STANFORD DONALDE,KNUTHProfessorEmeritusof

UNIVERSITY TheArtofComputerPrograsmning

ComputerScience

353 SaraMall~, StanfordCA 94305-9045PHONE (650)7234367

24 April 2009Aiison 3 BrimelowPresident,EuropeanPatentOffice80298Munich, Germany

DearMs Brimelow,

A Mend in Europejust told me that you are interestedin “amicuscurie” lettersto explainwhy somanycomputerscientistsaroundthe world havelong beenalarmedaboutpatenttrends,andthat you hopeto receivethemby 30 April. I hopethis letter reachesyou intime; I could not sendit by FedEx,having no completeaddress.

Enclosedis a copyof a letter that I wrote to the US Patent Commissionerin 1994; 1 believeit is selfexplanatory- Also enclosedis the transcriptof a talk I gaveat the TechnicalUniver-sity of Munich in 2001,whereI gavea somewhatmorenuancedview of extremelyunusualcasesin which algorithmsor evenmathematicalconstantsmight conceivablybe patentablein my view. IThe latter remarksoccur nearthe endof a rather longlecture; I havehigh-lighted the relevantinformation,on page324, for your conveniencel

Basically1 remainconvincedthat the patentpolicy most fair andmost suitablefor the worldwill regardmathematicalideas(suchasalgorithms) to benot subjectto proprietarypatentrights. For example,it would be terrible if somebodywere to havea patenton an integer,like say1009, sothat nobody would beableto usethat number “with further technicalef-fect” without paying for a license. Althoughmany softwarepatentshaveunfortunatelyal-readybeengrantedin the past,I hopethat this practicewill notcontinuein future. If Eu-ropeleadsthe way in this,I expectmany Americanswould want to emigratesothat theycould continueto innovatein peac&

Sincerely,

DonaldE KnuthProfessorof The Art of ComputerProgramming

- . STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305-2140

DONALD E. KNUTHProfessorEmeritus of The Art of

Computer ProgrammingDepartmentof Computer ScienceThlephone [4151723-4367

February23, 1994

Commissionerof Patentsand1~ademarksBox4Patentand T1’ademarkOfficeWashington,DC 20231

Dear Commissioner:

Along with many other computer scientists, I would like to askyou to reconsider the currentpolicy of giving patents fot computational processes.I find a considerableanxietythroughoutthe communityof practicing computer scientiststhat decisionsby the patent courtsand thePatent and ‘ftademarkOffice aremakinglife muchmore difficult for programmers.

In the period 1945—1980,it wasgenerally believed that patent law did not pertain to software.However, it now appears that somepeople have receivedpatents for algorithms of practicalimportance—e.g.,Lempel-Ziv compressionand lISA public key encryption—andare nowlegallypreventing other programmersfrom usingthosealgorithms.

This is a serious changefrom the previous policy under which thecomputer revolution becamepossible,and I fear this changewill beharmful for society. It certainly would have had pro-foundly negativeeffect on my own work: Forexample,I developedsoftwarecalled‘I~Xthat isnow usedto produce more than 90% of all books andjournalsin mathematicsandphysics andto produce hundredsof thousandsof technicalreportsin all scientific disciplines. If softwarepatents had beencommonplacein 1980, I would not have beenable to create such a system,nor would I probably have ever thought of doing it, nor can I imagine anyoneelsedoingso.

I am told that the courtsaretrying to make a distinction betweenmathematical algorithmsand nonmathematical algorithms. Tb a computer scientist, this makesno sense,becauseev-ery algorithm is asmathematicalas anything could be. An algorithm is an abstractconceptunrelatedto physical laws of the universe.

Nor is it possibleto distinguish between“numerical” and “nonnumerical” algorithms, as ifnumbersweresomehowdifferent from other kinds of preciseinformation. All dataarenumbers,andall numbers are data. Mathematicians work much more with symbolicentities than withnumbers.

To Commissionerof Patentsand T1’ademarks February23, 1994 Page2

Thereforetheideaof passinglawsthat saysomekindsof algorithmsbelongto mathematicsandsomedo not strikesme asabsurdasthe19thcenturyattemptsof the Indianalegislatureto passa law that theratio of a circle’s circumferenceto its diameteris exactly 3, not approximately3.1416. It’s like the medievalchurchruling that thesunrevolvesabout theearth. Man-madelaws canbesignificantly helpfulbut not whenthey contradictfundamentaltruths.

