Towards Formalizations in Case-Based Reasoning for Synthesis


  

 "!#$&%')(#$&% +*,%$&
- /.102
3-4 5% 76 8:9@? FHGHIKJML)NPORQTSUOWVYXDZNUQY[]\_^_A `aBDNUCbc^HE$ & '@?BA;C#D"@EGFC@H;IJ'KC#D'KLKMEONQP5I5R M'KS

TVUXWZY20K-\[^]_.`a0cb@UXWGdJWe-/,*f,^g5g*eW
hi,J.jWkbl-1Jblm

,We-/n/We-/1ol(.UXW
,*

[o@We*p1Wq-rd 1J0s(+WGdJ-ut
vwbY20K(.x, 1.Vy z zl{i|2n/,-/1c}
b@Ufgo@Uxg~bl(0K(€bl} (XWemg*eW
h+W.G]%,2-/1wd50K-0KU^,*eW.G]c‚0V‚;We*q*/o@(K0V,ƒ.‚;b@„X1 Wem 0K-(+Wb@-/,2* 1b@m

,2We-…bl}w(+Wem†g*0‡dr0iblm [email protected]+W
h%ˆd/o@U0s(‰,(0KŠ50Kmg*eWeˆ021)We-

‹ŒWGdŽy-‘.`5We(‰mcW
h+UKb@„X1Jb@m‰,2Wq-’1 We“Ž0sU0K-/.>(.x,J.p0s(hb@UXU0K„ (xg~bl-/1”.
b•,UXUi,2-dr0Km‰0K-/.(cbl}%b5[K–0—h.(tˆrdŽo@UK0K(c}pblUc(X`[email protected]| *eWe˜J0™h+WeUih+*0K(^nr0K*e*eWGg(0s(in5,2-/1w(iš5o,2U0K(f0@.h;›V*e*rb[K–02h.j(B,UK0B,2U+„ U^,-d5021ƒbl-,-/,J.Xo@Ui,2*@dU+W
1œ‹ab@*e*b‚;We-dƒ.`a0f,UXUb‚;(œWe-€‹ŒWGdŽ y

02,h`(.,.p0tjˆdŽo@UK0X|gUKbY2Wp10K(B,->WemgUKbY0Km‰0K-/.tjU02,2U+U^,-d„ We-db5[K–202h.(^nWe-/.UKb1oh+We-d‰-0@‚

b5[K–0—h.(^n [email protected] |

bl}ƒ.j`0%bl-5„

WeUih+*05ˆd/o@U0s„[ .‚;b>hb—gW0K( b@}  ˆdŽo@UK0K„X,  ,UXUi,-rdr021‰We-
 1 WeU02h.jWb@-;ˆdŽolU0K„Xh b@-0 hKb2g] bl}  ˆd/o@U0K„K[  ,UXUi,-rdr021)Wq- 1 WeU02h.jWb@-fˆdŽo@UK0K„X1 hbY0KU  ˆdŽo@UK0K„Xh  ˆd/o@U0s„Z0 .‚;b hb2gW0K(>b@}  ˆdŽo@UK0K„X1  ,U+U^,-d5021 We-‘  1 WeU02h2.Wb@-œ

d5b@We-d%b@-0—nr0@ d/ nˆdŽo@UK0K„X,

a

b

c

d

e ...

}Ublm

PO

m

4~

02š5o@WY2,*0K-/.n;,[@(.jUi, h2.

¼



,^g5gWe-d,dJWY0s-cU0gU0K„

b@} ,-)o@-5WY0KUX(0b@} 1 We(ihbo@U+(K0€b@-/.
b,2-b5.j`0KUB*b2d„

,*eW 021%[^] }Xo@-/h.jWkbl-u(x]rm

[b@*V,J[l(.U^, h.jWb@-5(

‚;`5W
h`

.j`0}+ol-/h.Wbl-c-/,2m 02nblUwh`/,2-dr0€bl} ‹ŒWGdŽ .jWb@-

{

€

,-/1

‹ŒWGdŽ

,*e*b‚

hb@*e*
,ig(K0cbl}

.j`0‡,U+W.G]5

PO

gUbY2W
1J0\0KŠa,2mg*0K(•b@}‰d50K-0KU^,*eW @,„

,2-/1‰,J[@(2.U^, h.jWkbl-o@(XWe-d.`0V(XW d-/,J.Xo@U0‡,2-/1 Y2,2U+W
,J[@*0K(

.`0

b@}f.`0DˆdŽo@UK0K(

inverse substitutions

copy(X,z,w)

