINES Books VARIABLES scheme INVARIANT scheme iseq(book ) ran(scheme) =BOOK read INITIALISATION scheme : perm(book ) OPERATIONS bk newbook = BEGIN bk := first(scheme) scheme := tail(scheme) read scheme scheme perm(s) s s scheme BOOK Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 14 scheme BOOK read read := scheme BOOK ran(scheme) =BOOK read read = scheme newbook Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 15

9 newbook read = scheme =[x,...] newbook a newbook r read = {x} scheme =[...] Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 16 REFINEMENT Books R R(BOOK ) REFINES Books R VARIABLES counter, bookarr INVARIANT counter 0.. card(book ) bookarr 1.. card(book ) BOOK (0..counter bookarr)^scheme = bookarr INITIALISATION counter := 0 bookarr : 1.. card(book ) BOOK bookarr Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 17

10 newbook OPERATIONS bk newbook = BEGIN counter := counter +1; bk := bookarr(counter) (0..counter bookarr)^scheme = bookarr scheme counter counter 0 newbook scheme = bookarr Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 18 newbook newbook (read BOOK ) scheme first tail counter card(book ) bookarr(counter) scheme newbook Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 19

11 sset slist MACHINE StBox SETS STORY CONSTANTS max score PROPERTIES max score N 1 VARIABLES score, sset INVARIANT score N score max score sset : P(STORY ) INITIALISATION score := 0 sset := {} Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 20 OPERATIONS gives(sc) = PRE sc N 1 THEN score := min({score + sc, max score}) ; penalty = SELECT score > 0 THEN score := score 1 WHEN sset {} THEN ANY st WHERE st sset THEN sset := sset {st} ELSE skip ; select(st) = PRE score > 0 st STORY THEN CHOICE score := score 1 sset := sset {st} OR sset := sset {st} ; st tell = IF sset {} THEN ANY st1 WHERE st1 sset THEN st := st1 sset := sset {st1} Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 21

12 select score scorer sset slist 1 Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 22 select free isfree free REFINEMENT SBox R REFINES SBox CONSTANTS free PROPERTIES free N 1 VARIABLES scorer, slist, isfree INVARIANT scorer N scorer = score slist iseq(story ) ran(slist) =sset isfree 0..free INITIALISATION scorer := 0 slist := [ ] isfree := 0 Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 23

13 OPERATIONS gives(sc) = BEGIN scorer := min({scorer + sc, max score}) ; st tell = BEGIN st := first(slist) slist := tail(slist) ; penalty = IF slist []THEN slist := tail(slist) ELSIF scorer > 0 THEN scorer := scorer 1 ; select(st) = BEGIN IF st / ran(slist) THEN slist := slist st ; IF isfree = free THEN CHOICE isfree := 0 OR isfree := 0 scorer := scorer 1 ELSE isfree := isfree + 1;scoreR := scorer 1 Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 24 free free IF IF isfree = free THEN isfree := 0 ELSE isfree := isfree + 1;scoreR := scorer 1 free Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 25

14 B Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 26 M M R M op m S m S m δ S r δ S r op r δ Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 27

15 Color MACHINE Colors SETS COLOR = {red, green, blue} VARIABLES colors INVARIANT colors COLOR INITIALISATION colors : P(COLOR {blue}) OPERATIONS add(cl) =PRE cl COLOR THEN colors := colors {cl} ; cc query = PRE colors THEN cc : colors ; change = LET clss BE clss = P(COLOR) {colors} IN colors : clss Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 28 Color colors COLOR colors {red, green} add colors query colors change LET clss = P(COLOR) {colors} clss COLOR colors change colors colors query Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 29

16 Color change add query colors REFINEMENT Colors R REFINES Colors VARIABLES color INVARIANT color COLOR color colors INITIALISATION color : COLOR {blue} OPERATIONS add(cl) =BEGIN color : {color,cl} ; cc query = cc := color ; change = LET cs BE cs = COLOR {color } IN color : cs Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 30 Color add add(cl) =BEGIN color : {color,cl} add color colors add color colors color : colors change change = LET cs BE cs = COLOR {color } IN color : cs change color colors cs colors color colors Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 31

