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Title:

Mathematical Logic

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Year of Publication:

2009-04-01

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http://www.mathematik.uni-muenchen.de/~schwicht/lectures/logic/ws03/ml.pdf

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Title:

Finite Notations for Infinite Terms

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In [1] Buchholz presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive (not " 0 -) recursive function its n-th predecessor and e.g. the last rule applied. Here we extend the method ...

In [1] Buchholz presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive (not " 0 -) recursive function its n-th predecessor and e.g. the last rule applied. Here we extend the method to the more general setting of infinite (typed) terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, we use the method to (1) give a new proof of a well-known trade-off theorem [6], which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and (2) construct a continuous normalization operator with an explicit modulus of continuity. It is well known that in order to study primitive recursion in higher types it is useful to unfold the primitive recursion operators into infinite terms. A similar phenomenon occurs in proo. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-11

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/recth96/fn3.ps.Z

http://www.mathematik.uni-muenchen.de/~schwicht/papers/recth96/fn3.ps.Z Minimize

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text

Language:

en

DDC:

515 Analysis *(computed)*

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Title:

Classifying Recursive Functions

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computability. With the preparations done it is now rather straightforward to define computability in our iterated function spaces C ae (based on N viewed as a flat information system). The tokens and finite sets of tokens are encodable by integers using sequence-coding. It is easy to see that the notions X 2 Con ae of consistency and X ` ae a o...

computability. With the preparations done it is now rather straightforward to define computability in our iterated function spaces C ae (based on N viewed as a flat information system). The tokens and finite sets of tokens are encodable by integers using sequence-coding. It is easy to see that the notions X 2 Con ae of consistency and X ` ae a of entailment correspond to recursive (in fact elementary) relations. Definition. A partial continuous functional ' of type ae is said to be computable if - when viewed as a set of (codes of) tokens - it is \Sigma 0 1 -definable. 5. Computability in higher types 27 Lemma. For all types ae; oe; ø the functionals eval ae;oe : (C ae ! C oe ) \Theta C ae ! C oe curry ae;oe;ø : (C ae \Theta C oe ! C ø ) ! (C ae ! (C oe ! C ø )) are computable. Proof. The tokens of eval ae;oe are of the form (W; X; a) with W 2 Con ae!oe , X 2 Con ae and a 2 C oe , and we have (W; X; a) 2 eval j a 2 W (X) j a 2 WX j WX ` a: Here WX is the application of fini. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-12

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/handbook97/classif2.ps.Z

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Title:

A Syntactical Analysis of Non-Size-Increasing Polynomial Time Computation

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A purely syntactical proof is given that all functions definable in a certain affine linear typed -calculus with iteration in all types are polynomial time computable. The proof also gives explicit polynomial bounds that can easily be calculated. 1 Summary In [6] Hofmann presented a linear type system for non-size-increasing polynomial time comp...

A purely syntactical proof is given that all functions definable in a certain affine linear typed -calculus with iteration in all types are polynomial time computable. The proof also gives explicit polynomial bounds that can easily be calculated. 1 Summary In [6] Hofmann presented a linear type system for non-size-increasing polynomial time computation allowing unrestricted recursion for inductive datatypes. The proof that all definable functions of type N ( N are polynomial time computable essentially used semantic concepts, such as the set-theoretic interpretation of terms. We present a different proof of the same result for a slightly modified version of the term system, which uses syntactical arguments only. However, this paper is more than a new proof of an already known result, as the method choosen has several benefits: ffl A reduction relation is defined on the term system such that the term system is closed under reduction. Therefore calculations can be done within the . Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-13

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/lics00/coin.ps.Z

http://www.mathematik.uni-muenchen.de/~schwicht/papers/lics00/coin.ps.Z Minimize

Document Type:

text

Language:

en

DDC:

511 General principles of mathematics *(computed)*

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Title:

Minlog - An Interactive Prover

Description:

normalization-by-evaluation : : : : : : : : : : : : : : : : : : : : : : : : 42 7.2 Implementation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 7.2.1 The model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 7.2.2 Interpretation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ...

