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Light Dialectica Program Extraction from a Classical Fibonacci Proof

2007
*
Electronical Notes in Theoretical Computer Science
*

The term

doi:10.1016/j.entcs.2006.10.050
fatcat:ozqg6y3sbne4xc3xpycrd32iea
*of**Gödel's**T*extracted by the LDI is, after strong normalization, exactly the usual recursive algorithm which defines the Fibonacci numbers (*in*pairs). ... This semi-classical proof is available*in*MinLog's library*of*examples. ... by terms*in**Gödel's**T*. ...##
###
Kurt Gödel and Computability Theory
[chapter]

2006
*
Lecture Notes in Computer Science
*

*In*particular,

*Gödel's*1931 paper on incompleteness and the methods developed therein were important for the early development

*of*recursive function theory and the

*lambda*calculus at the hands

*of*Church ... Seen

*in*the historical context, Gödel was an important catalyst for the emergence

*of*computability theory

*in*the mid 1930s. ... Unlike

*Gödel's*earliest work, his thoughts on proof

*complexity*and feasible computation

*in*the letter to von Neumann had no impact on the historical development

*of*computability and

*complexity*theory. ...

##
###
Synthesis of Moduli of Uniform Continuity by the Monotone Dialectica Interpretation in the Proof-system MinLog

2007
*
Electronical Notes in Theoretical Computer Science
*

We extract on the computer a number

doi:10.1016/j.entcs.2007.01.023
fatcat:mbhog62lvzdplpemv3mwh4lwgu
*of*moduli*of*uniform continuity for the first few elements*of*a sequence*of*closed terms*t**of**Gödel's**T**of*type (N → N) → (N → N). ... system Z*of**t*≈ (N→N)→(N→N)*t*. ... Schwichtenberg for having suggested to us that the already available*formulations**of*the Monotone (or Bounded) Dialectica may not be satisfying enough from the computer-applied viewpoint. ...##
###
Lambda Logic
[chapter]

2004
*
Lecture Notes in Computer Science
*

We use

doi:10.1007/978-3-540-25984-8_34
fatcat:vqfgfabukfhbdb6fp6ugxz45ka
*lambda*logic to state and prove a soundness theorem allowing the use*of*second order unification*in*resolution, demodulation, and paramodulation*in*a first-order context. ...*Lambda*logic is the union*of*first order logic and*lambda*calculus. We prove basic metatheorems for both total and partial versions*of**lambda*logic. ... That was done long ago*in*the case*of*typed logics; for example*Gödel's*theory*T*had what amounted to the ability to define functions by*lambda*-abstraction. ...##
###
Non-determinism, Non-termination and the Strong Normalization of System T
[chapter]

2013
*
Lecture Notes in Computer Science
*

As straightforward corollary

doi:10.1007/978-3-642-38946-7_5
fatcat:kosdctak6baaten2wo5dbfq2sq
*of*these results we obtain a new proof*of*strong normalization*of**Gödel's*System*T*by a simple translation*of*this latter system into the former. M. ... Then,*t*is a*lambda*abstraction or a pair or if or It , u has a subterm*of*the form xu 1 . . . u n and s = (xu 1 . . . u n )[*t*/x] = tt 1 . . .*t*n .*Of*course, n > 0, since*t*∈ SN and s / ∈ SN. ... Our work has also some aspects*in*common with the technique*of*Joachimski-Matthes [14] , which provides an adaptation*of*the technique*in*[18] that works for the*lambda**formulation**of*System*T*. ...##
###
Reflections on the Categorical Foundations of Mathematics
[chapter]

2011
*
The Western Ontario Series in Philosophy of Science
*

We examine

doi:10.1007/978-94-007-0431-2_9
fatcat:7raatkof5zewvaqvy7e52v4t6y
*Gödel's*completeness and incompleteness theorems for higher order arithmetic from a categorical point*of*view. ... The completeness theorem can be sharpened to represent any topos by continuous sections*of*a sheaf*of*local toposes. Research*of*P. J. ... Here is our*formulation**of**Gödel's*incompleteness theorem, which includes both the classical and intuitionist cases. ...##
###
The Slingshot Argument and Sentential Identity

2009
*
Studia Logica: An International Journal for Symbolic Logic
*

*In*this paper we examine the analysis

*of*the slingshot argument by means

*of*a non-Fregean logic undertaken recently by A. ... Nevertheless, the language

*of*non-Fregean logic can serve as a useful tool for representing the slingshot argument, and several versions

*of*the slingshot argument

*in*non-Fregean logics are presented. ... We will next reconstruct

*Gödel's*proof

*in*the form

*of*an "official" logical inference. ...

