DOCSLIB.ORG

Simplification Orders in Term Rewriting. Derivation Lengths

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Simplification Orders in Term Rewriting. Derivation Lengths Ingo Lepper Simplification Orders in Term Rewriting Derivation Lengths, Order Types, and Computability 2001 Mathematische Logik Simplification Orders in Term Rewriting Derivation Lengths, Order Types, and Computability Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik der Mathematisch-Naturwissenschaftlichen Fakult¨at der Westf¨alischen Wilhelms-Universit¨at Munster¨ vorgelegt von Ingo Lepper aus Munster¨ – 2001 – Typeset with LATEX 2ε using the document class KOMA-Script, the packages babel, AMS-LATEX, amsthm, hyperref, graphicx, url, color, natbib, backref, booktabs, array, ragged2e, acronym, nomencl, thumbpdf, nicefrac, currvita, titletoc, epigraph, and paralist, and the additional font packages amssymb, aeguill, stmaryrd, and pifont. The home page of this text is http://wwwmath.uni-muenster.de/logik/publ/diss/9.html. If you want to contact the author, this is possible by electronic mail via either [email protected] (expiration date approx. 12/2004) or [email protected]. Dekan: Prof. Dr. W. Lange Erster Gutachter: Prof. Dr. A. Weiermann Zweiter Gutachter: Prof. Dr. W. Pohlers Tage der mundlichen¨ Prufungen:¨ ❖ Mathematische Logik: 04.12.2001 ❖ Reine Mathematik: 29.11.2001 ❖ Angewandte Mathematik: 03.12.2001 Tag der Promotion: 04.12.2001 Fur¨ Anke und meine Familie ❦ Foreword Appreciate true friendship, it is for free! I would like to mention the persons who have significantly contributed to my work. First of all I wish to express my sincere thanks to my thesis supervisor, Prof. Andreas Weiermann, for invaluable help and encouragement. He arouse my interest in term rewriting, he benevolently answered nearly all of my ques- tions from various fields of mathematics, and in turn he asked questions which led to new or improved results. Without his support, this text would not exist. A further direct scientific contributor is Dieter Hofbauer. Besides cofounding the research concerning derivation lengths, his constant interest has been a main influence on my work. In various discussions he helped me gain insight into Knuth–Bendix orders, and his comments on preliminary versions of some of my results led to a much improved presentation. Everyone familiar with the work of H´el`ene Touzet will recognize that I also owe a lot to her. Just in time, Prof. Andrei Voronkov provided a Theorem which spared myself technical trouble and caused significant improvements in both presentation and generality of a part of this thesis. My results concerning computability via Knuth–Bendix orders partially emerged from a discussion with Guillaume Bonfante. The Institut fur¨ mathematische Logik und Grundlagenforschung has been of fundamental importance to my work. Its directors Prof. Justus Diller and Prof. Wolfram Pohlers provided one major part of my mathematical education. Their kind guidance and the friendly atmosphere at the institute made my work a pleasure. Considering the amount of time I have spent there, work may have been even more pleasant than it ought to be. Related to this are also many former and present members of the institute. I’d like to explicitely men- tion my dear friend Benjamin Blankertz, who managed to combine hospitality with a constant cheerful urge concerning the completion of my thesis, Arnold Beckmann, Wolfgang Burr, and Michael M¨ollerfeld, whose knowledge appears to have grown exponentially during the last years. Our secretaries Hildegard Brunstering and Martina Pfeifer, who often succeeded in confronting us mathe- maticians with the real life, constitute the secret heart of the institute. Looking v Foreword at the students which are affiliated to the institute at the moment, I am con- vinced it will remain as peaceful and open-minded a place as it has been for the past decade. On the private side, I have to commend my friends who managed, partially against my reluctance, to distract me from my work. This always turned out to be fruitful – apart from the headaches. The squirrels (sciurus vulgaris, the shy red kind) from my balcony have been a second constant source of amusement. In death-defying expeditions, which sometimes reached my desk, they discovered vast deposits of nuts and managed to transport and hide away large amounts of them. I wish to express my wholehearted gratitude to all these people and animals. Finally, there are the persons whose confidence and loving support have made my work possible. I dedicate the thesis to my love Anke and to my family. vi Contents 1 Introduction 1 2 Preliminaries 13 2.1 Conventions . 13 2.2 Basic definitions . 14 2.3 Orders and Order types . 15 2.4 Subrecursive Hierarchies . 27 2.5 Complexity Classes . 38 3 Term Rewriting 41 3.1 Basic Definitions . 