Metatheorems About Convertibility in Typed Lambda Calculi
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Lecture Notes on the Lambda Calculus 2.4 Introduction to Β-Reduction
2.2 Free and bound variables, α-equivalence. 8 2.3 Substitution ............................. 10 Lecture Notes on the Lambda Calculus 2.4 Introduction to β-reduction..................... 12 2.5 Formal definitions of β-reduction and β-equivalence . 13 Peter Selinger 3 Programming in the untyped lambda calculus 14 3.1 Booleans .............................. 14 Department of Mathematics and Statistics 3.2 Naturalnumbers........................... 15 Dalhousie University, Halifax, Canada 3.3 Fixedpointsandrecursivefunctions . 17 3.4 Other data types: pairs, tuples, lists, trees, etc. ....... 20 Abstract This is a set of lecture notes that developed out of courses on the lambda 4 The Church-Rosser Theorem 22 calculus that I taught at the University of Ottawa in 2001 and at Dalhousie 4.1 Extensionality, η-equivalence, and η-reduction. 22 University in 2007 and 2013. Topics covered in these notes include the un- typed lambda calculus, the Church-Rosser theorem, combinatory algebras, 4.2 Statement of the Church-Rosser Theorem, and some consequences 23 the simply-typed lambda calculus, the Curry-Howard isomorphism, weak 4.3 Preliminary remarks on the proof of the Church-Rosser Theorem . 25 and strong normalization, polymorphism, type inference, denotational se- mantics, complete partial orders, and the language PCF. 4.4 ProofoftheChurch-RosserTheorem. 27 4.5 Exercises .............................. 32 Contents 5 Combinatory algebras 34 5.1 Applicativestructures. 34 1 Introduction 1 5.2 Combinatorycompleteness . 35 1.1 Extensionalvs. intensionalviewoffunctions . ... 1 5.3 Combinatoryalgebras. 37 1.2 Thelambdacalculus ........................ 2 5.4 The failure of soundnessforcombinatoryalgebras . .... 38 1.3 Untypedvs.typedlambda-calculi . 3 5.5 Lambdaalgebras .......................... 40 1.4 Lambdacalculusandcomputability . 4 5.6 Extensionalcombinatoryalgebras . 44 1.5 Connectionstocomputerscience . -
Category Theory and Lambda Calculus
Category Theory and Lambda Calculus Mario Román García Trabajo Fin de Grado Doble Grado en Ingeniería Informática y Matemáticas Tutores Pedro A. García-Sánchez Manuel Bullejos Lorenzo Facultad de Ciencias E.T.S. Ingenierías Informática y de Telecomunicación Granada, a 18 de junio de 2018 Contents 1 Lambda calculus 13 1.1 Untyped λ-calculus . 13 1.1.1 Untyped λ-calculus . 14 1.1.2 Free and bound variables, substitution . 14 1.1.3 Alpha equivalence . 16 1.1.4 Beta reduction . 16 1.1.5 Eta reduction . 17 1.1.6 Confluence . 17 1.1.7 The Church-Rosser theorem . 18 1.1.8 Normalization . 20 1.1.9 Standardization and evaluation strategies . 21 1.1.10 SKI combinators . 22 1.1.11 Turing completeness . 24 1.2 Simply typed λ-calculus . 24 1.2.1 Simple types . 25 1.2.2 Typing rules for simply typed λ-calculus . 25 1.2.3 Curry-style types . 26 1.2.4 Unification and type inference . 27 1.2.5 Subject reduction and normalization . 29 1.3 The Curry-Howard correspondence . 31 1.3.1 Extending the simply typed λ-calculus . 31 1.3.2 Natural deduction . 32 1.3.3 Propositions as types . 34 1.4 Other type systems . 35 1.4.1 λ-cube..................................... 35 2 Mikrokosmos 38 2.1 Implementation of λ-expressions . 38 2.1.1 The Haskell programming language . 38 2.1.2 De Bruijn indexes . 40 2.1.3 Substitution . 41 2.1.4 De Bruijn-terms and λ-terms . 42 2.1.5 Evaluation . -
Plurals and Mereology
