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Fernando Ferreira - One of the best experts on this subject based on the ideXlab platform.
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spector s Proof of the Consistency of analysis
2015Co-Authors: Fernando FerreiraAbstract:The editors of this volume asked me to present and discuss Clifford Spector’s Proof of the Consistency of analysis. It is only fitting that, in a volume dedicated to Gerhard Gentzen, known for his epoch-making Consistency Proof of Peano arithmetic PA, Spector’s Proof of Consistency of analysis is discussed. Gentzen’s approach to Consistency Proofs has been systematically developed and generalized by the German school of Proof theory (Schutte, Pohlers, Buchholz, Jager, Rathjen, etc.) and others.
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Spector’s Proof of the Consistency of Analysis
Gentzen's Centenary, 2015Co-Authors: Fernando FerreiraAbstract:The editors of this volume asked me to present and discuss Clifford Spector’s Proof of the Consistency of analysis. It is only fitting that, in a volume dedicated to Gerhard Gentzen, known for his epoch-making Consistency Proof of Peano arithmetic PA, Spector’s Proof of Consistency of analysis is discussed. Gentzen’s approach to Consistency Proofs has been systematically developed and generalized by the German school of Proof theory (Schutte, Pohlers, Buchholz, Jager, Rathjen, etc.) and others.
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CiE - A short note on spector's Proof of Consistency of analysis
Lecture Notes in Computer Science, 2012Co-Authors: Fernando FerreiraAbstract:In 1962, Clifford Spector gave a Consistency Proof of analysis using so-called bar recursors. His paper extends an interpretation of arithmetic given by Kurt Godel in 1958. Spector's Proof relies crucially on the interpretation of the so-called numerical double negation shift principle. The argument for the interpretation is ad hoc. On the other hand, William Howard gave in 1968 a very natural interpretation of bar induction by bar recursion. We show directly that, within the framework of Godel's interpretation, numerical double negation shift is a consequence of bar induction.
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a short note on spector s Proof of Consistency of analysis
Conference on Computability in Europe, 2012Co-Authors: Fernando FerreiraAbstract:In 1962, Clifford Spector gave a Consistency Proof of analysis using so-called bar recursors. His paper extends an interpretation of arithmetic given by Kurt Godel in 1958. Spector's Proof relies crucially on the interpretation of the so-called numerical double negation shift principle. The argument for the interpretation is ad hoc. On the other hand, William Howard gave in 1968 a very natural interpretation of bar induction by bar recursion. We show directly that, within the framework of Godel's interpretation, numerical double negation shift is a consequence of bar induction.
W. Scherrer - One of the best experts on this subject based on the ideXlab platform.
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Consistency and relative efficiency of subspace methods
Automatica, 1995Co-Authors: Manfred Deistler, K. Peternell, W. ScherrerAbstract:We give a Consistency Proof for two subspace methods. We then show the asymptotic equivalence of a special subspace method and the initial estimate proposed by Hannan and Rissanen. Finally, a simulation study comparing two subspace methods and the maximum-likelihood method is performed.
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Consistency and Relative Efficiency of Subspace Methods
IFAC Proceedings Volumes, 1994Co-Authors: Manfred Deistler, K. Peternell, W. ScherrerAbstract:Abstract The paper consists of three parts. In part one we give a Consistency Proof for subspace methods, in part two we show the asymptotic equivalence of a special subspace method and the initial estimate proposed by Hannan and Rissanen. In part three a simulation study comparing two subspace methods and the maximum likelihood method is performed.
Carsten Schürmann - One of the best experts on this subject based on the ideXlab platform.
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Lexicographic Path Induction
2010Co-Authors: Jeffrey Sarnat, Carsten SchürmannAbstract:Abstract. Programming languages theory is full of problems that reduce to proving the Consistency of a logic, such as the normalization of typed lambda-calculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages researchers who often prefer alternatives such as Proofs by logical relations and model theoretic constructions. In this paper we harness the well-foundedness of the lexicographic path ordering to derive an induction principle that combines the comfort of structural induction with the expressive strength of transfinite induction. Using lexicographic path induction, we give a Consistency Proof of Martin-Löf’s intuitionistic theory of inductive definitions. The Consistency of Heyting arithmetic follows directly, and weak normalization for Gödel’s T follows indirectly; both have been formalized in a prototypical extension of Twelf.
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TLCA - Lexicographic Path Induction
Lecture Notes in Computer Science, 2009Co-Authors: Jeffrey Sarnat, Carsten SchürmannAbstract:Programming languages theory is full of problems that reduce to proving the Consistency of a logic, such as the normalization of typed lambda-calculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages researchers, who often prefer alternatives such as Proofs by logical relations and model theoretic constructions. In this paper we harness the well-foundedness of the lexicographic path ordering to derive an induction principle that combines the comfort of structural induction with the expressive strength of transfinite induction. Using lexicographic path induction, we give a Consistency Proof of Martin-Lof's intuitionistic theory of inductive definitions. The Consistency of Heyting arithmetic follows directly, and weak normalization for Godel's T follows indirectly; both have been formalized in a prototypical extension of Twelf.
Annika Siders - One of the best experts on this subject based on the ideXlab platform.
