Descriptive Set Theoretic Methods in Automata Theory: Decidability and Topological Complexity

212 Pages·2016·3.149 MB·English

by  Michał Skrzypczak (auth.)

Michał Skrzypczak
Descriptive
2
0 Set Theoretic Methods
8
9

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in Automata Theory
C
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Decidability and Topological Complexity
123
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ł
Micha Skrzypczak
Descriptive
Set Theoretic Methods
in Automata Theory
Decidability and Topological Complexity
123
Author
Michał Skrzypczak
University of Warsaw
Warsaw
Poland
Thiswork wascarried out at:
Faculty of Mathematics, Informatics, andMechanics,
University of Warsaw,
Warsaw,Poland
andaccepted there asaPhD thesis.
ISSN 0302-9743 ISSN 1611-3349 (electronic)
Lecture Notesin Computer Science
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Preface
This thesis is devoted to the study of problems of automata theory from the point of
view of descriptive set theory. The analyzed structures are x-words and infinite trees.
Mostoftheresultspresentedherehavetheformofaneffectivedecisionprocedurethat
operates on representations of regular languages.
Specialeffortisputintoprovidingeffectivecharacterizationsofregularlanguagesof
infinite trees that are definable in weak monadic second-order logic (WMSO). Although
no such characterization is known for all regular languages of infinite trees, the thesis
provides characterizations in some special cases: for game automata, for languages of
thin trees (i.e., trees with countably many branches), and for Büchi automata. Addi-
tionally,certainrelationsbetweenWMSO-definablelanguagesandBorelsetsareproved.
Another problem studied in the thesis is the alternating index problem (also called
the Rabin–Mostowski index problem). Again, the problem in its full generality seems
tobeoutofthereachofcurrentlyknownmethods.However,adecisionprocedurefor
the class of game automata is proposed in the thesis. These automata form the widest
class of automata for which the problem is currently known to be decidable.
The thesis also addresses the problem of providing an algebraic framework for
regular languages of infinite trees. For this purpose, the notion of prophetic thin
algebrasisintroduced.Itisprovedthatfinitepropheticthinalgebrasrecognizeexactly
thebi-unambiguouslanguages—languagesLsuchthatbothLandthecomplementLc
canberecognizedbyunambiguousautomata.Additionally,anewconjectureaboutthe
definability of choice functions is stated. It is proved that this conjecture is strongly
related to the class of prophetic thin algebras. In particular, the conjecture implies an
effective characterization of the class of bi-unambiguous languages.
Finally, the thesis studies contemporary quantitative extensions of the class of
regular languages. First, lower bounds (that match upper bounds) on the topological
complexity of MSO+U-definable languages of x-words are given. These lower bounds
can be used to prove that MSO+U logic is undecidable on infinite trees in a specific
sense. It is also shown that languages of x-words recognizable by certain counter
automata have a separation property with respect to x-regular languages. The proof
relies on topological methods in the profinite monoid.
June 2016 Michał Skrzypczak
Acknowledgements
IwouldliketoemphasizemygratitudetomysupervisorsMikołajBojańczykandIgor
Walukiewicz for their assistance, without which this thesis would not have been
possible. Both of them were always helpful and ready to discuss scientific problems,
share their experience, and exchange new ideas. Additionally, I would like to thank
themforencouragingmetoattendanumberofconferences,workshops,andscientific
visitsthatallowedmetoestablishinternationalcooperation.Moreover,Iwouldliketo
thank Henryk Michalewski for posing a number of inquiring and inspiring questions
thatprovidedgoodmotivationindevelopingsomeoftheresultspresentedinthethesis.
IowetheformationofmyscientificbackgroundtotheAutomataGroupinWarsaw,
especially to: Damian Niwiński, Mikołaj Bojańczyk, Henryk Michalewski, and Filip
Murlak. Additionally, I wish to thank Szczepan Hummel, Tomasz Idziaszek, Denis
Kuperberg, and Nathanaël Fijalkow for the time they devoted to our discussions.
During myPhDstudiesIhadtheopportunitytomakeseveralscientificvisits. First
and foremost I would like to thank my supervisor Igor Walukiewicz for a number of
invitations to LaBRI. Moreover, I would like to thank Jacques Duparc and Alex
Rabinovichforinvitingmetoworktogether(theresultsobtainedduringthesevisitsare
not included in the thesis).
I wish to express my gratitude to the reviewers of the thesis. Their suggestions
helped me to increase the readability of the final version of the document.
Last but not least, I list my co-authors, whom I would like to thank for the coop-
eration: M. Bilkowski, M. Bojańczyk, U. Boker, A. Facchini, O. Finkel, T. Gogacz,
S. Hummel, T. Idziaszek, D. Kuperberg, O. Kupferman, H. Michalewski, M. Mio,
F. Murlak, D. Niwiński, A. Rabinovich, A. Radziwończyk-Syta, and S. Toruńczyk.
My work during my PhD studies was supported by Poland’s National Science
Centre (decision DEC-2012/05/N/ST6/03254) and by ERC Starting Grant SOSNA.
Contents
1 Basic Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Finite Words and x-words. . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Infinite Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Positional Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Parity Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Automata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Parity Index of an Automaton. . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.1 Semigroups and Monoids. . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.2 Wilke Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.3 Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.4 Ramsey’s Theorem for Semigroups. . . . . . . . . . . . . . . . . . . 13