43 3.2 Abstract Orders on Terms . 48 3.3 Semantic Orders on Terms . 51 3.3.1 Interpretations. 52 3.3.2 Σ-algebras and the Termination Hierarchy . 54 3.4 Syntactic Orders on Terms . 62 3.4.1 Multiset Path Orders . 62 3.4.2 Lexicographic Path Orders . 63 3.4.3 Knuth–Bendix Orders . 64 3.5 Functions Computable by a TRS . 70 4 Order Types 73 4.1 Order Types of MPOs and LPOs . 74 4.2 Order Types of KBOs . 75 4.3 Order Types of Simplification Orders. 78 5 Derivation Lengths 79 5.1 ω-termination . 82 5.2 Termination via KBO . 84 5.3 Simple Termination is Complex. 93 5.3.1 The Fixed Point Free Veblen Function . 95 5.3.2 Fundamental Sequences and Hydrae below ∆k . 97 5.3.3 Encoding all Hydrae below ∆k . 99 5.3.4 Simulating all Hydra Battles below ∆k . 102 5.3.5 Proof of Total Termination . 108 vii Contents 5.4 A Digression: LPO-Controlled Derivations . 113 5.5 Another Digression: Ground Termination . 115 6 Computability 117 6.1 Computability via PT . 118 6.2 Computability via MPO and LPO . 119 6.3 Computability via KBO . 120 6.3.1 List Operations Compatible with KBO . 122 6.3.2 Simulating a Timebounded RM . 123 6.3.3 Long Linear Derivations. 124 6.3.4 Hard Computation via KBO . 129 6.4 Computability via Simple Termination . 132 7 Very Long Size-Controlled Derivations 135 7.1 Sequences of Iterated Logarithms. 139 7.2 Hydra Battles, Revisited . 144 7.3 Lengthened Derivations . 152 8 Conclusion 163 Bibliography 167 Glossary of Notation 177 Index 185 Curriculum Vitae 193 viii 1 Introduction A happy surprise is waiting for you. Concerning the future of proof theory and complexity in the 21st century, the following prediction has been made by Harvey Friedman (2000): “. there has recently been considerable work on the connections be- tween proof theory and term rewriting. This will also expand in the coming century.” This text is intended to be such a contribution. We approach complexity prob- lems of term rewriting theory using results and techniques from proof theory. The concept of term rewriting has been developed at the end of the 1960s. A term rewriting system (TRS) is a finite set R of directed equations of terms over some fixed signature Σ. The emerging rewrite relation →R is the closure of R under substitutions and contexts. TRSs provide a valuable tool in auto- mated deduction, and additionally they constitute an interesting paradigm for nondeterministic computation, importing the term notion for free. It should be mentioned that it is rather common to consider also infinite sets of rules, but for our purposes this makes not much sense. One major topic in term rewriting is termination.A TRS is said to terminate if all derivations induced by its rewrite relation are finite, i.e. if the rewrite relation does not admit infinite chains. Just as in computation theory, the problem of establishing termination of a given TRS is highly nontrivial. A vast variety of techniques for proving termination has evolved. Common to these approaches is the association of some well-founded order to the rewrite relation. Of particular importance in this field are the partial orders on terms whose well-foundedness is a consequence of Kruskal’s famous Tree Theorem. These orders are called simplification orders, and they are the key object of our studies. Prominent examples of such orders are the multiset path order (MPO) of Plaisted (1978), the lexicographic path order (LPO) of Kamin and L´evy (1980), and the Knuth–Bendix order (KBO) of Knuth and Bendix (1970). 1 1 Introduction A very convenient way of proving termination is by means of Σ-algebras. The basis of a Σ-algebra is a partial order (P, ≺). For any symbol of the signature, there is also a function on P of the appropriate arity. Roughly speaking, this corresponds to a Σ ∪ {<}-structure in first order logic. We call a Σ-algebra monotone if its interpreting functions are monotone. Termination of a TRS fol- lows from its compatibility with a monotone Σ-algebra whose underlying partial order is well-founded. Any TRS compatible with a monotone Σ-algebra which also satisfies the subterm property is called simply terminating. It turned out that a TRS is simply terminating if and only if its rewrite relation is contained in a simplification order. Thus simple termination implies termination. Zantema (1993, 1994) speaks of total termination if the monotone Σ-algebra is based on a well-order (being total by definition). This constitutes an im- portant (proper) subclass of simple termination. As any well-order is order- isomorphic to an ordinal, it is possible to classify totally terminating TRSs according to the ordinals α carrying compatible Σ-algebras. This leads to the notion of α-termination and unveils a rich structure of subhierarchies. Inside the collection of ω-terminating TRSs we can further differentiate according to the functions used for interpreting the function symbols of the Σ-algebra.
