Cartesian closed 2-categories and rewriting
|
|
- Florence Erica Moore
- 6 years ago
- Views:
Transcription
1 Cartesian closed 2-categories and rewriting Aurore Alcolei A brief presentation of Tom Hirschowitz s paper, Cartesian closed 2-categories and permutation equivalence in higher-order rewriting June 7, CCCAT and HORS June 7, / 21
2 Context Context Construction Représentation du λ-calcul par valeur 2-CCCAT and HORS June 7, / 21
3 Context Context A lot of calculi : λ-calcul in cbv, cbn, lazy, optimal, λ-calcul with let rec / refs / call/cc, π-calcul, etc. same kind of proofs again and again a common point: the abstractions (binding) Aim: Having a framework to specify the semantic of (any) programming language with binding. providing tools to specify/automate proofs and construction for these languages 2-CCCAT and HORS June 7, / 21
4 Context Previous Works What s already there: Higher-Order Rewrite Systems (HRSs) from T.Nipkow no notion of model, does not express reduction steps (binary relation) Categorical approach using Cartesian Closed Categories (CCC) by J.Lambek no notion of reductions, model for equational theories. Ideas: Making signatures for HRSs into a category (Sig) Adding a dimension to Lambek s approach using 2-Cartesian Closed Categories. 2-CCCAT and HORS June 7, / 21
5 Context In a nutshell Programming language/rewriting systems with binding as a 2-category where objects are types morphisms (1-cells) are terms morphisms between parallel morphisms (2-cells) are reductions What would that mean? l t t t t t t t. ev β a 2-CCCAT and HORS June 7, / 21
6 Construction Context Construction Représentation du λ-calcul par valeur 2-CCCAT and HORS June 7, / 21
7 Construction An example Pure λ-calculus: grammar: M, N Λ(Γ) = x Γ λx.m MN (Γ set of variables) reduction rules: (β) (λx.m)n M[N/x] (ξ) M M λx.m λx.m (R) N N MN MN (L) M M MN M N 2-CCCAT and HORS June 7, / 21
8 Construction Signature Example: 2-signature for pure λ-calculus Σ Λ = ({t}, { l [tt ] t a [t, t] t Three sets : 1. Basic types (sorts): X 0 = {t}. 2. Operations, l and a, with their type. }, { β a l x, y x(y) } ) 3. Rules β. redex and reduction are of the same type. H Sig 2-CCCat W 2-CCCAT and HORS June 7, / 21
9 Construction Step 1: 1-Signature Types of a signature: obtained by applying (the monad) on X 0 1-signature: L 0 Sets Sets X {A, B = x X 0 A B 1 B A } sequent := element of S 0 (X ) = L 0 (X ) L 0 (X ) 1-signature := (X 0,X 1 ) with ϕ 1 X 1 S 0 (X ) c (dom(c), cod(c)) 2-CCCAT and HORS June 7, / 21
10 Construction Step 2: 2-Signature (1) Terms of a signature: generated by + simply typed λ-calculus + pairing and projections +... Γ M i i... Γ c M 1,..., M n A c X 1 (,A) modulo β η reduction ( ) Examples: λx.m = l λx. M MN = a M, N x = x ( ) to get a structure close to the CCC 2-CCCAT and HORS June 7, / 21
11 Construction Step 2: 2-Signature (2) monad L 1 Sig 1 Sig 1 such that L 1 (X ) 0 = X 0 and L 1 (X ) 1 = terms of the signature L 1 (X ) := set of pairs of terms with same type 2-signature : (X,X 2 ) where X is a 1-signature and X 2 is equipped with ϕ 2 X 2 L 1 (X ) r (a term, its reduction by r) Ex : β (a l x, y, x[y]) L 1 (Sigma) 2-CCCAT and HORS June 7, / 21
12 Construction Step 3: The adjunction A similar approach: Defining a monad L over Sig Reductions are generated by a 2λ-calculus: Context rules : Γ, x A, x x x A Γ, x A P M N B Γ (λx A.P) λx A.M λx A.N B A Γ P 1 M 1 N 1 G 1... Γ P n M n N n G n Γ c P 1,..., P n c M 1,..., M n c N 1,..., N n A (c X1(G A))... Special rule :... Γ P i M i N i G i... Γ r P 1,..., P n M[M 1,..., M n ] N[N 1,..., N n ] A (r X (G M,N A)) Modulo some equations... to get a structure close to the 2-CCC 2-CCCAT and HORS June 7, / 21
13 Construction Example of equation Equivalence rules := β and η equivalences, equivalence by permutation,... (λx.m)n Example: (λx.m )N M[x N] M [x N ] Left reduction: a l λx t.p, Q ; β λx t.m, N β λx t.p, Q β λx t.m, N ; (λx t.p)q β λx t.m, N ; P[x Q] : Right reduction. 2-CCCAT and HORS June 7, / 21
14 Construction What we get An adjunction: H Sig 2-CCCat models for Σ : morphisms H(Σ) C in 2-CCCat What does that mean? l t W t t t t t t. ev β a 2-CCCAT and HORS June 7, / 21
15 Some Details composition of maps 2-CCCAT and HORS June 7, / 21
16 Représentation du λ-calcul par valeur Context Construction Représentation du λ-calcul par valeur 2-CCCAT and HORS June 7, / 21
