Table Of ContentLisbeth Fajstrup · Eric Goubault
Emmanuel Haucourt · Samuel Mimram
Martin Raussen
Directed Algebraic
Topology and
Concurrency
Directed Algebraic Topology and Concurrency
Lisbeth Fajstrup Eric Goubault
(cid:129)
Emmanuel Haucourt Samuel Mimram
(cid:129)
Martin Raussen
Directed Algebraic Topology
and Concurrency
123
Lisbeth Fajstrup Samuel Mimram
Department ofMathematical Sciences ÉcolePolytechnique
Aalborg University Palaiseau
Aalborg France
Denmark
Martin Raussen
EricGoubault Department ofMathematical Sciences
ÉcolePolytechnique Aalborg University
Palaiseau Aalborg
France Denmark
EmmanuelHaucourt
ÉcolePolytechnique
Palaiseau
France
ISBN978-3-319-15397-1 ISBN978-3-319-15398-8 (eBook)
DOI 10.1007/978-3-319-15398-8
LibraryofCongressControlNumber:2015960769
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Foreword
Computersciencehasaruggedtraditionofappropriatingconceptsandmechanisms
from classical mathematics, concepts once proudly considered pure and useless,
andadaptingthoseconceptstodescribeandanalyzephenomenathatoncecouldnot
have been imagined. Today, the most abstract discoveries of number theory form
thefoundations of moderncryptography and finance, graph theory lies attheheart
of communication networks, social networks, and search structures, and advanced
probability has myriad of everyday applications.
This book, too, presents novel adaptations and applications of basic concepts
fromcombinatorialandalgebraictopology:inthiscase,pathsandtheirhomotopies.
The intuition is appealing: one can visualize an execution of a concurrent,
two-process program as a path winding its way through a planar region, where
progress by one process nudges the arrow along the x-axis, and progress by the
other along the y-axis. Certain regions of the plane are forbidden: they correspond
tozonesofmutualexclusion.Twoexecutionsareconsidered equivalentifonecan
be continuously deformed to the other without crossing a forbidden zone, and the
different ways in which paths can snake around these zones define the different
ways synchronization mechanisms can shape computation.
Thesepaths,liketimeitself,cannotrunbackwards,sotheclassictheoryofpaths
andtheirhomotopiesmustbeadaptedtoincorporatearelentlesssenseofdirection,
yieldinganewtheoryofdirectedtopology.Thistheoryleadstonovelformulations
of program semantics, new algorithmic techniques, and a rich description in the
language of category theory, itself another area with few prior concrete applica-
tions. In the tradition of theoretical computer science, this book takes classic
mathematical ideas, shapes them to new and unforeseen applications, and from
these ingredients produces a new mathematics.
Providence Maurice Herlihy
June 2015
v
Preface
Fascinating links between the semantics of concurrent programs and algebraic
topology have been discovered and developed since the 1990s, motivated by the
hopethateachfieldcouldenrichtheother,byprovidingnewtoolsandapplications.
Soonenough,itturnedoutthatthisinteractionhadtoevolveintosomethingricher
than a simple dictionary: topological spaces were not exactly the right notion
(theymustberefinedinordertoincorporatethedirectionoftime).Thealgorithms
for verifying concurrent programs resulting from topological semantics were not
easytoinventwhileachievingreasonablecomplexity.Today,wethinkthatenough
material has been understood to justify a book on the topic. We felt the urge for a
coherent,exhaustive,andyetintroductorypresentationofthesubject,sothatitcan
gain a larger audience and constitute a panorama of the current knowledge upon
which future developments will be built.
The topic makes it natural to address both computer scientists and mathemati-
cians. We have done our best to write the book with this mixed audience in mind.
Except for the last chapters, we have tried to require few prerequisites, while
keeping a reasonable size for the text: only a general knowledge of semantics of
programminglanguagesisrequired,aswellasbasicnotionsof(algebraic)topology
and category theory.
We thank Thibault Balabonski, Uli Fahrenberg, Eric Finster, Philippe Gaucher,
RobvanGlabeek,MarcoGrandis,TobiasHeindel,MauriceHerlihy,KathrynHess
Bellwald, Mateusz Juda, Philippe Malbos, Nicolas Ninin, Sergio Rajsbaum,
Christine Tasson, Krzysztof Worytkiewicz, and Krzysztof Ziemiański for the
stimulating discussions without which this book could certainly not have been
written, and especially Jérémy Dubut, Sanjeevi Krishnan, and Tim Porter for a
carefulproofreadingofthebookandmanycomments.Wewouldalsoacknowledge
support from AS CNRS TAPESC, ACI project GEOCAL, ANR projects INVAL,
CHOCO, and CATHRE, and from the ESF research networking program ACAT.