Journal of Philosophical Logic (2021) 50:415–445 https://doi.org/10.1007/s10992-020-09570-9 Plurals and Mereology Salvatore Florio1 · David Nicolas2 Received: 2 August 2019 / Accepted: 5 August 2020 / Published online: 26 October 2020 © The Author(s) 2020 Abstract In linguistics, the dominant approach to the semantics of plurals appeals to mere- ology. However, this approach has received strong criticisms from philosophical logicians who subscribe to an alternative framework based on plural logic. In the first part of the article, we offer a precise characterization of the mereological approach and the semantic background in which the debate can be meaningfully reconstructed. In the second part, we deal with the criticisms and assess their logical, linguistic, and philosophical significance. We identify four main objections and show how each can be addressed. Finally, we compare the strengths and shortcomings of the mereologi- cal approach and plural logic. Our conclusion is that the former remains a viable and well-motivated framework for the analysis of plurals. Keywords Mass nouns · Mereology · Model theory · Natural language semantics · Ontological commitment · Plural logic · Plurals · Russell’s paradox · Truth theory 1 Introduction A prominent tradition in linguistic semantics analyzes plurals by appealing to mere- ology (e.g. Link [40, 41], Landman [32, 34], Gillon [20], Moltmann [50], Krifka [30], Bale and Barner [2], Chierchia [12], Sutton and Filip [76], and Champollion [9]).1 1The historical roots of this tradition include Leonard and Goodman [38], Goodman and Quine [22], Massey [46], and Sharvy [74]. Salvatore Florio [email protected] David Nicolas [email protected] 1 Department of Philosophy, University of Birmingham, Birmingham, United Kingdom 2 Institut Jean Nicod, Departement´ d’etudes´ cognitives, ENS, EHESS, CNRS, PSL University, Paris, France 416 S. -
Computational Lambda-Calculus and Monads
Computational lambda-calculus and monads Eugenio Moggi∗ LFCS Dept. of Comp. Sci. University of Edinburgh EH9 3JZ Edinburgh, UK [email protected] October 1988 Abstract The λ-calculus is considered an useful mathematical tool in the study of programming languages, since programs can be identified with λ-terms. However, if one goes further and uses βη-conversion to prove equivalence of programs, then a gross simplification1 is introduced, that may jeopardise the applicability of theoretical results to real situations. In this paper we introduce a new calculus based on a categorical semantics for computations. This calculus provides a correct basis for proving equivalence of programs, independent from any specific computational model. 1 Introduction This paper is about logics for reasoning about programs, in particular for proving equivalence of programs. Following a consolidated tradition in theoretical computer science we identify programs with the closed λ-terms, possibly containing extra constants, corresponding to some features of the programming language under con- sideration. There are three approaches to proving equivalence of programs: • The operational approach starts from an operational semantics, e.g. a par- tial function mapping every program (i.e. closed term) to its resulting value (if any), which induces a congruence relation on open terms called operational equivalence (see e.g. [Plo75]). Then the problem is to prove that two terms are operationally equivalent. ∗On leave from Universit`a di Pisa. Research partially supported by the Joint Collaboration Contract # ST2J-0374-C(EDB) of the EEC 1programs are identified with total functions from values to values 1 • The denotational approach gives an interpretation of the (programming) lan- guage in a mathematical structure, the intended model. -
Continuation Calculus
Eindhoven University of Technology MASTER Continuation calculus Geron, B. Award date: 2013 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Continuation calculus Master’s thesis Bram Geron Supervised by Herman Geuvers Assessment committee: Herman Geuvers, Hans Zantema, Alexander Serebrenik Final version Contents 1 Introduction 3 1.1 Foreword . 3 1.2 The virtues of continuation calculus . 4 1.2.1 Modeling programs . 4 1.2.2 Ease of implementation . 5 1.2.3 Simplicity . 5 1.3 Acknowledgements . 6 2 The calculus 7 2.1 Introduction . 7 2.2 Definition of continuation calculus . 9 2.3 Categorization of terms . 10 2.4 Reasoning with CC terms . -
Semantic Theory Week 4 – Lambda Calculus
