Combinators: A Centennial View

356 Pages·2021·63.318 MB·English

by  Stephen Wolfram

COMBINATORS
A CENTENNIAL VIEW
STEPHEN WOLFRAM
COMBINATORS
A CENTENNIAL VIEW
STEPHEN WOLFRAM
Combinators: A Centennial View
Copyright © 2021 Stephen Wolfram, LLC
W olfram Media, Inc. | wolfram-media.com
ISBN-13: 978-1-57955-043-1 (hardback)
ISBN-13: 978-1-57955-044-8 (ebook)

Mathematics/Science
Cataloging-in-publication data available at:
Library of Congress Cataloging-in-Publication Data
Names: Wolfram, Stephen, author.
Title: Combinators: a centennial view / Stephen Wolfram.
Description: Champaign: Wolfram Media, Inc., [2021] | Includes
bibliographical references and index.
Identifiers: LCCN 2021004419 (print) | LCCN 2021004420 (ebook) | ISBN
9781579550431 (hardback) | ISBN 9781579550448 (ebook)
Subjects: LCSH: Combinatorial analysis. | Computer science – Mathematics. |
Wolfram Language (computer program language) | AMS: Combinatorics –
Historical. | Combinatorics – Explicit machine computation and programs
(not the theory of computation or programming). | Computer science –
Discrete mathematics in relation to computer science – Combinatorics in
computer science. | Computer science – Discrete mathematics in relation
to computer science – Graph theory (including graph drawing) in
computer science.
Classification: LCC QA164.W65 2021 (print) | LCC QA164 (ebook) | DDC
511/.60285–dc23
LC record available at https://lccn.loc.gov/2021004419
LC ebook record available at https://lccn.loc.gov/2021004420
For information about permission to reproduce selections from this book, contact [email protected]. Sources for photos
and archival materials that are not from the author’s collection or in the public domain:
p. 167: Georg Olms Hildesheim; p. 168: Edizioni Cremonese; pp. 171, 173, 175–6, 178–9, 213, 215, 241–2, 244–7: Mathematische
Annalen, Springer Nature; p. 184: Monatshefte für Mathematik Physik, Springer Nature; pp. 186–7: Annals of Mathematics,
Johns Hopkins University Press; pp. 187, 268–71: Am. J. of Mathematics, Johns Hopkins University Press; p. 188: J. of Symbolic
Logic, Cambridge University Press; p. 193: Communications of the ACM, Cambridge University Press; p. 195: Raymond
Smullyan, Oxford U. Press; pp. 196, 272: North-Holland Pub. Co. (Cambridge University Press); pp. 213, 231–5, 240, 243,
254: Göttingen State and University Library; p. 217: Odessa State Archive, Ukraine; pp. 218, 264, 282: State Archives
of Dnepropetrovsk Region, Ukraine; p. 219: ©Map Data 2015 Google; pp. 220–1: Dneiper City Encyclopedia, Ukraine;
pp. 221–7: Odessa State Archive, Ukraine; p. 230: Cem Bozşahin, Stadtarchiv Göttingen; p. 238: Stadtarchiv Göttingen;
p. 239: Stadtarchiv Göttingen (modified); p. 240: Goettinger-Tageblatt.de; pp. 248–252: Dr. Ludwig Bernays and ETH
Zurich, Bernays Archive; p. 262: Deutsches Biographisches Archiv; p. 262: Staatsarchiv des Kantons Bern; p. 263: Semih
Baskan, Ankara University; p. 265: Kim Heffernan; pp. 265–8: Haskell P. Curry papers, Pennsylvania State University; p.
283: Elena Zavoiskaia and Central State Archive of Moscow; p. 283: gwar.mil.ru; p. 284: pamyat-naroda.ru; p. 285: Daily
Express/Reach Licensing.; p. 286: Osprey Publishing; pp. 287–8: world-war.ru/zabvenie-tradicii-baxrushinyx
Typeset with Wolfram Notebooks: wolfram.com/notebooks
Printed by Friesens, Manitoba, Canada. Acid-free paper. First edition. First printing.
00a-hbbook-copyright.pdf 1 4/30/21 8:36 AM
CONTENTS
Preface vii
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
Combinators: A Centennial View • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 3
Ultimate Symbolic Abstraction 3 • Computing with Combinators 6 • A Hundred Years Later... 12 •
Combinators in the Wild: Some Zoology 17 • Visualizing Combinators 34 • Updating Schemes and
Multiway Systems 54 • The Question of Evaluation Order 72 • The World of the S Combinator 87 •
Causal Graphs and the Physicalization of Combinators 102 • Combinator Expressions as
Dynamical Systems 120 • Equality and Theorem Proving for Combinators 125 • Lemmas and the
Structure of Combinator Space 133 • Empirical Computation Theory with Combinators 145 •
The Future of Combinators 157 • Historical & Other Notes 159
Combinators and the Story of Computation 165
• • • • • • • • • • • • • • • • • • • • • • • • •
The Abstract Representation of Things 165 • What Is Mathematics—and Logic—Made Of? 167 •
Combinators Arrive 171 • What Is Their Mathematics? 181 • Gödel’s Theorem and Computability 183 •
Lambda Calculus 186 • Practical Computation 191 • Combinators in Culture 195 • Designing Symbolic
Language 198 • Combinators in the Computational Universe 203 • Combinators All the Way Down? 208
Where Did Combinators Come From?
Hunting the Story of Moses Schönfinkel • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 213
December 7, 1920 213 • Who Was Moses Schönfinkel? 215 • The Beginning of the Story 217 •
