The Busy Beaver Game


This month, a bunch of ‘logic hackers’ have been seeking to determine the precise boundary between the knowable and the unknowable. The challenge has been around for a long time. But only now have people taken it up with the kind of world-wide teamwork that the internet enables.

A Turing machine is a simple model of a computer. Imagine a machine that has some finite number of states, say N states. It’s attached to a tape, an infinitely long tape with lots of squares, with either a 0 or 1 written on each square. At each step the machine reads the number where it is. Then, based on its state and what it reads, it either halts, or it writes a number, changes to a new state, and moves either left or right.

The tape starts out with only 0’s on it. The machine starts in a particular ‘start’ state. It halts if it winds up in a special ‘halt’ state.

The Busy Beaver Game is to find the Turing machine with N states that runs as long as possible and then halts.

The number BB(N) is the number of steps that the winning machine takes before it halts.

In 1961, Tibor Radó introduced the Busy Beaver Game and proved that the sequence BB(N) is uncomputable. It grows faster than any computable function!

A few values of BB(N) can be computed, but there’s no way to figure out all of them.

As we increase N, the number of Turing machines we need to check increases faster than exponentially: it’s

(4(n+1))^{2n}

Of course, many could be ruled out as potential winners by simple arguments. But the real problem is this: it becomes ever more complicated to determine which Turing machines with N states never halt, and which merely take a huge time to halt.

Indeed, matter what axiom system you use for math, as long as it has finitely many axioms and is consistent, you can never use it to correctly determine BB(N) for more than some finite number of cases.

So what do people know about BB(N)?

For starters, BB(0) = 0. At this point I should admit that people don’t count the halt state as one of our N states. This is just a convention. So, when we consider BB(0), we’re considering machines that only have a halt state. They instantly halt.

Next, BB(1) = 1.

Next, BB(2) = 6.

Next, BB(3) = 21. This was proved in 1965 by Tibor Radó and Shen Lin.

Next, BB(4) = 107. This was proved in 1983 by Allan Brady.

Next, BB(5). Nobody knows what BB(5) equals!

The current 5-state busy beaver champion was discovered by Heiner Marxen and Jürgen Buntrock in 1989. It takes 47,176,870 steps before it halts. So, we know

BB(5) ≥ 47,176,870.

People have looked at all the other 5-state Turing machines to see if any does better. But there are 43 machines that do very complicated things that nobody understands. It’s believed they never halt, but nobody has been able to prove this yet.

We may have hit the wall of ignorance here… but we don’t know.

That’s the spooky thing: the precise boundary between the knowable and the unknowable is unknown. It may even be unknowable… but I’m not sure we know that.

Next, BB(6). In 1996, Marxen and Buntrock showed it’s at least 8,690,333,381,690,951. In June 2010, Pavel Kropitz proved that

\displaystyle{ \mathrm{BB}(6) \ge 7.412 \cdot 10^{36,534} }

You may wonder how he proved this. Simple! He found a 6-state machine that runs for

