Irreducibility and Computational Equivalence: 10 Years After Wolfram's A New Kind of Science

353 Pages·2013·16.96 MB·English

by  Herbert W. Franke (auth.)

Hector Zenil
Editor
EMERGENCE, C
COMPLEXITY
C
AND
COMPUTATION E
Irreducibility
and Computational
Equivalence
10 Years
After Wolfram’s
A New Kind of Science
123
Emergence, Complexity and Computation 2
SeriesEditors
Prof.IvanZelinka
TechnicalUniversityofOstrava
CzechRepublic
Prof.AndrewAdamatzky
UnconventionalComputingCentre
UniversityoftheWestofEngland
Bristol
UnitedKingdom
Prof.GuanrongChen
CityUniversityofHongKong
Forfurthervolumes:
http://www.springer.com/series/10624
Hector Zenil (Ed.)
Irreducibility and Computational
Equivalence
10 Years After Wolfram’s A New Kind
of Science
ABC
Editor
Dr.HectorZenil
DepartmentofComputerScience
TheUniversityofSheffield
UK
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ISSN2194-7287 e-ISSN2194-7295
ISBN978-3-642-35481-6 e-ISBN978-3-642-35482-3
DOI10.1007/978-3-642-35482-3
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012954636
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Fourcenturiesago,telescopeswereturnedtotheskyforthefirsttime—
and what they saw ultimately launched much of modern science. Over
the pasttwenty yearsI havebegun to explorea new universe—thecom-
putational universe—made visible not by telescopes but by computers
Stephen Wolfram
A New Kind of Science, 2002
Foreword
It has been a pleasure and a privilege for me to discuss fundamental questions
withStephenovertheyears.InthisforewordIwanttoindicatewhatIregardas
some of the major contributions of A New Kind of Science (henceforth NKS).
In my opinion, NKS is a milestone work that will be appreciated more and
more with time. Now, ten years after its publication, some things already begin
to stand out.
Firstofall,Stephen’sbookiswonderfullyunconventional.Inanageinwhich
there are too many papers “filling in much needed gaps” (Stan Ulam’s classic
put-down), who can take the time from producing a constant stream of routine
papers, one that is required by the funding agencies, to write a conventional
book, let alone a magnum opus in the grand manner?
Stephen’sNKSisanexampletousall,abeaconofhighintellectualambition
shining through a fog of mediocrity and dispensable erudition.
IregardNKSasafundamentalworkonphilosophy,aworkonpossibleworlds
rather than merely about this one. Here are some of the topics that it touches
on:
– Ontology: the world may be built of discrete mathematical structures, out
of algorithms, not differential equations.
– Epistemology:evenknowingthefundamentallaws,theremaybenoshort-
cuts, no better way to deduce how a system will behave than to run it and
see.
– On randomness and pseudo-randomness: an entirely deterministic
world may seem to be random, but only be pseudo-random.
– Digital philosophy: what are the combinatorial bricks out of which the
world is built?
AnotherremarkablefeatureofNKSisthatitcanbe readbyanintelligenthigh-
schoolstudent, thuscontinuingthe fine traditionofNaturphilosophie. Theideas
are so basic that they do not require elaborate technical machinery.
Furthermore,Stephenbringstheplayfulexperimentalapproachofaphysicist
to combinatorial mathematics and theoretical computer science, fields formerly
not accustomed to such playfulness.
Newton developed the calculus, which was the tool he needed for his Prin-
cipia, andStephenhassimilarlydeveloped—andcontinuesdeveloping—thetool
he needs, the extremely powerful algorithmic language Mathematica.
And NKS is a kind of combinatorics art book, full of amazingly suggestive
high-resolutionillustrations, reminding me of works like D’Arcy Thompson, On
Growth and Form and Ernest Haeckel’s Art Forms in Nature. And this in an
age dominated by Bourbaki’s “death to triangles!” and totally opposed to any
visual images in mathematics.
As I said, I think the significance of NKS can only increase with time.
VIII Foreword
Let me illustrate this with a recentexample, involvingmy workon “metabi-
ology” that models biological evolution by studying the evolution of mutating
software.I amattempting to unify theoretical biology andtheoretical computer
science based on the ansatz that DNA is digital software.
Andhavingsetoutonthis path,I immediately discoverthatStephen’s NKS
is highly relevant. In particular, I think that the discussion of the Cambrian
explosion in Stephen Jay Gould’s Wonderful Life is illuminated by Wolfram’s
metaphor of mining the computational universe, and that the mysterious origin
oflife,whichisactuallythespontaneousdiscoveryofsoftwarebyNature,isillu-
minated by Wolfram’s thesis that it is easy to build a universal Turing machine
out of almost any discrete combinatorial components.1
So NKS is a book on biology and Stephen Wolfram is a biologist! Let us
celebrate the tenth anniversaryof this remarkable volume. We shall be lucky to
see its like again in this generation.
