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1.
Program development by proof transformation
Open Access
Title:
Program development by proof transformation
Author:
Ulrich Berger
;
Helmut Schwichtenberg
Ulrich Berger
;
Helmut Schwichtenberg
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We begin by reviewing the natural deduction rules for the!^8fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. We also consider two alternative methods to dea...
We begin by reviewing the natural deduction rules for the!^8fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. We also consider two alternative methods to deal with the strong or constructive existential quantifier 9\Lambda. Finally we discuss the wellknown notion of an extracted program of a derivation involving 9\Lambda, in order to set up a relation between the two alternatives. Section 2 deals with the computational content of classical proofs. As is wellknown a proof of a 89theorem with a quantifierfree kernel where 9 is viewed as defined by:8: can be used as a program. We describe a "direct method " to use such a proof as a program, and compare it with Harvey Friedman's Atranslation [3] followed by program extraction from the resulting constructive proof. It is shown that both algorithms coincide. In section 3 Goad's method of pruning of proof trees is introduced. It is shown how a proof can be simplified after addition of some further assumptions. In a first step some subproofs are replaced by different ones using the additional assumptions. In a second step parts of the proof tree are pruned, i.e. cut out. Note that the first step involves searching for new proofs using the new assumptions of formulas in the proof tree. Hence we also have to discuss proof search in minimal logic. Finally section 4 treats an example already considered by Goad in his thesis [5], the binpacking problem. The main difference to Goad's work is that he used a logic with the strong existential quantifier, whereas we work within the!8fragment. This example is particularly wellsuited to demonstrate that the pruning method can be applied to adapt programs to particular situations, and moreover that pruning can change the functions computed by programs. In this sense this method is essentially different from program development by program transformation. We would like to thank Michael Bopp and KarlHeinz Niggl for their help in preparing these notes.
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Year of Publication:
20090812
Source:
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod93/mod93.ps
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod93/mod93.ps
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DDC:
005 Computer programming, programs & data
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.134.6444
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod93/mod93.ps
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.134.6444
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod93/mod93.ps
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2.
A direct proof of the equivalence between Brouwer’s fan theorem and König’s lemma with a uniqueness hypothesis
Open Access
Title:
A direct proof of the equivalence between Brouwer’s fan theorem and König’s lemma with a uniqueness hypothesis
Author:
Helmut Schwichtenberg
Helmut Schwichtenberg
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Abstract: 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 ...
Abstract: 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. Key Words: Brouwer’s fan theorem, König’s lemma Category: F.1
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Year of Publication:
20090106
Source:
http://www.jucs.org/jucs_11_12/a_direct_proof_of/jucs_11_12_2086_2095_schwichtenberg.pdf
http://www.jucs.org/jucs_11_12/a_direct_proof_of/jucs_11_12_2086_2095_schwichtenberg.pdf
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.100.7737
http://www.jucs.org/jucs_11_12/a_direct_proof_of/jucs_11_12_2086_2095_schwichtenberg.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.100.7737
http://www.jucs.org/jucs_11_12/a_direct_proof_of/jucs_11_12_2086_2095_schwichtenberg.pdf
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3.
Finite Notations for Infinite Terms
Open Access
Title:
Finite Notations for Infinite Terms
Author:
Helmut Schwichtenberg
Helmut Schwichtenberg
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In [1] Buchholz presented a method to build notation systems for infinite sequentstyle derivations, analogous to wellknown 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 nth 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 sequentstyle derivations, analogous to wellknown 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 nth 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 prooftheoretic contexts as well as in recursion theory. As examples, we use the method to (1) give a new proof of a wellknown tradeoff 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.
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Year of Publication:
20090411
Source:
http://www.mathematik.unimuenchen.de/~schwicht/papers/recth96/fn3.ps.Z
http://www.mathematik.unimuenchen.de/~schwicht/papers/recth96/fn3.ps.Z
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515 Analysis
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.8357
http://www.mathematik.unimuenchen.de/~schwicht/papers/recth96/fn3.ps.Z
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.8357
http://www.mathematik.unimuenchen.de/~schwicht/papers/recth96/fn3.ps.Z
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4.