17 Représentation du λ-calcul par valeur 2-Signature du λ-calcul en CBV Σ Λ = ({t}, { l [tt ] t a [t, t] t Σ ΛCBV = ({t, v}, restreindre β l [t v ] v a [t, t] t r [v] t }, { β a l x, y x(y) } ), { β a r l x, r y x(y) }) valeur type v Λ(n) v Λ(n) constructeur r t v v l v v v r r ev β t t a t 2-CCCAT and HORS June 7, / 21
18 Représentation du λ-calcul par valeur Représentations équivalentes? Ce qu il fallait montrer 1. Λ CBV (n) L 1 (Σ ΛCBV ) 1 (x 1 v,..., x n v, t) et Λ v CBV (n) L 1(Σ ΛCBV ) 1 (x 1 v,..., x n v, v) 2. Commutativité avec la substitution par valeur 3. Fidélité à la réduction : bissimulation 2-CCCAT and HORS June 7, / 21
19 Sources T. Hirschowitz, Cartesian closed 2-categories and permutation equivalence in higher-order rewriting HAL : hal , version 2 (2011) R. Crole Categories for Types, Cambridge Mathematical Textbooks (1993) Chap.1-2,3-4 S. Awodey, Category Theory, Oxford Logic Guides 52 (2010) 2-CCCAT and HORS June 7, / 21
20 Idées Idée 1 Utiliser un cadre catégorique pour spécifier la sémantique des langages. Idée 2 Utiliser un λ-calcul simplement typé, paramétré, comme support de construction. D où on part : Spécification des syntaxes avec lieurs Syntaxe avec lieurs λ-calcul simplement typé paramétré par une 1-signature Catégorie Sig 1 de 1-signatures Monade L 1 sur Sig 1 : termes engendrés Ce qu on propose : Spécification des systèmes de récriture avec lieurs raffiner les 1-signatures 2-signatures Réductions 2λ-calcul simplement typé paramétré par Σ Catégories, monades, etc. 2-CCCAT and HORS June 7, / 21
21 Retour sur l exemple Plus de termes dans L 1 (Σ) que dans Λ Repose sur la correspondance: Exemples: λx.m = l λx. M Termes du λ-calcul pur avec n variables libres Termes sur Σ tels que x 1 t,..., x n t M t MN = a M, N x = x 2-CCCAT and HORS June 7, / 21
Modèles des langages de programmation Domaines, catégories, jeux. Programme de cette seconde séance:
Modèles des langages de programmation Domaines, catégories, jeux Programme de cette seconde séance: Modèle ensembliste du lambda-calcul ; Catégories cartésiennes fermées 1 Synopsis 1 the simply-typed λ-calculus,
More informationMathematical Synthesis of Equational Deduction Systems. Marcelo Fiore. Computer Laboratory University of Cambridge
Mathematical Synthesis of Equational Deduction Systems Marcelo Fiore Computer Laboratory University of Cambridge TLCA 2009 3.VII.2009 Context concrete theories meta-theories Context concrete theories meta-theories
More informationComplete Partial Orders, PCF, and Control
Complete Partial Orders, PCF, and Control Andrew R. Plummer TIE Report Draft January 2010 Abstract We develop the theory of directed complete partial orders and complete partial orders. We review the syntax
More informationCategories, Proofs and Programs
Categories, Proofs and Programs Samson Abramsky and Nikos Tzevelekos Lecture 4: Curry-Howard Correspondence and Cartesian Closed Categories In A Nutshell Logic Computation 555555555555555555 5 Categories
More informationA few bridges between operational and denotational semantics of programming languages
A few bridges between operational and denotational semantics of programming languages Soutenance d habilitation à diriger les recherches Tom Hirschowitz November 17, 2017 Hirschowitz Bridges between operational
More informationLazy Strong Normalization
Lazy Strong Normalization Luca Paolini 1,2 Dipartimento di Informatica Università di Torino (ITALIA) Elaine Pimentel 1,2 Departamento de Matemática Universidade Federal de Minas Gerais (BRASIL) Dipartimento
More informationA fully abstract semantics for a nondeterministic functional language with monadic types
A fully abstract semantics for a nondeterministic functional language with monadic types Alan Jeffrey 1 School of Cognitive and Computing Sciences University of Sussex, Brighton BN1 9QH, UK alanje@cogs.susx.ac.uk
More informationLambda-Calculus (I) 2nd Asian-Pacific Summer School on Formal Methods Tsinghua University, August 23, 2010
Lambda-Calculus (I) jean-jacques.levy@inria.fr 2nd Asian-Pacific Summer School on Formal Methods Tsinghua University, August 23, 2010 Plan computation models lambda-notation bound variables conversion
More informationCS 4110 Programming Languages & Logics. Lecture 16 Programming in the λ-calculus
CS 4110 Programming Languages & Logics Lecture 16 Programming in the λ-calculus 30 September 2016 Review: Church Booleans 2 We can encode TRUE, FALSE, and IF, as: TRUE λx. λy. x FALSE λx. λy. y IF λb.
