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1 chg hch cu AFDLING INFORMATICA W 52/75 NOVMBR L.G.L.T. MRTNS & J. C. VAN VLIT RPAIRING TH PARNTHSIS SKLTON OF ALGOL 68 PROGRAMS: PROOF OF CORRCTNSS P u b c 2 bhv 49 d 3 LI 0 T}:;. f 4 A T H M ). T ;. f.

2 Pd h Mhc Ck hvu Akd. Th Mhc Ck Pudd h h 06 Fbuky u g h u 06 uk hu d 44 c zd by h Nhd4 Gvk hkugh h Nhkd4 Ogz 6 0 h Advc 06 Puk R4ch (Z.W.0) by h Mucy 6 Akd b y h Uvkzy 06 Ad b y h F Uv4y Ad d by du44. AMS(MOS) u b c c f c ch ( ): 6 8 A20 68A0 ACMCug R v wc g :

3 R g h h k f ALGOL 6 8 g : f f c c ) by L. G. L. T. M & J. C. v V ABSTRACT Th c c b f u c u f d h b b c h b f f d g h h h d c d gh. N u g y h g h f d g h " " d f d c c ( 0 g b c hu g b c f h ud c c. Th c c f f h b c g h y h b g v w h y f w d u c v. KY WORDS & PHRASS: ALGOL 68 h uc u c c x gb d c d gh c c f duc v. Th f v w f u b c wh.

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5 O. INTRODUCTION I ALGOL 6 8 budc f h z d c u c v b wh c h y c c u d w h h. M f h c u c v h g. u c u. T h f uc f c g v u f f h f h h u c u f h u c g h b f u d c c. T h h h u cquc h h u y g v f u h g h y d u c g h ub f u b f h g y c c y c c. U v d b y c wh ch d g v u h " " c c u c x. I c v y h f h h f u c c v. O b v u wy ch v h " c c " h u c x c y w h h ch. S c h c c f c u f d u wh c c f h f h g w f h u g g v " g " f c c u c x. Th w y wh ch h g g h d h w b d k b ugh w d. D f d w h g h 2 w h c h d f c u v d v h u w g d c w d c c g z d. T h u d b cu w k f c u c ccd b y u. B y u d y g h f w f c w b u g h h w k h wh h y c u c u w c d. I f g c y h w v h g h vdd u hw c v c g y c c h b c k bg h f u f v d h w h c h g w g h h v. A b k h u g h ccu d wh w h u h d f v w g h g h u g g h b u f g x g h g h. W h u b h c c f u d u b f b. I. FORMULATION OF TH PROBLM I h w c c h h k.. h

6 2 u cc f h c c u g h u c x. I ud h h h ubd f. M v h d c " ch g " wh c h f w g v h ch ch h. F h k f c v c h d c d f d f d c d < h h h h h v. A f u h c c f h k d d b y h f w g y x : c c k : c u quc. c u : STYL g h c c k STYL c g h. H f c u STYL g h d S TY L c g h ch. I h u d b d h h f u d q u h h w f k g g d c g h d. (Th v b b g g ALGOL68' h h f w k; f h b g h w v w b c f h c c. ) A " " c f h k g f ub f h u ch h d f h h f h k u c c k. (Whh c c h h w b c u y d d ch g h u d b d wh ch f u d h c f h. ) A " " wh ch k h b ub f h. Th b d f g v h k. A " g " f k y b cdd f g ( b ) w h b d h c h h ubd f b ( h wh k g v b y ( 0 ) d ch g ( ) y ). I c d " u " f y b d u c d b y h bv y x f c c k. Th " v u " f g h ub f h h

7 h b d d f d u z. O b v u y h c c bw h v u f h g (0 ) d h. I h q u w c c g h f d g h v u ; h y b d b y d u c. 2. RROR VALU AS SHORTST PATH I d d h v u f g w y c d f w : I f h g y v u O. Oh w d w c v ch h d b w h g h f I. M v f ch f ch g h d h v u f h ( h ) g c d bw h h d d w c v h wh g c u d g h ch g h b d w h h v u. Th h v u f h g h g h f h h h b b b y f w g c f h b g g h d wh h g h f h h u f h g h f h d v d u c v v d. A x g v f g u. Fg. I. A g w h v u I. I c b u z d b y d g h kd h. Ob v u y h v u f u g 0 d h c f h h h quc f c u. T h g h