-1 θ 2 = { 2|z, Square|w }

-1 θ 1 = { 3|z, Circle|w}

copy(X,2,Square)

copy(X,3,Circle)

~´OWV2ÙD\_Nñ q„1…mq…cŽj&$†2ŽDŒ$‡1 åkˆ‰1Š ^ß k w ñ w Õ‹ åkŒC ã  åkˆ‰1Š ßjk w y w [ Ñ á1Žm‹  ã

 .x,˜J0 b@-0  1 WeU0—h+„
 .jWb@ ŒU0XgaU0K(K0K-/.( .x,˜0 bl-08Žšror,U0—n hKb2g]cW. .‚;bc.jWem 0K(B,2-/1c,2U+U^,-d50,*e*aWe-  1 WeU02h.jWb@-  œ›Qg„ ga* ]5We-d.`a0)Wq-/Y20KU+(K0(o[@(.jWZ.Xo5.Wbl-5( ( + × ,-/1 ( Ü + × hb@UXU0K„ × (xg~bl-/1 We-d* ]cUK0K(ol*Z.j(wWe-=DGFHF]o åkˆ‰Š ß^k w ² w ³ ãi -‹ŒW d/@{

hb@UXU0K(xg~bl-/1 (.pb

WeU^h+*02n;hb2g]

W.B.`5UK0i0c.jWem 0K(,2-/1

,UXUi,2-dr0>,2*e*œWe-

inverse substitutions

f(copy(X,2,Circle)) -1 θ4 = {divide | f}

-1 θ 3 = {cover | f} cover(copy(X,2Circle))

k

! "#%$& '@?

We-)Wq-~1oh.jWZY20€We-5}
0KU0K-~h0

^|

t ‰W.h`0K*e*;[email protected],*By zz s|—

~Y0KUX(+Wb@- y Ð Ü We(œb5[Y2Wb5o@(X* ](+Wemga*k0"ƒ ˜ Á  Ù o"- y Ð × o y Ð Ü wy Ð × o y Ð × wy Ð Ü o y Ð Ü 0  ›V(w,2-c0KŠa,mga*k0 }pblU;go@UK0fdr0^b@m 0l.U+W
hi,2*œ.U^,-5(X}pb@UXm ,.Wb@-c‚;0 m ,K] We-5(2.,-/.jWp,.p0>.`Wq(B(^h`0Km 0>[^]cdUbo@-/1 .p0KUXmc(ƒ y Ð × o kå ˆ‰1Š ßjk w y w Õ ‹ kå ŒC ã w,y Ð Ü o kå ˆ‰Š ß w y w Õ ‹ kå ŒC ãi [^]We-/hb@Uxg~blUi,.We-d

E - equational knowledge E

f(copy(X,2,Circle))

f(copy(Y,2,Circle)) -1

θ4

-1

θ3

copy(Y,1,(copy(X,2,Circle)))

inverse substitutions

divide(copy(Y,2,Circle))

~.pb>We-/hblUgœb@Ui,.p0Qdr0s-0KUi,2*eW l,J.Wbl-/n/,J[@(2.U^, h.jWkbl-/n

copy(Y,z,w)

θ2-1

inverse substitutions

θ2-1

divide(copy(Y,2,Square))

,2-/1>dr0iblm [email protected]+W
hi,2*.jUi,-(+}pblU+m

θ1-1

f(copy(Y,z,w))

~ o  A >30 >1 €D$ & '@? C&ORL+E CF€LKMk"F2SHZ'2M"F