More informationA quoi ressemblerait une géométrie du raisonnement?
Journée de la Fédération Charles Hermite A quoi ressemblerait une géométrie du raisonnement? Paul-André Melliès CNRS & Université Paris Diderot LORIA Nancy Mardi 14 octobre 2013 La prose du monde «La parole
More informationNon deterministic classical logic: the λµ ++ -calculus
Paru dans : Mathematical Logic Quarterly, 48, pp. 357-366, 2002 Non deterministic classical logic: the λµ ++ -calculus Karim NOUR LAMA - Equipe de Logique, Université de Savoie 73376 Le Bourget du Lac
More informationCall-by-value non-determinism in a linear logic type discipline
Call-by-value non-determinism in a linear logic type discipline Alejandro Díaz-Caro? Giulio Manzonetto Université Paris-Ouest & INRIA LIPN, Université Paris 13 Michele Pagani LIPN, Université Paris 13
More informationM, N ::= x λp : A.M MN (M, N) () c A. x : b p x : b
A PATTERN-MATCHING CALCULUS FOR -AUTONOMOUS CATEGORIES ABSTRACT. This article sums up the details of a linear λ-calculus that can be used as an internal language of -autonomous categories. The coherent
More informationLambda Calculus. Andrés Sicard-Ramírez. Semester Universidad EAFIT
Lambda Calculus Andrés Sicard-Ramírez Universidad EAFIT Semester 2010-2 Bibliography Textbook: Hindley, J. R. and Seldin, J. (2008). Lambda-Calculus and Combinators. An Introduction. Cambridge University
More informationThe category of open simply-typed lambda terms
The category of open simply-typed lambda terms Daniel Murfet based on joint work with William Troiani o Reminder on category theory Every adjunction gives rise to a monad C F / G D C(Fx,y) = D(x, Gy) C(FGx,x)
More informationConstructive approach to relevant and affine term calculi
Constructive approach to relevant and affine term calculi Jelena Ivetić, University of Novi Sad, Serbia Silvia Ghilezan,University of Novi Sad, Serbia Pierre Lescanne, University of Lyon, France Silvia
More informationThe Safe λ-calculus. William Blum. Joint work with C.-H. Luke Ong. Lunch-time meeting, 14 May Oxford University Computing Laboratory
The Safe λ-calculus William Blum Joint work with C.-H. Luke Ong Oxford University Computing Laboratory Lunch-time meeting, 14 May 2007 Overview Safety is originally a syntactic restriction for higher-order
More informationUniversity of Oxford, Michaelis November 16, Categorical Semantics and Topos Theory Homotopy type theor
Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011 November 16, 2011 References Johnstone, P.T.: Sketches of an Elephant. A Topos-Theory Compendium.
More informationQuantum groupoids and logical dualities
Quantum groupoids and logical dualities (work in progress) Paul-André Melliès CNS, Université Paris Denis Diderot Categories, ogic and Foundations of Physics ondon 14 May 2008 1 Proof-knots Aim: formulate
More informationType Systems. Lecture 2 Oct. 27th, 2004 Sebastian Maneth.