74120785350949561017417256114460496971828161169529
80089256690109516566242803284854935655097454968325
70980660176344529980240910175257246046044979228025
75771151854805208765058993515648321741354119777796
52792554935324476497129358489749784615398677842157
90591584199376184970716056712502662159444663041207
99923528301392867571069769762663780101572566619882
47506945421793112446656331449750558811894710601772
36559599539650767076706500145117029475276686016750
65295229541290448711078495182631695097472697111587
32776867610634089559834790714490552734463125404673
70809010860451321212976919019625935072889346503212
31429040253205457633515626107868771760119916923828
74680371458459216127416939655179359625797982162506
60314494227818293289779254614732935080486701454668
21939225145869908038228128045266110256571782631958
92689852569468996669587422751961691118179060839800
19742149153987715916968833647534774800748757776661
12880815431005997890623859440699663891591940342058
44534513595160016891606589527654143070266884025253
13506538908970519826326303859380836606399479857223
51182179370081120817877269116559398341767052162655
91720120243332830032375287823064283197180507239529
73532295517310483710218115976193011673158268680311
96305710080419119304779715796727850295295594397972
94500432643483372677378872480519292318129075141594
30017042374851948272409014919776223009518089664572
07992743507711148214208698063203898384663486444006
34378985820511533007233636175063664244348363429381
71686527767592717386843513084430760538597788497854
57039288736337621694394446002685937650424904397470
70469396499307353236961551408770635423051623551580
20502046433028802860624333066826817081190086396068
05449212705508208665911831651713621990622680221463
40355100698032539208500321737935916572824167109243
92179632770888698806462239286922365720234049353262
29319760109739336919551555856149717013936570835108
29337138019755841416703048425473095781543947284695
30557891048303296887105203445899799657005379321565
69516567462536723341045028533527590132444550041637
29329181785691615066366432555395455966924963740865
92347905851808900172725987566577427348338512074400
14671953393767922133480111674181667283600828299423
61956450241322000295203110123701834947807654204715
50872484529282734610014634854736508912653344685939
21925381546951641870418349782007870841424352960246
81621927943512800969148833336725244172361946188547
20963814880877462649154982581243612288332193203522
41878334479833109122807808425070272194370533803098
15576207436359412405607991428582586135325000600450
34044570755614842353801605755138514728637444538199
91270653752636827482388619627545586769702982563550
21579190594347001678124794147520737780545710725238
09263070578948251221705378404200557518123183429763
74391628221317569903581457033268409573939140317537
92951945222572832854376667354589981221872208463941
92173302390371597480313550832469764638860694385735
23920879420538715366934472880272772745254215764827
22658077210282649639911775387884460117481542574020
98604710597497363577816224906240529468176628001243
37027642430572009172724680494845807607875336391296
35595374936463756499152341721363955306855005063147
84058597424341392606532443545414393065035952175386
45638222669677018885647203581909266795843083183075
45078527771378321186170870661268143673661410440879
12940056479135404302810843179830761186081727156785
64098233869131431441387859096996327035057252704630
66502399928551829284912464275322457081097679293349
77256939108943396587781783827866809713105339479801
94252766851530607523746692117596241149131822801952
28443629054406029354014078168448839468854310977774
64971341943282207403959764566321636691138805567444
40040338242074160211898209536897949768268405300638
55020960995862149067127133242923955216689288381118
44058888209904636044250765206694970737949801463627
09477533118591401481473656166664409698471099509772
67427126852419476331478775678441642258692918630399
93094799728916927267392317125315003396591007151226
51178203821487177649041575923087270542299624201754
57070699334124035606469963629320951287401378625068
24738877579310019018071588005151785675631912063264
63879171290239696789072427830303321073398269847363
45629019926879365533487397619450023732220399774409
78878227032599708324913637134795947392057672257001
88982988598790346622775744604841009275542606005747
73489847857869077885071196141198763333605601699259
29619179821052298166718147760782068394323831299733
35022262733114475600303463447691380426320406458971
00672747110856632877598317707068109285992387448288
96303378384828434300860309575950421131368727514681
55719991530038357093718116282958853868614897146782
54967080333475258187514533923483088945298272830364
47705805899342687014494268126923276698359373141482
44674928751199880396209243410394589961770757345685
64543015202758133002674347175029218497929457573786
65467651289853262700253064391422801251703856704304
39933674129974352639333328279021853998732340493391
52439866339607669375777654460893584609291320819482
35450294931218519646257691473024784635292462805654
60565812545710189564825444516533104654549626307774
13759501791681585406819992391995410832229305735031
79743073679478401659929359532688412917629928632646
23531586811022948074724601440807980838830594224020
70309042157187760781779401504832688794481346645989
97848941467191367820110325917723306165886986506294
31831412363348418517790881203602332829570523258317
17949497171355624217480958863040591015617418162750
63724305524405091305861072355204855928475793642357
60246280346642123778396019497710295060768286004460
92308617300735231303196433523457326103470236858178
28414431828300312373660803542368392168889742383761
80821770347653926560938368076716961022633531512872
88504183987441820108573621090001961860938961602460
20885687763639795010335265588767970024085673382770
49445342795888876360045283740672969599921612120088
19088433242165085295954910707655578576243692034551
97614559215669274189870537223818067080510833662246
14856941296324679132997700908284769865178966760885
53161005611550761795254034078533136648007541637608
33678935806596748974138821535451231444972327449291
10024213166459016651306588184925060362024845259323
65173413805791200232437686840453953297502340211872
17360259379142737381790192340837264502917404521851
49249430081886421605763562692675510215455806969064
63544907025795646739155477402570724663876748602049
93436166161865545758100837460724945336759484902958
05760448798855031700989947223661484740412761111279
72682354237393559488179028185454100490417580488953
17216043021530545928659613553982781316650550537867
00952857558642344328764468061901146269419736379089
31879223575291135204858555802576520665674856000998
87879817374651045887072894889964276716532631796097
12769213622199519840449246106705404347452689144525
02026902997192020285019028617442866626487265731300
40640551447578058727968270741870141068657514616959
20677169731946885529018487968298771083015182717866
76088530960537199256279472232540485815576475023989
54150471113235298546458042675161613703655941414055
14600875711454906941647699297619365796793820814580
39814831564180619061865499399314083189554983356803
30767015598546845567749092553092817193708115207411
30039726042496587009377011208504194686832850560577
71332112325173133436614303950199710563675659743576
95759634767858347057063619337247953842775381829735
01515839943757098551455066444703952999001985213970
88527752763275564055679719982684702573490505326753
98021282801078182123611019529338931475797349672464
44059385046714080201178146810297989489194773283941
93746291180257355629914922000638130878140351659284
17485358899365619763286647381859607448645462954784
59233969830735095006570618760591874509688179105104
12456865774195864509969928029317962965086359995739
53548814859217251629847330330353163380838028768031
61203651855417256064885345072718378483364209631654
63908303387213995060644412231834039370317361626721
33451703923209110038936674869051927213642317876528
16991320616980772154778362044248178351052875315790
59407440453142692201342020196027082359311172043450
63647014817599680233754740166551904281645680384785
43484207291054858039349689807941261676589504440279
34675739151384342020909767349894791903987566481434
15238843747734338550634527977710083665707008945548
12980194777531956507697526221024482444025315826484
41683017177169605153673188813829296522594387128245
65901287931783268300595479085143271190752306807114
75945173848401996529051879487394089712075068830376
53688488908938496328770371731709863096656986444104
08201803469169112306001808844056491446464723441228
80657997624356240757329628378856760617602118493595
76037880180779827784647086182400197605967361763950
33673997643549829889211843819703562151131479726729
01802880267974602161706988320836675081916966310882
49095313995134342308211792489590022680899453895917
74944902836882433137054714577056337316253774263170
52294019030895743857006850581035142676723449462789
28264714750222749382953079695438542590392058673291
39096257478555758969160065468207714202648605111186
23874795826486193186947393782106560542813451140273
43780784300043577743478580825356921145531672555409
70149321650226039685363112051645618451238774970868
00014499436813245398575403118912847356591290588765
75653595733135742217401557961347275793910801273534
29151807248390239645759760752048078546363519519946
78919336290268584412830837345667768883056990324807
63173327386242877190676977493730442619179902126530
09880564208648705195942161723657549836039821614124
19940671663992530293860119765744117402843445712267
13596618543665796686329880389747123731107419115403
45207462458782741411300537230539221887374778233703
83605096596611557973677807766580326262439332267121
11801923744981394260402383843712773562047935390674
42122563315255005117319806544749303592608240355114
26104287190561845186438803878522806658022994173893
84778533884131972707349318882891948374079553673814
52850343823429237584760779384386062926213863866427
81961360138375377545545650428193002638139604593657
31337406162040202984032689042877403357513000261872
62895135097140548323692495424233162737319444152387
76746662103742710883076108190383757894072689353642
64969030654139724556329796612143291052231412044970
37933420967501497840698712388746483202475604070974
28823745682524686454563102344797165959894997090034
44051049030017408384964948728694659966590270138458
13453290239761803750653458018811684218923119008085
91762200196368137317672026076675105255423940046735
45014486702306358119905764811368254565294469758077
56929475913717952306191477036094256081328646465135
88173202952685000478508674285854433060409037157131
34973953490623757649011875332905719971957353757223
09860503547981930039783388507068159541449119588220
38967528182569374339481592073458636154289283223650
19534546781485375722855718519447096773869485755174
38100581579701010217915862939108556064322256968608
64061646106615054106258657747708850915917029017031
45625886387599673950676041820159666498173631174677
64716193096172970604794524963250374628801095983779
56073534151057391381495922022764751197295965014861
95636807693589605464925125373492393655880278853499
02202446706284772379836648849167504828201710381073
03329458670141724685763992293288349334389759164917
97309423042332708701046852013961258231103600450165
34505411303924779865224301545810028949776070035913
31854675753493005328299653077777661036596594334161
18324140736770437225608805982478257555946825524468
43743031331166759021115160275501148125345230987606
77278160953638658876659824121654739540845026103104
56833241900758722085690632764275681521803243648281
75973088546131339174823173625957848652825117498954
13479595716866331789714691850880571150460499972976
43306369801233196879814397180168695393291032751573
92506158006680468011940918143780196654320279894118
99753676684493284932246345070256837568979217094824
13674789294795851372211654038911266565104851609104
36913412156057173851727044502460820221614272608195
13894166831606579837662570513633907356874954240367
61054951791262883573121076674516756368643088470606
54365581172433912025679081223772154694705809408962
69112615546800941866814965362918061068014102459091
46087743968661858764328075373663888207948725625707
08747688827166843119576034872969317512228990778772
87050904881869406146583157468517895291237675578225
89024394102022506915147947735521950610377154115619
18117495558879264747451407598816062773916529397563
83562243551858599818978259130319451464037816343233