Load more

Recommended publications
  • The Interplay Between Computability and Incomputability Draft 619.Tex
    The Interplay Between Computability and Incomputability Draft 619.tex Robert I. Soare∗ January 7, 2008 Contents 1 Calculus, Continuity, and Computability 3 1.1 When to Introduce Relative Computability? . 4 1.2 Between Computability and Relative Computability? . 5 1.3 The Development of Relative Computability . 5 1.4 Turing Introduces Relative Computability . 6 1.5 Post Develops Relative Computability . 6 1.6 Relative Computability in Real World Computing . 6 2 Origins of Computability and Incomputability 6 2.1 G¨odel’s Incompleteness Theorem . 8 2.2 Incomputability and Undecidability . 9 2.3 Alonzo Church . 9 2.4 Herbrand-G¨odel Recursive Functions . 10 2.5 Stalemate at Princeton Over Church’s Thesis . 11 2.6 G¨odel’s Thoughts on Church’s Thesis . 11 ∗Parts of this paper were delivered in an address to the conference, Computation and Logic in the Real World, at Siena, Italy, June 18–23, 2007. Keywords: Turing ma- chine, automatic machine, a-machine, Turing oracle machine, o-machine, Alonzo Church, Stephen C. Kleene,klee Alan Turing, Kurt G¨odel, Emil Post, computability, incomputabil- ity, undecidability, Church-Turing Thesis (CTT), Post-Church Second Thesis on relative computability, computable approximations, Limit Lemma, effectively continuous func- tions, computability in analysis, strong reducibilities. 1 3 Turing Breaks the Stalemate 12 3.1 Turing’s Machines and Turing’s Thesis . 12 3.2 G¨odel’s Opinion of Turing’s Work . 13 3.3 Kleene Said About Turing . 14 3.4 Church Said About Turing . 15 3.5 Naming the Church-Turing Thesis . 15 4 Turing Defines Relative Computability 17 4.1 Turing’s Oracle Machines .
    [Show full text]
  • The Realm of Ordinal Analysis
    The Realm of Ordinal Analysis Michael Rathjen Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom Denn die Pioniere der Mathematik hatten sich von gewissen Grundlagen brauchbare Vorstellungen gemacht, aus denen sich Schl¨usse,Rechnungsarten, Resultate ergaben, deren bem¨achtigten sich die Physiker, um neue Ergebnisse zu erhalten, und endlich kamen die Techniker, nahmen oft bloß die Resultate, setzten neue Rechnungen darauf und es entstanden Maschinen. Und pl¨otzlich, nachdem alles in sch¨onste Existenz gebracht war, kamen die Mathematiker - jene, die ganz innen herumgr¨ubeln, - darauf, daß etwas in den Grundlagen der ganzen Sache absolut nicht in Ordnung zu bringen sei; tats¨achlich, sie sahen zuunterst nach und fanden, daß das ganze Geb¨audein der Luft stehe. Aber die Maschinen liefen! Man muß daraufhin annehmen, daß unser Dasein bleicher Spuk ist; wir leben es, aber eigentlich nur auf Grund eines Irrtums, ohne den es nicht entstanden w¨are. ROBERT MUSIL: Der mathematische Mensch (1913) 1 Introduction A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their ‘rank’ or ‘complexity’ in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of ‘proof the- oretic ordinals’ to theories, gauging their ‘consistency strength’ and ‘computational power’. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen’s head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories.