8 4 v u y d d b y h c d u c c y y b vd ( b y d u c h g h f h g ) b y b v g f h u g y d d b b d b y d g h ud v g h c d b y u z g h w y g f g c w h c h d b v u y h d f x c y h d c d h ch u z g c d g d h d h f g h d d. I c b h h c d u c f h c u c f c y c c d c d g h wh dg ( h c ) b d w h h g h f h h h u b g h. I d f f c u f u g h wh ch v d c u g h v u f ubg. H w v f u h z b ( d d b ) d f h u f b h g h c c f h z d f f d u c d. 3. ( )A L GBRA IC NOTATION A c v w y x g h f f d g h h h d c d g h h g h b y x. L ch dg f h g h b b d w h g h ( g v c g u b ). I d b u f c u u ccd v c c d d c c d b y dg f g h h h f b g h g v b y {c I R : x 0 } u f. A ch v x c c d f b y dg f z g h. N w w c h g h ( 4 0 d k g y A u h g h f h dg f v x ) x ( 4. ) v x. A x x f g h d x g v A f g u 2 d 2 b. b y u b g h v c f

9 Fg. 2 A d c d g h. Fg. 2 b. M x f h gh f g. 2. Th g h : k g h u d k g h u. F h w d u c h f w g : F w g h x d y h u f x d y dd x + y d h u d d x x Y wh f cu x + + x x d x x... x x..) I y ch ck d h h + d x b y h w f cu v y c v y d v d b u v y wh y h c d g b c : x + y y + x x y) + z x x y y x x x x ( y x z ) ( x x y ) x z xx (y z ) x x y x x z F h dx+. xx0 xdxx. h. k h f h z d 0 h f h u y. F d c " +" c v g. I w w y b c f h c x wh c h d d.

10 6 I u f u k h f w g f c d : I f x + y z h z x d z y ; v z x z y I f x y h x + y x. 4. MATRIX OPRATIONS Ug h w c w d f " d d " d " u c " f c : A + B dfd by (A + B) +. A x B d f d b y ( A x B ) 3 I A k x B k wh h u X f d u g h + d u c d c 3. A b x c d x d d c v d c u v x u c c v b u c u v d g h h d b u v. A h x I h v g 0 h d g d c wh f I x A A x I A d v h d y x. F h f c h c h v 0 h d g f w h I + A A. Th w f x c b d f d u u b y A 0 I + A A x A. T h AP A. Th c b d : A f b y k g S c h h h bw y hw v c g h w h + v c c f dg A g h h g h f h h h f. Th f u u h y c d " v c u ". hf I + A A f w h A + A A A x ( I + A ) + + A A h I x h g h A f u c h v dg f 3 + f+ A ua +

11 f > h w g f h c f d w h. (Th q u v h h h g h y c y c c b u g c y d. ) T h y wh c h v d b y h 4. x d w b c y ud f w. ( I f c h g h f c v d w h >. ) F h y w d v LMMA I A ' d d y h u b PROOF. Sx c A A : A A : : d B I : :. :. u (A f x fb) X AI x k A I k x c k0 k c w h > k k > h v f c q u 00 d h f d c b u h u. v h (3) ( A x k ) w h Sc A A A L c b x d A. x A _ wh h u f d v ( 4 ) u P f u c f c w h q u d c y b d w b d 5 5 A x (P ) I wh d 2 h u 2 f d v ( ) u ( k 5 0 < ) 5 w h x < 0 I. (W d 5. h h h < c h w < f c w u d b d f h d u c f v g " h g " b yud.) _ X I h q u wh v h bv u d h u c v 5 d h u d d. F h x u f h f h bv u b u w x c u d g h < x h S f c A x 2

12 8 + A I X A x 22 ( ) x A w b u d. S w _ h v A h A F h + w c v A LMMA u 2. I f A c b x c f d q h A + +. B PROOF. A A. ' 4?.2 ( ) q f / X ( g Z2 ( ) X + A A +. X X A. x A u.?_2 () k. 2 () + F + 7 ( ) f h c d f h u c v h c A f wh ch w c cu d h x P q X X ) ( B x 2 wx x ( P ) ' ( x. Y x. P _ A K ) : x h x B xl x _ ( A q X ( B. x B " ).?.2 _ ( " ) P ' ; (Th u q u ) c c f x x + y ). S w y w f d + A + A+. ' h + h ) ) Th A h u d f d d f y I + A A.+ h x f g h. I u y d u c d f f c h c.