Type Systems Lecture 2 Oct. 27th, 2004 Sebastian Maneth http://lampwww.epfl.ch/teaching/typesystems/2004 Today 1. What is the Lambda Calculus? 2. Its Syntax and Semantics 3. Church Booleans and Church
More informationDenoting computation
A jog from Scott Domains to Hypercoherence Spaces 13/12/2006 Outline Motivation 1 Motivation 2 What Does Denotational Semantic Mean? Trivial examples Basic things to know 3 Scott domains di-domains 4 Event
More informationNotes on Logical Frameworks
Notes on Logical Frameworks Robert Harper IAS November 29, 2012 1 Introduction This is a brief summary of a lecture on logical frameworks given at the IAS on November 26, 2012. See the references for technical
More informationA probabilistic lambda calculus - Some preliminary investigations
A probabilistic lambda calculus - Some preliminary investigations Ugo Dal Lago, Margherita Zorzi Università di Bologna, Università di Verona June, 9-11, 2010, Torino Introduction: Λ P We present some results
More informationExercise sheet n Compute the eigenvalues and the eigenvectors of the following matrices. C =
L2 - UE MAT334 Exercise sheet n 7 Eigenvalues and eigenvectors 1. Compute the eigenvalues and the eigenvectors of the following matrices. 1 1 1 2 3 4 4 1 4 B = 1 1 1 1 1 1 1 1 1 C = Which of the previous
More informationParameterizations and Fixed-Point Operators on Control Categories
Parameterizations and Fixed-Point Operators on Control Categories oshihiko Kakutani 1 and Masahito Hasegawa 12 1 Research Institute for Mathematical Sciences, Kyoto University {kakutani,hassei}@kurims.kyoto-u.ac.jp
More informationNon-Standard Multiset. Jean-Louis Giavitto IRCAM umr 9912 CNRS UPMC & INRIA projet MuSync
1 Non-Standard Multiset Jean-Louis Giavitto IRCAM umr 9912 CNRS UPMC & INRIA projet MuSync Gamma and beyond Gamma considers seriously multiset (rewriting) for programming However, sometimes even multisets
More informationFive Basic Concepts of. Axiomatic Rewriting Theory
Five Basic Concepts of Axiomatic Rewriting Theory Paul-André Melliès Institut de Recherche en Informatique Fondamentale (IRIF) CNRS & Université Paris Denis Diderot 5th International Workshop on Confluence
More informationLocally cartesian closed categories
Locally cartesian closed categories Clive Newstead 80-814 Categorical Logic, Carnegie Mellon University Wednesday 1st March 2017 Abstract Cartesian closed categories provide a natural setting for the interpretation
More informationSequent Combinators: A Hilbert System for the Lambda Calculus
Sequent Combinators: A Hilbert System for the Lambda Calculus Healfdene Goguen Department of Computer Science, University of Edinburgh The King s Buildings, Edinburgh, EH9 3JZ, United Kingdom Fax: (+44)
More informationAbout Typed Algebraic Lambda-calculi
About Typed Algebraic Lambda-calculi Benoît Valiron INRIA Saclay/LIX Palaiseau, France valiron@lix.polytechnique.fr Abstract Arrighi and Dowek (2008) introduce an untyped lambdacalculus together with a
More information(La méthode Event-B) Proof. Thanks to Jean-Raymond Abrial. Language of Predicates.
CSC 4504 : Langages formels et applications (La méthode Event-B) J Paul Gibson, A207 paul.gibson@it-sudparis.eu http://www-public.it-sudparis.eu/~gibson/teaching/event-b/ Proof http://www-public.it-sudparis.eu/~gibson/teaching/event-b/proof.pdf
More informationProbabilistic Applicative Bisimulation and Call-by-Value Lam
Probabilistic Applicative and Call-by-Value Lambda Calculi Joint work with Ugo Dal Lago ENS Lyon February 9, 2014 Probabilistic Applicative and Call-by-Value Lam Introduction Fundamental question: when
More informationType Systems. Today. 1. What is the Lambda Calculus. 1. What is the Lambda Calculus. Lecture 2 Oct. 27th, 2004 Sebastian Maneth
Today 1. What is the Lambda Calculus? Type Systems 2. Its Syntax and Semantics 3. Church Booleans and Church Numerals 4. Lazy vs. Eager Evaluation (call-by-name vs. call-by-value) Lecture 2 Oct. 27th,
More informationProgramming Languages in String Diagrams. [ one ] String Diagrams. Paul-André Melliès. Oregon Summer School in Programming Languages June 2011
Programming Languages in String Diagrams [ one ] String Diagrams Paul-ndré Melliès Oregon Summer School in Programming Languages June 2011 String diagrams diagrammatic account o logic and programming 2
More informationMonads and Adjunctions for Global Exceptions
MFPS 2006 Monads and Adjunctions for Global Exceptions Paul Blain Levy 1 School of Computer Science University of Birmingham Birmingham B15 2TT, U.K. Abstract In this paper, we look at two categorical
More informationExtraction from classical proofs using game models
1/16 Extraction from classical proofs using game models Valentin Blot University of Bath research funded by the UK EPSRC 2/16 The computational content of classical logic Griffin, 1990: Computational content
More informationOperational semantics for disintegration
Operational semantics for disintegration Chung-chieh Shan (Indiana University) Norman Ramsey (Tufts University) Mathematical Foundations of Programming Semantics 2016-05-25 1 What is semantics for? 1.