Congresswisely decidedlong ago that mathematicalthingscannotbepatented.Surelynobodycould apply mathematicsif it were necessaryto pay a license fee wheneverthe theoremofPythagorasis employed.The basicalgorithmicideasthat peoplearenowrushing to patentaresofundamental,theresultthreatensto be like whatwould happenif we allowedauthorsto havepatentson individual words and concepts.Novelistsor journalistswould be unableto writestoriesunlesstheir publishershad permissionfrom theownersof the words. Algorithms areexactlyasbasicto softwareaswordsareto writers,becausetheyarethefundamentalbuildingblocks neededto makeinterestingproducts. What would happenif individual lawyerscouldpatenttheir methodsof defense,or if SupremeCourtjusticescouldpatenttheir precedents?

I realizethat thepatentcourtstry their bestto servesocietywhentheyformulatepatentlaw.The PatentOffice has fulfilled this missionadmirablywith respectto aspectsof technologythat involve concretelaws of physicsratherthan abstractlaws of thought. I myself haveafew patentson hardwaredevices. But I strongly believe that the recent trend to patentingalgorithmsis of benefit only to a very small numberof attorneysand inventors, while it isseriouslyharmfulto thevast majority of peoplewho want to do usefulthingswith computers.

WhenI think of thecomputerprogramsI requiredaily to getmy own work done,I cannothelpbut realizethat noneof them would exist today if softwarepatentshadbeenprevalentin the1960sand 1970s.Changingtherulesnow will havetheeffect of freezingprogressat essentiallyits current level. If presenttrendscontinue, the only recourseavailable to the majority ofAmerica’sbrilliant softwaredeveloperswill be to give up softwareor to emigrate.TheU.S.A.will soonloseits dominantposition.

Pleasedo whatyou canto reversethis alarmingtrend. Therearefar betterways to protectthe intellectual property rights of softwaredevelopersthan to take away their right to usefundamentalbuilding blocks.

Sincerely,

Donald K KnuthProfessor

All QuestionsAnsweredDonaldKnuth

On October5, 2001,at the TechnischeUniversitat (1979), theAdeiskOldMedalfromtheRoyalSwedishMünchen1DonaldKnuth presentedalectureentztled AcademyofSciences(1994),the HarveyPrize from“All QuestionsAnswered”.The lecturedrewan au- the TechnionofIsrael (1995),theJohn vonNeumanndienceof around 350people.This article contains Medalfrom theInstitute ofElectricalandElectron-the textof the lecture, editedby Notices senior ics Engineers(1995),and the Kyoto Prize from thewriterand deputyeditorAllyn Jackson. Inamori Foundation (1996).Since 1968Knuth has

Originally trainedasa mathematician,Donald beenon the facultyof Stanford University, whereKnuth is renownedfor his researchin computersci- he currentlyholdsthe title ofProfessorEmeritusofence,especiallythe analysisofalgorithms. He is a TheArt of ComputerProgramming.prolific author, with 160 entries in MathSciNet. —AllynJacksonAmong his manybooksis the threevolurneseriesTheArt of ComputerProgrammingITAOCPL for Knuth: In everyclassthat I taught at Stanford,which he receivedthe AMSSteelePrize for E.xposi- the last day was devotedto “all questionsan-Lion in 1986.Thecitation for the prizestatedthat swered”.The studentsdidn’t haveto come to classTAOCP “has madeasgreat a contribution to the if they didn’t want to, but If they did, they couldteachingofmathematicsfor thepresentgeneration askanyquestionon anysubjectexceptreligion orof studentsasanybook in mathematicsproper in politics or the final exam. I got the idea fromrecentdecad~~”The long awaitedfourth volumeis RichardFeyriman,who did the samething in hisin preparationandsomepartsareavailablethrough classesatCaltech,andit wasalwaysinterestingtoKnuth ‘S websire, http //www-cs -faculty. seewhat thestudentsreallywantedtoknow.Todaystanford. edu/-knuth/. I’ll answer any questionon any subject.Do we

Knuth is the creatorof the TEX and METAFONT allow religion orpolitics?I don’t know. But therelanguagesfor computertypesetting,which have is rio final examto worry about. nil try to answerrevolutionizedthe preparation and distribution of without takingtoomuchtime sothatwecangetatechnicaldocumentsin manyfields, including math- lot of questionsin.ematics.In 1978hepresentedtheAMSGibbsLecture So, whowantsto askthefirst question?...Well,entitled “Mathematical Typography”. The lecture if there areno questions...(Knuth makesasif towassubsequentlypublishedin the Bulletin of the leave.)AMS [MTJ. Question.’Therewasaspecialreportto theArner-