    [Show full text]
  • Effective Descriptive Set Theory
    Effective Descriptive Set Theory Andrew Marks December 14, 2019 1 1 These notes introduce the effective (lightface) Borel, Σ1 and Π1 sets. This study uses ideas and tools from descriptive set theory and computability theory. Our central motivation is in applications of the effective theory to theorems of classical (boldface) descriptive set theory, especially techniques which have no classical analogues. These notes have many errors and are very incomplete. Some important topics not covered include: • The Harrington-Shore-Slaman theorem [HSS] which implies many of the theorems of Section 3. • Steel forcing (see [BD, N, Mo, St78]) • Nonstandard model arguments • Barwise compactness, Jensen's model existence theorem • α-recursion theory • Recent beautiful work of the \French School": Debs, Saint-Raymond, Lecompte, Louveau, etc. These notes are from a class I taught in spring 2019. Thanks to Adam Day, Thomas Gilton, Kirill Gura, Alexander Kastner, Alexander Kechris, Derek Levinson, Antonio Montalb´an,Dean Menezes and Riley Thornton, for helpful conversations and comments on earlier versions of these notes. 1 Contents 1 1 1 1 Characterizing Σ1, ∆1, and Π1 sets 4 1 1.1 Σn formulas, closure properties, and universal sets . .4 1.2 Boldface vs lightface sets and relativization . .5 1 1.3 Normal forms for Σ1 formulas . .5 1.4 Ranking trees and Spector boundedness . .7 1 1.5 ∆1 = effectively Borel . .9 1.6 Computable ordinals, hyperarithmetic sets . 11 1 1.7 ∆1 = hyperarithmetic . 14 x 1 1.8 The hyperjump, !1 , and the analogy between c.e. and Π1 .... 15 2 Basic tools 18 2.1 Existence proofs via completeness results .
    [Show full text]
  • Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
    Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory Robert I. Soare∗ January 3, 2009 Contents 1 Introduction 4 1.1 Terminology: Incompleteness and Incomputability . 4 1.2 The Goal of Incomputability not Computability . 5 1.3 Computing Relative to an Oracle or Database . 5 1.4 Continuous Functions and Calculus . 6 2 Origins of Computability and Incomputability 7 2.1 G¨odel'sIncompleteness Theorem . 7 2.2 Alonzo Church . 8 2.3 Herbrand-G¨odelRecursive Functions . 9 2.4 Stalemate at Princeton Over Church's Thesis . 10 2.5 G¨odel'sThoughts on Church's Thesis . 11 3 Turing Breaks the Stalemate 11 3.1 Turing Machines and Turing's Thesis . 11 3.2 G¨odel'sOpinion of Turing's Work . 13 3.2.1 G¨odel[193?] Notes in Nachlass [1935] . 14 3.2.2 Princeton Bicentennial [1946] . 15 ∗Parts of this paper were delivered in an address to the conference, Computation and Logic in the Real World, at Siena, Italy, June 18{23, 2007. Keywords: Turing ma- chine, automatic machine, a-machine, Turing oracle machine, o-machine, Alonzo Church, Stephen C. Kleene, Alan Turing, Kurt G¨odel, Emil Post, computability, incomputability, undecidability, Church-Turing Thesis, Post-Turing Thesis on relative computability, com- putable approximations, Limit Lemma, effectively continuous functions, computability in analysis, strong reducibilities. Thanks are due to C.G. Jockusch, Jr., P. Cholak, and T. Slaman for corrections and suggestions. 1 3.2.3 The Flaw in Church's Thesis . 16 3.2.4 G¨odelon Church's Thesis . 17 3.2.5 G¨odel'sLetter to Kreisel [1968] .
