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Title:

A direct proof of the equivalence between Brouwer’s fan theorem and König’s lemma with a uniqueness hypothesis

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From results of Ishihara it is known that the weak (that is, binary) form of König’s lemma (WKL) implies Brouwer’s fan theorem (Fan). Moreover, Berger and Ishihara [MLQ 2005] have shown that a weakened form WKL! of WKL, where as an additional hypothesis it is required that in an effective sense infinite paths are unique, is equivalent to Fan. Th...

From results of Ishihara it is known that the weak (that is, binary) form of König’s lemma (WKL) implies Brouwer’s fan theorem (Fan). Moreover, Berger and Ishihara [MLQ 2005] have shown that a weakened form WKL! of WKL, where as an additional hypothesis it is required that in an effective sense infinite paths are unique, is equivalent to Fan. The proof that WKL! implies Fan is done explicitely. The other direction (Fan implies WKL!) is far less directly proved; the emphasis is rather to provide a fair number of equivalents to Fan, and to do the proofs economically by giving a circle of implications. Here we give a direct construction. Moreover, we go one step further and formalize the equivalence proof (in the Minlog proof assistant). Since the statements of both Fan and WKL! have computational content, we can automatically extract terms from the two proofs. It turns out that these terms express in a rather perspicuous way the informal constructions. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-08-01

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/bridges05/bridges05.pdf

http://www.mathematik.uni-muenchen.de/~schwicht/papers/bridges05/bridges05.pdf Minimize

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text

Language:

en

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

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Title:

Mathematical Logic

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-05-28

Source:

http://www.mathematik.uni-muenchen.de/%7Eschwicht/lectures/logic/ss10/ml.pdf

http://www.mathematik.uni-muenchen.de/%7Eschwicht/lectures/logic/ss10/ml.pdf Minimize

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text

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en

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Title:

A syntactical analysis of non-size-increasing polynomial time computation

Description:

A syntactical proof is given that all functions definable in a certain affine linear typed λ-calculus with iteration in all types are polynomial time computable. The proof also gives explicit polynomial bounds that can easily be calculated.

A syntactical proof is given that all functions definable in a certain affine linear typed λ-calculus with iteration in all types are polynomial time computable. The proof also gives explicit polynomial bounds that can easily be calculated. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-10-10

Source:

http://arxiv.org/pdf/cs/0011037v1.pdf

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Document Type:

text

Language:

en

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Recursion on the partial continuous functionals

Description:

We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the well-known abbreviation

We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the well-known abbreviation Minimize

Publisher:

Springer

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-03-24

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/athen05/total06.pdf

http://www.mathematik.uni-muenchen.de/~schwicht/papers/athen05/total06.pdf Minimize

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text

Language:

en

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Title:

Constructive Solutions of Continuous Equations

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We modify some seminal notions from constructive analysis, by providing witnesses for (strictly) positive quantifiers occurring in their definitions. For instance, we understand.

We modify some seminal notions from constructive analysis, by providing witnesses for (strictly) positive quantifiers occurring in their definitions. For instance, we understand. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-17

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/russell02/constr.ps

http://www.mathematik.uni-muenchen.de/~schwicht/papers/russell02/constr.ps Minimize

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text

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en

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Title:

Proofs, Lambda Terms and Control Operators

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ed M : V and typed by M A : V :A ffi ) and context unwrapping (denoted V E and typed by requiring V to be of type :B ffi and the evaluation context E[] to be of type B with the `hole' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and extensions based on work of Sabry and Felleisen [18]. In part...

ed M : V and typed by M A : V :A ffi ) and context unwrapping (denoted V E and typed by requiring V to be of type :B ffi and the evaluation context E[] to be of type B with the `hole' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and extensions based on work of Sabry and Felleisen [18]. In particular we stress its connection with questions of termination of different normalization strategies for minimal, intuitionistic and classical logic, or more precisely their fragments in implicational propositional logic. We also give some examples (due to Hirokawa) of derivations in minimal and classical logic which reproduce themselves under certain reasonable conversion rules. This work clearly owes a lot to other people. Robert Const Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-13

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mod95/mod95.ps.Z

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text

Language:

en

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Title:

Classical Proofs and Programs

Description:

Contents 1 Introduction 1 2 General Background 2 2.1 Godel's System T . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Intuitionistic Arithmetic for Functionals . . . . . . . . . . . . . . 6 2.3 Program Extraction from Constructive Proofs . . . . . . . . . . . 7 2.4 Example: Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . 13...

Contents 1 Introduction 1 2 General Background 2 2.1 Godel's System T . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Intuitionistic Arithmetic for Functionals . . . . . . . . . . . . . . 6 2.3 Program Extraction from Constructive Proofs . . . . . . . . . . . 7 2.4 Example: Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . 13 3 Computational Content of Classical Proofs 14 3.1 Definite and Goal Formulas . . . . . . . . . . . . . . . . . . . . . 14 3.2 Computational Content . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Example: Fibonacci Numbers Again . . . . . . . . . . . . . . . . 23 3.4 Example: Integer Square Roots . . . . . . . . . . . . . . . . . . . 26 3.5 Example: The Greatest Common Divisor . . . . . . . . . . . . . 28 3.6 Example: Dickson's Lemma . . . . . . . . . . . . . . . . . . . . . 35 3.7 Towards More Interesting Examples . . . . . . . . . . . . . . . . 38 1 Introduction It is Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-13

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mod99/wm.ps.Z

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text

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en

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Title:

DECORATING PROOFS

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Dedicated to Grigori Mints on occasion of his 70th birthday Abstract. The programs synthesized from proofs are guaranteed to be correct, however at the cost of sometimes introducing irrelevant computations, as a consequence of the fact that the extracted code faithfully reflects the proof. In this paper we extend the work of Ulrich Berger [2], w...

Dedicated to Grigori Mints on occasion of his 70th birthday Abstract. The programs synthesized from proofs are guaranteed to be correct, however at the cost of sometimes introducing irrelevant computations, as a consequence of the fact that the extracted code faithfully reflects the proof. In this paper we extend the work of Ulrich Berger [2], which introduces the concept of “non-computational universal quantifiers”, and propose an algorithm by which we identify at the proof level the components- quantified variables, as well as premises of implications- that are computationally irrelevant and mark them as such. We illustrate the benefits of this (optimal) decorating algorithm in some case studies and present the results obtained with the proof assistant Minlog. We consider proofs in minimal logic, written in natural deduction style. The only rules are introduction and elimination for implication and the universal quantifier. The logical connectives ∃, ∧ are seen as special cases of inductively defined predicates, and hence are defined by the introduction and elimination schemes Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-03-20

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mints09/deco20090728.pdf

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mints09/deco20090728.pdf Minimize

Document Type:

text

Language:

en

DDC:

005 Computer programming, programs & data *(computed)*

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Title:

Proof Theory

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-04-28

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/lectures/proofth/ss06/s.pdf

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en

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Title:

∀x(A → ∃xA)

Description:

Dedicated to Grigori Mints on occasion of his 70th birthday We consider proofs in minimal logic, written in natural deduction style. The only rules are introduction and elimination for implication and the universal quantifier. The logical connectives ∃, ∧ are seen as special cases of inductively defined predicates, and hence are defined by the i...

Dedicated to Grigori Mints on occasion of his 70th birthday We consider proofs in minimal logic, written in natural deduction style. The only rules are introduction and elimination for implication and the universal quantifier. The logical connectives ∃, ∧ are seen as special cases of inductively defined predicates, and hence are defined by the introduction and elimination schemes Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-04-12

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mints09/deco.pdf

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en

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