Going to College in Odessa 222 • Göttingen, Center of the Mathematical Universe 228 •
Problems Are Brewing 234 • The 1924 Paper 241 • The “1927” Paper 247 • To Moscow and
Beyond… 257 • Other Schönfinkels... 262 • Haskell Curry 265 • Schönfinkel Rediscovered 274 •
What Should We Make of Schönfinkel? 275
A Little Closer to Finding What Became of
Moses Schönfinkel, Inventor of Combinators • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 281
1920, 2020 and a $20,000 Prize:
Announcing the S Combinator Challenge • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 297
Hiding in Plain Sight for a Century? 297 • The Basic Setup 300 •
The Operation of the S Combinator Challenge 303
Excerpts from A New Kind of Science (2002)• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 307
Section 10: Symbolic Systems 307 • Notes 310 • Section 12: Universality in Turing Machines and
Other Systems 315 • Notes 319
A Bibliography of Combinators
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 325
Foundational Documents 325 • Books 325 • Surveys & Summaries 326 • Combinators as
Symbolic Expressions 327 • Combinators as Mathematical Constructs 334 • Combinator Computation 341 •
Extensions & Applications 350 • Confusing Issues 352
Index • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 357
Preface
Iʼdhaditinmycalendarformorethantwentyyears:December7,2020—thecente‐
naryofcombinators.Buthowshoulditbecelebrated?Iʼdfirstlearnedabout
combinatorsmorethanfortyyearsago—andafewtimesIʼdexplicitlyusedthem.
Butthey—andtheirorigins—hadalwaysmaintainedacertainairofmysteryforme.
SoafewmonthsagoIdecidedthatthewayIwantedtocelebratetheircentenary
wasfinallytodomybesttounderstandhowcombinatorswork,andhowthey
cametobeinvented.Thisbookistheresult.
AndontheactualdayofDecember7,2020—exactlyatthehourahundredyears
a�erMosesSchönfinkelgavehistalkaboutcombinators—Iwasverypleasedto
beabletohostanonlineeventwhichbroughttogetheralargefractionofthe
livingworldexpertsoncombinators.
Combinators—andtheirinvention—areataleofbothtriumphandtragedy.They
definedavisionofcomputationthathasaprofoundandtimelessabstractelegance.
Butevenacenturylatertheycontinuetoseemexoticandabstruse—andsomehow
fundamentallyunsuitableforhumanconsumption.Inadifferentversionofhistory
thecomputerrevolutionmighthavebeenbasedonmolecularratherthanelec‐
troniccomputing,andcombinatorsmighthavebeenatthecenterofit.Butasitis,
ourcomputersaremadeoflogicgates,andtherearenʼtanycombinatorsinsight.
Butisthatreallytrue?AsIworkedoncombinatorsinpreparingfortheircente‐
nary,Irealizedthatcombinatorshavehadamuchmoreprofoundeffectonat
leastmythinkingaboutcomputationthanIʼdeverimagined.Forinthemorethan
fortyyearsthatIʼveworkedtowardsthefull‐scalecomputationallanguagethatis
ourmodernWolframLanguage,thefoundationofitallhasbeentheideaof
representingeverythingintermsoftransformationsofsymbolicexpressions.And
Inowrealizethatultimatelythatidea—andeverythingthathasflowedfromit—
canbetracedbacktocombinators,andtheirintroductiononDecember7,1920.
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COMBINATORS:ACentennialView
Combinatorsforeshadowedanotherbigpartofmylifetoo.Intryingtogener‐
alizetheconceptofmathematicalmodels,Ibeganmanyyearsagotoexplore
thecomputationaluniverseofsimpleprograms—anddiscoveredthateven
extremelysimpleprogramscangenerateimmenselycomplexbehavior,ofthe
kindwesee,forexampleinnature.MostoftheprogramsIʼvestudiedoverthe
yearsarenʼtspecificallylikecombinators.Butinpreparingforthecentenaryof
combinatorsIdecidedtoapplythemethodsIʼvehonedinexploringthecompu‐
tationaluniversetocombinators—anddiscoveredthatmanyofthestrongest
phenomenaIʼvefoundinthecomputationaluniversehavebeenlurkinglargely
unnoticedincombinatorsforacentury.
WhenIfirstputthecombinatorcentenaryonmycalendartwentyyearsago—even
thoughIdidnʼtconnectthemasmuchwithcombinatorsasIdonow—Ialready
knewaboutsymbolicexpressionsandcomputationallanguage,andaboutwhat
happensinthecomputationaluniverseofsimpleprograms.But—thoughIhad
somepremonitionsmorethantwentyyearsago—itʼsonlybeeninthelastyearand
ahalfthatIʼvecometorealizewithincreasingcertaintythatourphysicaluniverse
isfundamentallycomputational.
Itʼsgivenmeanewwaytothinkaboutcomputation,deeplyinterwovennot
onlywithphysicsbutalsowithareaslikemetamathematics.Anditʼsmademe
seethatcombinators—withsomeoftheirhard‐to‐understandfeatures—
fundamentallyreflect someofhowcomputationrelatestophysicsandour
experienceofthephysicalworld.
Iʼmsogladtohavehadtheimpetusfromtheircentenarytogettoknowcombina‐
torsalittlebetter,andtobeabletosharewhatIʼvelearnedinthisbook.Ithink
thereʼslittledoubtthatthegeneraldevelopmentofcomputationhasbeenthe
greatestintellectualachievementofthepastcentury.Butinthatendeavor,
combinatorshavebeentheroadnottaken.IhopeherethatIʼmabletoexplorea
littleoftheremarkableworldofcombinators,andwhatmightlieaheadwith
themnowthatwebegintoseewhatʼspossible.
April30,2021
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Combinators:

A CentennialView

Ultimate Symbolic Abstraction

BeforeTuringmachines,beforelambdacalculus—evenbeforeGödel’s theorem—

therewerecombinators.Theyweretheveryfirstabstractexamplesevertobe

constructedofwhatwenowknowasuniversalcomputation—andtheywerefirst

presentedonDecember7,1920.Inanalternativeversionofhistoryourwhole

computinginfrastructuremighthavebeenbuiltonthem.Butasitis,fora

century, theyhaveremainedforthemostpartakindofcuriosity—andapinnacle

ofabstraction,andobscurity.

It’s nothardtoseewhy. Intheiroriginalformfrom1920,thereweretwobasic

combinators, sandk,whichfollowed thesimplereplacement rules(nowrepre-

sentedverycleanly intermsofpatternsintheWolfram Language):
s[x_][y_][z_]x[z][y[z]]
k[x_][y_]x

Theideawasthatanysymbolicstructurecouldbegeneratedfromsomecombina-

tionofs’sandk’s.Asanexample, considera[b[a][c]].We’re notsayingwhata,b

andcare;they’rejustsymbolicobjects.Butgivena,bandchowdoweconstruct

a[b[a][c]]?Well, wecandoitwiththes,kcombinators.

Considerthe(admittedlyobscure)object
s[s[k[s]][s[k[k]][s[k[s]][k]]]][s[k[s[s[k][k]]]][k]]

(sometimesinsteadwrittenS(S(KS)(S(KK)(S(KS)K)))(S(K(S(SKK)))K)).
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COMBINATORS:ACentennialView

