Closed and Open Recursion RALF HINZE Introduction Recursive functions Recursive objects - PDF document

ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Trinity I functionalcore, I imperativecore, I object-orientedcore, I [modulesystem]. 2/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Dynamicsemantics (fun(x1: 1 ))e)+(fun(x1: 1 ))e)e2+(fun(x1: 1 ))e)letvalx1=e1ineend+ e2e1+ 5/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Recursivefunctions recfun(n: Nat ))ifn 0then1elseself(n�1)n +selfreferstothefunctionitself. 6/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Dynamicsemantics (recfun(x1: 1 ))e)+(fun(x1: 1 ))e)fself7!recfun(x1: 1 ))eg+Therecursiveknotistiedattheearliestpossiblepointintime:theintroductionform. 7/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Objects - object M end - valuesobjectendintroductionobjectmendeliminatione:x 8/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Openfunctions valfac-base=funopen(0: Nat ))1valfac-step=recfunopen(n+1: Nat ))selfn(n+1)valfactorial=fac-baseorfac-step +Anopenfunctionisapartialfunction.+Thecombinatororcombinestwopartialfunctions. 15/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Abstractsyntax e::=


openfunctions: jfunopenrnon-recursiveopenfunctionjrecfunopenrrecursiveopenfunctionje1ore2alternationr::=emptyrulejp)esinglerulejr1jr2sequencesofrules 16/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Dynamicsemantics (recfunopenr)+(recfunopenr)e1+(recfunopenr1)e2+(recfunopenr2) e1ore2+recfunopen(r1jr2)e2+(recfunopenr)casee1ofrfself7!recfunopenrgend+ e2e1+ 17/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Application|genericprogramming valsum-nat=funopen h Nat i )funx)xvalsum-pair=recsumfunopen h ( a1 ; a2 ) i )fun(x1;x2))sum h a1 i x1+sum h a2 i x2 +Insteadofacasewehaveatypecase. 18/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Conclusion I Openfunctionsareusefulinconjunctionwithopendatatypes. I Vision:replacementforoverloadingandtypeclasses. I Openrecursivemodules? 19/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Objects|abstractsyntax m Method methoddeclaration m::=emptydeclarationjmethodx=emethoddenitionjm1m2sequenceofdeclarationsjlocaldinmendlocaldeclaration 21/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Objects|dynamicsemantics + (methodx=e)+fx7!egm1+1m2+2 m1m2+1;2d+m+ localdinmend+ 22/23 ClosedandOpenRecursion RALFHINZE Introduction Recursivefunctions Recursiveobjects Recursivefunctionsrevisited Conclusion Appendix Finitemaps WhenXandYaresetsX!nYdenotesthesetof nitemaps fromXtoY.Thedomainofanitemap'isdenoteddom'. I the singletonmap iswrittenfx7!yg I domfx7!yg=fxg I fx7!yg(x)=y I when'1and'2arenitemapsthemap'1;'2called'1 extended by'2isthenitemapwith I dom('1;'2)=dom'1[dom'2 I ('1;'2)(x)='2(x)ifx2dom'2'1(x)otherwise 23/23