—Gregory Chaitin
Federal University of Rio de Janeiro
1 For more on such topics, please see G. Chaitin, Proving Darwin: Making Biology
Mathematical, New York:Pantheon, 2012.
Preface
In 2005, I began a project on mathematical logic at the then called NKS Sum-
mer School (today the Wolfram Science Summer School), a project close to my
heart but rather far afield of my then interest in Computability theory. Quite
surprisingly, Stephen Wolfram would listen to my stuff on neural networks, real
numbers and computing at different levels of the arithmetical hierarchy with
great interest, even as I myself was becoming increasingly aware that he was
very skeptical about it. The project I started at Stephen’s suggestion was a
natural extension of the exploration of the “mathematical universe” of propo-
sitional sentences he had undertaken in A New Kind of Science (chapter 12,
section 9 “Implications for Mathematics and Its Foundations”) with very inter-
esting results. Among his discoveries is the shortest single axiom equivalent to
theaxiomsofBooleanalgebrawithaformulaofsix“Nand”’sandtwovariables:
((a.b).c).(a.((a.c).a))=c. He alsofoundthatBooleanformulasthathadaname
in, for example textbooks, suchasthe LawsofTautology,Commutativity, Dou-
ble Negation, etc., were exactly the formulas independent of (that cannot be
derived from) the set of shorter formulas by number of variables and symbols.
The very few exceptions, such as De Morgan’s Laws, enable the shortening of
proofs that would otherwise be significantly longer.
SoIundertookthechallengeofmakingpredicatecalculusmy“mathematical
universe” of investigation during the school at Brown, which eventually gener-
ated severalresearchideas, some of which people from Carnegie Mellon Univer-
sity(CMU)foundinterestingenoughtoinviteme toCMUinPittsburghforthe
Spring semester of 2008as a visiting scholar. My time at Brown and CMU ulti-
mately led to a paper I published in CristianCalude’s festschrift under the title
“From Computer Runtimes to the Length of Proofs”1 with a potential applica-
tion to waiting times useful in automatic theorem proving (and perhaps more
usefulinthefieldofverificationofprograms,amatternowbeinginvestigatedby
a student of mine). At CMU I also encountered Klaus Sutner, a researcherwho
had been interested in the possible connections between Wolfram’s Principle of
Computational Equivalence and intermediate Turing degrees almost since the
publication of the ANKS book and contributor to this volume summarising for
the readers some of his latest findings and ideas in this direction.
While at the NKS Summer School, as participant and then instructor, I
encountered a thriving community of remarkably smart people who I hold in
the highest esteem. Two of them (Bolognesi and Joosten) report some of their
currentandfuturedirectionsinconnectiontoNKSquestions,whotogetherwith
1 H. Zenil, From Computer Runtimes to the Length of Proofs: With an Algorithmic
Probabilistic ApplicationtoWaitingTimes in AutomaticTheorem Proving.InM.J.
Dinneen, B. Khousainov, and A. Nies (Eds.), Computation, Physics and Beyond:
Theoretical Computer Science and Applications, WTCS 2012, LNCS 7160, pp. 223–
240, Springer, 2012.
X Preface
Beckage and Maymin are NKS-SS alumni contributing to this volume. My own
interestschangedsoonafter participatingatWolfram’s ScienceSummer School,
notonlybecauseofmyexperiencewithandatWolframbutbecauseoftheinsight
of my PhD thesis advisor Jean-Paul Delahaye, at Lille, who immediately saw
thequestionofhyper-computationasa“fauxprobl`eme”giventhatthequestion
belongstophysicsandnottocomputerscience.FromthenonIdevelopedagreat
interest in the intersection of Wolfram’s experimental techniques with another
exciting area of computer science, Algorithmic Information Theory (AIT), as
developed by Kolmogorov, Chaitin, Levin and Solomonoff, among others. Two
of the aforementioned founders of AIT (Chaitin and Solomonoff) have been
relatively close to Wolfram’s NKS, attending conferences and contributing from
their particular vantage points and insightful perspectives.