More informationSequents are written as A, where A; B represent formulae and ; represent multisets of formulae. Where represents the multiset A 1 ; : : : ; A n, then!
What is a Categorical Model of ntuitionistic Linear Logic G.M. Bierman University of Cambridge Computer Laboratory Abstract. This paper re-addresses the old problem of providing a categorical model for
More informationGeometry of Interaction
Linear logic, Ludics, Implicit Complexity, Operator Algebras Geometry of Interaction Laurent Regnier Institut de Mathématiques de Luminy Disclaimer Syntax/Semantics Syntax = nite (recursive) sets Semantics
More informationLambda calculus L9 103
Lambda calculus L9 103 Notions of computability L9 104 Church (1936): λ-calculus Turing (1936): Turing machines. Turing showed that the two very different approaches determine the same class of computable
More informationSubtyping and Intersection Types Revisited
Subtyping and Intersection Types Revisited Frank Pfenning Carnegie Mellon University International Conference on Functional Programming (ICFP 07) Freiburg, Germany, October 1-3, 2007 Joint work with Rowan
More informationVariations on a theme: call-by-value and factorization
Variations on a theme: call-by-value and factorization Beniamino Accattoli INRIA & LIX, Ecole Polytechnique Accattoli (INRIA Parsifal) Variations on a theme: call-by-value and factorization 1 / 31 Outline
More informationOverview. Systematicity, connectionism vs symbolic AI, and universal properties. What is systematicity? Are connectionist AI systems systematic?
Overview Systematicity, connectionism vs symbolic AI, and universal properties COMP9844-2013s2 Systematicity divides cognitive capacities (human or machine) into equivalence classes. Standard example:
More informationA Terminating and Confluent Linear Lambda Calculus
A Terminating and Confluent Linear Lambda Calculus Yo Ohta and Masahito Hasegawa Research Institute for Mathematical Sciences, Kyoto University Kyoto 606-8502, Japan Abstract. We present a rewriting system
More informationThe algebraicity of the lambda-calculus
The algebraicity of the lambda-calculus André Hirschowitz 1 and Marco Maggesi 2 1 Université de Nice Sophia Antipolis http://math.unice.fr/~ah 2 Università degli Studi di Firenze http://www.math.unifi.it/~maggesi
More informationAdjunction Based Categorical Logic Programming
.. Wesleyan University March 30, 2012 Outline.1 A Brief History of Logic Programming.2 Proof Search in Proof Theory.3 Adjunctions in Categorical Proof Theory.4 Connective Chirality and Search Strategy
More informationIntroduction to λ-calculus
p.1/65 Introduction to λ-calculus Ken-etsu FUJITA fujita@cs.gunma-u.ac.jp http://www.comp.cs.gunma-u.ac.jp/ fujita/ Department of Computer Science Gunma University :Church 32, 36, 40; Curry 34 1. Universal
More informationEquivalence of Algebraic λ -calculi extended abstract
Equivalence of Algebraic λ -calculi extended abstract Alejandro Díaz-Caro LIG, Université de Grenoble, France Alejandro.Diaz-Caro@imag.fr Christine Tasson CEA-LIST, MeASI, France Christine.Tasson@cea.fr
More informationFull abstraction for nominal exceptions and general references
Full abstraction for nominal exceptions and general references Nikos Tzevelekos Oxford University N.Tzevelekos GALOP 08 1 Summary Summary Nominal games Further directions Semantics of nominal computation.
More informationInterpreting the Full λ-calculus in the π-calculus
Interpreting the Full λ-calculus in the π-calculus Xiaojuan Cai Joint work with Yuxi Fu BASICS Lab October 12, 2009 Motivation The λ-calculus: sequential model; The π-calculus: concurrent model A deep
More informationProofs in classical logic as programs: a generalization of lambda calculus. A. Salibra. Università Ca Foscari Venezia
Proofs in classical logic as programs: a generalization of lambda calculus A. Salibra Università Ca Foscari Venezia Curry Howard correspondence, in general Direct relationship between systems of logic
More informationGrammatical resources: logic, structure and control
Grammatical resources: logic, structure and control Michael Moortgat & Dick Oehrle 1 Grammatical composition.................................. 5 1.1 Grammar logic: the vocabulary.......................
