This thesis focuses on the computational interpretation of Cohen's forcing through the classical Curry-Howard correspondence, using the tools of classical realizability. In a first part, we start by a general introduction to classical realizability in second-order arithmetic (PA2). We cover the description of the Krivine Abstract Machine (KAM), the construction of the realizability models, the realizers for arithmetic and the main two computational topics: specification and witness extraction. To illustrate the flexibility of this approach, we show that it can be effortlessly adapted to several extensions such as new instructions in the KAM or primitive datatypes like natural, rational and real numbers. These various works are formalized in the Coq proof assistant.In the second part, we redesign this framework in a higher-order setting and compare it to PA2.This change is necessary to fully express the forcing transformation, but it also allows us to uniformize the theory and integrate all datatypes. We present forcing in classical realizability, initially due to Krivine, and extend it to generic filters whenever the forcing conditions form a datatype. We can then see forcing as a program transformation adding a memory cell with its access primitives. Our aim is to find more efficient realizers rather than independence results, which are the common use of forcing techniques. The methodology is illustrated on the example of Herbrand's theorem, the proof by forcing of which gives a much more efficient program than the usual proof. Furthermore, we can recover the natural algorithm that one can write to solve the underlying computational problem if we use a datatype as forcing poset.

https://tel.archives-ouvertes.fr/tel-01061442/document

On Forcing and Classical Realizability

A traditional random variable X is a function that maps from a stochastic process to the real line. Here, “real line” refers to the structure (R, |⋅|, ≤) , where R is the set of real numbers, ≤ is the standard linear order relation on R , and d (x, y) ≜ |x − y| is the usual metric on R . The traditional expectation value E(X) of X is thenoften apoor choice of a statisticwhen the stochastic process that X maps from is a structure other than the real line or some substructure of the real line. If the stochastic process is a structure that is not linearly ordered (including structures totally unordered) and/or has a metric space geometry very different from that induced by the usual metric, then statistics such as E(X) are often of poor qualitywith regards to qualitative intuition and quantitative variance (expected error) measurements. For example, the traditional expected value of a fair die is E(X) = 1 6 (1 + 2 +⋯+ 6) = 3.5. But this result has no relationship with reality or with intuition because the result implies that we expect the value of ⚂ or ⚃ more than we expect the outcome of say ⚀ or ⚁. The fact is, that for a fair die, we would expect any pair of values equally. The reason for this is that the values of the face of a fair die are merely symbols with no order, and with no metric geometry other than the discrete metric geometry. On a fair die, ⚁ is not greater or less than ⚀ ; rather ⚁ and ⚀ are simply symbols without order. Moreover, ⚀ is not “closer” to ⚁ than it is to ⚂ ; rather, ⚀ , ⚁ , and ⚂ are simply symbols without any inherit order or metric geometry. This paper proposes an alternative statistical system, based somewhat on graph theory, that takes into account the order structure and metric geometry of the underlying stochastic process. 2010Mathematics SubjectClassification 60B99,60G05 (primary); 60E15,60G20,62C20,62F03,62P10,92D20 (secondary)

Order and metric geometry compatible stochastic processing

Lob's theorem and Godel's theorems make predictions about the behavior of systems capable of self-reference with unbounded computational resources with which to write and evaluate proofs. However, in the real world, systems capable of self-reference will have limited memory and processing speed, so in this paper we introduce an effective version of Lob's theorem which is applicable given such bounded resources. These results have powerful implications for the game theory of bounded agents who are able to write proofs about themselves and one another, including the capacity to out-perform classical Nash equilibria and correlated equilibria, attaining mutually cooperative program equilibrium in the Prisoner's Dilemma. Previous cooperative program equilibria studied by Tennenholtz (2004) and Fortnow (2009) have depended on tests for program equality, a fragile condition, whereas "Lobian" cooperation is much more robust and agnostic of the opponent's implementation.

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Parametric Bounded Löb's Theorem and Robust Cooperation of Bounded Agents.

The concept of reducibility in the sense of computation is a central theme of computer science. Classic set theory, however, does not fully reflect this discrete notion. In Martin-Lof type theory, a set is viewed from a type-centric perspective; allowing more explicit structures to be considered. In this thesis we explore basic countable discrete sets from said type-centric perspective. While a keen focus on the notion of set is maintained, we also discuss the basic outline of the intuitionistic logic which the theory is based on.

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Elementary Discrete Sets in Martin-Löf Type Theory

Copyright © 2019 by author(s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/ by/4.0) ABSTRACT In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n-th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed; notwithstanding, heuristic-based iterative methods are additionally normal.

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An Analysis and Study of Iteration Procedures

Introduction 5. Recursion theory 5.1 Primitive recursive functions and sets 5.2 Recursive functions 5.3 Turing machines 5.4 Recursively enumerable sets 5.5 Exercises for Chapter 5 6. Formalization of arithmetic, Godel's theorems 6.1 Peano's axioms 6.2 Representable functions 6.3 Arithmetization of syntax 6.4 Incompleteness and undecidability theorem 7. Set theory 7.1 The theories Z and ZF 7.2 Ordinal numbers and integers 7.3 Inductive proofs and definitions 7.4 Cardinality 7.5 The axiom of foundation and the reflections schemes 7.6 Exercises for Chapter 7 8. Some model theory 8.1 Elementary substructures and extensions 8.2 Construction of elementary extensions 8.3 The interpolation and definability theorems 8.4 Reduced products and ultraproducts 8.5 Preservations theorems 8.6 -categorical theories 8.7 Exercises for Chapter 8 Solutions to the exercises of Part II Chapter 5 Chapter 6 Chapter 7 Chapter 8 Bibliography Index

Recursion theory, Gödel's theorems, set theory, model theory

Foreword Introduction How to use the book 1. Propositional Calculus 2. Boolean Algebras 3. Predicate Calculus 4. Completeness Theorems Answers to the exercises of chapters 1-4 Bibliography Notations Index

Propositional calculus, Boolean algebras, predicate calculus

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200030292

D. Lascar

Papers

Recursion theory, Gödel's theorems, set theory, model theory

Propositional calculus, Boolean algebras, predicate calculus

European Summer Meeting of the Association for Symbolic Logic, Paris, 1985