Knuth earnedhis Ph.D. in mathematicsin 1963 ican president,the PJTAC report (PJTACL contain-from the California Instituteof Technologyunder ing somerecommendations.I am wonderingthe directionof Marshall Hall. He hasreceivedthe whetheryouwouldbewilling tocommenton thepri-Turing Awardfrom theAssociation(or Computing orities outlined in theserecommendations:Machinery (1974), the National Medal of Science bettersoftwareengineering, building a teraflop

318 NoncEsor iiir AMS VOLUME 49, Nui~s~3

computer,improvementsin theInternetincludinghighersecurityand higherbandwidth, and thesocio-economicimpactsofmanaginginformationavailablevia computernetworks.

Knuth I think that’sa brilliant solutionof theproblemof what to presentto apresident.But infactwhatI would like to seeis thousandsof com-puterscientistslet looseto do whatevertheywant.That’swhatreally advancesthefield. Frommy ex-periencewriting TheArtofComputerProgramming,If you askedme any yearwhat was the most im-portantthing thathappenedincomputersciencethatyear,I probablywouldhavenoan-swerfor thequestion,butoverfive years’timethewholefieldchanges.Computerscienceisa tremendouscollaborationof peoplefrom all over theworldaddinglittle brickstoamassivewall. The IndividualbricksarewhatmakeIt work,andnot themilestones.

Next question?Questio,vMathematicians

saythatGodhasthe “BookofProofs”, whereall the mostaestheticproofs are written.Can you recommendsomealgorithmsfor the “BookofAl-gorithms”?

KnuthThat’saniceques-tion. It was Paul ErdOswhopromulgatedthe Ideathat Godhasabook con-tainingthebestmathematicalproofs,andI guessmy friend GünterZiegler in Berlin hasrecentlywritten aboutIt [PPM.

I rememberthat mathematicianswere tellingmein the 1960sthat theywould recognizecom-puterscienceasamaturedisciplinewhenit had1,000deepalgorithms.I think we’ve probablyreached500.Therearecertainlylotsof algorithmsthatI thinkhavetobeconsideredabsolutelybeau-tiful andimmortal, in somesense.Two examplesaretheEuclidalgorithmandacorrespondingonethatworks in binary notationandthatmayhavebeendevelopedindependentlyin China,almostasearlyasEudid’salgorithmwasinventedinGreece.Inmy booksI ammostlyconcernedwith thealgo-rithmsthatareclassicalandthathavebeenaroundfor a longtime.But still, everyyearwefindbrandnewideasthatI thinkaregoingto remainforever.

QuestiotvDo you have thoughts on quantumcomputing?

Knutfr Yes,butI don’t knowagreatdealaboutit. It’s quiteadifferentparadigmfromwhatI’m usedto. It haslotsof thingsin commonwith thekindof computingI know,but it’s alsoquitemysteriousin that youhaveto getall the answersatthe end;

youdon’tmakeatestandthenhavethatdeterminewhatyoudo next.A lotof youhaveseenthemovieLola rennt(calledRunLolaRunin theU.S.),in whichtheplot is playedout threedifferenttimes,with theoutcometakingthreedifferentbranches.Quantumcomputingis somethinglike that: Theworldgoesinto manydifferentbranches,andwe’reinterestedin theonewheretheoutcomeis thenicest.

I’m good atnonquantumcomputingmyself,soit’s quitepossiblethatif quantumcomputingtakesover, I won’t beableto do the newstuff My life’s

work is with computersnotbecauseI’m interestedsomuchin computation,but be- ~causeI happento begood at ~thiskind of computing.For-innatelyfor me,I foundthatthethinglcoulddowellwas ,

interestingto otherpeople.Ididn’t developan ability to tthink aboutalgorithmsbe- ~ acauseI wantedto helppeople.~ ~

solveproblems.Somehow,by ~the time I wasa teenager,I Phadapeculiarway of think- —!ingthatmademegoodatpro-~ agramming.ButI mightnotbegoodat quantumprogram-ming. It seemstobeadiffer-entworld frommy own. ~

I’ll take a questionfrom Z

theback uQuestioirlam working in

theoremproving,andoneofthe mostimportantpa-pen is yourpaper“Simple word problemsin uni-versalalgebra” (KB] from 1970,written withP.B. BendkI havetwoquestions.Thefirstis, doyoustill follow thisareaandwhatdoyouthink ofit?Andthesecondis~whois andwhatbecameofP. B. Ben-dix?

