Automata for relation algebra and formal proofs

107 Pages·2017·1.73 MB·English

by  Damien Pous

Automata for relation algebra and formal proofs
Damien Pous
To cite this version:
Damien Pous. Automata for relation algebra and formal proofs. Computer Science [cs]. ENS Lyon,
2016. ￿tel-01445821￿
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Damien POUS
Automata
for relation algebra and formal proofs
devantlacommissiond’examenforméede
Luca ACETO rapporteur
Arnaud DURAND examinateur
Georges GONTHIER examinateur
Peter JIPSEN rapporteuretexaminateur
Olivier LAURENT examinateur
François POTTIER rapporteuretexaminateur
Igor WALUKIEWICZ examinateur

Acknowledgements
Jetienstoutd’abordàremercierLucasAceto,PeterJipsenetFrançoisPot-
tier, qui ont accepté de relire ce manuscrit puis fait de nombreuses remar-
ques afin de l’améliorer. Je remercie également Arnaud Durand, Georges
Gonthier, Olivier Laurent et Igor Walukiewicz qui ont accepté de prendre
partaujury.
Viennentensuitelescollègues,étudiants,anciensprofesseursouamis,
hurluberlussansquicetravailn’auraitjamaisvulejour:deJacquesSauloy
àFilippoBonchi,enpassantparDanielHirschkoff,Tom(deSavoie),Mon-
strencageetIouri.
Je suis également redevable aux institutions: le CNRS pour la liberté
qu’il m’accorde, Plume pour sa convivialité inégalée, et le LIP pour l’air
frais et le nouveau point de vue qu’apportent chaque déménagement sur
lesproblèmesquinousrésistenttrop.
MercienfinàPamitosetnosjoyeuxtrublions,LucasetHugo.

Contents
Introduction 1
1 Relationalgebra 3
1.1 The(positive)calculusofrelations . . . . . . . . . . . . . . . 3
1.2 Theidealfragment:Kleenealgebra . . . . . . . . . . . . . . . 5
1.3 Thestrangefragment:allegories . . . . . . . . . . . . . . . . 8
1.4 Puttingitalltogether:Kleeneallegories . . . . . . . . . . . . 12
1.5 Kleenealgebrawithtests . . . . . . . . . . . . . . . . . . . . . 16
2 Automataalgorithms 23
2.1 Deterministicautomata . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Non-deterministicautomata . . . . . . . . . . . . . . . . . . . 28
2.3 Automatawithalargealphabet . . . . . . . . . . . . . . . . . 40
3 AutomationintheCoqproofassistant 55
3.1 RelationalgebraandKATinCoq . . . . . . . . . . . . . . . . 56
3.2 Casestudies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Appendix:overallstructureofthelibrary . . . . . . . . . . . 64
4 Abstractcoinduction 67
4.1 Notationandpreliminarymaterial . . . . . . . . . . . . . . . 68
4.2 Knaster-TarskiandCompatibility . . . . . . . . . . . . . . . . 70
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Compatibilityup-to . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Symmetryarguments . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 Example:up-tocongruenceforCCS . . . . . . . . . . . . . . 80
4.7 Respectfulvs.compatible . . . . . . . . . . . . . . . . . . . . 82
4.8 Parameterizedcoinduction . . . . . . . . . . . . . . . . . . . 84
4.9 Extensionalcharacterisationofthecompanion . . . . . . . . 86
4.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Notes 89