    [Show full text]
  • An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations∗
    An ordinal analysis of admissible set theory using recursion on ordinal notations¤ Jeremy Avigad May 23, 2001 Abstract The notion of a function from N to N defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well-orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke-Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal analysis. 1 Introduction In informal proof-theoretic parlance, the definition of a set of objects is said to be impredicative if it makes reference to a collection of sets that includes the set being defined. A classic example arises if one takes the real numbers to be lower Dedekind cuts of rationals, and then defines the least upper bound of a bounded set of reals to be the intersection of all the upper bounds. A theory is said to be (prima facie) impredicative if its intended interpretation depends on such a definition. The circularity implicit in an impredicative theory poses problems for its ordinal analysis, since the goal of ordinal analysis is to measure the theory’s strength in terms of well-founded ordinal notations — that is, “from the bot- tom up.” For that reason, the first ordinal analyses of impredicative theories, due to Takeuti, Buchholz, and Pohlers were a landmark (see the discussion in the introduction to [7]). Another important step was the move to studying fragments of set theory instead of second-order arithmetic, carried out by J¨ager [12, 13, 14], providing a more natural framework for the analysis of impredica- tivity.
    [Show full text]
  • Cardinals and Intro to Ordinals Notations
    Cardinals and Intro to Ordinals Notations Pete Manolios Northeastern Formal Methods, Lecture 5 September 2008 1 Last Time: Axiomatic Set Theory Axiom 0. Set Existence: ⟨∃x :: x=x⟩ Axiom 1. Extensionality: ⟨∀x,y::⟨∀z :: z∈x ≡ z∈y⟩ ⇒ x=y⟩ Axiom 3. Comprehension Scheme: For each formula φ without y free, the universal closure of the following is an axiom: ⟨∃y :: ⟨∀x :: x∈y ≡ x∈z ∧ φ⟩⟩ Axiom 4. Pairing: ⟨∀x,y :: ⟨∃z :: x∈z ∧ y ∈z⟩⟩ Axiom 5. Union: ⟨∀F :: ⟨∃A :: ⟨∀x,Y :: (Y∈F ∧ x∈Y) ⇒ x∈A⟩⟩⟩ Axiom 6. Replacement Scheme. For each formula φ without Y free, the universal closure of the following is an axiom: ⟨∀x∈A :: ⟨∃!z:: φ(x,z)⟩⟩ ⇒ ⟨∃Y:: ⟨∀x∈A :: ⟨∃z∈Y :: φ(x,z)⟩⟩⟩ Axiom 7. Infinity: ⟨∃x:: 0∈x ∧ ⟨∀y∈x :: y+ ∈ x⟩⟩ (y+ = y ⋃ {y}) Axiom 9. Choice. ⟨∀A :: ⟨∃R :: R well-orders A⟩⟩ 2 Hey, What About Axiom 2? Axiom 2: Foundation. Sets are well-founded under ∈ ⟨∀x :: ⟨∃y∈x⟩ ⇒ ⟨∃y∈x :: ¬⟨∃z∈x :: z∈y⟩⟩⟩ y is a minimal element above See handout for an alternate definition of foundation There are set theories where foundation is omitted So, we can have x∈x The problem is: when are two sets equal? What would you do? 3 Cardinals (& Axiom 8) |A| is the least ordinal α s.t. α≈A (≈ means bijection) α is a cardinal iff α = |α| A cardinal number is an ordinal number whose size is > any of its elements A is finite iff |A| < ω A is infinite iff it is not finite A is countable iff |A| ≤ ω A is uncountable iff it is not countable We can’t prove that uncountable sets exist Axiom 8.
    [Show full text]
  • To Infinity and Beyond
    To Infinity and Beyond Some Notes on Set Theory and Transfinite Numbers Prerna Nadathur August 15, 2008 Contents 1 Introduction 1 2 The Axioms 2 2.1 The Axioms . 2 2.2 Some Consequences . 4 2.2.1 Intersections, Unions and Complements . 4 2.2.2 Ordered Pairs and Cartesian Products . 9 2.2.3 Relations and Functions . 11 2.3 An Axiom for the Natural Numbers . 13 3 Order 14 4 Ordinals and Cardinals 16 4.1 Ordinals . 16 4.1.1 Ordinal Numbers . 16 4.1.2 Ordinal Arithmetic . 18 4.1.3 Countability . 19 4.2 Cardinals . 22 4.3 Cofinality . 26 5 The Axiom of Choice, Zorn's Lemma, and the Well-Ordering Theorem 27 1 Introduction Perhaps the first problem we encounter in set theory is that of the definition of a set. It is a remarkable (although not altogether curious) fact that most books on set theory either actively decline to define a set, or else neglect to mention the problem of definition altogether. Intuitively, we know that sets contain things{elements, or objects, and so in some sense, we 1 can define a set as a collection of objects. Yet this can lead us into trouble: we shall see that there are collections of objects that are not sets, that are \too big" to be sets, that create paradoxes if we regard them as sets. For example, although Cantor's original development of set theory was not axiomatic, it is clear that he relied implicitly on three assumptions: the axiom of extension (more later), the axiom of abstraction, and the axiom of choice.