More informationA Taste of Categorical Logic Tutorial Notes
A Taste of Categorical Logic Tutorial Notes Lars Birkedal birkedal@cs.au.dk) Aleš Bizjak abizjak@cs.au.dk) October 12, 2014 Contents 1 Introduction 2 2 Higher-order predicate logic 2 3 A first set-theoretic
More informationCompleteness and Partial Soundness Results for Intersection & Union Typing for λµ µ
Completeness and Partial Soundness Results for Intersection & Union Typing for λµ µ Steffen van Bakel Department of Computing, Imperial College London, 180 Queen s Gate, London SW7 2BZ, UK Abstract This
More informationFrom syntax to semantics of Dependent Type Theories - Formalized
RDP 2015, Jun. 30, 2015, WCMCS, Warsaw. From syntax to semantics of Dependent Type Theories - Formalized by Vladimir Voevodsky from the Institute for Advanced Study in Princeton, NJ. I will be speaking
More informationProgramming Language Concepts: Lecture 18
Programming Language Concepts: Lecture 18 Madhavan Mukund Chennai Mathematical Institute madhavan@cmi.ac.in http://www.cmi.ac.in/~madhavan/courses/pl2009 PLC 2009, Lecture 18, 30 March 2009 One step reduction
More informationOn the Semantics of Parsing Actions
On the Semantics of Parsing Actions Hayo Thielecke School of Computer Science University of Birmingham Birmingham B15 2TT, United Kingdom Abstract Parsers, whether constructed by hand or automatically
More informationAlgebraic theories in the presence of binding operators, substitution, etc.
Algebraic theories in the presence of binding operators, substitution, etc. Chung Kil Hur Joint work with Marcelo Fiore Computer Laboratory University of Cambridge 20th March 2006 Overview First order
More informationSHARING IN THE WEAK LAMBDA-CALCULUS REVISITED
SHARING IN THE WEAK LAMBDA-CALCULUS REVISITED TOMASZ BLANC, JEAN-JACQUES LÉVY, AND LUC MARANGET INRIA e-mail address: tomasz.blanc@inria.fr INRIA, Microsoft Research-INRIA Joint Centre e-mail address:
More informationUniform Schemata for Proof Rules
Uniform Schemata for Proof Rules Ulrich Berger and Tie Hou Department of omputer Science, Swansea University, UK {u.berger,cshou}@swansea.ac.uk Abstract. Motivated by the desire to facilitate the implementation
More informationOrigin in Mathematical Logic
Lambda Calculus Origin in Mathematical Logic Foundation of mathematics was very much an issue in the early decades of 20th century. Cantor, Frege, Russel s Paradox, Principia Mathematica, NBG/ZF Origin
More informationNormalisation by evaluation
Normalisation by evaluation Sam Lindley Laboratory for Foundations of Computer Science The University of Edinburgh Sam.Lindley@ed.ac.uk August 11th, 2016 Normalisation and embedded domain specific languages
More informationA categorical model for a quantum circuit description language
A categorical model for a quantum circuit description language Francisco Rios (joint work with Peter Selinger) Department of Mathematics and Statistics Dalhousie University CT July 16th 22th, 2017 What
More informationApprentissage automatique Méthodes à noyaux - motivation
Apprentissage automatique Méthodes à noyaux - motivation MODÉLISATION NON-LINÉAIRE prédicteur non-linéaire On a vu plusieurs algorithmes qui produisent des modèles linéaires (régression ou classification)
More informationThe equivalence axiom and univalent models of type theory.