06633058393645640234765483567475994622676485484999
55277646683491016566937482924707993950782403274121
53574422245717762195720278348883131018490842980937
95687038636668821642422745084346013789736450796552
77272084564445898912543737560152633299759359959420
51990771767713693321032039215107832734360378720851
09136762349886344362420984720955074780889202541797
70896763846953214720553922486992302733707196483348
49045969833114793301657429813958969539406732142636
92502178082237337231815692752660602500625642690902
48328985111428648135448597379991004313275485637303
77657758862782442714471712782977203952113306637505
99526051279198553751017345873305211333857760858608
28684951160042807909556692892506555761946160549835
20303885701870763423255286037095591483157305753323
90742350781364515849011549346797178301358198056103
42477861647880999927308727218642361092720037983209
92492109020448284198786534092136303978056649046760
46986040724567578002859838619110484477846477503720
50610100383123165074971031850256994835659647733530
18102379912398920890647330875072013095255495682868
99218106145284129097154350663342836841523804131925
94548014661761166732470886782425501725751052498528
19766225123357293850179242144805633465265465905934
85940544983902174680151695063515178026513270513373
14567430101451394436010539789612192204784076741162
02598379558248660340254801433826543073346880809175
33794003463907669978751710212827335152650286849811
73373300353615348808238739424609173004246887262560
12109804136735339867925129597497616688796356759848
54311863756767089107147427840772086928447453283837
70900225008489928559780170100832519186908501709113
03429578260203366213647163963514273177141793258212
70129903691271912759011710088979532621678654470430
81224819170877249988068180433429813000194708364122
65880853306383812583157641813642029350388222120649
08993143533172049134282598134601427505321082386102
96641697114646047681037048275294879590675546238961
22511345901203153781514214990931579674719005995452
69360291389396593588517951759097436505189999310858
97899228473407418165051239458663407082579219063351
69221909938428450917040668276528126542834183887723
37308687968877323296188638808928460012302773700078
70065837663746381648888008236867292692703324810208
21191152713868278958122711338568954056108558339496
39688210557308597484533729528356864688107193747735
65682131726787955429457687066390524734027375900484
94014381838353463379587602239201589869366921779214
00650533231084382211721311712842354720530958884987
55043951760209301641240570251935929483202398985064
01127949408135378762836687221506379804091561420489
95319428463913516967083275489061836976589369361669
92232599237862885826800721062057774065285577658956
08567726912177628446395314028140718885884417469719
40995239084087377128760976350345980702249228280657
48985114060542624187997015459894041486547255798132
59016156893986505363351455934516022571657000511969
63364078453953442051819839062916836973503211443235
36474735357860570085136285632030631500478179833304
99800682580344897169351621775022943484116507698528
54499366926348099904655866479825346766284305965206
58548938005306528588644226075291218639460859881010
53843832526089853965121985896567112431020946618961
60263246569884467142995224200214985903732185320564
13645583944220113320571903734419681519647268176440
72268766271760668375452471249796875977741923733307
01446073918712502971204655640711174439878554539601
21956175234574438805654552770685162438998516074279
89283178151266467822868370900746468416658852633006
42510798459220886536081340421882720060398236598449
13841586932985432819518800346922653513675772280557
58466448137203029042196278568835120842164396569247
33319030990329406645032723523253309176296741474039
69491394804908661080074948439992404585193352085053
09330332512851734540875138034948953970107554992837
25469264930630900931579777556892751293755577688575
75357774429395052634349211740548847867692095519205
17699024598176549937631734736458995116976781129272
63202646609659513696443442087992621407581818380904
14744945867220301128198294741475121079329121236086
41323148039824447971931349360127464522185082768744
03180156798758715856884779512012055254508937236140
78939505326048765866117108935218142707910114350460
74353501694827332583764635697133794254802641888996
87913418861391825942398310699443294378064810421916
36641257207248956877756345317415840161437244655168
75304856274379634975849417916789483691875055902865
18398650158930683922703754119022369253524344945915
72155384819819961291221629082483354519614388928289
11148160489016637902881331691919347083264362558491
33955648462626291884858567370226518877896536048408
73744522109056681738413460309253412525678038038964
72501859470776876779209754549293563256277168162177
52667449731255992079828345654966814836569388956343
42904345058260363639446282809052087932690706797000
74240602327232553076811874826485551456431407715195
46095671918646666760239836781094793929785589321896
60105515770494189377077318842984857150688833466597
67873006834384199643085303023369579906932498775902
98298391813274179420482526911798729149720989952114
72503417489527247775311783591486298722278199904522
91317033049722862054204732050563957980743822757495
41170686057631593358748161282948487565036294435952
43802641826408557340015251538440908071972626469253
35669966286640071824184801009495201185896687340306
46711830686618180737245339363712327731476370208255
37354634496570455353470479079995111794315681691296
26684518805748751642847384627370638321580719294749
19778254463611140131322463114379893205517832231772
81658157349479812300869694103735382886808072545111
54980271730008673426149474830482385937373996048689
70676203938236347012289061965532898796528960064177
72592915524362701132166185105888514998283778233792
90683631740502877246805331618901384782387956924470
20430891474567033737676759364696487984277760319017
54124070029194325093600825837936415834980674275196
25567523839068737435232552721973963767167401533932
55164265174152406859802799708331567170008010888109
85633153836876717009088587847684157861719851890067
80601296164015477525961334994818419517562211350086
32568919833666392153524361473386539741964955984720
37448859861264953986984660523954497130466421921294
34352011971909145010038990480537675490804792798960
02335487981061723356671856618419111379350218406183
60014249461759955259245055371126596518713470579754
43833478288862803623021917201984348951536123102836
48317702248444053032752947990790393950632070253139
72349953184406861226132851468607167036974898023131
58647946195859301983184566013688170241533801527247
57011566582912032538695530313579328051739626585030
94812965683415715842324324576448306610483102691986
50122809162200977042104273569907096975843914647362
76686555266544360523628469436056445811575098216374
74650286221778007278738282039500933724220757198566
63808427422727677846423529592696120773721931205951
57552021549753607600855673968249852015294426475521
53999285492450486367556298997990634291469802900197
51054224681502319297181372108063268533489382824589
50809414454870209670524032770690571247423406528636
34733767166335777635943719498773398479154565023031
60854295033213421642837903387899795782579128072673
31503512096661498826616948582559435768175108716922
94224762979998504772149467096685583864190281516967
81779668091097801140992680071084513522266799153637
46461649141077551468131132063337637827362591828848
70277014659501753332070729727520807182116526879225
16203992098883918006142068688375571834634988318331
27687883618850734782939117519577624789959269179476
40872808507667548587249382853072119804227594688696
30698618906631685300257429734614228164160174853134
89712770338297146503773683082925441028484520858091
25796032704398598828302033251004062051314827302245
35015670388517203516426505963764666476551390887086
89180474461807362368873178978001667165892881421488
16167123834546655723492211848290831098097567359535
81721182323815119656265560935653361656815499161813
37807958386329323377296558389531527819719654901224
46058863417452675741010825146473478017543955579942
93030502263028520112745647216501845004319033153126
40371572959976276990660248497949726695076901784395
13662456254441684664490915853787199152311813509376
33218526610085160073867815791780631467048848912653
43577978665784964015551083273542683527148588024446
53500520955111872202196242770734919030521968391768
76189131349176826737907636627825350426345483048015
83056559247046664024511272868516591991130061230889
00768815881536735390908055571891184596448949113205
38012098809875184013749087806248011864416900847625
89290233796121789219019136221872572530277024302878
13598322471103754045955640650664405395216317543343
76356257437836390341758928019613119394530675826328
81699566127350927274329031500791260707239914957174
72342249190062454784782496508606891242518376613090
89810037493784203099325680519186496917658479437634
62474815362448251341106923834255537054760626728367
84353501846445608591994708782490713452600754620212
37654458397397721985685664235339024412284487030078
81366561248579779108938180966300065805899231548006
40281431634764708308767699284267674863099374345555
14448145668824159080936881969206315703386025432855
34122191331935947544223496910646305888272768507449
57407678258447637704095407352062349994042470258984
80500745308544533680238231870524495823289886265644
45483339843093524183279678300528876209455482159209
81025371405999175784413871996829844219531364845461
72478302176269972592695347623168082186092141710963
69966742354198221603563833915170134203447117943290
22675807314744316867133134879664829520234325266411
54874433858728761623211243214927783400558098748738
32413005042751326844407910029374506542074345413639
92806155664034624944129299554544315672751326958262
22958629783603884369424985934080781181681534347061
66252353598635687448737089275281731808347458828818
81798108175504255176583933602675803276606597052890
22401556052554902399988250724615836510691518810625
19396468323865173197592819431166745335939340346029
42406877883634318008197456920759904661940295501974
41351516607613670668265669570800796898495300707479
85746577660570305353376394163675575724153525786910
72101785199939424200395541386155250427435810475404
74266307925462192126166099869832098095253680036965
59071509487671036599777983368361311057929815125953
03576629595660226203279779989568868521030237646474
85882097407730271561694861947452566776214029789052
64707296525947137093271267011937747113840860616968
69663548225972442703314913144011499100926569149203
79129503359479838920742529432704431262443617230949
91813796420707303625383578796693661964853048148509
85738217203512564120768025383484445153971635623430
74971994537849060453745299007465225973424539915610
56004083714384400768047255172493253205806162075641
27165266908610075409838165767502239507666797334818
04200946983934751625339660004763238002545306283362
37620552930583636902933504511521335469194932829108
47433207804468636210836676395788367169193401870353
66281527039401611108473953726486066451730262680605
30951657640772770504057469408853154978103950989430
12288969060965013714183010710246952729794637740634
74650649505427764478357727163654262729228469677452
53866472835271598739571911536260545809120793301375
94385061132511498328259666284170774982319343715698
62596091372657528693640001255925324053452604625210
02192134608756468302251329172661133599304107462656
76421345618822019839480230808737713008574851995557
42902025967654909244725979460267807505055355544577
88183455568184623716406738484674881216810438947129
42424253822337942342396684870143701961847122730006
97527463110809874668231405700676923117228143434800
79705259561363371191589216659596460387088282208647
46659166755140183778373106734797631382064700839797
23608865167350848607794665023847421547511130680349
21836254329898910151704652266789364917284374866172
21409415510005400095663128111630786452888317869078
86705623439746521289503824460489905596734600860962
65869115157904993399593615143266963994605010126303
65264851279569995793606443577504674189470139948827
80732701526257819098598077288033523150930630993640
28554012527540423824478945178779671260604681648840
79243261254097596633683779443753577000979402674481
55411218315026598694889295524421405575254221786788
12689568960142588606563462298201550569280235135408
91551869416490705739820870162793930431835953020411
75538916881133186575549973802728942306922816124893
