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Title:

Proof-Theoretic Analysis of Termination Proofs

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Introduction In [Cichon 1990] the question has been discussed (and investigated) whether the order type of a termination ordering places a bound on the lengths of reduction sequences in rewrite systems reducing under . It was claimed that at least in the cases of the recursive path ordering rpo and the lexicographic path ordering lpo the followi...

Introduction In [Cichon 1990] the question has been discussed (and investigated) whether the order type of a termination ordering places a bound on the lengths of reduction sequences in rewrite systems reducing under . It was claimed that at least in the cases of the recursive path ordering rpo and the lexicographic path ordering lpo the following theorem holds. (0) If is the order type of a termination ordering for a nite rewrite system R then the function G from the Slow-Growing Hierarchy bounds the lengths of reduction sequences in R. From (0) together with Girard's Hierarchy Comparison Theorem one derives (I) If the rules of a nite rewrite system R are reducing under rpo then the lengths of reduction sequences in R are bounded by some primitive recursive function. (II) If the rules of a nite rewrite system R are reducing under lpo then the lengths of reduction sequences in R are bounded by some function F from the fast-growing hierarchy below ! . Unfort Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-18

Source:

http://www.mathematik.uni-muenchen.de/~buchholz/articles/ppc94a.ps.gz

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Dedicated to Wolfram Pohlers on his retirement

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One of the major problems in reductive proof theory in the early 1970s was to give a proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. This problem was solved in [BFPS] in various ways which all where based on the method of cut-elimination (normalization, reps.) ...

One of the major problems in reductive proof theory in the early 1970s was to give a proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. This problem was solved in [BFPS] in various ways which all where based on the method of cut-elimination (normalization, reps.) for infinitary Tait-style sequent calculi (infinitary systems of natural deduction Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-02-01

Source:

http://www.mathematik.uni-muenchen.de/~buchholz/buchholz_neu.pdf

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text

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en

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Title:

Bar recursive encodings of tree ordinals

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-11-22

Source:

http://www.phil.uu.nl/preprints/lgps/authors/bezem/bar-recursive-encodings-of-tree-ordinals/pdf/

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text

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Title:

Relating Ordinals to Proofs in a Perspicious Way

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this paper, we omit it here. (a) Let c = D c 0 c 1 ::: cm , a = D a 0 a 1 ::: an with principal terms c 1 ; :::; c m ; a 1 ; :::; an . 1. < : From c m : : : c 1 D c 0 we get by IH o(c m ) : : : o(c 1 ) o(D c 0 ) = o(c 0 ) < +1 and thus o(c) < +1 o(D a 0 ) o(a). 2. = and c 0 a 0 : By IH o(c 0 ) < o(a 0 ). Since D c 0 2 OT, we have G c 0 c 0 and t...

this paper, we omit it here. (a) Let c = D c 0 c 1 ::: cm , a = D a 0 a 1 ::: an with principal terms c 1 ; :::; c m ; a 1 ; :::; an . 1. < : From c m : : : c 1 D c 0 we get by IH o(c m ) : : : o(c 1 ) o(D c 0 ) = o(c 0 ) < +1 and thus o(c) < +1 o(D a 0 ) o(a). 2. = and c 0 a 0 : By IH o(c 0 ) < o(a 0 ). Since D c 0 2 OT, we have G c 0 c 0 and thus by IH o(c 0 ) 2 C(o(c 0 ); o(c 0 )). Hence o(c 0 ) < o(a 0 ) by Theorem 1.2(c). Now o(c) o(a) follows as in 1. (using that o(a) is additively closed). 3. = & c 0 = a 0 & c 1 ::: cm a 1 ::: an : Immediate by IH. (b) 1. c = c 0 ::: c k 1 with k 6= 1: Then G c i a and thus (by IH) o(c i ) 2 C := C(o(a); o(a)) for i < k Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-16

Source:

http://www.mathematik.uni-muenchen.de/~buchholz/articles/f7june.ps.gz

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Title:

A note on SLDNF-resolution

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1 Introduction In this paper, starting from Definition 8.8 in [3], we design a new (and as it seems to us rather compact and elegant) notion of SLDNF-tree together with the appropriate definition of fairness such that the following "strong completeness theorem " can be established: Theorem Let S be an input/output specification, P an S-correct l...

1 Introduction In this paper, starting from Definition 8.8 in [3], we design a new (and as it seems to us rather compact and elegant) notion of SLDNF-tree together with the appropriate definition of fairness such that the following "strong completeness theorem " can be established: Theorem Let S be an input/output specification, P an S-correct logic program, T a fair SLDNFtree for G w.r.t. P. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://www.mathematik.uni-muenchen.de/~buchholz/articles/sldlapre.ps.gz

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Title:

Dedicated to Wolfram Pohlers on his retirement

Description:

One of the major problems in reductive proof theory in the early 1970s was to give a proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. This problem was solved in [BFPS] in various ways which all where based on the method of cut-elimination (normalization, reps.) ...

One of the major problems in reductive proof theory in the early 1970s was to give a proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. This problem was solved in [BFPS] in various ways which all where based on the method of cut-elimination (normalization, reps.) for infinitary Tait-style sequent calculi (infinitary systems of natural deduction, resp.). Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-03-19

Source:

http://wwwmath.uni-muenster.de/logik/Personen/rds/pohlers_volume/buchholz.pdf

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Title:

Bar Recursive Encodings of Tree Ordinals

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this paper are. It should be remarked that we are only using bar recursion of lowest type. It is not obvious how the generalization to bar recursion of higher type should be done. For metamathematical reasons the overall limitation of such generalizations is given by the (largely unknown) so-called ordinal of analysis, i.e. the ordinal which is ...

this paper are. It should be remarked that we are only using bar recursion of lowest type. It is not obvious how the generalization to bar recursion of higher type should be done. For metamathematical reasons the overall limitation of such generalizations is given by the (largely unknown) so-called ordinal of analysis, i.e. the ordinal which is related to analysis in the same way as " 0 is related to arithmetic. Most of the material presented in this paper can be found elsewhere in the literature, though sometimes in a different presentation. A paper which is particularly close to ours is Vogel [14], which came to our notice only in the final stage of the completion of this paper. We have tried to give proper credits and adequate references. 2. Preliminaries Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-11

Source:

ftp://ftp.phil.ruu.nl/logic/PREPRINTS/preprint68.ps.Z

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Title:

Refined Program Extraction from Classical Proofs

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this paper we develop a refined method of extracting reasonable and sometimes unexpected programs from classical proofs.

this paper we develop a refined method of extracting reasonable and sometimes unexpected programs from classical proofs. Minimize

Publisher:

Springer Verlag

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-13

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/troelstra00/rpe4.ps.Z

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Title:

Refined program extraction from classical proofs

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2014-02-25

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/troelstra00/rpe5.ps

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Title:

Re ned Program Extraction from Classical Proofs

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It is well known that it is undecidable in general whether a given program meets its speci cation. In contrast, it can be checked easily by amachine whether a formal proof is correct, and from a constructive proof one can automatically

It is well known that it is undecidable in general whether a given program meets its speci cation. In contrast, it can be checked easily by amachine whether a formal proof is correct, and from a constructive proof one can automatically Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://www.cs.swan.ac.uk/reports/yr2002/CSR14-2002.pdf

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