The equivalence axiom and univalent models of type theory. (Talk at CMU on February 4, 2010) By Vladimir Voevodsky Abstract I will show how to define, in any type system with dependent sums, products and
More informationThe Safe Lambda Calculus
The Safe Lambda Calculus William Blum Linacre College Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy Oxford University Computing Laboratory Michaelmas 2008 Abstract
More informationBinding in Nominal Equational Logic
MFPS 2010 Binding in Nominal Equational Logic Ranald Clouston 1,2 Computer Laboratory, University of Cambridge, Cambridge CB3 0DF, United Kingdom Abstract Many formal systems, particularly in computer
More informationDenotational semantics of linear logic
Denotational semantics of linear logic Lionel Vaux I2M, Université d Aix-Marseille, France LL2016, Lyon school: 7 and 8 November 2016 L. Vaux (I2M) Denotational semantics of linear logic LL2016 1 / 31
More informationCurry-Howard Correspondence for Classical Logic
Curry-Howard Correspondence for Classical Logic Stéphane Graham-Lengrand CNRS, Laboratoire d Informatique de l X Stephane.Lengrand@Polytechnique.edu 2 Practicalities E-mail address: Stephane.Lengrand@Polytechnique.edu
More informationLambek Grammars as Combinatory Categorial Grammars
Lambek Grammars as Combinatory Categorial Grammars GERHARD JÄGER, Zentrum für Allgemeine Sprachwissenschaft (ZAS), Jägerstr 10/11, 10117 Berlin, Germany E-mail: jaeger@zasgwz-berlinde Abstract We propose
More informationSimply Typed λ-calculus
Simply Typed λ-calculus Lecture 1 Jeremy Dawson The Australian National University Semester 2, 2017 Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, 2017 1 / 23 A Brief History of Type Theory First developed
More informationTyped Abstract Syntax
Typed Abstract Syntax Julianna Zsido To cite this version: Julianna Zsido. Typed Abstract Syntax. Mathematics [math]. Université Nice Sophia Antipolis, 2010. English. HAL Id: tel-00535944
More informationCOMP6463: λ-calculus
COMP6463: λ-calculus 1. Basics Michael Norrish Michael.Norrish@nicta.com.au Canberra Research Lab., NICTA Semester 2, 2015 Outline Introduction Lambda Calculus Terms Alpha Equivalence Substitution Dynamics
More informationAbout categorical semantics
About categorical semantics Dominique Duval LJK, University of Grenoble October 15., 2010 Capp Café, LIG, University of Grenoble Outline Introduction Logics Effects Conclusion The issue Semantics of programming
More informationThe lambda calculus with constructors
The lambda calculus with constructors Categorical semantic and Continuations Barbara Petit Focus - Univ. Bologna CaCos 2012 Barbara Petit (Focus - Univ. Bologna) The lambda calculus with constructors 1
More informationTowards a Flowchart Diagrammatic Language for Monad-based Semantics
Towards a Flowchart Diagrammatic Language or Monad-based Semantics Julian Jakob Friedrich-Alexander-Universität Erlangen-Nürnberg julian.jakob@au.de 21.06.2016 Introductory Examples 1 2 + 3 3 9 36 4 while
More informationBisimulation and coinduction in higher-order languages
Bisimulation and coinduction in higher-order languages Davide Sangiorgi Focus Team, University of Bologna/INRIA ICE, Florence, June 2013 Bisimulation Behavioural equality One of the most important contributions
More informationCompleteness and Partial Soundness Results for Intersection & Union Typing for λµ µ
Completeness and Partial Soundness Results for Intersection & Union Typing for λµ µ (Annals of Pure and Applied Logic 161, pp 1400-1430, 2010) Steffen van Bakel Department of Computing, Imperial College
More informationRandom variables. Florence Perronnin. Univ. Grenoble Alpes, LIG, Inria. September 28, 2018
Random variables Florence Perronnin Univ. Grenoble Alpes, LIG, Inria September 28, 2018 Florence Perronnin (UGA) Random variables September 28, 2018 1 / 42 Variables aléatoires Outline 1 Variables aléatoires
More informationand one into the linear calculus, yielding a semantics in L. For we use a trivial mapping into the monadic calculus and Girard's translation into the
Linear Logic, Monads and the Lambda Calculus Nick Benton University of Cambridge Computer Laboratory New Museums Site Pembroke Street Cambridge CB2 3QG, UK Nick.Benton@cl.cam.ac.uk Philip Wadler University
More informationRPO, Second-Order Contexts, and λ-calculus
RPO, Second-Order Contexts, and λ-calculus Pietro Di Gianantonio, Furio Honsell, and Marina Lenisa Dipartimento di Matematica e Informatica, Università di Udine via delle Scienze 206, 33100 Udine, Italy
More informationPartially commutative linear logic: sequent calculus and phase semantics
Partially commutative linear logic: sequent calculus and phase semantics Philippe de Groote Projet Calligramme INRIA-Lorraine & CRIN CNRS 615 rue du Jardin Botanique - B.P. 101 F 54602 Villers-lès-Nancy
More informationMeta-reasoning in the concurrent logical framework CLF
Meta-reasoning in the concurrent logical framework CLF Jorge Luis Sacchini (joint work with Iliano Cervesato) Carnegie Mellon University Qatar campus Nagoya University, 27 June 2014 Jorge Luis Sacchini
More informationHenk Barendregt and Freek Wiedijk assisted by Andrew Polonsky. Radboud University Nijmegen. March 5, 2012
1 λ Henk Barendregt and Freek Wiedijk assisted by Andrew Polonsky Radboud University Nijmegen March 5, 2012 2 reading Femke van Raamsdonk Logical Verification Course Notes Herman Geuvers Introduction to
More informationSémantique des jeux asynchrones et réécriture 2-dimensionnelle
Sémantique des jeux asynchrones et réécriture 2-dimensionnelle Soutenance de thèse de doctorat Samuel Mimram Laboratoire PPS (CNRS Université Paris Diderot) 1 er décembre 2008 1 / 64 A program is a text
More informationThe Lambda Calculus is Algebraic
Under consideration for publication in J. Functional Programming 1 The Lambda Calculus is Algebraic PETER SELINGER Department of Mathematics and Statistics University of Ottawa, Ottawa, Ontario K1N 6N5,
More informationTopos Theory. Lectures 17-20: The interpretation of logic in categories. Olivia Caramello. Topos Theory. Olivia Caramello.