##
###
Combinators and the Story of Computation
[article]

2021
*
arXiv
*
pre-print

We discuss the role

arXiv:2102.09658v1
fatcat:jikqdq4lu5bmvim5g5ethyzhpm
*of*combinators*in*the development*of*the modern conception*of*computation over the course*of*the past century. ... We then discuss how combinators informed*lambda*calculus and symbolic computation, and their relationship to the development*of*practical computation. ... He justifies the idea*of**T*by saying that "The function*T*makes it possible to alter the order*of*the terms*of*an expression, and*in*this way it compensates to a certain extent for the lack*of*a commutative ...##
###
The Limits of Computation

2021
*
Axiomathes
*

Moreover, it is claimed that typed systems

doi:10.1007/s10516-021-09561-8
fatcat:w6nlldlpxvblpjgxp37u6jejke
*of*the*lambda*calculus give rise naturally to a functional interpretation*of*rich systems*of*types and to a hierarchy*of*ordinal recursive functionals*of*arbitrary ... It is argued that the power*of*algorithms is at least as strong as functions that can be proved to be totally computable*in*type-theoretic translations*of*subsystems*of*second-order Zermelo Fraenkel set ...*of**Gödel's*System*T**of*primitive recursive functionals*of*finite type led the way as a foundation for computing. ...##
###
Reasoning about Iteration in Gödel's Class Theory
[chapter]

2003
*
Lecture Notes in Computer Science
*

A computer implementation

doi:10.1007/978-3-540-45085-6_18
fatcat:4qjtzv3wu5dm3akk2dgbzlarjq
*of**Gödel's*algorithm for class formation*in*Mathematica TM was used to*formulate*definitions and theorems about iteration*in**Gödel's*class theory. ... The applications include the theory*of*transitive closures*of*relations, the arithmetic*of*natural numbers, construction*of*invariant subsets, and the Schröder-Bernstein theorem. ... Much*of*the*complexity**of*the output*of**Gödel's*original algorithm stems from his use*of*Kuratowski's definition for ordered pairs. ...##
###
Lexicographic Path Induction
[chapter]

2009
*
Lecture Notes in Computer Science
*

The consistency

doi:10.1007/978-3-642-02273-9_21
fatcat:mvdw7j7e6zhldpom5ms3myk2zq
*of*Heyting arithmetic follows directly, and weak normalization for*Gödel's**T*follows indirectly; both have been formalized*in*a prototypical extension*of*Twelf. ... 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 ... We would like to thank Michael Rathjen and Georg Moser for their helpful answers to our questions regarding large ordinals, and Søren Debois for his helpful comments on an earlier draft*of*this paper. ...##
###
Coinductive Models of Finite Computing Agents

1999
*
Electronical Notes in Theoretical Computer Science
*

Non-well-founded sets are the interactive analog

doi:10.1016/s1571-0661(05)80270-1
fatcat:aqfntxxu2jcxxhwsdmgj4bc5za
*of*recursively enumerable sets for Turing machines, while coalgebras play the role*of*the*lambda*calculus. ... Maximal fixed points are shown to play a role*in*models*of*observation that parallels minimal fixed points*in*inductive mathematics. ... Non-Well-Founded Sets and Coalgebras Though questions*of*expressiveness can be*formulated*and proved entirely*in*terms*of*machine-based notions, reformulation*in*terms*of*non-well-founded set theory and ...##
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Optimized programs from (non-constructive) proofs by the light (monotone) Dialectica interpretation

2007
*
Zenodo
*

Moreover, its presentation is given

doi:10.5281/zenodo.5258446
fatcat:27uccunrnzer3cj2jk7keg7hqi
*in*Natural Deduction style, as an improvement*of*Jørgensen's recent adaptation*of*pure*Gödel's*Dialec- tica. ... The machine comparison with the more established program-synthesis technique*of*refined A-translation shows a very good per- formance*of*Light Dialectica, which is outperformed only*in*the case*of*Dick ... The term system, which we denote by*T*, is a variant*of**Gödel's**T**formulated*over the finite types with*lambda*-abstraction as primitive. This is most appropriate*in*a Natural Deduction context. ...##
###
Normalization for the Simply-Typed Lambda-Calculus in Twelf

2008
*
Electronical Notes in Theoretical Computer Science
*

*In*this case study, some boundaries

*of*Twelf current capabilities are touched and discussed. ... Normalization for the simply-typed λ-calculus is proven

*in*Twelf, an implementation

*of*the Edinburgh Logical Framework. ... He is indebted to Thierry Coquand for comments on a draft, to Andrej Filinski and Carsten Schürmann to comments on the workshop version

*of*this article, and to Chung-Chieh Shan for improvements

*of*the ...

##
###
An Interaction Net Encoding of Gödel's System $$\mathcal {T}$$
[chapter]

2016
*
Lecture Notes in Computer Science
*

*In*this paper we show how the ideas can be extended and simplified to encode

*Gödel's*System

*T*-the simply typed λ-calculus extended with numbers. ... The graph rewriting system

*of*interaction nets has been very successful for the implementation

*of*the

*lambda*calculus. ... A first step

*in*this direction is the extension

*of*the language with data-types,

*in*particular lists, and investigate if other algorithms can also benefit from the techniques used here. ...

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