13 5. NOTATIONS FOR INDUCTIV ASSRTIONS AND VRIFICATION RULS Th c c f f h g h g v b y g ub f h x d b y v f y g x ub f v f c c d g d b y v f c u. Th u h v b ch uch h h ub f d d. ch c d bw { d }. Cud w. g. { A B C } g { A A B A C }. I f A h A c ( v : ) d f A w h u b u d f v. T h u b u h u d b k y h h y. F x f A d f 84 x y z h A c ( 2 x 2 : x + I ) d f ( x + ) x y z. I h d b y g yb b u b y. Th c c f ch ck d b y v f y g h c c f ch w h c u u d g h f h v f c u :... {A } S {A I { A P P f w g v f c { u : A IA v : {3}: A { B c ( v : ) }. _ A f C h S S2 f 0 3 : } {A 0 S IA f v f f b y 0f 3 I { { B } A d {A I } B c ( v : A S B v f v S {B c O d c ( v :} I ) I C /. 0. S 2 I B. F h h y f c c u g u g h w h v h Th u h v b d d y u h c h ( h u f g u ). Th v f c c d h h h f ( " c d " ) ) chd wh

14 0 h cd (" c q u " ). I h u d b k d h h u g v h g b u h d h c u w h f d w h f. A h (0 f cu u h u d h f H ' x c hd c f b u d f d v g c c f h wh x b y b u h u c c f d u v f c c d g d b y d w c. ) I d b f f u y f h u f g c g k y v : v : ' 2 2 g v u d w h h y c c f. I w y b u d f v f y f A c ( 7 : A 0 c ( v 2: c : v v V : q : v 2 ): : 2 u d w h h g c d () y A ' ya v: v d c ( TH v ALGORITHM : k ): v. I d q h g h f d g h v u c b ; F : f y v d c c f d f f v u dd. u) Ug d c h ( 0 g b c w c w : ( v h 2 : y d 2 ) v 2 2 h v u Q 0 f w h f _4. Q + h w. f + A c h g (+

15 O b v u y Q h x f h g h d u c d c 2 wh h dg d c d f f g h. T h h c u h d f w gudd f w f L u g d u c Th g h cu c u b y c u x R u ch h R Q. I u f v f x f w : RO O : 0 {RO : 0 0 : O Q0 : 0 0 : f f b y {RO : d A d d f h ( ) d 0 { R N : 0 : QO : 0 : } : R: R Q {R Q 0 I h " A d d f h ()" h w b w k d u f u h. : Th v f c c d f h u g h b v u q u v f + f h c u u g v b v 0 {RO : 0 : Q0 : 0 : : A d d f h ( ) { R N : } 0 F c : y w h h w b k g b f c h d d g g h u g h h f. Q Th c c f g v b w u h R ' Q ' d R. Th 0 d u c d h f d d u b d u y ; f h : b g h y h u d b d R Q d R c v y. L RQ d 0 f : W : 0 : W0 : 0 : d PT f. : Q I V k : : : + X R k z x R zk+ L z R k.

16 2 (I v w f h f c h h w g u d d c 0. RQ c b c d d c v b b v f : R Th " A d d f h ( ) " y h b w f w : q O QR : : ( I 0 ) V k : 2 : T : T : 0 { N P T c ( f : f b y _ PT} q f d 0 } R : ( c h ) T I c ) V k : I : T k : T M + Ik x T d IRO c ( : + ) } 7. VRIFICATION q By y g h v f c u f g u b u d u f y g c h u wh w b h f w g f v v f c h c d ch f w d b y f f h b v u : h V I : M O N ( 0 : _ V k : 2 : + R k z x R q zk+ z Q _ R 0 + X R z x R k z : z S c R. R ) d f I : : w h c h f u b u f R ) ( I 0 ) f R I : d ( I 0 ) O :. T h /. c Qd H d Q 0 f w d y f h d f f ( Q : Q h :

17 R k c b w X C k. x x. () Sc k < w h v k < f c h. B y u g f v c h z k < z I d f c g u R k z w b V2: I. R c X I k z x ( X X R z zk+ () X z I x z. zk+ Th f w f h g b f c h I V3: I RQ P T c ( : ) {RQ P T c ( : ) }. V4: I RQ R ' : ( f f x R ) (Th c g VI d c b u h u. ) Sc B 0 d 0 + x 0 f y x h c f V I cqu h d V k : : T k + R k z x R z R k zk+ /? / R ' I ( R : ( c h g ( ) R ' : W : V k c : : T k R k x T + R c z x R z zk+ R k }. F T + R x R I R I w d d u c z + I T R I ( h u Sc R' R d 0 Q w h v hw 3 H f

18 4 Q Q ( c h g ( ) I c h g ( R'ILO : Q I V k ) 0 :... O I : T U g T R R d R 0 : 0 : Q 0 : 0 : (R0 ) w b h c d wh ch f w d y f h d f f Q. Th c f V 4 ' cqu f w f h h d c f c d g h w h T R V5: I M 0 :. 0 : 0 0 : 0 : R 0 8. OPTIMIZATION OF TH ALGORITHM : 0 : Q : 0 :. Ju k g h y b d b y x w y x b y g h. U u y h g h f f c f h x.. c y q u c. S c h c c c u g h c d d g h u d h g h w v f b c wh x A d b y c g w h h y h (A ) f ( ) w h < d < 00. S h h c f g u g b ckw d h h f w d. L g c y h d f f b u h f f c h h c c f h b c g h uch f f c h w u d b h w. F x h g d V k : : T k : T k + R k x T V ( k ) ( R ) : T k : T k + x T.

