BiGUL’sBidirectionality

ZhenjiangHu&JoshKoNationalInstituteofInformatics,Japan

OxfordSummerSchoolonBidirectionalTransformations 27July2016

Plan• Reviewwell-behavedness,withpartialityexplicitlyrepresented.

• Getatasteofputback-baseddesignbylookingatseveralBiGULconstructs.

• Startwithanaturalputsemantics,andthenreviseitsothatacorrespondinggetexists.

(Well-behaved)Lensput::s->v->Maybesget::s->Maybev

PutGet

GetPut

UniquenessofgetForanylenseslandr,putl=putr==>getl=getr

Proof

Replace::BiGULvvputReplacesv=Justv

Skip::BiGULsv(?)putSkipsv=Justs

Skip::(s->v)->BiGULsvput(Skipf)sv=ifv==fsthenJustselseNothing

Prod::BiGULslvl->BiGULsrvr->BiGUL(sl,sr)(vl,vr)

put(l`Prod`r)(sl,sr)(vl,vr)=dosl’<-putlslvlsr’<-putrsrvrreturn(sl’,sr’)

get(l`Prod`r)(sl,sr)=dovl<-getlslvr<-getrsrreturn(vl,vr)

PutGetforProdAssumeput(l`Prod`r)(sa,sb)(va,vb)=Just(sa’,sb’)

Proveget(l`Prod`r)(sa’,sb’)=(va,vb)

(dosl’<-putlsavasr’<-putrsbvbreturn(sl’,sr’))=Just(sa’,sb’)

(dovl<-getlsa’vr<-getrsb’return(vl,vr))=Just(va,vb)

PutGetforProd(dosl’<-putlsavasr’<-putrsbvbreturn(sl’,sr’))=Just(sa’,sb’)

∃sl’.putlsava=Justsl’∧∃sr’.putrsbvb=Justsr’∧Just(sl’,sr’)=Just(sa’,sb’)

putlsava=Justsa’∧putrsbvb=Justsb’

PutGetforProdAssumeputlsava=Justsa’putrsbvb=Justsb’

Provegetlsa’=Justvagetrsb’=Justvb

Case(binary)

put(Case(pl,l)(pr,r))sv=ifplsvthenputlsvelseifprsvthenputrsvelseNothing

typeCaseBranchsv=(s→v→Bool,BiGULsv)

Case::CaseBranchsv->CaseBranchsv->BiGULsv

Case(withexitconditionsandadaptation)typeCaseBranchsv=(s→v→Bool,BiGULsv,s->Bool)

typeCaseAdaptiveBranchsv=(s->v->Bool,s->v->s)

Case::CaseBranchsv->CaseBranchsv->CaseAdaptiveBranchsv->BiGULsv

Rearrangement• $(rearrV[|\(Left(x,y))->(y,x)|])…

• Key:patternmatchingandexpressionevaluationareinvertible

• Type-safeimplementationwithHaskell’sgeneralisedalgebraicdatatypes