84328618794345894041147858850703842815435695955268
54604636601831720513984432052505883454118854211409
62277433851221051432369107658497370446927143429634
32741216798855689801402749961021214489079686952173
74816330003249658220010815208739003557117090058915
39260941647522717101001822461469198454404259691368
11160528258042790396182432982255302576424018487611
82108811718205167340130206882625265533608582477936
40996990541316946145071012448731864534347341565881
45952426468147168769076238720670076693996518128972
17661485695611456289875945575536246110132460342695
13949813045952981045256199502552725006062775090950
21112996278631717344378151528446608295456563985535
02863219231501970059165209018705673346441956078354
10978596286215047498049495879254996517157922009547
13467505279056362907045367065463054391330939433419
32561753337708995115018216500103133436822925515528
72470951859706413346437315061034278142063895247807
79638191708822808641043224144057548480066262263622
84865108600166566552198738373021273054540786632857
10237864460334373105775448913780176468769360231405
44809965105445004587746851390928690562100862404738
25351078769295888504866076475066807117317619797258
70216271969664859447019374500056442787633659026784
88468325873593027240906584026524160395492209392478
24998320213447280415528464974717582766127914504769
78394726173591548842520880672777589569857846665575
09359090955333626345267284481392018113699807502917
78270693692625828316377680602959448213821780078202
64095567581992141153215589034041050436693161866352
03375197313604619175053865815748662228641715403512
47176917247555655158933344596036086599469086198096
32887651893720128619987149952015200673431254965712
78516247107068007704240582566192894463092650904663
67920658737849797778283817394980396961898938022447
74033643867327139927264647335898116535609813029651
61318661546436639774687265777784346842026627963848
57747240127921180120799971167234434034055984879310
06316787429449101749577889555663153277445508121297
97893231849524617938827213409843654638497892446504
73899879967895057997435232065101118524861572857845
15334562067709732148950491306421966867582410017358
99912362374319420643955424880431537060598320632545
01556148893822628560302757136541039243390348268825
92297721155902090201990832611774621362130994392758
37332612891833890592973681544564182393712211047159
32804246255929723935396648344357182519336263026317
83406631373471407037905149014119097283352862388764
81302570090572565407034180269740775532616223613864
46417948717813908193216479439000687756820066709452
28599718946445752303322392055083844853454302374759
20681171563187929477821332835065499926274436773936
94186415342682904785931018063421143017613836359930
01007511264125201640231577870379328925877581650481
24708409654308131745014606613308037095758248803838
53703094351328499011986696617756357203317205482638
98018585859130767851534607651147641778753812085838
96509166738189451920411415869541186904652986515240
42746288863339499157257060527793168087082446034471
43167833456601302629965035077092142438499381068287
10596669075348236542190810034528337520828595848843
51797054043516394986261271860645371135668378744705
74439684442037114924923886084313957103406519224407
47022132643598215293327338839416343363171329200897
09730771857207311526370010430111977075941812602307
72376014916565189341889310193270446357645932285406
51675603176960581128605870085156306964338652918116
47470819414013086540664201041699067627359002476183
97441734031754830743452227112683715049815859152908
02862983168967228070833913519337198385661136554339
62851438948135974452279480136078914693182156818148
16297396478610087712467686268064886982681725715164
24929395190591315386573936767998999968870705845576
08893755357532620316432446928838057972884766228974
10131776538813050682735963180571500151178162311474
69990036265878868229927231767265567356432956517643
59103619542851170727473847169547858711173803896624
78801571561070123296147380608747007717085842780541
44314696513348875950813445481618617390462875640560
14223487405828994564351656056960117293612995285025
95365239779744792189608943593074512316856916984777
96309674486800136913387853328160702324267484311332
24204455544008718250784737266968027073730861657613
29014341798367958342778305666662518827856377071193
54004210396531585096287782401454883328134534630356
21887634170142130804926549347082096161768826532124
20946016310337676277784171017740238812684225399008
58075983464299660971438560230143092504564484906894
33859515325280873687522552440873389822259898572887
08616952425456830986513532384167684218033165913388
75193709124723078541516745612984826500055683548584
17852290843312089512362627423048016086699676471637
35147854111012464519653717464329869869173930623135
45980351110901360322158544586508241657603119286683
49558826020916663125429973494194055461581225633949
80703308633040999177305264352224174430826697774318
76369465566079898997510774644419064529504930651379
13908840809054542426918377991067744703455451227335
16122974857001011282482364400153502188778628986672
52014237402183844341392004432201798203108406472425
35265458437883895286526648490617313714847219300499
14728265224929738624430756990275707141233922070270
33003862028855177775324754018761369511084610482939
99966110560450228658375682663588269830548281134843
48593533341354255022753142468562466603367442116939
49111012266149800196830812162487975146084789262372
22250383995968198718352419711082802784003163572887
61976823017819013623813583857687425342963348092655
82299549543848306228309087395785215163121420804111
85414674824399818902039299598141119367854509189125
82509411930329585586279087316095471630032607570795
41798965557328074489221493475705896364007850102309
98608982137394898962877566829533222382120121277885
46286354339766910445850887107156199679184646499267
51994585788293143788095067363774602774785316545938
80586516906808069516586376041874835414910416438046
98020384720894486533059556453463277404919782891158
67249118745843018273227887282763135556994164394750
79236624098209838711196489146565110755004581971099
36519894915141581119449211885137869996513445924335
74179788395256600003697852315986680131174264650635
55422498449995716648520919145951012273733741715265
61886397525516905954642193344472414059672799512206
51343669584112913727903560817212679255453578098847
29619778656382752283703523073380043645305772938365
71382902731985761581552942189374425534261116939281
46939822047487536381747440387576628596190610523132
62526398714808427005031359405815456065589000738436
84875317248057834596408531829111584209126136971880
29730134243256309343213621070504158874966417544034
40637823514645022279848007360189376647467363278724
44164801997811912761889162074192834565275689346835
60397590580503621377726704540986029025118559388222
08650976977926409452158196552468257766254708041097
17625576428249908865033360778907337875043811148414
66965978224351886674427982057073646781057580910376
82496118558637199155904663531858354485468312877361
59016065276103471553526718665313781441486352447115
75933405758127776214650925846732615829278713641448
78065303195660876027013972950344124525648658281202
98644663126368482770781755280015743411976115069754
58885197224051756810220210344457105752159620520010
58901976514020860989473496121460553667610009208155
60641902463244064221782103519393453021339137535920
18219814516400137820973109685531195344121241497589
32648947132549657653880734314031250318397656798122
67800306483529552691880979740190819931928334439138
31036854955282952306101854660968166155042764270651
02479337471160875801455244803852479812754986345246
26439730846216309598240053255131870073015249288288
44150272316980035035371688072705937991233573103320
76810393162230532462449856153185470499629581779256
29737290506005000054376861872612136884705332860904
26237702503255705858711275434358903547610345778147
56042042166725129709808043190716588723380785336603
22966448060374352642763233712959166808891103525594
17462453611474123311826492400534657790529695637970
14940034362660479559404460065139243888015015079321
43881759139711791807591446947442401651321344563206
38703939480221794666579368899985060132503516636784
54849052507185881787900024915433075335630318016881
19535355859169933891361128774659358105685232413330
18206162177873663026713933468489708054602754322993
09338706902030891605919350179098941371298507942794
63178175281493035748693516046930020542835004496284
12437420573192673009625527488432138708155169004508
33487458990723468801118556524546582638316269527209
27569251690991029393597687960780818916005205512001
67977712443572110745259310850411512558482821094977
86737098252380505747502563657220320175399483837098
94871855977863419223979999039165469590933447472735
64390946693354609937656222506242038767369337494021
99273573670542945658981146062475248967563610321708
62760505309923165497906143508759602888497331721652
29370337857080366232139446761828793609673433032334
32585557536041970672179308034877273668488251635719
14432495327697411636009656282211309130203882825069
26676570351832671688238787401929833790450599655059
71332476576201495740387143194066286128191290185098
78041988386520042018089673233535007442736556268111
24565699383811116389175299087394994079229065025901
09932362839121349567025369695891762635747444753948
11323396655839159294220494170850891175875434376079
00912177562705938184442800001221268949974532769905
85186469046165318475130195596186195695693301361574
13092936825139939155744166752791806298991315142756
04751619971580109933777371206938349703659450545917
99086435789023264890032497734833661139947200260743
05805938712242853874676609731910812703733209567890
91914646607426909877903750638235352500143012922346
84207321742378981403305477034941755260627158862866
39547771864848075851449765843196982379504784274547
10166411077534765361442880635680994357957101057192
74648013603032719711174406107922040006156900695275
92189653858205255927880288280372077585984295829915
70356175070958821709085211649615848293841308860736
89513009850391223097508490164263520408717702361215
96530251705369377644672577502898564911413385213577
59655576051348180738326237828923361030508930002718
76498693801052511910466326786799575951334548987940
94767081107273610897653963259507032202194746026667
53556501284564352923685161312692205113923111965659
23809253379322987640218108966330037544515321606450
76565942988897950720262586988263749869310709817026
33625926769445884564762341943691593418228788258329
66608714346918244407463976272653026631998188220588
63943436931229964879033230091279601828273066188517
38227218390715490370390317107411134038467560864087
09529898615806032736061553938080612875467642224957
58609812612669466849788545193495345754592869245291
04827616552643571689209316367953691428498500020434
18738942352565376594741749369768853312898488149905
59914195951380269708251544052057214524548773659044
84797633817530336049853924925315149276545906051913
70313566666349292144343487834912587102830709775895
86302329433440274220160031259093667954707442785996
09895426614987890087606066195547820996024984086973
40197915444342987229441664529015714679475579971637
96345053175305295311487784317564157202424808898125
74030925458897973764040363008447298817079317321949
35828013381138804774850874697541473568955917253382
90875162194816133883653251152435404821931403571934
23083405143009825506375030953405042506624062447165
17113728845547477715226484387676705672853467409579
77663621447167449425124941685624394269650531262340
06717273036326293435607524697042540723616179395393
33893301349198923975607651537354928215382694266011
41757508273896892391736574632063784909558019481334
00865655804108829123890977650354951367777973842170
75868014247898823178741452147122239560266142666409
52374887226747227761434033018712705272938680809111
85289596582679848237670101390122656182959680517361
46076619939139036787876060299301996426764725427403
64011297421238171255442319681637731281152043413431
85060894391599524452878044334616995055050095307491
88403737341503747816446538950384710455426239295288
28998531209709108913722340683342671325423513366061
63474488522182030819462026736771976453958939935828
32050459497790182712782471389976977879426526570877
30339646201949991550918186788970663139748645058559
72718469140463791494759075349756092667385993283854
20778335173462088786041537266587943086744729710814
62579154103447702796046390902553975412328182470276
13052411058118309311147944260478608

