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Light Dialectica Program Extraction from a Classical Fibonacci Proof

Mircea-Dan Hernest
2007 Electronical Notes in Theoretical Computer Science  
The term 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.  ... 
doi:10.1016/j.entcs.2006.10.050 fatcat:ozqg6y3sbne4xc3xpycrd32iea

Kurt Gödel and Computability Theory [chapter]

Richard Zach
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.  ... 
doi:10.1007/11780342_59 fatcat:mi2qey4monflhagqyijp3wqjfi

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

Mircea-Dan Hernest
2007 Electronical Notes in Theoretical Computer Science  
We extract on the computer a number 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.  ... 
doi:10.1016/j.entcs.2007.01.023 fatcat:mbhog62lvzdplpemv3mwh4lwgu

Lambda Logic [chapter]

Michael Beeson
2004 Lecture Notes in Computer Science  
We use 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.  ... 
doi:10.1007/978-3-540-25984-8_34 fatcat:vqfgfabukfhbdb6fp6ugxz45ka

Non-determinism, Non-termination and the Strong Normalization of System T [chapter]

Federico Aschieri, Margherita Zorzi
2013 Lecture Notes in Computer Science  
As straightforward corollary 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.  ... 
doi:10.1007/978-3-642-38946-7_5 fatcat:kosdctak6baaten2wo5dbfq2sq

Reflections on the Categorical Foundations of Mathematics [chapter]

Joachim Lambek, Philip J. Scott
2011 The Western Ontario Series in Philosophy of Science  
We examine 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.  ... 
doi:10.1007/978-94-007-0431-2_9 fatcat:7raatkof5zewvaqvy7e52v4t6y

The Slingshot Argument and Sentential Identity

Yaroslav Shramko, Heinrich Wansing
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.  ... 
doi:10.1007/s11225-009-9182-5 fatcat:jk4mz6w36fe2pldqede4xwzk2i

Combinators and the Story of Computation [article]

Stephen Wolfram
2021 arXiv   pre-print
We discuss the role 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  ... 
arXiv:2102.09658v1 fatcat:jikqdq4lu5bmvim5g5ethyzhpm

The Limits of Computation

Andrew Powell
2021 Axiomathes  
Moreover, it is claimed that typed systems 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.  ... 
doi:10.1007/s10516-021-09561-8 fatcat:w6nlldlpxvblpjgxp37u6jejke

Reasoning about Iteration in Gödel's Class Theory [chapter]

Johan Gijsbertus Frederik Belinfante
2003 Lecture Notes in Computer Science  
A computer implementation 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.  ... 
doi:10.1007/978-3-540-45085-6_18 fatcat:4qjtzv3wu5dm3akk2dgbzlarjq

Lexicographic Path Induction [chapter]

Jeffrey Sarnat, Carsten Schürmann
2009 Lecture Notes in Computer Science  
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.  ...  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.  ... 
doi:10.1007/978-3-642-02273-9_21 fatcat:mvdw7j7e6zhldpom5ms3myk2zq

Coinductive Models of Finite Computing Agents

Peter Wegner, Dina Goldin
1999 Electronical Notes in Theoretical Computer Science  
Non-well-founded sets are the interactive analog 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  ... 
doi:10.1016/s1571-0661(05)80270-1 fatcat:aqfntxxu2jcxxhwsdmgj4bc5za

Optimized programs from (non-constructive) proofs by the light (monotone) Dialectica interpretation

HERNEST MIRCEA DAN, Jean-Pierre JOUANNAUD, Helmut SCHWICHTENBERG
2007 Zenodo  
Moreover, its presentation is given 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.  ... 
doi:10.5281/zenodo.5258446 fatcat:27uccunrnzer3cj2jk7keg7hqi

Normalization for the Simply-Typed Lambda-Calculus in Twelf

Andreas Abel
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  ... 
doi:10.1016/j.entcs.2007.11.009 fatcat:vgf7bxyus5cpldkgxvwdou4rc4

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

Ian Mackie, Shinya Sato
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.  ... 
doi:10.1007/978-3-319-29604-3_6 fatcat:jrsk6jbitjfklfq7grqsnntd5i
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