KnuthiThiswork waspublishedin 1970,but Iactuallydid it in 1967while I was at Caltech.Itwasasimple idea,but fortunatelyif s turnedoutto be verywidelyapplicable.The ideais to takeaset of mathematicalaxiomsandfind all theimplicationsof thoseaxioms.If I haveacertainset of axiomsandyou havea possibletheorem,you ask,doesthis theoremfollow from thoseaxiomsor not?I calledmy paper“Simple wordproblemsin universalalgebra”,and I saidaproblemwas “simple” if my methodcouldhandleit. Now peoplehaveextendedthemethodquite a lot, so that a lot moreproblemsare“simple”. I thinktheir workis beautifuL

The year1967 was the mostdramaticyearofthy life by far. I hadno time for research.I hadtwo children lessthantwo yearsold; I hadbeenscheduledto be a lecturerfor ACM (Associationfor ComputingMachinery)for threeweeks;I had

MARcH 2002 Ncrricrsori~n~AMS 319

to give lecturesIn aNATO summerschoolin Copenhagen;I hadtospeakIn aconferenceatOxford; andso on. AndI wasgettingthe pageproofs for The Art ofComputer Program-ming, of which the firstvolume was beingpublished in 1968. Allof t’ was in addition

c1as~esI wasan attack

atput me inic hosi , andbeing

an editor for twelvejournals.That year I

- thoughtof two littleideas.Onehasbecome

knownastheKnuth-Bendixalgorithm; theotheroneis known asattributegrammars(AGJ. Thatwasthe mostcreativeyearof my life, andit wasalsothemosthectic.

YouaskedaboutPeterBendix.Hewasa sopho-more in a class I taughtat Caltech~“Introductionto Algebra”. Everystudentwassupposedto do aclassproject,andPeterdid histermpaperon theimplementationofthealgorithm.Hewasaphysicsmajor.Thiswasthetimeof theVietnamWar,andbe becamean objector. He went to Canadaandworkedasa high school teacherfor about fiveyearsandlatergotadegreein physics.I foundhewasliving nearStanforda coupleofyearsago,soI calledhim up andfoundout thathe hasbadafairly happylife in recentyears.

In the1960s,If! wroteajoint paperwithmyad-visorMarshallHall, it meantthathedidthetheoryandI did theprogramming.But If I wroteapaperwith anybodyelse,it meantthat I did thetheoryandtheotherpersondid the programming.So PeteBendixwasagoodprogrammerwho implementedthemethod,

Question It seems to me it’s easier to revise abook than the huge software programs we see dayto day. How can we apply theory to improve soft-ware?

KnuthiCertainlyerrorsin softwarearemore dif-ficult to fix than errorsin book& In fact,my mainconclusionafterspendingtenyearsofmy life work-ing on theT~Cprojectis that softwareis hard.It’sharderthananythingelseI’ve everhadtodo.WhileI wasworkingon the TJ??C program,I wasunabletodo full-time teaching.AlthoughI loveteaching,Ihadtotakeayearoff fromit becausetherewasjusttoomuchtokeepinmy headatonetime.Writing abookis alittle moredifficult thanwritingatechni-calpaper,butwritingsoftwareisalotmoredifficultthanwilting abook.

In mybooksI offer rewardsfor the first personwhofinds anyparticularerror,and I mustsaythatI’ve written more checks to people In Germanythanin anyother cotmiryIn theworld. I getlettersfromall over,but my German readersarethebestnltplckersthatFveeverhad!In softwareI similarlypayforerrorsin theT~CandMETAFONTprograms.Therewardwas doublingeveryyear:It startedoutat $2.56,then it went to $5.12,and soon,until itreached$327.68,at which time1 stoppeddou-bling.There hasbeennoerrorreportedInT~(since1994or 1995,although thereIs arwnor thatsome-body hasrecentlyfound one.Fmgoingto have tolook at It again in ayear or two.!do everything Inbatchmode,by the way.I am goingto look againat possibleerrors in TI~Cin, say, theyear 2003.