steps and then halts!

Of course, I’m just kidding when I say this was simple. The machine is easy enough to describe, but proving it takes exactly this long to run takes real work! You can read about such proofs here:

• Pascal Michel, The Busy Beaver Competition: a historical survey.

I don’t understand them very well. All I can say at this point is that many of the record-holding machines known so far are similar to the famous Collatz conjecture. The idea there is that you can start with any positive integer and keep doing two things:

• if it’s even, divide it by 2;

• if it’s odd, triple it and add 1.

The conjecture is that this process will always eventually reach the number 1. Here’s a graph of how many steps it takes, as a function of the number you start with:

Nice pattern! But this image shows how it works for numbers up to 10 million, and you’ll see it doesn’t usually take very long for them to reach 1. Usually less than 600 steps is enough!

So, to get a Turing machine that takes a long time to halt, you have to take this kind of behavior and make it much more long and drawn-out. Conversely, to analyze one of the potential winners of the Busy Beaver Game, people must take that long and drawn-out behavior and figure out a way to predict much more quickly when it will halt.

Next, BB(7). In 2014, someone who goes by the name Wythagoras showed that

\displaystyle{ \textrm{BB}(7) > 10^{10^{10^{10^{10^7}}}} }

It’s fun to prove lower bounds on BB(N). For example, in 1964 Milton Green constructed a sequence of Turing machines that implies

\textrm{BB}(2N) \ge 3 \uparrow^{N-2} 3

Here I’m using Knuth’s up-arrow notation, which is a recursively defined generalization of exponentiation, so for example

\textrm{BB}(10) \ge 3 \uparrow^{3} 3 = 3 \uparrow^2 3^{3^3} = 3^{3^{3^{3^{\cdot^{\cdot^\cdot}}}}}

where there are 3^{3^3} threes in that tower.

But it’s also fun to seek the smallest N for which we can prove BB(N) is unknowable! And that’s what people are making lots of progress on right now.

Sometime in April 2016, Adam Yedidia and Scott Aaronson showed that BB(7910) cannot be determined using the widely accepted axioms for math called ZFC: that is, Zermelo—Fraenkel set theory together with the axiom of choice. It’s a great story, and you can read it here:

• Scott Aaronson, The 8000th Busy Beaver number eludes ZF set theory: new paper by Adam Yedidia and me, Shtetl-Optimized, 3 May 2016.

• Adam Yedidia and Scott Aaronson, A relatively small Turing machine whose behavior is independent of set theory, 13 May 2016.

Briefly, Yedidia created a new programming language, called Laconic, which lets you write programs that compile down to small Turing machines. They took an arithmetic statement created by Harvey Friedman that’s equivalent to the consistency of the usual axioms of ZFC together with a large cardinal axiom called the ‘stationary Ramsey property’, or SRP. And they created a Turing machine with 7910 states that seeks a proof of this arithmetic statement using the axioms of ZFC.

Since ZFC can’t prove its own consistency, much less its consistency when supplemented with SRP, their machine will only halt if ZFC+SRP is inconsistent.

Since most set theorists believe ZFC+SRP is consistent, this machine probably doesn’t halt. But we can’t prove this using ZFC.

In short: if the usual axioms of set theory are consistent, we can never use them to determine the value of BB(7910).

The basic idea is nothing new: what’s new is the explicit and rather low value of the number 7910. Poetically speaking, we know the unknowable starts here… if not sooner.

However, this discovery set off a wave of improvements! On the Metamath newsgroup, Mario Carneiro and others started ‘logic hacking’, looking for smaller and smaller Turing machines that would only halt if ZF—that is, Zermelo–Fraenkel set theory, without the axiom of choice—is inconsistent.

By just May 15th, Stefan O’Rear seems to have brought the number down to 1919. He found a Turing machine with just 1919 states that searches for an inconsistency in the ZF axioms. Interestingly, this turned out to work better than using Harvey Friedman’s clever trick.

Thus, if O’Rear’s work is correct, we can only determine BB(1919) if we can determine whether ZF set theory is consistent. However, we cannot do this using ZF set theory—unless we find an inconsistency in ZF set theory.

For details, see:

• Stefan O’Rear, A Turing machine Metamath verifier, 15 May 2016.

I haven’t checked his work, but it’s available on GitHub.

What’s the point of all this? At present, it’s mainly just a game. However, it should have some interesting implications. It should, for example, help us better locate the ‘complexity barrier’.

I explained that idea here:

• John Baez, The complexity barrier, Azimuth, 28 October 2011.

Briefly, while there’s no limit on how much information a string of bits—or any finite structure—can have, there’s a limit on how much information we can prove it has!

This amount of information is pretty low, perhaps a few kilobytes. And I believe the new work on logic hacking can be used to estimate it more accurately!

51 Responses to The Busy Beaver Game

  1. You mentioned you wanted to see an analysis of the 6 state record holder. I don’t have one, but toward the bottom of this page, there are analyses of two previous 6-state record holders. You get a feeling at least for the kind of thing that is going on in these long-lasting state machines.

    • Robert Munafo, Large numbers – notes, 27 November 2001.

    • John Baez says:

      Cool! I think these contestants should all be put on the same Github site and checked by people who want to play this game; otherwise we just need to trust individual contestants.

      • Stefan O'Rear says:

        I think you’re slightly mischaracterizing it to say there are “contestants”. Scott Aaronson and Adam Yedidia published the 7918-state TM using Friedman’s statements and the program which generated it, and made claims about the minimum complexity of a TM using FOL directly that I disagreed with. So I wrote a proof of concept of a direct version, that couldn’t actually be evaluated because I misjudged how Adam’s compiler worked; but it still sparked interest, and Mario suggested improvements to it. I was able to get it working using a compiler of my own, resulting in the 5349-state version, then after spending a few days improving it got it to 1919 states; Peter Taylor made a few small improvements to get to 1895 states. I still haven’t used Mario’s recommendations; once I do that, it’ll shrink quite a bit (I’m guessing 1500 states, but we won’t know until it happens).

        To say “contestants” implies we are competing, but this is a cooperative project using a single codebase.

        Terminological nitpick: “Metamath” is not the name of a newsgroup. It is the name of a software program (a computer proof verifier, very broadly similar to Coq, HOL Light, Mizar, etc); the newsgroup does not have a proper name, but we call it “the Metamath group” because it exists to discuss Metamath.

        Also, ZF and ZFC are proof-theoretically essentially the same system. Each is consistent iff the other is. I tend to say ZF because I’m one of those people who gets a bad aftertaste from the axiom of choice, and the consistency checker omits it because it serves no purpose, but to describe it as “a simpler system” is a bit odd.

        As far as I am aware, nobody has tried to optimize the original SRP version using the tricks that worked well for me, so we don’t really know how wide the gap is, or in which direction(!). Harvey Friedman’s work area of “natural arithmetical statements that imply Con(ZF)” is also, IIUC, in its infancy, so there could also be substantial simplicity improvements on that front.

        I’m flattered by the continuing interest in what I did there. Let me know if I can help with general understanding of the results or anything else.

        • John Baez says:

          Thanks! I’ve tried to fix some of my slips.

          (I only described ZF as simpler because it has fewer axioms. But what the heck, I might as well tell everybody what it is.)

  2. John Baez says:

    If anybody made the mistake of reading this article more than an hour ago, I apologize: it wasn’t quite done. It should be better now.

    (The time stamps here use Greenwich Mean Time, which is basically the same as Coordinated Univeral Time, or UTC.)

  3. Layra Idarani says:

    BB(n) as described here is actually counting the number of 1s the machine prints starting from all 0s, rather than the number of steps it takes. For instance, the 2 state machine given in the post takes 13 steps and prints 6 1s.

