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Journal ArticleDOI

Some results on cut-elimination, provable well-orderings, induction and reflection

05 Nov 1998-Annals of Pure and Applied Logic (North-Holland)-Vol. 95, pp 93-184
TL;DR: The attic is gathered to gather miscellaneous results in proof theory from the attic, including an equivalence between transfinite induction rule and iterated reflection schema over IΣn, and proof theoretic strengths of classical fixed points theories.
About: This article is published in Annals of Pure and Applied Logic.The article was published on 1998-11-05 and is currently open access. It has received 28 citations till now. The article focuses on the topics: Proof theory & Bar induction.
Citations
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Book ChapterDOI
01 Aug 2015
TL;DR: Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory.
Abstract: Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work.

339 citations

Posted Content
TL;DR: The aim of this introduction is to present the main ideas in an easily accessible fashion to make the result presented accessible to the general public.
Abstract: This thesis is concerned with investigations into the "complexity of term rewriting systems". Moreover the majority of the presented work deals with the "automation" of such a complexity analysis. The aim of this introduction is to present the main ideas in an easily accessible fashion to make the result presented accessible to the general public. Necessarily some technical points are stated in an over-simplified way.

28 citations


Additional excerpts

  • ...Let s, t be 3...

    [...]

Journal ArticleDOI
TL;DR: The derivation length function of a finite term rewriting system terminating via a Knuth–Bendix order is shown to be bounded by the Ackermann function applied to a single exponential function.

26 citations

Journal ArticleDOI
TL;DR: This paper defines dynamic ordinals – they will be sets of number theoretic functions measuring the amount of sΠb1(X) order induction available in a theory, and compares order induction to successor induction over weak theories.
Abstract: Dynamic ordinal analysis is ordinal analysis for weak arithmetics like fragments of bounded arithmetic. In this paper we will define dynamic ordinals – they will be sets of number theoretic functions measuring the amount of sΠ b 1(X) order induction available in a theory. We will compare order induction to successor induction over weak theories. We will compute dynamic ordinals of the bounded arithmetic theories sΣ b n (X)−L m IND for m=n and m=n+1, n≥0. Different dynamic ordinals lead to separation. In this way we will obtain several separation results between these relativized theories. We will generalize our results to further languages extending the language of bounded arithmetic.

18 citations


Cites background or methods or result from "Some results on cut-elimination, pr..."

  • ...de (1)The results for sΣbm(X)-L IND are part of the authors dissertation [3]; the results for sΣbm(X)-L IND base on results of Arai [1]....

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  • ...One half of these results are part of the author’s dissertation [3], the other half bases on results due to Arai [1]....

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  • ...We will start this section computing upper bounds on dynamic ordinals for theories sΣm(X)-L IND by applying results from Arai [1], section 2....

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  • ...Furthermore, we will use results from Arai [1] to obtain DO(T 0 2 (X)) ≡ {λn....

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  • ...By investigations from Arai [1], section 2....

    [...]

Journal ArticleDOI
TL;DR: Epsilon substitution method for theories (H)α0 of absolute jump hierarchies, and two termination proofs of the H-process, based on transfinite induction up to an ordinal θ(α0+ ω)0, which is best possible.
Abstract: We formulate epsilon substitution method for theories (H)α0 of absolute jump hierarchies, and give two termination proofs of the H-process: The first proof is an adaption of Mints M, Mints-Tupailo-Buchholz MTB, ie, based on a cut-elimination of a specially devised infinitary calculus The second one is an adaption of Ackermann Ack Each termination proof is based on transfinite induction up to an ordinal θ(α0+ ω)0, which is best possible

18 citations


Cites background or methods from "Some results on cut-elimination, pr..."

  • ...3 of [2] by a given arithmetic formula A....

    [...]

  • ...This is seen as in [5] and [2]: Replace the parameter variable X in Subsection 6....

    [...]

References
More filters
Book
19 Dec 1990
TL;DR: The Handbook of Theoretical Computer Science provides professionals and students with a comprehensive overview of the main results and developments in this rapidly evolving field.
Abstract: "Of all the books I have covered in the Forum to date, this set is the most unique and possibly the most useful to the SIGACT community, in support both of teaching and research.... The books can be used by anyone wanting simply to gain an understanding of one of these areas, or by someone desiring to be in research in a topic, or by instructors wishing to find timely information on a subject they are teaching outside their major areas of expertise." -- Rocky Ross, "SIGACT News" "This is a reference which has a place in every computer science library." -- Raymond Lauzzana, "Languages of Design" The Handbook of Theoretical Computer Science provides professionals and students with a comprehensive overview of the main results and developments in this rapidly evolving field. Volume A covers models of computation, complexity theory, data structures, and efficient computation in many recognized subdisciplines of theoretical computer science. Volume B takes up the theory of automata and rewriting systems, the foundations of modern programming languages, and logics for program specification and verification, and presents several studies on the theoretic modeling of advanced information processing. The two volumes contain thirty-seven chapters, with extensive chapter references and individual tables of contents for each chapter. There are 5,387 entry subject indexes that include notational symbols, and a list of contributors and affiliations in each volume.

3,089 citations

Book
30 Apr 2012
TL;DR: In this article, the Lambda-Calculus has been studied as a theory of composition and reduction, and the theory of reduction has been used to construct models of Lambda Theories.
Abstract: Towards the Theory. Introduction. Conversion. Reduction. Theories. Models. Conversion. Classical Lambda Calculus. The Theory of Combinators. Classical Lambda Calculus (Continued). The Lambda-Calculus. Bohm Trees. Reduction. Fundamental Theorems. Strongly Equivalent Reductions. Reduction Strategies. Labelled Reduction. Other Notions of Reduction. Theories. Sensible Theories. Other Lambda Theories. Models. Construction of Models. Local Structure of Models. Global Structure of Models. Combinatory Groups. Appendices: Typed Lambda Calculus. Illative Combinatory Logic. Variables. References.

2,632 citations

Book ChapterDOI
01 Jan 1990
TL;DR: In this paper, the authors focus on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained.
Abstract: Publisher Summary This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

1,551 citations

Book
02 Jan 1991
TL;DR: In this article, the authors focus on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained.
Abstract: Publisher Summary This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

1,381 citations

Journal ArticleDOI
TL;DR: Model Theory (Contributors: J.J. Jech, I.C. Juhasz, K. Kunen, M.E. Rudin, J.R. Smorynski, R.S. Statman, A.A. Shore, S.G. Troelstra).
Abstract: Model Theory (Contributors: J. Barwise, P.C. Eklof, H.J. Keisler, A. Kock, A. Macintyre, M. Makkai, M. Morley, G.E. Reyes, K.D. Stroyan). Set Theory (Contributors: J.P. Burgess, K.J. Devlin, T.J. Jech, I. Juhasz, K. Kunen, M.E. Rudin, J.R. Schoenfield). Recursion Theory (Contributors: P. Aczel, M. Davis, H.B. Enderton, A. Kechris, D.A. Martin, Y.N. Moschovakis, M.O. Rabin, R.A. Shore, S.G. Simpson). Proof Theory and Constructive Mathematics (Contributors: H. Barendregt, S. Feferman, M.P. Fourman, L. Harrington, J. Paris, H. Schwichtenberg, C. Smorynski, R. Statman, A.S. Troelstra). Indices.

909 citations