Introduction
We review in this manuscript several results we obtained since our PhD.
Weorganisedthoseresultsintofourchapterscorrespondingtofourdistinct
butrelatedfieldsincomputerscience.
Thefirstchapterisabiasedintroductiontothecalculusofrelations.This
fieldwasinitiatedbyDeMorgan,PeirceandthenSchröderinthelateXIXth
century,andstudiedquiteextensivelybyTarskiinthe1940’s.Itconsistsin
understanding the algebraic and algorithmic properties of operations on
binary relations. Our introduction is biased in the sense that we remove
the most problematic operation from the beginning, set-theoretic comple-
ment, and that we consider reflexive transitive closure. This brings us to
theconceptsofallegories(FreydandScedrov,AndrékaandBredikhin)and
Kleene algebra (Kleene, Conway, Kozen); and this allows us to state our
recentresultswithPaulBrunetontheirleastcommongeneralisation.
The second chapter is about algorithms for checking equivalence of
finite automata. Such algorithms provide decision procedures for Kleene
algebra;theyarealsoabasicblockforverificationsoftware:programsthat
makeitpossibletotestthevalidityofotherprograms.Therewepresentin
detail a new algorithm we have discovered with Filippo Bonchi, based on
a proof technique from concurrency theory: bisimulations up to congruence.
WealsopresentanextensionofastandardalgorithmbyHopcroftandKarp
todealwithsymbolicautomata.Suchautomataareusefulwhenworkingon
largealphabets.
Thethirdchapterpertainstothedomainofformalmechanisedproofs,
whereoneusesthecomputertowriteandproofcheckmathematicalproofs
(be they proofs of mathematical theorems, or correctness proofs for pro-
gramsorsystems).There,wediscusssomeapplicationsofalibrarywede-
velopedfortheCoqproofassistant,whereweprovidetoolsforautomated
reasoning in the calculus of relations, notably for Kleene algebra. These
automationtoolswereobtainedbyimplementingandcertifyingautomata
algorithmsaswellasimportantresultsaboutKleenealgebra.
Inthelastchapterwepresentanabstracttheoryofcoinduction,amath-
ematicaldevicecoinedbyMilnerinconcurrencytheory.Thistoolprovides
2 Contents
powerful proof methods, especially for the study of state-based systems;
it was a surprise to discover that it could also be used to obtain efficient
algorithms, as in the second chapter. Our abstract theory of coinduction is
a refinement of the work of Sangiorgi in the 1990’s; it makes it simpler to
provideenhancementsofthecoinductiveproofmethod,andtomechanise
theminproofassistants.Webelievethatitcouldhelpustofindnewdeci-
sionproceduresforthecalculusofrelations,andthentocertifythem.
We point to the publications we assembled to obtain this manuscript
inthenotesonpage89.
Notation
We denote sets by capital letters X,Y,S,T ... and functions by lower case
lettersf,g,... GivensetsXandY,X×Y istheirCartesianproduct,X(cid:93)Y is
theirdisjointunionandXY isthesetoffunctionsf: Y → X.Thecollection
of subsets of X is denoted by P(X). For a set of letters Σ, Σ(cid:63) denotes the
set of all finite words over Σ; (cid:15) the empty word; and uv the concatenation
ofwordsu,v ∈ Σ(cid:63).Weuse2fortheset{0,1}.
Chapter 1
Relation algebra
We consider algebraic and algorithmic questions related to binary rela-
tions. On the algebraic side, we want to understand and characterise the
lawsgoverningthebehaviourofstandardoperationsonrelations:union,in-
tersection, composition, converse, etc.... On the algorithmic side, we look
fordecisionproceduresforequalityorinclusionofrelations.
Westartbydefiningformallythecalculusofrelations;thenwefocuson
twowell-studiedfragmentsofparticularimportance:Kleenealgebrasandal-
legories.TryingtounifythosefragmentsleadustoanewresultwithBrunet,
andseveralopenquestions.
We also define Kleene algebra with tests, a framework introduced by
Kozenmakingitpossibletodealwithbothrelationsandpredicates.
1.1 The (positive) calculus of relations
Given a set P, a relation on P is a set of pairs of elements from P. For in-
stance, the usual order on natural numbers is a relation. In the sequel, re-
lationsarerangedoverusinglettersR,S,theirsetiswrittenP(P×P),and
wewritepRq for(cid:104)p,q(cid:105) ∈ R.
The set of relations is equipped with a partial order, set-theoretic in-
clusion(⊆),andthreebinaryoperations:set-theoreticunion,writtenR+S,
set-theoreticintersection,writtenR∩S,andrelationalcomposition:
R·S (cid:44) {(cid:104)p,q(cid:105) | ∃r ∈ P, pRr ∧ rS q} .
It also contains three specific relations: the empty relation, written 0, the
universalrelation,written(cid:62),andtheidentityrelation:
1 (cid:44) {(cid:104)p,p(cid:105) | p ∈ P} .
Lastly,onecanconsiderthreeunaryoperations:set-theoreticcomplement,
writtenRc,converse(ortranspose),R◦,andreflexive-transitiveclosure,R(cid:63),