I think letting usersknow thatyou welcomere-ports of errorsIs oneImportant techniquethatcould be usedin the software industry. I thinkMicrosoft should say,“You’ll getacheckfrom BillGateseverytimeyou find an error?

Qpestio,vWhatimportancedoyougivetothede-signofefficientalgorithms,andwhatemphasisdoyousuggestgiving this areain the future?

Knut Ithink thedesignofefficientalgorithmsis somehowthe coreof computerscience.It’s atthe center of ourfield. Computersareincrediblyfast now comparedto what they werebefore, sofor manyproblems thereis no needto havean ef-ficient algorithm.I canwrite programs that areinsomesenseextremely Inefficient, but if It’s onlygoingto take onesecondto get the answer,whocares?Still, somethingswehave to domillions orbillions of times, andjust knowingthat the num-ber of timesisfinite doesn’t tell usthatwecanhan-dle it. Sothenumber of problems that are In needof efficient algorithmsishuge.For example,manyproblems are NP complete, and NP complete isjust a small level of complexity.Therefore I seeanalmost Infinite horizon for the needfor efficientalgorithms. And that makesme happy becausethoseare the kinds of problemsI like thebest

320 NoncrsOF ThE MIS VOLUME 49,Nlmtha3

Question:You havea big interest in puzzles,in- Question: Youcluding the “Tower ofHanoi” puzzleon more than spent a lot of time on3 pegs. I won’t ask a harder question—what is the computer typesetting.shoiiestsolution7—because lam not sure everyone What are your reflec-knowsthis puzzle.But I will ask a more philosoph- tions on the impactofical question: Is it possible oshow this canneverbe this work?solved2 Knuth: I am ex-

Knuth: Do peopleknowthe “Tower of Hanoi” tremely happythatproblem7You have3 pegs,andyou havedisksof my work was in thedifferentsizes.You’resupposedtotransferthedisks public domain andfrom onepegto another,andthe diskshaveto be madeit possibleforsortedon eachpegsothat thebiggestisalwayson people on all plat-thebottom.You canmoveonly onediskat a time. forms to communi-HenryDudeneyinventedtheideaof generalizing catewith eachothervia the Internet. Espe-thispuzzletomorethan3pegs,andthetaskof fmd- cially now! m thrilleding theshortestsolutionto the 4-pegproblemhas by somerecentpro-beenanopenquestionfor more thanahundred jects.lwo weeksagoyears.The3-pegproblemisverysimple;weteachit ~heard agreat lecturetofreshmen. by BerndWegnerfrom

But takeanother,more famousproblem,the the Technical IGoldbachconjecturein mathematics:Everyevenifl~ sity of’ Berlintegeris the sum of two odd primes. Now, I think the for cthat’s a problem that’s never goingto be solved.I Jour:think it might not even have a proof. It might be ~oneof theunprovable theoremsthat Godel showed c ~ Suciexist.In fact,wenowknowthat in somesenseal- would s’—’; -

mostall correctstatementsaboutmathematicsare beenImpossiblewith-unprovable.Goldbach’sconjectureis just,sortof, out the open sourcetrue becauseit can’t be false. There aresomany software that camewaysto representan even number as the sum of out ofmy ~‘ i I’mtwo odd numbers, that as the numbers grow the extremely de -

number of representationsgrows bigger andbig- this 15 1ger.Takea lO’°’°-digit evennumber,andimagine V~1~”howmanywaysthereare to writethat as the sumof two oddintegers.For an n-bit oddnumber,the ‘~chancesareproportionalto 1/n thatit’s prime.How ~ - -

are all of thosepairsof odd numbersgoingto be startednonprlme?It just can’t happen. But it doesn’tfol- bookson rlow thatyou’ll findaproof,becausethedefinition icswerelookingworseofprimality ismultiplicative,while Goldbach’scon andworse from year• to year. It took a lotjecturepertainstoanadditiveproperty.Soit might of skilled handworkvery well be that the conjecturehappensto be to do it in theold sys-true, but there is no rigorous way to prove It. tern.The peoplewho

In the caseof the4-peg“Tower of Hanoi”, there could do that werearemany, manyways to achievewhat we think is dying out, and highthe minimum number of moves, but we have no ~ did not go to -igood way to characterize all those solutions. So mathematical books.that’s why I personallycameto theconclusionthat I never expectedthatI was never going to be able to solve it, and I T~wouJdtakeovertheennreworldofpublishing.stoppedworking on it in 1972. But I spent a solid i’m not a very competitive person, and I did notweekworkmg on it prettyhard. want to takejobs away from anybodywho was