normalization-by-evaluation : : : : : : : : : : : : : : : : : : : : : : : : 42 7.2 Implementation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 7.2.1 The model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 7.2.2 Interpretation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 7.2.3 Quote and unquote : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 43 7.2.4 Animation of program-constants and function-symbols : : : : : : : : : : 43 7.2.5 Normalization-by-evaluation : : : : : : : : : : : : : : : : : : : : : : : : : 43 7.2.6 Normalization-by-evaluation for proof terms : : : : : : : : : : : : : : : : 44 7.2.7 Internals : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 8 The T E X-output 45 8.1 How to output : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45 8.2 How to modify the output of types, terms and formulas : : : : : : : : : : : : : 46 8.2.1 add-groun. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-14

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/minlog/doc/manual.ps.gz

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text

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en

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Title:

Feasible Programs from Proofs

Description:

We restrict induction and recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types oe ( and formulas A ( B as well as 8x A with "complete" variables x, and by adding linear concepts to the lambda calculus (fo...

We restrict induction and recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types oe ( and formulas A ( B as well as 8x A with "complete" variables x, and by adding linear concepts to the lambda calculus (for object terms and proof terms). For the arithmetical system we define modified realizability and show that the programs extracted from proofs of \Pi 2 -theorems characterize the polynomial time computable functions. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-16

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/specker00/fp.ps.Z

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Title:

Constructive Analysis with Witnesses

Description:

Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. Non-Countability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completeness 11 2.2. Limits and Inequalities 13 2.3. Series 1...

Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. Non-Countability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completeness 11 2.2. Limits and Inequalities 13 2.3. Series 13 2.4. Redundant Dyadic Representation of Reals 14 2.5. Convergence Tests 15 2.6. Reordering Theorem 17 2.7. The Exponential Series 18 3. The Exponential Function for Complex Numbers 21 4. Continuous Functions 23 4.1. Suprema and In ma 24 4.2. Continuous Functions 25 4.3. Application of a Continuous Function to a Real 27 4.4. Continuous Functions and Limits 28 4.5. Composition of Continuous Functions 28 4.6. Properties of Continuous Functions 29 4.7. Intermediate Value Theorem 30 4.8. Continuity of Functions with More Than One Variable 32 5. Dierentiation 33 5.1. Derivatives 33 5.2. Bounds on the Slope 33 5.3. Properties of Derivatives 34 5 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-18

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/seminars/prosemss04/constr.ps

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text

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en

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Title:

Termination of permutative conversions in intuitionistic Gentzen calculi

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It is shown that permutative conversions terminate for the cut-free intuitionistic Gentzen (i.e. sequent) calculus; this proves a conjecture by Dyckhoff and Pinto. The main technical tool is a term notation for derivations in Gentzen calculi. These terms may be seen as -terms with explicit substitution, where the latter corresponds to the left i...

It is shown that permutative conversions terminate for the cut-free intuitionistic Gentzen (i.e. sequent) calculus; this proves a conjecture by Dyckhoff and Pinto. The main technical tool is a term notation for derivations in Gentzen calculi. These terms may be seen as -terms with explicit substitution, where the latter corresponds to the left introduction rules. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-03-24

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/gentzen97/perm5.ps.Z

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Program extraction in constructive analysis. Submitted to: Logicism, Intuitionism, and Formalism – What has become of them

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We sketch a development of constructive analysis in Bishop’s style, with special emphasis on low type-level witnesses (using separability of the reals). The goal is to set up things in such a way that realistically executable programs can be extracted from proofs. This is carried out for (1) the Intermediate Value Theorem and (2) the existence o...

We sketch a development of constructive analysis in Bishop’s style, with special emphasis on low type-level witnesses (using separability of the reals). The goal is to set up things in such a way that realistically executable programs can be extracted from proofs. This is carried out for (1) the Intermediate Value Theorem and (2) the existence of a continuous inverse to a monotonically increasing continuous function. Using the Minlog proof assistant, the proofs leading to the Intermediate Value Theorem are formalized and realizing terms extracted. It turns out that evaluating these terms is a reasonably fast algorithm to compute, say, approximations of √ 2. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-12-04

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/uppsala05/uppsala05.pdf

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Title:

Realizability interpretation of proofs in constructive analysis

Description:

We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language – a version of Gödel’s T – evaluation is reasonably efficient.

We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language – a version of Gödel’s T – evaluation is reasonably efficient. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-07

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/swansea06/cie06.pdf

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