    • John Baez says:

      There are two functions people look at:

      \Sigma(n) is the maximum number of 1’s printed by an n-state Turing machine that halts.

      S(n) is the maximum number of steps an n-state Turing machine can take before it halts.

      People usually use BB(n) as an abbreviation for \Sigma(n), but Scott Aaronson has taken to using BB(n) as an abbreviation for S(n), including in his recent blog article, where he writes:

      Recall that BB(n), or the nth Busy Beaver number, is defined to be the maximum number of steps that any n-state Turing machine takes when run on an initially blank tape, assuming that the machine eventually halts.

      So that’s what I’m trying to do in this article, but I may have gotten mixed up at times. If so, I’ll try to fix it.

      • John Baez says:

        Actually it looks like the values of BB(n) that I listed here are consistent with the definition I gave, namely BB(n) = Σ(n). If there are inconsistencies, I’d like to hear about them, so I can fix them.

        • Layra Idarani says:

          So there must be an issue somewhere, because the machine you printed definitely does not halt in 6 steps. It halts in 13 steps, and prints 6 1s, making it the 3-state BB machine by most definitions I’ve come across. In particular, I don’t think the C as denoted by the machine you’ve drawn counts as the halt state.

        • John Baez says:

          Oh! The problem is that I put the picture of the 3-state Turing machine in the wrong place! I said it has 3 states, but I stuck it after my discussion of the 2-state winner. I’ll fix that.

        • Layra Idarani says:

          There’s still an issue, in that the machine listed wins the “most 1s” game, but not the “longest running” game! It prints 6 1s in 13 steps. In fact, according to Pascal Michel’s page, the machines that run for 21 steps only print 5 1s.

        • John Baez says:

          Okay. Darn, then I need to remove that picture.

  4. Todd Trimble says:

    It looks like one of your formulas should read BB(2N) \geq 3 \uparrow^{N-2} 3, not BB(2N) \geq 3 \uparrow^{k-2} 3.

  5. Craig says:

    When Scott published his blog post, I thought a bit about how one might go about explaining this puzzle to people unfamiliar with Turing machines (e.g., smart high school students, since I had just such a talk coming up). Unfortunately, I couldn’t think of a good way to do it.

    I think it’s easy to explain the Halting Problem in a high-level way, since it can be re-expressed in a high-level programming language. But as defined, the Busy Beaver problem depends more directly on the “hardware” of Turing Machines.

    Is there a compromise? What’s the most natural way to get a BB-like problem in, say, the Lambda Calculus? Or can I define a tiny subset of a language like Python and define \mathrm{BB}_\mathrm{py}(N) to be the most output I can create from an N-line program in this Python subset?

    • John Baez says:

      Hi! You can take whatever programming language you like, from COBOL to Python to the lambda calculus, and define a version of BB(n) to be either the maximum number of steps that a halting program written with N symbols can run, or the maximum number of symbols that a halting program written with N symbols can print. As long as your programming language lets you compute all Turing-computable functions and nothing more—and all the ones I listed have this property—the fundamental theorems will always be true:

      • BB(n) will grow faster than any computable function.

      • You’ll eventually hit a value of n such that BB(n) cannot be proved to have any particular value using the usual axioms of set theory… or indeed, any recursively enumerable axiom system in first-order logic… as long as those axioms are a consistent extension of Peano arithmetic.

      So yes, for this you can use whatever programming language your audience likes best!

      But the exact value of BB(n) will, of course, depend on the details. And this particular post was all about the exact value when we use 2-symbol Turing machines: what we know about it, what we don’t know about it, and what we ‘can’t know’ about it. So I had to give a crash course on Turing machines.

      By the way, you have to be careful about “the most output you can create from an n-line program”, since if the lines can be arbitrarily long, this could be undefined.

      • Craig says:

        Yes, that’s sort of what I’m getting at. Obviously there’s a busy beaver style problem for every (Turing complete) programming language, expressed very simply in terms of the number of symbols you use. But that’s not as satisfying to contemplate as it is with pure Turing machines. To begin with, it takes a lot longer before you get any output at all, because of the overhead of writing any non-trivial program. That doesn’t invalidate the process, but it makes it less fun to play with. Second, with an expressive language I feel like it becomes more of an exercise of “who can name the largest number” than a process of unraveling the esoteric behaviour of (say) a 6-state TM that nevertheless manages to run on and on.

        Inasmuch as Busy Beaver is a game or puzzle, it needs to be expressed in a language that makes the game fun to play.

        • John Baez says:

          Yes, and because there’s a long tradition of playing this game with certain rules, it’s less satisfying to play it when you change the rules. It’s sort of like doing the 85-meter dash.

          This paper has a quite complete history up to February 2016:

          • Pascal Michel, The Busy Beaver Competition: a historical survey.

          The author keeps updating it, and it’s on its fourth update.

  6. John Baez says:

    Ah, I love it when Hacker News notices my blog. The number of hits per hour spikes up:

    I get the feeling there’s a vast horde of bored programmers out there, just waiting for entertainment.

  7. Dave Tweed says:

    So this is a bit nit picky but: if we’ve got an n-state Turing machine encoding of a ZFC contradiction searcher, isn’t it the case that BB(n) is unknowable if all we know is ZFC unless we run it and it actually stops within an amount of time we’re prepared to wait (which it can only do it ZFC is inconsistent of course). If that happens we likely still don’t know BB(n) since there are probably lots of other programs if length n we don’t know how to prove termination of, but one obstacle is gone.

    • Stefan O'Rear says:

      That’s an important nit, and a well-known one. Almost everything Gödelian requires an assumption of “if X, and Y, and your logic is consistent”, because if ZFC is inconsistent, then ZFC will prove Con(ZFC) (and everything else).

      The OP gave that caveat twice that I could find:

      Indeed, matter what axiom system you use for math, as long as it has finitely many axioms and is consistent,

      However, we cannot do this using ZF set theory—unless we find an inconsistency in ZF set theory.

    • John Baez says:

      Dave wrote:

      So this is a bit nit picky but:if we’ve got an n-state Turing machine encoding of a ZFC contradiction searcher, isn’t it the case that BB(n) is unknowable if all we know is ZFC unless we run it and it actually stops within an amount of time we’re prepared to wait (which it can only do it ZFC is inconsistent of course).

      Yes, I think I made that type of point here:

      Thus, if O’Rear’s work is correct, we can only determine BB(1919) if we can determine whether ZF set theory is consistent. However, we cannot do this using ZF set theory—unless we find an inconsistency in ZF set theory.

      I should add that most mathematicians are far more scared of being hit by an asteroid when they walk out the door than discovering an inconsistency in ZFC, so you’ll often find statements that omit the clause “unless ZFC is inconsistent”. I did that here:

      In short, BB(7910) is unknowable if all we know is ZFC.

      Perhaps you are reacting against that momentary lack of caution. Since I pride myself on being clear about precisely this issue, I’ll change my statement to this:

      In short: if the usual axioms of set theory are consistent, we can never use them to determine the value of BB(7910).

      Less flashy, but more accurate.

      By the way, it’s even possible that the propositional calculus is inconsistent. There are proofs that it’s not… but if the propositional calculus is inconsistent, we can’t trust anything, not even these proofs!

      In practice, every logician I know wisely acts like the propositional calculus is consistent—and most mathematicians scold me for even mentioning the possibility that it’s not. It’s not something I lose sleep over, but I think it’s interesting.

      • davetweed says:

        Thank Stefan and John. My point was more along the lines that running the machine might also give you some information (but was unlikely to since it would only do that if ZFC was inconsistent and that inconsistency was small.) I’m not a good enough logician to know if there are any of the statments which are independent of ZFC which can actually be phrased as search problems which might stop in finite time, but if they exist they’d be unprovable-to-terminate if you only have ZFC unless running them actually reveals a stop after a short time.

      • John Baez says:

        Dave wrote:

        My point was more along the lines that running the machine might also give you some information (but was unlikely to since it would only do that if ZFC was inconsistent and that inconsistency was small.)

        True. Part of the problem is that in this game people are trying to minimize the number of states in their Turing machines, not minimize the run time before a contradiction is spotted if one exists. If you actually wanted to find a contradiction before our civilization ends, you’d write a different sort of program.

        Here’s something else I wanted to add. If we really believe ZFC is consistent, we can add the axiom Con(ZFC) to ZFC and get an axiom system that proves that a Turing machine that searches for an inconsistency in ZFC never halts. This does not necessarily enable us to determine the value of BB(1919) or BB(7910)! However, it raises a curious question: could some set of plausible extra axioms enable us to determine the value of BB(n) for some n where we’d otherwise be stuck?

