Contents

pp vii-xii

Preface

pp xiii-xiv

Acknowledgements

pp xv-xvi

Introduction

pp 1-6

PART I - BASICS

pp 7-8

1 - The definition of the models

pp 9-15

2 - Measure on ℬ

pp 16-18

3 - Witnessing quantifiers

pp 19-33

4 - The truth in N and the validity in K(F)

pp 34-36

PART II - SECOND-ORDER STRUCTURES

pp 37-38

5 - Structures K(F, G)

pp 39-46

PART III - AC0 WORLD

pp 47-48

6 - Theories IΔ0, IΔ0(R) and V01

pp 49-51

7 - Shallow Boolean decision tree model

pp 52-54

8 - Open comprehension and open induction

pp 55-59

9 - Comprehension and induction via quantifier elimination: a general reduction

pp 60-62

10 - Skolem functions, switching lemma and the tree model

pp 63-69

11 - Quantifier elimination in K(Ftree, Gtree)

pp 70-74

12 - Witnessing, independence and definability in V01

pp 75-80

PART IV - AC0(2) WORLD

pp 81-82

13 - Theory Q2V01

pp 83-84

14 - Algebraic model

pp 85-88

15 - Quantifier elimination and the interpretation of Q2

pp 89-95

16 - Witnessing and independence in Q2V01

pp 96-98

PART V - TOWARDS PROOF COMPLEXITY

pp 99-100

17 - Propositional proof systems

pp 101-104

18 - An approach to lengths-of-proofs lower bounds

pp 105-109

19 - PHP principle

pp 110-112

PART VI - PROOF COMPLEXITY OF Fd AND Fd (⊕)

pp 113-114