1.6 Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6.1 Borel and Projective Hierarchy. . . . . . . . . . . . . . . . . . . . . . 14
1.6.2 Topological Complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6.3 Ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.4 The Boundedness Theorem . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.5 Co-inductive Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Regular Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7.1 Classes of Regular Tree Languages. . . . . . . . . . . . . . . . . . . 21
1.7.2 Index Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7.3 Topological Complexity of Regular Languages. . . . . . . . . . . 23
1.7.4 The languages Wi;j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.5 Separation Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7.6 Deterministic Languages . . . . . . . . . . . . . . . . . . . . . . . . . . 26
X Contents
Subclasses of Regular Languages
2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.1 Definability in WMSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.2 Index Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.3 Bi-unambiguous Languages . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.4 Borel Languages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.5 Topological Complexity vs. Decidability . . . . . . . . . . . . . . . 32
2.1.6 Separation Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Overview of the Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 Part I: Subclasses of Regular Languages . . . . . . . . . . . . . . . 34
2.2.2 Part II: Thin Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.3 Part III: Extensions of Regular Languages . . . . . . . . . . . . . . 36
3 Collapse for Unambiguous Automata . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Unique Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Construction of the Automaton. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 When a Büchi Language Is Definable in WMSO . . . . . . . . . . . . . . . . . 45
4.1 The Ordinal of a Büchi Automaton. . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.1 Truncated Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.2 The Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.3 Ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Extending Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 K-reach Implies Big Rank . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 Big Rank Implies K-reach . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Automata for K-safe Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Boundedness Game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 Rules of the Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.2 Winning Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Equivalence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5.1 Implication (1) ⇒ (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5.2 Implication (2) ⇒ (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Index Problems for Game Automata . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 Runs of Game Automata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Non-deterministic Index Problem . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Partial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.1 Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.2 Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.3 Automata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Contents XI
5.3.4 Composing Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.5 Resolving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4 Alternating Index Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4.2 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4.3 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Thin Algebras
6 When a Thin Language Is Definable in WMSO . . . . . . . . . . . . . . . . . . 93
6.1 Basic Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.1.1 Thin Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.1.2 Automata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1.3 Examples of Skurczyński. . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1.4 Ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1.5 Skeletons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Thin Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.1 The Automaton Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.1 Embeddings and Quasi-skeletons . . . . . . . . . . . . . . . . . . . . 106
6.3.2 Proof of Proposition 6.13. . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4 Characterisation of WMSO-definable Languages . . . . . . . . . . . . . . . . 110
6.4.1 Implication (2) ⇒ (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.4.2 Implication (5) ⇒ (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Recognition by Thin Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1 Prophetic Thin Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2 Bi-unambiguous Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2.1 Prophetic Thin Algebras Recognise only
Bi-unambiguous Languages . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2.2 Markings by the Automaton Algebra for an Unambiguous
Automaton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2.3 Every Bi-unambiguous Language is Recognised
by a Prophetic Thin Algebra . . . . . . . . . . . . . . . . . . . . . . . 129
7.3 Consequences of Conjecture 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.4 Decidable Characterisation of the Bi-unambiguous Languages . . . . . 131
7.4.1 Pseudo-syntactic Morphisms. . . . . . . . . . . . . . . . . . . . . . . . 132
7.4.2 Decidable Characterisation. . . . . . . . . . . . . . . . . . . . . . . . . 134
7.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8 Uniformization on Thin Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.1 Basic Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2 Transducer for a Uniformized Relation . . . . . . . . . . . . . . . . . . . . . 139