17 M T I R T 1 J R T 1 T J T T 1 R M T 1 T T 1 J T Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 32 M R M s R s (s, s ) M R R M / J J(x, x ) M s R s I M J R M R T 1 R R s M T s J(s, s ) J T J T J Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 33

18 T J J J [T ] J [T ] J T J [T ] J T J T [T ] J T J T J R T 1 T T 1 [T ] J T 1 [T ] J [T 1 ] [T ] J Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 34 Colors ColorsR J color colors T = colors : P(COLOR {blue}) [T ] J J T color = blue [T ] J T colors color = blue color = red color = green T [T ] J [T ] J Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 35

19 [colors : P(COLOR {blue})] (color colors) = ( colors. (colors P(COLOR {blue}) (color colors))) = colors. (colors P(COLOR {blue}) (color colors)) = colors. (colors (COLOR {blue}) color colors) COLOR {blue} colors color colors color blue color blue [T 1 ] [T ] J Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 36 ColorsR T 1 = color : COLOR {blue} [T 1 ] [T ] J = [T 1 ](color blue) = [color : COLOR {blue}](color blue) = color. (color (COLOR {blue}) color blue) = true ColorsR Colors I T I I T Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 37

20 [T ] J T J T J T T [T ] I T [T ] J Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 38 M R PRE P THEN S PRE P 1 THEN S 1 P 1 P 1 S 1 S [S 1 ] [S] J J I I J I J P [S 1 ] [S] J Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 39

21 I J P [S 1 ] [S] J [S 1 ] [S] J S 1 S [S 1 ] [S] J S 1 S [S 1 ] [S] J Colors ColorsR I J colors COLOR color COLOR color colors add add(cl) =PRE cl COLOR THEN colors := colors {cl} add(cl) =BEGIN color : {color, cl} Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 40 I J P colors COLOR color COLOR color colors cl COLOR add [S 1 ] [S] J = [S 1 ] [S] (color colors) = [S 1 ] [colors := colors {cl}] (color colors) = [S 1 ] ( (color colors {cl})) = [S 1 ](color colors {cl}) = [color : {color, cl}](color colors {cl}) = x.(x {color, cl} color colors {cl}) = (color colors {cl}) (cl colors {cl}) = color colors {cl} I J P [S 1 ] [S] J J = color colors color colors color colors {cl} Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 41

22 M R P 1 PRE P THEN S PRE P 1 THEN S 1 P P 1 P 1 P P 1 M P R P 1 P P P P 1 P 1 I J P P 1 Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 42 Colors ColorsR query ColorsR query green green Colors J Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 43

23 o o out out out out out) S 1 out out S 1 [out /out] out = out out 1 = out 1... out n = out n out out [S 1 [out /out]] [S] (out = out) Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 44 P 1 I J P [S 1 [out /out]] [S] (J out = out) I J P P 1 Colors ColorsR J color colors query cc query = PRE colors THEN cc : colors cc query = cc := color Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 45

24 [S 1 [cc /cc]] [S] (J cc = cc) = [S 1 [cc /cc]] [cc : colors] (J cc = cc) = [S 1 [cc /cc]] ( cc. (cc colors (J cc = cc))) = [S 1 [cc /cc]] ( cc. ( (cc colors) (J cc = cc))) = [S 1 [cc /cc]]( cc. ( (cc colors) (J cc = cc))) = [S 1 [cc /cc]]( cc. (cc colors J cc = cc)) = [cc := color ]( cc. (cc colors J cc = cc)) = J cc. (cc colors color = cc)) J color colors = J I J I P [S 1 [cc /cc]] [S] (J cc = cc) Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 46 SETS CONSTANTS PROPERTIES CONSTRAINTS CONSTRAINTS PROPERTIES PROPERTIES PROPERTIES Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 47

25 St 1,C 1,B 1 PROPERTIES St,C,B,Cn PROPERTIES CONSTRAINTS Cn St 1,C 1,St,C. B 1 B T 1 J T Cn B B 1 [T 1 ] [T ] J out op(in) =PRE P THEN S out op(in) =PRE P 1 THEN S 1 Cn B B 1 I J P [S 1 [out /out]] [S] (out = out) Cn B B 1 I J P P 1 Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 48 Formal Methods: Software Development in B Qiu Zongyan (May 11, 2010) 49

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