Nowtreatthislikeafunctionandapplyittoa,b,c [a][b][c].Then
s[s[ k[s] ][s[k[k]][s[k[s]][k]]]][s[k[s[s[k ][k]]]][k]]
watchwhathappenswhenwerepeatedlyusethes,kcombinatorreplacementrules:
s[s[k[s]][s[k[k]][s[k[s]][k]]]][s[k[s[s[k][k]]]][k]][a][b][c]
s[k[s]][s[k[k]][s[k[s]][k]]][a][s[k[s[s[k][k]]]][k][a]][b][c]
k[s][a][s[k[k]][s[k[s]][k]][a]][s[k[s[s[k][k]]]][k][a]][b][c]
s[s[k[k]][s[k[s]][k]][a]][s[k[s[s[k][k]]]][k][a]][b][c]
s[k[k]][s[k[s]][k]][a][b][s[k[s[s[k][k]]]][k][a][b]][c]
k[k][a][s[k[s]][k][a]][b][s[k[s[s[k][k]]]][k][a][b]][c]
k[s[k[s]][k][a]][b][s[k[s[s[k][k]]]][k][a][b]][c]
s[k[s]][k][a][s[k[s[s[k][k]]]][k][a][b]][c]
k[s][a][k[a]][s[k[s[s[k][k]]]][k][a][b]][c]
s[k[a]][s[k[s[s[k][k]]]][k][a][b]][c]
k[a][c][s[k[s[s[k][k]]]][k][a][b][c]]
a[s[k[s[s[k][k]]]][k][a][b][c]]
a[k[s[s[k][k]]][a][k[a]][b][c]]
a[s[s[k][k]][k[a]][b][c]]
a[s[k][k][b][k[a][b]][c]]
a[k[b][k[b]][k[a][b]][c]]
a[b[k[a][b]][c]]
a[b[a][c]]

Or, atinybitlessobscurely:
s s[k[s]][s[k[k]][s[k[s]][k]]] s[k[s[s[k][k]]]][k] a [b][c]
s k[s] s[k[k]][s[k[s]][k]] a [s[k[s[s[k][k]]]][k][a]][b][c]
k s a [s[k[k]][s[k[s]][k]][a]][s[k[s[s[k][k]]]][k][a]][b][c]
s s[k[k]][s[k[s]][k]][a] s[k[s[s[k][k]]]][k][a] b [c]
s k[k] s[k[s]][k] a [b][s[k[s[s[k][k]]]][k][a][b]][c]
k k a [s[k[s]][k][a]][b][s[k[s[s[k][k]]]][k][a][b]][c]
k s[k[s]][k][a] b [s[k[s[s[k][k]]]][k][a][b]][c]
s k[s] k a [s[k[s[s[k][k]]]][k][a][b]][c]
k s a [k[a]][s[k[s[s[k][k]]]][k][a][b]][c]
s k[a] s[k[s[s[k][k]]]][k][a][b] c
k a c [s[k[s[s[k][k]]]][k][a][b][c]]
a s k[s[s[k][k]]] k a [b][c]
a k s[s[k][k]] a [k[a]][b][c]
a s s[k][k] k[a] b [c]
a s k k b [k[a][b]][c]
a k b k[b] [k[a][b]][c]
ab k a b [c]
a[b[a][c]]

Afteranumberofsteps,wegeta[b[a][c]]!Andthepointisthatwhateversymbolic

constructionwewant,wecanalwayssetupsomecombinationofs’sandk’sthat

willeventuallydoitforus—andultimatelybecomputationuniversal.They’re
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UltimateSymbolicAbstraction

equivalenttoTuringmachines,lambdacalculusandallthoseothersystemswe

knowareuniversal.Buttheywerediscoveredbeforeanyofthesesystems.

Bytheway, here’stheWolframLanguagewaytogettheresultabove(//.repeatedly

appliestherulesuntilnothingchangesanymore):
In[]:= s[s[k[s]][s[k[k]][s[k[s]][k]]]][s[k[s[s[k][k]]]][k]][a][b][c]//.{s[x_][y_][z_]x[z][y[z]],k[x_][y_]x}
Out[]= a[b[a][c]]

And,yes,it’snoaccidentthatit’sextremelyeasyandnaturaltoworkwithcombi-

natorsintheWolfram Language—becauseinfactcombinatorswerepartofthe

deepancestryofthecoredesignoftheWolfram Language.

Forme,though,combinatorsalsohaveanotherprofoundpersonalresonance.

They’reexamplesofverysimplecomputationalsystemsthatturnout(aswe’llsee

atlengthhere)toshowthesameremarkablecomplexityofbehaviorthatI’vespent

somanyyearsstudyingacrossthecomputationaluniverse.

Acenturyago—particularlywithoutactualcomputersonwhichtodoexperi-

ments—theconceptualframeworkthatI’vedevelopedforthinkingaboutthe

computationaluniversedidn’t exist.ButI’vealwaysthoughtthatofallsystems,

combinatorswereperhapstheearliestgreat“nearmiss”towhatI’veendedup

discoveringinthecomputationaluniverse.
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