Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen

Diller, Justus

doi:10.1007/bf02025118

Cited by 56 publications

(44 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In other words one must have a non-constructive ("choice") operator for bounded quantification (of the sort provided in the system T V by the bounded choice operator C). In order to avoid this, Burr (1998Burr ( , 2000 gives a further Diller-Nahm (1974) ∀∃-variant interpretation of KPω in a QF theory of primitive recursive set functionals of finite type. It is quite different from the interpretation given here in sec.…”

Section: Burr's Interpretation Of Kpωmentioning

confidence: 99%

On the Strength of some Semi-Constructive Theories

Feferman¹

2012

Logic, Construction, Computation

View full text Add to dashboard Cite

abstract. Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic. But a natural conception of the settheoretic universe is as an indefinite (or "potential") totality, to which intuitionistic logic is more appropriately applied, while each set is taken to be a definite (or "completed") totality, for which classical logic is appropriate; so on that view, set theory should be axiomatized on some correspondingly mixed basis. Similarly, in the case of predicative analysis, the natural numbers are considered to form a definite totality, while the universe of sets (or functions) of natural numbers are viewed as an indefinite totality, so that, again, a mixed semi-constructive logic should be the appropriate one to treat the two together. Various such semiconstructive systems of analysis and set theory are formulated here and their proof-theoretic strength is characterized. Interestingly, though the logic is weakened, one can in compensation strengthen certain principles in a way that could be advantageous for mathematical applications. 1

show abstract

Section: Burr's Interpretation Of Kpωmentioning

confidence: 99%

On the Strength of some Semi-Constructive Theories

Feferman¹

2012

Logic, Construction, Computation

View full text Add to dashboard Cite

show abstract

“…This choice for the treatment of the exponentials corresponds to a variant of Gödel's Dialectica interpretation due to Diller and Nahm [6].…”

Section: Interpretation 1: Kreisel's Modified Realizabilitymentioning

confidence: 99%

Computational Interpretations of Classical Linear Logic

Oliva

2007

Logic, Language, Information and Computation

View full text Add to dashboard Cite

Abstract. We survey several computational interpretations of classical linear logic based on two-player one-move games. The moves of the games are higherorder functionals in the language of finite types. All interpretations discussed treat the exponential-free fragment of linear logic in a common way. They only differ in how much advantage one of the players has in the exponentials games. We discuss how the several choices for the interpretation of the modalities correspond to various well-known functional interpretations of intuitionistic logic, including Gödel's Dialectica interpretation and Kreisel's modified realizability.

show abstract

“…It extends intuitionistic logic in finite types with appropriate combinators 13 , a cases operator D and some very basic arithmetic needed to define characteristic functionals for quantifier-free formulas.…”

Section: The Weak Base System Eil ωmentioning

confidence: 99%

“…17 In a natural deduction context, it might be more natural to treat λ-abstraction as a primitive concept. Natural deduction formulations of functional interpretation are provided by Diller-Nahm [13] (see also [46,51]) and Joergensen [25]. In the former all cases definitions for realizing terms of contractions are postponed to the end by collecting all candidates and making a single final global choice.…”

Section: The Weak Base System Eil ωmentioning

confidence: 99%

A Complexity Analysis of Functional Interpretations

Hernest¹,

Kohlenbach²

2003

BRICS

View full text Add to dashboard Cite

We give a quantitative analysis of Gödel's functional interpretation and its monotone variant. The two have been used for the extraction of programs and numerical bounds as well as for conservation results. They apply both to (semi-)intuitionistic as well as (combined with negative translation) classical proofs. The proofs may be formalized in systems ranging from weak base systems to arithmetic and analysis (and numerous fragments of these). We give upper bounds in basic proof data on the depth, size, maximal type degree and maximal type arity of the extracted terms as well as on the depth of the verifying proof. In all cases terms of size linear in the size of the proof at input can be extracted and the corresponding extraction algorithms have cubic worst-time complexity. The verifying proofs have depth linear in the depth of the proof at input and the maximal size of a formula of this proof.

show abstract

Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen

Cited by 56 publications

References 5 publications

On the Strength of some Semi-Constructive Theories

On the Strength of some Semi-Constructive Theories

Computational Interpretations of Classical Linear Logic

A Complexity Analysis of Functional Interpretations

Contact Info

Product

Resources

About