        A concrete example might be BB(5). The obstacle to determining its value is that there are about 40 Turing machines with 5 states that seem to run forever… but we don’t know how to prove they do. If we thought hard about these, might we either settle these cases using ZFC or invent plausible new axioms that settle them?

        Possibly… but I doubt it will go that way. We could, if we wanted, add the Riemann Hypothesis or the Collatz Conjecture as an extra axiom to ZFC. But we probably won’t—not unless they’re proved independent of ZFC.

        I’m not a good enough logician to know if there are any of the statements which are independent of ZFC which can actually be phrased as search problems which might stop in finite time.

        The only statements of this form that leap to mind are “ZFC + X is inconsistent” for various axioms X. We can always search for a proof of 0=1 in ZFC + X.

      • John Baez says:

        John wrote:

        If we really believe ZFC is consistent, we can add the axiom Con(ZFC) to ZFC and get an axiom system that proves that a Turing machine that searches for an inconsistency in ZFC never halts. This does not necessarily enable us to determine the value of BB(1919) or BB(7910)! However, it raises a curious question: could some set of plausible extra axioms enable us to determine the value of BB(n) for some n where we’d otherwise be stuck?

        The answer is yes! By starting with ZFC and repeatedly adding axioms that say “all my previous axioms are consistent”, and doing this infinitely many times, we can eventually get axioms that determine BB(n) for all n!

        In fact we don’t even need to start with ZFC: Peano arithmetic will suffice!

        There’s a nice discussion of this here:

        Π1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.?, MathOverflow.

        It’s fairly subtle, since we need to use transfinite induction and choose an indexiing scheme for countable ordinals, and the results can depend on our indexing scheme. I would like to learn this and explain it sometime.

        • Dave Tweed says:

          Wow! Throughout this whole topic I’d been unsure about the dual role being played by ZFC (or ZF, or PA) as both the means by which we prove termination/non-termination and also being encoded in what the Turing machine is computing. It wasn’t clear to me if this was considering excessively complete programs and that there might not be much simpler programs which aren’t provable to terminate under any reasonable theory.

          But the above is talking about determining BB(n) — ie behaviour of all programs of length n — from this infinitely augmented theory, not just the behaviour of a program of length n derived from ZFC (or one of its augments).

        • John Baez says:

          Dave wrote:

          It wasn’t clear to me if this was considering excessively complete programs and that there might not be much simpler programs which aren’t provable to terminate under any reasonable theory.

          The ‘logic hackers’ have found a 1919-state Turing machine whose termination cannot be decided using ZFC. There might be much simpler programs whose termination cannot be decided using ZFC. Somewhere between 5 and 1919 lies the smallest N such that there’s an N-state Turing machine whose termination is undecidable in ZFC. This is also the smallest N such that BB(N) cannot be proven to have the value it has using ZFC. (All this is assuming ZFC is consistent!)

          It’s an annoyingly large gap, but shrinking it a lot will probably require new ideas, not just writing more clever programs that check ZFC for contradictions.

          As far as I can tell, nobody has a clue whether or not we’ll ever be able to find the smallest N such that there’s an N-state Turing machine whose termination is undecidable in ZFC.

          But all this is about ZFC, not “any reasonable theory”.

          The logic hackers tried asking the same questions for PA. To find some number N such that BB(N) cannot be proved to have the value it has using PA, it suffices to create an N-state Turing machine that seeks a contradiction in PA. However, even though PA seems simpler than ZFC in some respects, it’s apparently more complicated in other ways, which have so far made this number N larger than 1919.

          Some of the theories I described, where we take ZFC or PA and repeatedly add axioms that say “all my previous axioms are consistent”, infinitely many times, are not exactly what I’d call “reasonable” axioms. They’re believable, but I don’t think it’s possible to work with them in practice. The fact that they can be used to settle the Busy Beaver problem is shows that no computer program can enumerate all the consequences of these axioms.

        • Royce Peng says:

          Hmm, if I understand the discussion correctly, the problem is that once you get to transfinite ordinals the method you use to code up the ordinal can contain lots of information, so you can do some prestidigitation to be able to prove every \Pi_2 statement. I guess one might then ask how much one can prove if one uses “natural” representations of ordinals, and don’t use any “tricks to code up extra information”. But then the problem is that the terms in quotes are not well-defined.

      • John, the statement that the propositional calculus could be inconsistent is very interesting (almost disturbing :-)). Is there anything more to be said about it, aka are there any papers about it?

        • John Baez says:

          I don’t know any papers about it. There should be some philosophy and logic papers about the concept of “logical certainty”, and the question of when we can trust a consistency proof: they are well-known issues that have been discussed for a long time. But I don’t know any papers raising doubts about the usual consistency proofs for the propositional calculus.

          Personally I think the consistency of very fundamental principles like the propositional calculus, the predicate calculus and Peano arithmetic is widely believed for the following reasons: 1) they haven’t yet led to any contradictions, and 2) we can prove certain subsets of these principles won’t lead to contradictions if we assume certain other subsets won’t lead to contradictions. So, everyone has decided that these principles are consistent. And I think that’s fine — as long as we notice what we are doing!

        • Bhupinder Singh Anand says:

          The following paper, due to appear in the December issue of Cognitive Systems Research, gives a finitary proof of consistency for the first-order Peano Arithmetic PA as sought by Hilbert in the second of his twenty three Millenium ICM-1900 problems:

          The truth assignments that differentiate human reasoning from mechanistic reasoning: The evidence-based argument for Lucas’ Gödelian thesis

          Ergo, the first-order predicate calculus FOL too is consistent in Hilbert’s sense, which is how I believe the term ‘consistency’ is usually understood when referring to a formal mathematical system.

          If the propositional calculus is ‘inconsistent’, it may be in some sense of the word ‘consistent’ other than that which I believe Hilbert had in mind.

        • John Baez says:

          I’m just saying that if there’s a derivation of F = T in the predicate calculus, we can probably derive almost anything in any system of logic, so we can’t trust the arguments usually given to prove the consistency of mathematical systems.

  8. […] The Busy Beaver Game, and the limits between the knowable and the unknowable […]

  9. Ah the Busy Beaver. My favorite beaver. I did an article about Busy Beaver about 10 years ago with graphs of how it walks and other interesting info:

    http://www.catonmat.net/blog/busy-beaver/

  10. arch1 says:

    “…there are about 40 [BB(5)] machines that do very complicated things that nobody understand. It’s believed they never halt, but nobody has been able to prove this.”

    I see from the referenced historical survey that the number of “holdouts” had been reduced to ~100 (out of ~63 trillion, right?) by 1990 via a combination of machine and manual analyses, so presumably an additional 60 machines were proven not to halt in the ensuing 26 years. Can anyone describe the basis for the belief that the remaining 40 never halt? Also I am curious as to recent progress in identifying non-halters: Is it fair to say that it’s stalled, or are successive cases still being resolved?

    • John Baez says:

      Thanks for digging into this. I think the details will actually be more interesting than the “world records” I listed in my post. What have people done to study those 40 holdouts that prevent us from computing BB(5)? It’s bound to be interesting.

      Unfortunately many of the people trying to understand BB(n) seem to be “hackers”, more interested in solving problems than writing nice expository papers.

      However, this survey:

      • Pascal Michel, The Busy Beaver Competition: a historical survey.

      says:

      The study of Turing machines with 5 states and 2 symbols is still going on. Marxen and Buntrock (1990), Skelet, and Hertel (2009) created programs to detect never halting machines, and manually proved that some machines, undetected by their programs, never halt. In each case, about a hundred holdouts were resisting computer and manual analyses. See Skelet’s study in

      http://skelet.ludost.net/bb/index.html

      Daniel D. Briggs created a website and a forum dedicated to this study: see

      http://web.mit.edu/~dbriggs/www

      So those are the places to start! Unfortunately I can’t find Briggs’ website or forum, even with a little creative googling and the Wayback Machine. So, Skelet is the go-to guy for this.

    • John Baez says:

      Skelet writes:

      My program, called bbfind enumerates the Turing machines and then tries to prove their infinitness. As a side effect stoping machines are identified and checked for BB record.

      Current version properly evaluates S(n) for class TM(4) and important subclass of TM(5), called RTM(5) (reversal turing machines with 5 states).

      The function S(n) is the more technical name for what I’m calling BB(n) in this post. TM(5) is the set of 5-state Turing machines, which is what arch1 was asking about.

      For class TM(5) the program leaves 164 machines unproven (some are isomorphic to other by trivial reasons).