Paris Samuel Mimram
September 2015
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 A Toy Language for Concurrency . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 A Toy Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Semantics of Programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 The Transition Graph. . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Operational Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Verifying Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Correctness Properties . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Reachability in Concurrent Programs. . . . . . . . . . . . . . . 21
3 Truly Concurrent Models of Programs with Resources . . . . . . . . . 25
3.1 Modeling Resources in the Language. . . . . . . . . . . . . . . . . . . . 25
3.1.1 Taming Concurrency. . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Extending the Language with Resources. . . . . . . . . . . . . 27
3.2 State Spaces for Conservative Resources . . . . . . . . . . . . . . . . . 29
3.2.1 Conservative Programs. . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Transition Graphs for Conservative Programs . . . . . . . . . 31
3.3 Asynchronous Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Toward True Concurrency . . . . . . . . . . . . . . . . . . . . . . 36
3.3.2 Asynchronous Semantics . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.3 Coherent Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.4 Programs with Mutexes Only . . . . . . . . . . . . . . . . . . . . 46
3.4 Cubical Semantics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.1 Precubical Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.2 The Geometric Realization. . . . . . . . . . . . . . . . . . . . . . 54
3.5 Historical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
ix
x Contents
4 Directed Topological Models of Concurrency . . . . . . . . . . . . . . . . 57
4.1 Directed Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 A Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.2 Limits and Colimits. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.3 Directed Geometric Semantics. . . . . . . . . . . . . . . . . . . . 61
4.1.4 Simple Programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Homotopy in Directed Algebraic Topology. . . . . . . . . . . . . . . . 67
4.2.1 Classical Homotopy Theory . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Homotopy Between Directed Paths in Dimension 2. . . . . 68
4.2.3 Dihomotopy and the Fundamental Category . . . . . . . . . . 70
4.2.4 Simple Programs with Mutexes. . . . . . . . . . . . . . . . . . . 76
4.2.5 D-Homotopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Constructions on the Fundamental Category . . . . . . . . . . . . . . . 80
4.3.1 The Seifert–Van Kampen Theorem . . . . . . . . . . . . . . . . 80
4.3.2 The Universal Dicovering Space. . . . . . . . . . . . . . . . . . 81
4.4 Historical Notes and Other Models . . . . . . . . . . . . . . . . . . . . . 89
5 Algorithmics on Directed Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1 The Boolean Algebra of Cubical Regions. . . . . . . . . . . . . . . . . 92
5.2 Computing Deadlocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Factorizing Programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 The Category of Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1 Weak Isomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1.1 Systems of Weak Isomorphisms . . . . . . . . . . . . . . . . . . 108
6.1.2 A Maximal System . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1.3 Quotienting by Weak Isomorphisms . . . . . . . . . . . . . . . 111
6.1.4 Other Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Examples of Categories of Components . . . . . . . . . . . . . . . . . . 117
6.2.1 Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2.2 Cubical Regions in Dimension 2. . . . . . . . . . . . . . . . . . 118
6.2.3 The Floating Cube and Cross . . . . . . . . . . . . . . . . . . . . 119
6.3 Computing Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3.1 The Case of One Hole. . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.3 The Seifert–Van Kampen Theorem . . . . . . . . . . . . . . . . 124
6.4 Historical Notes, Applications, and Extensions . . . . . . . . . . . . . 125
6.4.1 Categories with Loops. . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4.2 Past and Future Components. . . . . . . . . . . . . . . . . . . . . 127
7 Path Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.1 An Algorithm for Computing Components of Trace Spaces . . . . 130
7.1.1 Path Spaces for Simple Programs . . . . . . . . . . . . . . . . . 130
7.1.2 The Index Poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.1.3 Determination of Dipath Classes. . . . . . . . . . . . . . . . . . 134
7.1.4 An Efficient Implementation. . . . . . . . . . . . . . . . . . . . . 138
Contents xi
7.2 Combinatorial Models for Path Spaces. . . . . . . . . . . . . . . . . . . 139
7.2.1 Contractibility of Restricted Path Spaces . . . . . . . . . . . . 139
7.2.2 Presimplicial Sets and the Nerve Theorem . . . . . . . . . . . 141
7.2.3 Path Space as a Prod-Simplicial Complex. . . . . . . . . . . . 146
8 Perspectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Chapter 1
Introduction
Concurrentprogramsconsistofmultipleprocessesrunninginparallel.Theirusehas
become more and more widespread in order to efficiently exploit recent architec-
tures(processorswithmanycores,clouds,etc.),buttheyarenotoriouslydifficultto
designandtoreasonabout:onehastoensurethattheprogramwillnotgowrong,
regardlessofthewaythedifferentprocessesconstitutingtheprogramarescheduled.
Inprinciple,inordertoachievethistaskwiththehelpofacomputer,wecouldapply
traditional verification techniques for sequential programs on each of the possible
executions of the program. But this is not feasible in practice because the number
of those executions, or schedulings, may grow exponentially with the size of the
program. Fortunately, it can be observed that many of the schedulings are equiva-
lentinthesensethatonecanbeobtainedfromtheotherbypermutingindependent
instructions:suchequivalentexecutionswillalwaysleadtothesameresult.Hence,
ifoneofthoseexecutionscanbeshownnottoleadtoanerror,neitherwillanyother
executionwhichisequivalenttoit.
Thissuggeststhatamodelforconcurrentprogramsshouldincorporatenotonly
thepossibleexecutionsoftheprogram(asintraditionalinterleavingsemantics),but
alsothecommutationsbetweeninstructions,followingtheprincipleofwhatisnow
calledtrueconcurrency.Interestingly,theresultingmodelsarealgebraicstructures
whichcanbeinterpretedgeometrically:roughlyastopologicalspacesinwhichpaths
correspondtoexecutionsandtwoexecutionsareequivalentwhenthecorresponding
paths are homotopic, i.e., connected by a continuous deformation from one to the
other.Inordertomakethisconnectionprecise,itturnsoutthattopologicalspaces
arenotexactlytherightnotionforourpurposes.Oneneedstouseadirectedvariant,
i.e.,toincorporateanotionofirreversibletime.
Mutatis mutandis, starting from very practical motivations (the verification of
concurrentprograms),questionsofamoretheoreticalnaturearise.Whatisagood
notionofadirectedspace,andhowdoclassicaltechniquesfromalgebraictopology
apply to this setting? What is the geometry of concurrent programs? How can a
geometricallyrefinedunderstandingofconcurrencybeusedinordertodesignnew
©SpringerInternationalPublishingSwitzerland2016 1
L.Fajstrupetal.,DirectedAlgebraicTopologyandConcurrency,
DOI10.1007/978-3-319-15398-8_1