∀x(A → ∃xA)
Open Access
Title:
∀x(A → ∃xA)
Author:
Diana Ratiu
;
Helmut Schwichtenberg
Diana Ratiu
;
Helmut Schwichtenberg
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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
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Year of Publication:
20120412
Source:
http://www.mathematik.unimuenchen.de/~schwicht/papers/mints09/deco.pdf
http://www.mathematik.unimuenchen.de/~schwicht/papers/mints09/deco.pdf
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.219.3380
http://www.mathematik.unimuenchen.de/~schwicht/papers/mints09/deco.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.219.3380
http://www.mathematik.unimuenchen.de/~schwicht/papers/mints09/deco.pdf
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5.
Constructive Solutions of Continuous Equations
Open Access
Title:
Constructive Solutions of Continuous Equations
Author:
Peter Schuster
;
Helmut Schwichtenberg
Peter Schuster
;
Helmut Schwichtenberg
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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.
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Year of Publication:
20090417
Source:
http://www.mathematik.unimuenchen.de/~schwicht/papers/russell02/constr.ps
http://www.mathematik.unimuenchen.de/~schwicht/papers/russell02/constr.ps
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.12.3554
http://www.mathematik.unimuenchen.de/~schwicht/papers/russell02/constr.ps
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.12.3554
http://www.mathematik.unimuenchen.de/~schwicht/papers/russell02/constr.ps
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6.
Proofs, Lambda Terms and Control Operators
Open Access
Title:
Proofs, Lambda Terms and Control Operators
Author:
Helmut Schwichtenberg
;
P
Helmut Schwichtenberg
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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
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Year of Publication:
20090413
Source:
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod95/mod95.ps.Z
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod95/mod95.ps.Z
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.4125
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod95/mod95.ps.Z
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.4125
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod95/mod95.ps.Z
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7.
Classical Proofs and Programs
Open Access
Title:
Classical Proofs and Programs
Author:
Helmut Schwichtenberg
Helmut Schwichtenberg
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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
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20090413
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http://www.mathematik.unimuenchen.de/~schwicht/papers/mod99/wm.ps.Z
http://www.mathematik.unimuenchen.de/~schwicht/papers/mod99/wm.ps.Z
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http://www.mathematik.unimuenchen.de/~schwicht/papers/mod99/wm.ps.Z
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.9857
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8.
DECORATING PROOFS
Open Access
Title:
DECORATING PROOFS
Author:
Diana Ratiu
;
Helmut Schwichtenberg
Diana Ratiu
;
Helmut Schwichtenberg
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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 “noncomputational 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
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Year of Publication:
20100320
Source:
http://www.mathematik.unimuenchen.de/~schwicht/papers/mints09/deco20090728.pdf
http://www.mathematik.unimuenchen.de/~schwicht/papers/mints09/deco20090728.pdf
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http://www.mathematik.unimuenchen.de/~schwicht/papers/mints09/deco20090728.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.156.9681
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9.
Mathematical Logic
Open Access
Title:
Mathematical Logic
Author:
Helmut Schwichtenberg
Helmut Schwichtenberg
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Year of Publication:
20090401
Source:
http://www.mathematik.unimuenchen.de/~schwicht/lectures/logic/ws03/ml.pdf
http://www.mathematik.unimuenchen.de/~schwicht/lectures/logic/ws03/ml.pdf
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http://www.mathematik.unimuenchen.de/~schwicht/lectures/logic/ws03/ml.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.7394
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10.