    [Show full text]
  • Proof Theory of Martin-Löf Type Theory – an Overview
    Proof Theory of Martin-L¨of Type Theory { An Overview Anton Setzer∗ February 22, 2004 Abstract We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsis- tent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of Martin-L¨of type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of Martin-L¨of type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally we repeat the con- cept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength. Keywords: Martin-L¨of type theory, proof theory, Kripke-Platek set theory, W-type, well-founded trees, Kleene's O, Mahlo universe, inductive-recursive definitions, generic programming. 1 Introduction The goal of this article is to introduce the reader, who is not necessarily an expert in proof theory, into the current state of the art of proof theory of Martin-L¨of type theory and the techniques used there. We start by giving a brief overview over the contents of this article.
    [Show full text]
  • Ordinal Notation
    Dmytro Taranovsky Last Update: December 31, 2018 Ordinal Notation Abstract: We introduce a framework for ordinal notation systems, present a family of strong yet simple systems, and give many examples of ordinals in these systems. While much of the material is conjectural, we include systems with conjectured strength beyond second order arithmetic (and plausibly beyond ZFC), and prove well-foundedness for some weakened versions. Contents: 1 Introduction 1.1 Introduction 1.2 Prior Work 1.3 Goals of Ordinal Analysis 2 A Framework for Ordinal Notations — defines the general structure for the notations. 2.1 Definition of the General Notation 2.2 Basic Properties 2.3 Bachmann-Howard Ordinal 2.4 One Variable C 2.5 Reflection Configurations 3 Degrees of Recursive Inaccessibility — presents a notation system for ATR0 plus "for every ordinal a, there is recursively a-inaccessible ordinal". 3.1 Definition of the Notation 3.2 Comparison Relation 3.3 Examples 4 Degrees of Reflection — presents a notation system for KP + Πn reflection and somewhat stronger systems, with the canonical assignment for nonrecursive ordinals. 4.1 Definition of Degrees of Reflection 4.2 Examples and Additional Properties 4.3 Assignment of Degrees 4.4 A Step towards Second Order Arithmetic 5 Main Ordinal Notation System — presents the main notation system, which plausibly reaches second order arithmetic (Z2) or higher. 5.1 Definition and Basic Properties 5.2 Strength of the Main System 5.3 Old Analysis and Examples 6 Beyond Second Order Arithmetic — gives insights for going beyond second order arithmetic and ZFC 6.1 Old Analysis 6.2 New Analysis 7 Iteration of n-built from below — proposes a strengthening of the main system.
    [Show full text]
  • A Tutorial Overview of Ordinal Notations
    A Tutorial Overview of Ordinal Notations Jacques Bailhache ([email protected]) January-April 2018 1 Interest of transfinite ordinal numbers The domain of transfinite ordinal numbers, or ordinals, has the particularity of being the only mathematical domain that cannot be automated. In all other domains of mathematics, it is at least theoretically possible to deduce the theorems automatically from a formal system consisting of a finite set of axioms and rules. But G¨odelproved that given any formal system of a theory sufficiently powerful to contain arithmetics, it is possible to build a proposition that expresses its own unprovability in this formal system. This proposition, which is very huge, has also a meaning as an ordinary arithmetic proposition, but is very useless in ordinary arithmetics. If the formal system is consistent, then this proposition is undecidable. At first sight one could think that we just have to add this proposition to the system as an axiom, but this augmented system also have its own G¨odelianproposition. By adding successively G¨odelianpropositions, we obtain an infinite sequence of systems, and the system defined as the union of all these systems also has its G¨odelianproposition, and so on. But according to Solomon Feferman in "Penrose's G¨odelianargument" http://math.stanford.edu/ feferman/papers/penrose.pdf p.9 : "one obtains completeness for all arithmetical sentences in a progression based on the transfinite iteration of the so-called global or uniform reflection principle" The uniform reflection principle, which is something similar to adding the G¨odelianproposition as an axiom, is described for example in John Harrison's paper "Metatheory and Reflection in Theorem Proving: A Survey and Critique" http://www.cl.cam.ac.uk/ jrh13/papers/reflect.ps.gz p.18 : ` 8n:P r(dφ[n]e) ) φ[n] Harrison also says p.19 : "Feferman showed that a transfinite iteration based on it proves all true sentences of number theory".