      This is larger than the number “about 40” that I mentioned in my blog post, which I got from the Wikipedia article, which says: “The current 5-state busy beaver champion produces 4,098 1s, using 47,176,870 steps (discovered by Heiner Marxen and Jürgen Buntrock in 1989), but there remain about 40 machines with non-regular behavior which are believed to never halt, but which have not yet been proven to run infinitely”.

      The Wikipedia article refers to Skelet for this fact, and indeed, later on this page Skelet mentions that he’s been unable to determine whether 43 machines halt or not. I guess he uses his program bbfind for an initial analysis and then works harder!

      Current version works slowly, and full scanning for TM(5) may take 1 or 2 weeks.

      For class TM(5) the program enumerates about 150M machines.

      I don’t know where that number comes from. The total number of n-state Turing machines should be

      (4(n+1))^{2n}

      and thus for n = 5,

      24^{10} = 63,403,380,965,376

      which is indeed about 63 trillion, as arch1 wrote. However, experts know ways to rule out most of these right away!

      • arch1 says:

        Thanks John. I don’t know where the 150M comes from but Lin and Rado gave the formula (2N-1)4N^(2N-2) for a smaller subset one can focus on for BB purposes.

        For BB(5) this gets one from ~6.3 trillion machines down to (exactly) 230.4 billion (2.304e11). Some of the details are sketched here under Robert Munafo’s Large Numbers page.

        (OFF-TOPIC: Munafo’s home page looks pretty amazing; avoid if looming deadlines:-).

    • John Baez says:

      On his webpage, Skelet groups the 5-state Turing machines into various kinds, which I don’t understand, because he uses a lot of Turing machine notation to define them.

      1. k-linear sequences.

      a. Linearly increasing sequences: the non-halting of some of these machines can be proved by his procedure TryL1Cnt, resolving about 95% of the nontrivial machines.

      b. Collatz-like sequences: the non-halting of some of these machines can be proved by his procedure TryModFin, but all the record-holding machines (which are known to halt) are in this class.

      c. Linear-binary sequences: the non-halting of some of these machines can be proved by his procedure TryBL_Proof.

      1. Nonlinear sequences.

      a. Fixed positions counter: the non-halting of some of these machines can be proved by his procedure TryPFin_2.

      b. Swimming head counter: whether or not these machines halt can be proved by his procedure “Chaotic Huffman Test”, which has something to do with Huffman codes.

      1. Other types (nonregular).

      These are machines that do mysterious things. There are 164 of them. Of the machines whose halting behavior is still unknown, all 43 lie in this class, which he discusses on another page.

  11. domenico says:

    I don’t understand, the halt command of the Turing machine is obtained with an inaccessible tape with only 0, and a terminal tape with 2^N possible simbols with last symbol printed that is 0 or 1, and a second last state.
    If it is chosen a single state, and a simbol, then there is a Turing machine halt; but if the terminal tape is enough great, then can be impossible the halt state for each possible string of symbol on the tape, and for each initial state (if this happen the machine does not stops): I think that the number of possible terminal string is less that the number of possible steps of the Turing machine, so that can be tested if it stops; but there must be a my mistake somewhere.

    • John Baez says:

      I don’t quite understand what you’re saying, but in the Busy Beaver Game the tape starts with all 0’s on it. Does that help?

      • domenico says:

        I try to think on the final state of the Turing machine, then the final tape is a string filled of 1s and 0s until the terminal symbol that is 1 or 0, then the remaining tape is full of 0s (the position where the head of Turing machine don’t write before the halt); then the Turing machine halt, in a possible inner point.
        It is possible to study the final part of the Turing machine (like an asymptotic behaviour of a function), then it is possible to choose a initial cell (near the terminal part of the tape), an initial state, and to verify if the Turing machine halt (to search if the final string is a terminal string) I think that the number of steps is lower of the number of the steps of the full sequence (although the halting cell could be in the middle of the tape): it is not possible to verify the number of steps of the Turing machine, but it is possible to verify if there are sequences that give the halt (and one of this is a possible busy beaver game), and to know the final values (like the asymptotes).
        The symbol on the tape change with the dynamic, but if one consider all the possible final strings, one of these must be the terminal string (with different initial state) with minimum step to obtain the halting (for example, if one chooses a cell, a terminal string, a inner state, then there may be a few of step to verify if there is the halt).

        • domenico says:

          I am thinking that there are many Turing machine that have the same BB(N). For example a machine that make the same operation to the right instead of the left, or if there is an initial tape full of 1s (an arbitrary redefinition of starting tape and there is a busy beaver game that have the same dynamic with opposite symbols), or a Turing machine with permutation of state symbols (or a combination of all these conditions); but if this is a dynamic, so is possible increase the BB(N) with an arbitrary initial condition, an initial string with some 1s and 0s, of finite lenght? If the initial string is not a string that the Turing machine write on the tape starting from the empty tape, then it is possible to increase the length of BB(N).
          If the Turing machine does not halt, then there is a periodicity in the written string with written states (another tape).

  12. Greg Egan says:

    Slightly off topic, but the latest article in Quanta describes a recent result showing that anything that can be proved using the infinite Ramsey theorem applied to pairs can also be proved without assuming the existence of any infinite sets. But for the infinite Ramsey theorem applied to triples, this isn’t true!

    • John Baez says:

      That’s interesting! The article says:

      But then in January, Patey and Yokoyama, young guns who have been shaking up the field with their combined expertise in computability theory and proof theory, respectively, announced their new result at a conference in Singapore. Using a raft of techniques, they showed that \mathrm{RT}_2^2 is indeed equal in logical strength to primitive recursive arithmetic, and therefore finitistically reducible.

      The article nicely explains what this statement \mathrm{RT}_2^2 says; since it concerns arbitrary infinite sets I presume it’s the consequences of \mathrm{RT}_2^2 for arithmetic that when combined with very weak axioms are equivalent to primitive recursive arithmetic.

      Primitive recursrive arithmetic or PRA is a set of axioms for arithmetic much weaker than Peano arithmetic, which doesn’t even include the quantifiers \forall and \exists.. Let me take this opportunity to help myself remember it. According to Wikipedia:

      The language of PRA consists of:

      • A countably infinite number of variables x, y, z,….

      •The propositional connectives;

      • The equality symbol =, the constant symbol 0, and the successor symbol S (meaning add one);

      • A symbol for each primitive recursive function.

      The logical axioms of PRA are the:

      • Tautologies of the propositional calculus;

      • The usual axiomatization of equality as an equivalence relation.

      The logical rules of PRA are modus ponens and variable substitution. The non-logical axioms are:

      • S(x) ≠ 0;

      • S(x)=S(y) ⇒ x=y,

      and recursive defining equations for every primitive recursive function as desired.

      So, very simple!

      The Ackermann function is computable but not primitive recursive; no primitive recursive function can grow that fast, so PRA is not only finitistic but also limited in what functions it can talk about.

      In fact, I just read that a function

      f : \mathbb{N} \to \mathbb{N}

      is primitive recursive iff there is a natural number m such that f(n) can be computed by a Turing machine that always halts within A(m,n) or fewer steps, where A(m,n) is the Ackermann function! So, there’s a nice connection between primitive recursive arithmetic and the Busy Beaver Game!

      The idea of having a separate symbol for every primitive recursive function sounds like a mess but it’s just a way of avoiding the need to set up a framework to define new functions. There’s even a way to describe PRA that doesn’t mention logical connectives at all!

      We can prove PRA is consistent if we can do induction up to \omega^\omega. By comparison, to prove Peano arithmetic consistent we need to do induction up to \epsilon_0.

      The article touches on, but doesn’t dwell on, the big project of classifying which statements can be proved in which kinds of arithmetic. It’s called reverse mathematics, and among people working on ‘conventional’ foundations of mathematics—that is, not topos theory, homotopy type theory or other category-flavored approaches—this project has become quite popular.

  13. nad says:

    We have a sort of ongoing project where we explore the use of the human body for “programming” or “steering” tasks, which is intended to go beyond that what you can do with a screen and typical input devices (which mainly use audio and visual interaction and “finger tapping or swishing” i.e. interactions with keyboard or mouse or mousepad) ). In our recent experiment you can program a 2 symbol Turing machine with hand gestures:
    http://www.randform.org/blog/?p=6184
    The application itself (programmed by Tim) is however intended to be also used with other sensory input (the most banal is of course again the keyboard, or buttons and mouse, and in fact those were of course used during prototyping the application). The experiment is summarized in a blog post:
    http://www.randform.org/blog/?p=6184
    Some example of “dance gestures” are also given there.

You can use Markdown or HTML in your comments. You can also use LaTeX, like this: $latex E = m c^2 $. The word 'latex' comes right after the first dollar sign, with a space after it.

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s