In 2007, Stephen Wolfram introduced me to Greg Chaitin, whom I visited
at his office in the IBM TJ Watson Research Center in Yorktown, NY. Chaitin
and Wolfram have been discussing about NKS and AIT for many years, both
their points of possible contact and of possible stress with always interesting,
and often unexpected, outcomes2 in part motivating the beginning of my own
researchprogramme. Chaitin has an advantageousfirst hand view of Wolfram’s
work that shares with us in his foreword, once more with an unexpected, but
provocative,outcome(thatWolframis ratherabiologist!).Calude,onthe other
hand, has approached Wolfram’s ideas more recently and brings a fresh but in-
sightfuloutlookofWolfram’sworkinhisafterword,equilibratedwithreferences
to some of the reviews of Wolfram’s book in the course of the last ten years.
While the cellular automaton is among the preferred computing models in
ANKS (also the model of choice for most of the contributors to this volume,
see e.g. the opening chapter by Franke), the book’s main concepts are the con-
cepts of Irreducibility and Computational Equivalence that Wolfram claims to
beofessentialvalueforunderstandingnatureandhasconsequentlyadvancedin
the form of guiding general principles. Cellular automata are therefore merely
a means to arrive at and illustrate these central concepts. Zwirn and Delahaye
contribute an interesting approach to one aspect of Wolfram’s notion of irre-
ducibility from the perspective of traditional computational complexity. Sutner
investigatesthe formal side of the Principle of Computational Equivalence from
the perspectiveofclassicalRecursiontheory,alsothe directionofJoosten’scon-
tribution in connection to the question of the pervasiveness of complexity in
nature. Rucker explores the aspect of Wolfram’s ideas on computation that are
more connected to the material and everyday world, comparing different sys-
tems and suggesting that both universality and undecidability are everywhere.
Beckage,Kauffman,Gross,Zia,VattayandKolibaexplorethenatureofcompu-
tational irreducibility in connection to limits in physical and biologicalsystems.
Maymin and Velupillai connect concepts of computational irreducibility and
2 You can find their publicdebates together with other personalities in the transcrip-
tions(byAdrianGerman)publishedin:H.Zenil,RandomnessThroughComputation,
WorldScientific,2011;andH.Zenil(ed),AComputableUniverse:Understandingand
ExploringNatureAsComputation,WorldScientific,2013.
Preface XI
universality to algorithmic finance and computable economics, fields inaugu-
rated by the authors themselves. Burgin explores the consequences for tech-
nology and engineering. Dowek, Arrighi, Bolognesi and Vidal discuss the
foundationsofphysicsfromthebottomup,fromparticlestocosmology.Baetens
and De Baets; Margenstern; Sapin; Ruivo and Oliveira; Mainzer; Mart´ınez and
Seck Tuoh Mora; and Reisinger, Martin, Blankenship, Harrison, Squires and
Beavers, investigate cellular automata in connection to their computational be-
haviouralproperties, their limits and the notion of computational irreducibility.
BaileyalsoexplorestheconnectionsofWolfram’spragmaticapproachestoexper-
imentalmathematicsandthediscoveryofformulaswiththehelpofcomputersin
a sort of NKS exploration fashion in traditional mathematics. Dodig-Crnkovic,
Tagliabue and Bringsjordenlighten us by dissecting Wolfram’s most fundamen-
tal assumptions and the philosophical consequences of his work.
Everything came together for this book at the right time. Since Wolfram’s
book had been published ten yearspreviously, the authors had enough material
topresentandprogresstoreportvis-`a-visnewdirectionsrelatedtoormotivated
by the subjects covered in Wolfram’s work. I had established connections with
manyoftheleadingresearchersinthefieldat,amongotherevents,theWolfram
ScienceconferencesheldinBoston,WashingtonandBurlington,thetwoMidwest
NKS conferences organised by Adrian German at the University of Indiana,
Bloomington in 2005 and 2008, and the JOUAL workshop held at the CNR in
Pisa,Italy,organisedbyBolognesiin2009where,amongthe speakers,werealso
contributors to this volume (Dowek, Burgin, Sapin and Dodig-Crnkovic).
I want to thank all the aforementioned individuals, and also specially to
Matthew Szudzik, Joost Joosten, Todd Rowland, Catherine Boucher, Adrian
German, Paul-Jean Letourneau, Genaro J. Mart´ınez, Vela Velupillai, Gre-
gory Chaitin, Cristian Calude, Jean-Paul Delahaye, Gordana Dodig-Crnkovic,
Cl´ement Vidal and Stephen Wolfram. Thanks also to Elena Villarreal for her
continuous support and understanding, to Ivan Zelinka, the series editor of
“Emergence, Complexity and Computation”, who supported this project, and
to Thomas Ditzinger, the publishing editor of Springer.
—Hector Zenil
The University of Sheffield, United Kingdom