Proof Theory
Open Access
Title:
Proof Theory
Author:
Helmut Schwichtenberg
Helmut Schwichtenberg
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Year of Publication:
20100428
Source:
http://www.mathematik.unimuenchen.de/~schwicht/lectures/proofth/ss06/s.pdf
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http://www.mathematik.unimuenchen.de/~schwicht/lectures/proofth/ss06/s.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.137.1959
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(1) Holger Benl
(1) Jäger, G.
(1) Karl Stroetmann
(1) Kenji Miyamoto
(1) KleineBüning, H.
(1) Kripke Structure
(1) Letouzey, Pierre
(1) Ludwig Bauer
(1) Ludwig Maximilians
(1) Ludwigmaximiliansuniversität München
(1) Müller, G.H.
(1) P
(1) Peter Schuster
(1) Pierre Letouzey
(1) Prof Helmut Schwichtenberg
(1) Richter, M.M.
(1) Rose, H.E.
(1) Rödding, Dieter
(1) Scharlau, W.
(1) Schütte, Kurt
(1) Shepherdson, J.C.
(1) Simmons, H.
(1) Simon Huber
(1) Stefan Berghofer
(1) Stephen J. Bellantoni
(1) Troelstra, A.S.
(1) Törnig, W.
(1) Ulrich Berger Helmut
(1) Universität Ws
(1) Vermuri, R.
(1) Wainer, S.S.
(1) Wainer, Stanley S.
(1) Wenzel, Markus M.
(1) van Dalen, D.
Author:
Subject
(26) ddc 510
(26) mathematik
(25) informatik und statistik
(5) lambda calculus
(4) higher types
(4) linear logic
(4) nc
(4) parallel computation
(4) recursion
(3) 510 mathematics
(3) implicit computational complexity
(3) mathematica
(2) ddc 000
(1) 03f60
(1) beweisterme curry howard isomorphismus...
(1) categories and subject descriptors
(1) complexity
(1) computer science logic in computer science
(1) dialectica interpretation
(1) extensionality
(1) f 2 2
(1) f 2 2 analysis of algorithms and problem...
(1) f 4 1
(1) f 4 1 mathematical logic and formal languages
(1) hereditarily majorizable functionals
(1) info info lo computer science logic in computer...
(1) languages additional key words and phrases
(1) mathematical logic lambda calculus and related...
(1) mathematics logic
(1) mechanized theorem proving readable formal...
(1) monotone functionals
(1) monotone majorizable functionals
(1) nonnumerical algorithms and problems general terms
(1) normalization by evaluation
(1) program extraction from proofs
(1) proof terms curry howard isomorphism...
(1) r τ l ρ
(1) realizability
(1) simply typed lambda calculus
(1) theory
(1) typed lambda calculus
Subject:
Dewey Decimal Classification (DDC)
(11) Computer science, knowledge & systems [00*]
(2) Mathematics [51*]
Dewey Decimal Classification (DDC):
Year of Publication
(36) 2009
(8) 2008
(6) 2013
(6) 2014
(5) 2010
(4) 1972
(4) 1975
(4) 1991
(4) 2011
(3) 1969
(3) 1986
(2) 1979
(2) 2007
(2) 2012
(1) 1968
(1) 1971
(1) 1973
(1) 1974
(1) 1976
(1) 1977
(1) 1978
(1) 1982
(1) 1987
(1) 1988
(1) 1989
(1) 1990
(1) 1992
(1) 1993
(1) 1995
(1) 2000
(1) 2002
(1) 2006
(1) 2015
Year of Publication:
Content Provider
(67) CiteSeerX
(26) Munich LMU: Open Access
(4) Project Euclid
(3) DigiZeitschriften
(3) Göttingen Center for Retrospective Digitization...
(2) ArXiv.org
(2) Munich TU: mediaTUM
(1) HAL  Hyper Article en Ligne
Content Provider:
Language
(82) English
(22) Unknown
(4) German
Language:
Document Type
(77) Text
(10) Article, Journals
(9) Books
(8) Reports, Papers, Lectures
(4) Theses
Document Type:
Access
(71) Open Access
(37) Unknown
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