    [Show full text]
  • Formalizing Knuth-Bendix Orders and Knuth-Bendix Completion∗
    Formalizing Knuth-Bendix Orders and Knuth-Bendix Completion∗ Christian Sternagel1 and René Thiemann2 1 School of Information Science Japan Advanced Institute of Science and Technology, Japan [email protected] 2 Institute of Computer Science University of Innsbruck, Austria [email protected] Abstract We present extensions of our Isabelle Formalization of Rewriting that cover two historically related concepts: the Knuth-Bendix order and the Knuth-Bendix completion procedure. The former, besides being the first development of its kind in a proof assistant, is based on a generalized version of the Knuth-Bendix order. We compare our version to variants from the literature and show all properties required to certify termination proofs of TRSs. The latter comprises the formalization of important facts that are related to completion, like Birkhoff’s theorem, the critical pair theorem, and a soundness proof of completion, showing that the strict encompassment condition is superfluous for finite runs. As a result, we are able to certify completion proofs. 1998 ACM Subject Classification F.3.1 Specifying and Verifying and Reasoning about Pro- grams, F.4.2 Grammars and Other Rewriting Systems Keywords and phrases certification, completion, confluence, termination Digital Object Identifier 10.4230/LIPIcs.RTA.2013.287 1 Introduction In their seminal paper [10], Knuth and Bendix introduced two important concepts: a procedure that allows us to solve certain instances of the word problem – (Knuth-Bendix) completion – as well as a specific order on terms that is useful to orient equations in the aforementioned procedure – the Knuth-Bendix order (or KBO, for short). Our main contributions around KBO and completion are: There are several variants of KBO (e.g., incorporating quasi-precedences, infinite signatures, subterm coefficient functions, and generalized weight functions).
    [Show full text]
  • Well-Partial-Orders and Ordinal Notation Systems
    Faculty of Sciences Department of Mathematics Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems Jeroen Van der Meeren Supervisor: Prof. Dr. Andreas Weiermann Dissertation submitted in fulfillment of the requirements for the degree of Doctor (Ph.D.) in Sciences, Mathematics 2015 Copyright. The author and the supervisor give the authorization to consult and to copy parts of this work for personal use only. Any other use is limited by the laws of copyright. Permission to reproduce any material contained in this work should be obtained from the author. This does not include the Ghent University logo on the front page, which remains under full copyright of Ghent University. Das Unendliche hat wie keine andere Frage von jeher so tief das Gem¨utdes Menschen bewegt; das Unendliche hat wie kaum eine andere Idee auf den Verstand so anregend und fruchtbar gewirkt; das Unendliche ist aber auch wie kein anderer Begriff so der Aufkl¨arungbed¨urftig. From time immemorial, the infinite has stirred men's emotions more than any other question. Hardly any other idea has stimu- lated the mind so fruitfully. Yet, no other concept needs clarifi- cation more than it does. - David Hilbert, Uber¨ das Unendliche (On the infinite ) [39] Preface Kruskal claims in his now-classical 1972 paper [47] that well-partial-orders are among the most frequently rediscovered mathematical objects. Well- partial-orders have applications in many fields outside the theory of orders: computer science, proof theory, reverse mathematics, algebra, combinatorics, etc. The maximal order type of a well-partial-order characterizes that order's strength.
    [Show full text]