Question:Whatarethefivemostimportantprob- doing anotherwayof printing. But! foundthat no-lemsin computerscience? body wanted to do mathematical publishing well,

Knuth: I don’t like this “top ten” business.It’s somath was something I could improve withoutthe bottom ten that 1 like. I think you’ve got to getting anybodyupsetaboutme bemgan upstart.go for the little things, the stonesthat make up The downside isthat I’m too sensitiveto thingsthewall, now. I can’t go to a restaurantandorder food

MAR01 2002 NoncrsOF mr AMS 321

Questlo,vCanyou giveus an outline for com-puterscience,somemilestonesfor the next ten ortwentyyears?

Knuth: You’re asking for milestonesagain.ThereIs a lot of Interestinapplicationsto biologybecausesomanythingshaveopenedup in thatdomain, with chancesto cure diseases.The factthat humanbeingsarebasedon a discretecodemeansthat peoplelike you and me,who are goodat discreteproblems,areable to dorelevantworkfor this area.The problems arevery difficult andchallenging,andthat’s why I foreseeanImportantfuturethere.

But In all aspectsof ourfield, I really don’t seeany slowing down. Every thneI think Fve discov-ered somethinginteresting, I look on the Internetandfind that somebodyelsehasdoneit too.Sowehavea field that at the moment still seemsto belike a boiling kettle, where you can’t keep the lidon-

In thefield of biology,I thinkwecanconfidentlypredict that It’s going to have rich problems tosolvefor at least500years.I can’t makethat claimfor computer science.

QuestiowWhatIs theconnectionbetweenmath-ematicsandcomputerprogrammingviewedasanan?

Knuth: Art is Kunst, The AmericanmovieArtificial Intelligenceis calledKunstlicherlntelligenzin Germany—thatis, artificial aswellasartistic.Ithink of programmingwith beautyIn mind, asbeingsomethingelegant,somethingthatyoucanbeproud offor the wayit fits together.Mathematicsin the samewayhaselegance.Both fields, com-puting and mathematics,are different fromother sciencesbecausethey areartificial; theyarenot in nature.They’retotally underour owncontrol. We make up the axioms,andwhenwe

solveaproblem,wecanprovethatwe’ve solvedit.Noastronomerwill everknowwhetherhistheoriesofastronomyarecorrect.You can’tgouptothesunandmeasureit.

Sothesearemy first thoughtsonthat connec-tion. But thereIs adifferencebetweenmathemat-icsandcomputerprogramming,and sometimesIcan feelwhenFmputtingon onehat or theother.Somepartsof me like mathematics,and somepartsof me like emacshacking.I think theygotogetherokay,but!don’tseethatthey’rethesamepare

Question:Whatisthe relationshipbetweenGodandcomputers?

Knutlu In one of my books,3:18 BIble TextsIlluminatedfBTIJ, I usedrandom samplingto studysixtydifferentversesof theBible andwhatpeoplefrom all differentreligious persuasionsanddif-ferentcenturieshavesaidaboutthoseversesJdidthestudyatfirst on my own,andthenI founditwasinterestingenoughthatI oughtto makeabookabout it. I gotsixtyof thebestartistsin the worldto illustratethe book,manyof them in Germany.The artwork wasexhibitedtwice in Germany,andIn othercountriesaroundthe world. It wasalsoshownin the National Cathedralin Washington,DC. In that book I used methodologythatcom-puter scientistsoftenusefor understandingacomplicatedsubject,to seeIf thatmethodwouldgivesomeinsight into theBible, which is acom-plicatedsubject.In thebook,I don’tgiveanswers.I just sayI think it’s good that life shouldbe anongoing search.The journeyIs moreimportantthanthedestination.

Question:Doyou knowwhether“P equalsNP”hasbeenproved?I hearda rumorthat It has.

Knutlr Whichrumordid you hear?Question:OnefromRussia.Knuth From Russia?That’snewto me.Well, I

don’t think anybodyhasprovedthatPequalsNPyet. But I know that Andy Yao hasretiredandhopesto solve the problem in the next five toten years.He is inspired by Andrew Wiles, who

becauseI keep looking atthe fontson the menu.Five minutes later I realize that it’s alsotalkingaboutfood.If I hadneverthoughtaboutcomputertypesetting,I mighthavehadahappierlife In someways.