19 5 I y b h h g h f c R y f f c c b u. Nw w c k h f w g b v. O u g h cu R Q b u w uch d Q b u h Q. F h y R u ch h R w v u u q u y w. H f u y u c h R c b d v h Q. Nw f w k f g u 3 w k wh c f u d f v u x b x g v h f c h h w g d c d b y ". " h v v u f d b c v y. T h c cd f y Q. A c b f f g u 3 b h w v h y u f u u R f w y d R h Q y R f h g d y x b x. M v h w h cc d d f f ( d c d b y k g h c c b c w h k ) d f h f hu f v w h f g u 3 b f b h f g u 3. Fg. 3. A g w h v u _ x b x. Fg. 3 b A d f f w y f x h b g h h g f g u 3 h v u 5 _ x b x I.

20 6 I d h v u g g d b y h c d w y hv d f h g f R:. R q f 0 h w. + H " q u " y d c u ch h ch g ( ) d q u (k ) c h g ( k ) f < k. q Th g h w h h v f c c d h c d g c w b u R ' h w b d R d O b v u y d h Q v ' f c f g Q h u g h c wh u. d f h g f R I h c h v b ch ck d. Q ' c R Th f c c h h v f c f V I. A w h v c 4 f y h d y A h A w f g + C q u y h g f AR " C c ( q u h d f W : h v f c f V I f c. + w h A : Th cd cc V4 wh w w hv v f y ( k g h C h d u ) g T R f ) : w f h q R u 0 ( R d : L O ch g ( ) I R ( : f q fu g R I ++. h. cqu ) c b w ). R 0 ) + : ( c h g ( ) I R Q Q Q Q3 + d S c R 0 f I q uch h 3 q wh R h b 0 x: c f + d q w h v f L 2 '. F h R d f f Q w h v Q 4 Q c ( 0 R d : I g I

21 7 O c h ) I I 0. ). U g h d w y w h cqu ( c h g ( ) I R ( c h g ( ) I Q I c ) c h + I g ( c) h Q u h f h b v u. W h f c c 4 h) c wh c h g ( ) d h v h w + R N. I I u f f c + b h h x + c f v u x u c h h + R + x + Q 0 Q c h c w f d R + ( x R R Q + Th cd x + q u v v g x u g g h x h u d b k g b ). F h f q u w h w c k f x h + u f h R R + R (P ) d wh ch h v " d d " IR+ ( + x + QX + R X R x (P ) x R _

22 8 by h f R R. Th u v v f c R kd d d d ). T h b y h d f f R h _ P k Tkg k g h h h v g f x d f c R z z c w P b k 0 P R P " k 3 (P ) x 4 ) ( X x R R x x R k _ T z ; d 0 x ( R z x x ) z P? _ k ) x )z zx ( c R I h QI P) x R x z d z _ R I gf c h g. (z x ) Q z Q ( h z z. z A ' x Q Cz x x z ) Q x Q f x z Q zq x q x R ( P ) u Q( z z R X / X z x Q P k.2 ( ) ( Q. k _ S y zf q u (z ) wh c h c h g ( w f d _ k. x x Q z x x Q z ) x k z ( Q g I z x Q c _ z S f c z hi g (z ) q u (z ) x + Q Q. x Sc x h u f h x z ' w h v I Q U ) x Q h z I

23 9 x + x ) Th c h v f c f h g h u d h z d u c d b y h w d f f R. 9. FURTHR OPTIMIZATIONS Ah q u z c b b d f w c d g u h bw g d c g h. L d c "yb g h " d "yb c g h " b d f d w h h : f yb g h ( ) h f k > ; c h g ( k f yb ) c g h () h ch g (k f k < Tw d c h w h d c d + y y b d f h k w h u chg h v u f ch g ( + ) I yb g h ( + ). Th c b y v d f w : b Tk y f h k. F w hw h h h cw h c h g d wh c h h + k d. F u h b h k d. O b v u y u k g b h g v b. I f h h h d y k d y k h h u ch g f h c c k b d b y d g k d h h d. L h ch g h b d. F h f h d c g v b v f w h < k < B u h u c d x c y h d h v u c ( f g u 3 d 3 b ) d h b d b y h (