logic s Lectures 17-20: logic in 2 / 40 logic s Interpreting first-order logic in In Logic, first-order s are a wide class of formal s used for talking about structures of any kind (where the restriction
More informationIntensionality, Extensionality, and Proof Irrelevance in Modal Type Theory
Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory Frank Pfenning LICS 01 Boston, Massachusetts June 17, 2001 Acknowledgments: Alberto Momigliano,... 1 Outline 1. Introduction 2.
More informationHow to combine diagrammatic logics
How to combine diagrammatic logics Dominique Duval To cite this version: Dominique Duval. How to combine diagrammatic logics. 2009. HAL Id: hal-00432330 https://hal.archives-ouvertes.fr/hal-00432330v2
More informationMechanizing Metatheory in a Logical Framework
Under consideration for publication in J. Functional Programming 1 Mechanizing Metatheory in a Logical Framework Robert Harper and Daniel R. Licata Carnegie Mellon University (e-mail: {rwh,drl}@cs.cmu.edu)
More informationNames and Symmetry in Computer Science
1/27 Names and Symmetry in Computer Science Andrew Pitts Computer Laboratory LICS 2015 Tutorial 2/27 An introduction to nominal techniques motivated by Programming language semantics Automata theory Constructive
More informationJOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX MASANOBU KANEKO Poly-Bernoulli numbers Journal de Théorie des Nombres de Bordeaux, tome 9, n o 1 (1997), p. 221-228
More informationThe Curry-Howard Isomorphism
The Curry-Howard Isomorphism Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) The Curry-Howard Isomorphism MFES 2008/09
More informationGeneralized Ehrhart polynomials
FPSAC 2010, San Francisco, USA DMTCS proc. (subm.), by the authors, 1 8 Generalized Ehrhart polynomials Sheng Chen 1 and Nan Li 2 and Steven V Sam 2 1 Department of Mathematics, Harbin Institute of Technology,
More informationMary Southern and Gopalan Nadathur. This work was funded by NSF grant CCF
A Translation-Based Animation of Dependently-Typed Specifications From LF to hohh(and back again) Mary Southern and Gopalan Nadathur Department of Computer Science and Engineering University of Minnesota
More informationThe Lambda-Calculus Reduction System
2 The Lambda-Calculus Reduction System 2.1 Reduction Systems In this section we present basic notions on reduction systems. For a more detailed study see [Klop, 1992, Dershowitz and Jouannaud, 1990]. Definition
More informationA Taste of Categorical Logic Tutorial Notes
A Taste of Categorical Logic Tutorial Notes Lars Birkedal (birkedal@cs.au.dk) Aleš Bizjak (abizjak@cs.au.dk) July 10, 2017 Contents 1 Introduction 2 2 Higher-order predicate logic 2 3 A first set-theoretic
More informationReview. Principles of Programming Languages. Equality. The Diamond Property. The Church-Rosser Theorem. Corollaries. CSE 230: Winter 2007
CSE 230: Winter 2007 Principles of Programming Languages Lecture 12: The λ-calculus Ranjit Jhala UC San Diego Review The lambda calculus is a calculus of functions: e := x λx. e e 1 e 2 Several evaluation
More informationWhich types have a unique inhabitant? Focusing on pure program equivalence
Which types have a unique inhabitant? Focusing on pure program equivalence Gabriel Scherer, under the supervision of Didier Rémy Gallium INRIA March 30, 2016 1 1 Background 2 Overview 3 Focusing and saturation
More informationTOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES. Sheaves and Cauchy-complete categories
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES R. F. C. WALTERS Sheaves and Cauchy-complete categories Cahiers de topologie et géométrie différentielle catégoriques, tome 22, n o 3 (1981),
More information