322 NonctsormtAMS Voumw49,Nuz’wn 3

devotedseveralyearsto proving Fermat’sLastTheorem.They’rebothPrincetonpeople.Andycando it If anybodycan.

Threeor four yearsago, there was apaper in aChinesejournalof computer scienceandtechnol-ogyby a professorwhoclaimedbe couldsolveanNP-hardproblemin polynomialtime.The problemwasabout cliques,andhe hadaverydeverwaytorepresent cliques. The method was supposedlypolynomialtime,but it actuallytooksomethingliken12 steps,soyou couldn’t evencheckIt whennequals5. SoIt wasvery hardtoseethebugin hisproof. I went to Stanfordandsatdownwith ourgraduate students,andwe neededa couple ofhoursbeforewefoundtheflaw~I wrotetheauthora letterpointingout the error,andhe wrotebacka coupleof monthslater,saying,“No, no, thereisno error.” I decidednot to pursueit anyfurther.Ihaddonemy part. But I don’t believeit’s beensolved.That’s the mostmind-bogglingproblemfacing theoreticalcomputerscience,andmaybeall of scienceat themoment.

Question:What do you think of research incryptographIcalgorithms?Andwhat do youthink ofefforts bypoliticianstodayto put limits oncryptographyresearch?

Knuthi Certainlythewholeareaofcryptographicalgorithmshasbeenoneof the mostactiveand ex-citing areasIn computersciencefor the pasttenyears,andmanyof theresultsarespectacularandbeautiful. I can’t claimthat I’m good at thatpar-ticular subject,though,becauseI can’t think ofsneakyattacksmyself. But the key problemis,whataboutthe abuseof securemethodsof com-munication?I don’t want criminalsto usethesemethodsto becomebetter criminals.

I’m a religious person,and I think that Godknowsall mysecrets,solalwaysfeel thatwhateverI’m thinkingis public knowledgein someway. Icome from this kind of background.1 don’t feelI haveto encrypteverythingI do. On the otherhand, I would certainly feel quite differently ifsomebodystartedto usesuchopennessagainstme,

by stealingmybankaccountsorwhatever.SoJamsupportiveof ahighlevelof secrecy.But whetherit shouldbe impossiblefor the authoritiestodecodethingsevenin criminal investigations,inextremecases—thereI tendto comedownon theside of wanting to havesomeway to breaksomekeys sometimes.

Question:Will wehaveIntelligentmachinessome-timein thefuture?Shouldwehavethem?

Knuthi Therehavealwaysbeeninflatedesti-matesasto how soonwe are going to have amachinethat’s intelligent. I still seeno signsofgettingaroundthe centralproblemof under-standingwhatiscognition,whatit meansto think.Neurologistsaremaking bettermeasurementsthantheyeverhavebefore,but wearestill sofarfrom finding an answerthat I can’t yet rankneuroscienceasoneof the mostactive fields ofcurrentwork. Biology hasbeengetting answers,with DNA and stemcelisandsoon.But with cog-nition wearestill lookingfor thesecret

Someverythought-provokingbookscameoutayearor twoago, onebyHansMoravec [Mo], andoneby RayKurzweilfKu]. Bothof them are sayingthatin twentyor thirty yearsweare goingto havemachinessmarterthanhumans.Somepeoplewereworried about that. My attitudeis, if that’s true,more power to them, If theyaresmarterthanus,sowhat?Thenwecanlearnfrom them.But I seenosigns that there areany breakthroughsaroundthe corner.

Twoweeksagoin GreeceI wasattheinaugura-tionof anewbookbyChristosPapadimitriou, whois chairmanof computerscienceat Berkeley.Hepublishedanovelin theGreeklanguagecalledTheSmileof Turing [Pal. I don’twant to give awaythestory, but when it gets published in GennanorEnglish,you’ll find thatit hasa very nice discus-sionof artificial intelligenceandtheTuringtestforintelligence.

The most promisingmodel of how the brainworksthatI’ve seensaysthatthebrainisadynamicgeneticalgorithmthat operatesall the time.As I

MARcH 2002 No’nctsor nit AMS 323

am talkingto you now, your brains have a lot ofcompetingtheoriesaboutwhat I’m goingto say.It’s

the survival of the fittest, a continualbattleamongthe competingtheories.Somecometo the surfaceandactuallyenteryour consciousness,but theothers areall there.Somekind of mat-ingof conceptsmightbegoingonin ourheadsall thetime.Thismodelseemstohavethe right properties to accountforhow we can do what we do with therelatively slowresponsetime that ourneuronshave.But lamby nomeansanexperton this.