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from stenius Consistency Proof to schutte s cut elimination for ω arithmetic
Review of Symbolic Logic, 2016Co-Authors: Annika SidersAbstract:The book Das Interpretationsproblem der Formalisierten Zahlentheorie und ihre Formale Widerspruchsfreiheit by Erik Stenius published in 1952 contains a Consistency Proof for infinite ω -arithmetic based on a semantical interpretation. Despite the Proof’s reference to semantics the truth definition is in fact equivalent to a syntactical derivability or reduction condition. Based on this reduction condition Stenius proves that the complexity of formulas in a derivation can be limited by the complexity of the conclusion. This independent result can also be proved by cut elimination for ω -arithmetic which was done by Schutte in 1951. In this paper we interpret the syntactic reduction in Stenius’ work as a method for cut elimination based on invertibility of the logical rules. Through this interpretation the constructivity of Stenius’ Proof becomes apparent. This improvement was explicitly requested from Stenius by Paul Bernays in private correspondence (In a letter from Bernays begun on the 19th of September 1952 (Stenius & Bernays, 1951–75)). Bernays, who took a deep interest in Stenius’ manuscript, applied the described method in a Proof Herbrand’s theorem. In this paper we prove Herbrand’s theorem, as an application of Stenius’ work, based on lecture notes of Bernays (Bernays, 1961 ). The main result completely resolves Bernays’ suggestions for improvement by eliminating references to Stenius’ semantics and by showing the constructive nature of the Proof. A comparison with Schutte’s cut elimination Proof shows how Stenius’ simplification of the reduction of universal cut formulas, which in Schutte’s Proof requires duplication and repositioning of the cuts, shifts the problematic case of reduction to implications.
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FROM STENIUS’ Consistency Proof TO SCHÜTTE’S CUT ELIMINATION FOR ω -ARITHMETIC
The Review of Symbolic Logic, 2015Co-Authors: Annika SidersAbstract:The book Das Interpretationsproblem der Formalisierten Zahlentheorie und ihre Formale Widerspruchsfreiheit by Erik Stenius published in 1952 contains a Consistency Proof for infinite ω -arithmetic based on a semantical interpretation. Despite the Proof’s reference to semantics the truth definition is in fact equivalent to a syntactical derivability or reduction condition. Based on this reduction condition Stenius proves that the complexity of formulas in a derivation can be limited by the complexity of the conclusion. This independent result can also be proved by cut elimination for ω -arithmetic which was done by Schutte in 1951. In this paper we interpret the syntactic reduction in Stenius’ work as a method for cut elimination based on invertibility of the logical rules. Through this interpretation the constructivity of Stenius’ Proof becomes apparent. This improvement was explicitly requested from Stenius by Paul Bernays in private correspondence (In a letter from Bernays begun on the 19th of September 1952 (Stenius & Bernays, 1951–75)). Bernays, who took a deep interest in Stenius’ manuscript, applied the described method in a Proof Herbrand’s theorem. In this paper we prove Herbrand’s theorem, as an application of Stenius’ work, based on lecture notes of Bernays (Bernays, 1961 ). The main result completely resolves Bernays’ suggestions for improvement by eliminating references to Stenius’ semantics and by showing the constructive nature of the Proof. A comparison with Schutte’s cut elimination Proof shows how Stenius’ simplification of the reduction of universal cut formulas, which in Schutte’s Proof requires duplication and repositioning of the cuts, shifts the problematic case of reduction to implications.
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A Direct Gentzen-Style Consistency Proof for Heyting Arithmetic
Gentzen's Centenary, 2015Co-Authors: Annika SidersAbstract:Gerhard Gentzen was the first to give a Proof of the Consistency of Peano Arithmetic and in all he worked out four different Proofs between 1934 and 1939. The second Proof was published as [1], the third as [2], and the fourth as [3]. The first Proof was published posthumously in English translation in [4] and in the German original as [5].
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Gentzen's Consistency Proof without heightlines
Archive for Mathematical Logic, 2013Co-Authors: Annika SidersAbstract:This paper gives a Gentzen-style Proof of the Consistency of Heyting arithmetic in an intuitionistic sequent calculus with explicit rules of weakening, contraction and cut. The reductions of the Proof, which transform derivations of a contradiction into less complex derivations, are based on a method for direct cut-elimination without the use of multicut. This method treats contractions by tracing up from contracted cut formulas to the places in the derivation where each occurrence was first introduced. Thereby, Gentzen’s heightline argument, which introduces additional cuts on contracted compound cut formulas, is avoided. To show termination of the reduction procedure an ordinal assignment based on techniques of Howard for Godel’s T is used.
Manfred Deistler - One of the best experts on this subject based on the ideXlab platform.
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Consistency and relative efficiency of subspace methods
Automatica, 1995Co-Authors: Manfred Deistler, K. Peternell, W. ScherrerAbstract:We give a Consistency Proof for two subspace methods. We then show the asymptotic equivalence of a special subspace method and the initial estimate proposed by Hannan and Rissanen. Finally, a simulation study comparing two subspace methods and the maximum-likelihood method is performed.
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Consistency and Relative Efficiency of Subspace Methods
IFAC Proceedings Volumes, 1994Co-Authors: Manfred Deistler, K. Peternell, W. ScherrerAbstract:Abstract The paper consists of three parts. In part one we give a Consistency Proof for subspace methods, in part two we show the asymptotic equivalence of a special subspace method and the initial estimate proposed by Hannan and Rissanen. In part three a simulation study comparing two subspace methods and the maximum likelihood method is performed.