24 20 f g h k f k w h c h v u g d. Nw h b v u c d c bw wh ch h + k d d f h k b d by g d + d h c d c v h ub f kd h. F h h c f w d y. A f g h d + w h wh c h w y d c y bc d y b y b d g h. By g h c w b " x c " f h k wh ch h g g h h b f d. T h x c c b d d by c k g h : ck : y; f d f ( y f c h g ( ( ) ) A I yb c g h ( ( )) h () u h ( ) f y b d. A h d h c k c h x c. F h w h g g v c c f : h v v g u ( k 0 : ) v u ( c k + k : ). Th c f h z f y u b c g h x c h g g h bc b v u f h c f "yb g " d "yb c g " h d. I h c c c k h h y x c h h wh g c k. A f z f u d b y b v g h ( ) )

25 2 A d d f h ( ) y b w f yb c g h ( ) h A d d f h ( ) A d d f h ( ) f wh h h c f g d v d c 6 y b g y f d u g v v d f bu f c f 2. F h c c h g ( ) c h d h g R c b f d R :. Sc h h v u f T g u d h g f T c b d. F c g u h c f h h d h d f g b v u b b v w w hv b d ( w g g g yb ): : : ( 0 ) ; f yb c g h ( ) h V k : 2 : T k : T : f f b y d R I : ( c h g ( ) I I I ' ) d V k : : T k : T k + R k x T V k 0 : 2 : : c f. Ud h d cu d c 8 h bv c. 0. APPLICATION TO ALGOL 68_ Th h z d c u c f ALGOL 6 8 : BIBLIOTHK AATH7/AT:S C 7 : ' T R

26 22 bg.. d f h... f... h {.} f ( ) c.... {u u. c ( I { I : I.} {I..} ) $ $ { f.} {f.} {by.} {...} {wh...} d H f... d {... } b y ub f. A f c c h d u b u c g d g f u ch h h. I y vc b y c d g h f d f f h c g f w d b y g. Ah b f h u f yb w h w g (... ) cc f h h ucu d d c f h h c u g h. A h c d u d h d f f Q d h cud f h f T u ch wy h h c d g c c h w b c u c d. u Th f c h c c u b h g d c g h b c h g g h f c y b. d A w h u ch c. I y h w $ ' d y c c ch h d g h x c ( c 9 ). H w v c c b d d f h d c x f g v $ g c g h. y Th f c c b x d b y x : w ) y w h + c v d d ; + g H d d ; b) u w h + g v d d ; + c w d d. I + c v q u y c u wh w v d d d c d g w h c d h h u c u h u g g d b y ( w h d d ) ( d d ). I b v h v d d c d c d h h u c u c d ( w h (d d ) d d ). S h q u c h g ( ) f h h dd h c x f h. F u y h d c c b k h bd f h w c k

27 23 wh T f b y w h d 7 g h y f d y f f h k 9 b g f h f c u u w h d f II b d TA G k v d k d A f h k c h f ) c fx. I 2 h " h " w u d f w d c f yb wh ch c d " # c c d g. Th g h d h u h h k f h " w c h " h d y b d. A f f c g h g v 3. I b d h g h u ch w y h c d w h h w d c f h b u h w u d u g f f f c c y c h g h x c f c 9 b c uch g.. SPD OF TH PROCSS F h g h h u d b b v u h h k b y h c h u bud f h f 0 ( 3 ). W h v b b d v h bud. H w v w h v d d c f u x g h dd f c u k b y f h g h w ALPH d u CYBR 7 3. Th k w b d b y g g w h d c c c k d d g h w h b b y c. I 84 k w g d w h v y g f d c f 0. d f u f h f c u. A gd f w b d b y P c. q + c w f d A y c 4 b v d f 000 d c c. h

28 24 RFRNCS H A R C. A. R. A x c b f c u g g C. ACM 2 ( ) MRTNS L. G. L. T. & J. C. VAN VLIT R g h h k f ALGOL 6 8 g R I W 2 / 7 3 M h c C Ad ( ). 3 MRTNS L. G. L. T. & J. C. VAN VLIT R g h w c h k f ALGOL 6 8 g R I C Ad ( ). 5 / 7 4 M h c ON' AMGN 3 4

Book of Plans. Application for Development Consent. Thames Tideway Tunnel Thames Water Utilities Limited. Application Reference Number: WWO10001

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