Question.I r~o1rwan

means~tomakepublic”.I wastrainedin thecultureof mathematics,so

I’m notusedtochargingpeopleapennyeverytimetheyuseatheoremI proved.But I chargesornehod~for the lime I spendtelling them which theoremto apply. It’s okay to chargefor servicesandcustomizanonandimprovement,but don’t makethe algorithmsthemselvesproprietary

There’saninterestingissue,though.Couldyoupossiblyhavea patentan apositiveinteger?ft isnot inconceivablethatif we took a million of thegreatestsupercomputerstodayandsetthemgoing,theycould computea certain 3OO~digitconsiantthat would solveany NP-hardproblemby takingtheGCD of thisconstantwith aninputnumber,orby someother funny cumbmation.Thts integerwould requiremassiveamountsof computationtime tofind,andit ~ouknewthat integer,thenyoucould do afl kindsof usefulthings. Now, is that

integerrealil diskmetedb~man?Uris it ‘~ominhmgthat is (rod given2i~henwestartthinkingof om~pIeidt~issues,weha’e tochangeourviewpoint asto whatis in natureandwhat is invented.

Question.-You havebeenwriting checksto peo-ple whopointouterrorsin yourbooks.I haveneverheardofanyonecashingthesechecks.Doyouknowhowmuchmoneyyouwouldbeout of, if everyonesuddenlycashedthe checks?

Knuth There’s onemanwho livesnearFrank-furt who would probably havemore than$1,000if hecashedall thechecksI’ve senthim.There’s amanin Los Gatos,California,whomI’ve nevermet,whocashesacheckfor $236about oncea month,and that’s been going on for someyears now.AltogetherI’ve written more than2,000checksover the years,andthe averageamount exceeds$8.00per check.Even if everybodycashedtheirchecks,it would still bemorethanworthit to meto know that my booksaregettingbetter.

References[TAOCPJTheArtofComputerProgramming,byDonaldE

Knuth. Volume1: FundamentalAlgorithms(thirdedition, Addison-Wesley,1997). Volume2: Semi-numericalAlgorithms(thirdedition,Addison-Wesley,1997).Volume3: SortingandSearching(secondedition,Addison-Wesley,1998).Volume4: Combina-torialAlgorithms(In preparation).

[MTJ Mathematicaltypography,by DonaldE. Knuth. Bull.Amer. Math. Soc. (N.S.) 1 (1979), no. 2, 337-372.Reprintedin Digital Typography(Stanford,Califor-nia: CSLIPublications,1998),pp. 19-65.

[P1TAC] President’sinformationtechnologyadvisorycorn-ininee.See Pittp://www.itrdgov/ac/.

(PFBI Proofsfrom TheBook,by MartinAignerandGün-terZiegler.Secondedition,SpringerVerlag,2001.

IKBI Simple word problemsIn universalalgebras,byPeter B. BendixandDonaldKnuth. ComputationalProblemsinAbstractAlgebra,J. Leech,ed.(OxfordtPergamon,1970),pp.263-297.Reprintedin Au-tomationofReasonmg,JörgH. SiekmannandGrahamWrightson.eds.(SprInger,1983),pp. 342-376.

[BTIJ 3:16 Bible TextsIlluminated,by Donald E. Knuth.A-R Editions,Madison.Wisconsin,1990.

[AG) Semanticsof context-freelanguages,by Donald E.Knuth. MathematicalSystemsTheory2 (1968),127-145.Seealso The genesisof atthbutegram-mars,in LectureNotesin ComputerScIence461(1990), 1-12.

[Pa) TO XAMOGELOTOY TOYRJNGK(The Smile of Tur-ing), by ChristosPapadiuiltriou.Livani Publishers,Athens,Greece,2001.

[Ku] TheAgeofSpiritualMachines: WhenComputersEx-ceedHuman Intelligence,by RayKurzweil.PenguinliSA, 2000.

[Moj Robot:Mere Machine to TranscendentMind, byHansP.Moravec.OxfordUniversityPress,2000.

Photographsusedin this article are courtesyofAndreasJung, TechnischeUniversitdtMdnchen.

324 NoiictsOF ThE AMS VOLUME 49, NUMBER 3