Semantic Theory Week 4 – Lambda Calculus Noortje Venhuizen University of Groningen/Universität des Saarlandes Summer 2015 1 Compositionality The principle of compositionality: “The meaning of a complex expression is a function of the meanings of its parts and of the syntactic rules by which they are combined” (Partee et al.,1993) Compositional semantics construction: • compute meaning representations for sub-expressions • combine them to obtain a meaning representation for a complex expression. Problematic case: “Not smoking⟨e,t⟩ is healthy⟨⟨e,t⟩,t⟩” ! 2 Lambda abstraction λ-abstraction is an operation that takes an expression and “opens” specific argument positions. Syntactic definition: If α is in WEτ, and x is in VARσ then λx(α) is in WE⟨σ, τ⟩ • The scope of the λ-operator is the smallest WE to its right. Wider scope must be indicated by brackets. • We often use the “dot notation” λx.φ indicating that the λ-operator takes widest possible scope (over φ). 3 Interpretation of Lambda-expressions M,g If α ∈ WEτ and v ∈ VARσ, then ⟦λvα⟧ is that function f : Dσ → Dτ M,g[v/a] such that for all a ∈ Dσ, f(a) = ⟦α⟧ If the λ-expression is applied to some argument, we can simplify the interpretation: • ⟦λvα⟧M,g(A) = ⟦α⟧M,g[v/A] Example: “Bill is a non-smoker” ⟦λx(¬S(x))(b’)⟧M,g = 1 iff ⟦λx(¬S(x))⟧M,g(⟦b’⟧M,g) = 1 iff ⟦¬S(x)⟧M,g[x/⟦b’⟧M,g] = 1 iff ⟦S(x)⟧M,g[x/⟦b’⟧M,g] = 0 iff ⟦S⟧M,g[x/⟦b’⟧M,g](⟦x⟧M,g[x/⟦b’⟧M,g]) = 0 iff VM(S)(VM(b’)) = 0 4 β-Reduction ⟦λv(α)(β)⟧M,g = ⟦α⟧M,g[v/⟦β⟧M,g] ⇒ all (free) occurrences of the λ-variable in α get the interpretation of β as value. -
Labelled Λ-Calculi with Explicit Copy and Erase
Labelled λ-calculi with Explicit Copy and Erase Maribel Fern´andez, Nikolaos Siafakas King’s College London, Dept. Computer Science, Strand, London WC2R 2LS, UK [maribel.fernandez, nikolaos.siafakas]@kcl.ac.uk We present two rewriting systems that define labelled explicit substitution λ-calculi. Our work is motivated by the close correspondence between L´evy’s labelled λ-calculus and paths in proof-nets, which played an important role in the understanding of the Geometry of Interaction. The structure of the labels in L´evy’s labelled λ-calculus relates to the multiplicative information of paths; the novelty of our work is that we design labelled explicit substitution calculi that also keep track of exponential information present in call-by-value and call-by-name translations of the λ-calculus into linear logic proof-nets. 1 Introduction Labelled λ-calculi have been used for a variety of applications, for instance, as a technology to keep track of residuals of redexes [6], and in the context of optimal reduction, using L´evy’s labels [14]. In L´evy’s work, labels give information about the history of redex creation, which allows the identification and classification of copied β-redexes. A different point of view is proposed in [4], where it is shown that a label is actually encoding a path in the syntax tree of a λ-term. This established a tight correspondence between the labelled λ-calculus and the Geometry of Interaction interpretation of cut elimination in linear logic proof-nets [13]. Inspired by L´evy’s labelled λ-calculus, we define labelled λ-calculi where the labels attached to terms capture reduction traces. -
Lambda Λ Calculus
Lambda λ Calculus Maria Raynor, Katie Kessler, Michael von der Lippe December 5th, 2016 COSC 440 General Intro ❏ 1930s - Alonzo Church ❏ 1935 - Kleene and Rosser say inconsistent ❏ 1936 - published lambda calculus relevant to computation, untyped ❏ 1940 - published typed lambda calculus ❏ Language that expresses function abstraction and application Definition “A formal system in mathematical logic for expressing computation based on abstraction and application using variable binding and substitution” ∈ ● Variables v1, v2, v3, ... , vn Var ● Parenthesis () ● Functions-take 1 argument,return 1 value Syntax Everything is an expression 1. E ID 2. E λ ID. E 3. E E E 4. E (E) Examples and Non-Examples ❏ x ❏ λ x . x ❏ x y ❏ λ λ x . y ❏ λ x . y z What about ambiguous syntax? There is a set of disambiguation rules: ❏ E E E is left associative ❏ x y z becomes (x y) z ❏ w x y z becomes ((w x) y) z ❏ λ ID . E extends as far right as possible ❏ λ x . x y becomes λ x . (x y) ❏ λ x . λ x . x becomes λ x . (λ x . x) ❏ λ a . λ b . λ c . a b c becomes λ a . (λ b . (λ c . ((a b) c))) Simple Examples Church Booleans: if p then x, else y ❏ TRUE = λ x . λ y . x ❏ FALSE = λ x . λ y . y Identity function λ x . x Constant function λ x . k What about λ x . + x 1? How it relates to theoretical computer science Mathematical Composition vs Machine Composition a. Lambda calculus is mathematical by nature. b. Turing Machines are difficult to applied to programming because there is no natural notation for machines Curry-Howard Correspondence This foundation in math allows us to establish functional programming proofs of formal logic. -
An E Cient and General Implementation of Futures on Large
An Ecient and General Implementation of Futures on Large Scale SharedMemory Multipro cessors A Dissertation Presented to The Faculty of the Graduate School of Arts and Sciences Brandeis University Department of Computer Science James S Miller advisor In Partial Fulllment of the Requirements of the Degree of Doctor of Philosophy by Marc Feeley April This dissertation directed and approved by the candidates committee has b een ac cepted and approved by the Graduate Faculty of Brandeis University in partial fulll ment of the requirements for the degree of DOCTOR OF PHILOSOPHY Dean Graduate School of Arts and Sciences Dissertation Committee Dr James S Miller chair Digital Equipment Corp oration Prof Harry Mairson Prof Timothy Hickey Prof David Waltz Dr Rob ert H Halstead Jr Digital Equipment Corp oration Copyright by Marc Feeley Abstract An Ecient and General Implementation of Futures on Large Scale SharedMemory Multipro cessors A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University Waltham Massachusetts by Marc Feeley This thesis describ es a highp erformance implementation technique for Multilisps future parallelism construct This metho d addresses the nonuniform memory access NUMA problem inherent in large scale sharedmemory multiprocessors The technique is based on lazy task creation LTC a dynamic task partitioning mechanism that dramatically reduces the cost of task creation and consequently makes it p ossible to exploit ne grain parallelism In LTC idle pro cessors get work to do -
The Complexity of Flow Analysis in Higher-Order Languages
The Complexity of Flow Analysis in Higher-Order Languages David Van Horn arXiv:1311.4733v1 [cs.PL] 19 Nov 2013 The Complexity of Flow Analysis in Higher-Order Languages A Dissertation Presented to The Faculty of the Graduate School of Arts and Sciences Brandeis University Mitchom School of Computer Science In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by David Van Horn August, 2009 This dissertation, directed and approved by David Van Horn’s committee, has been accepted and approved by the Graduate Faculty of Brandeis University in partial fulfillment of the requirements for the degree of: DOCTOR OF PHILOSOPHY Adam B. Jaffe, Dean of Arts and Sciences Dissertation Committee: Harry G. Mairson, Brandeis University, Chair Olivier Danvy, University of Aarhus Timothy J. Hickey, Brandeis University Olin Shivers, Northeastern University c David Van Horn, 2009 Licensed under the Academic Free License version 3.0. in memory of William Gordon Mercer July 22, 1927–October 8, 2007 Acknowledgments Harry taught me so much, not the least of which was a compelling kind of science. It is fairly obvious that I am not uninfluenced by Olivier Danvy and Olin Shivers and that I do not regret their influence upon me. My family provided their own weird kind of emotional support and humor. I gratefully acknowledge the support of the following people, groups, and institu- tions, in no particular order: Matthew Goldfield. Jan Midtgaard. Fritz Henglein. Matthew Might. Ugo Dal Lago. Chung-chieh Shan. Kazushige Terui. Christian Skalka. Shriram Krishnamurthi. Michael Sperber. David McAllester. Mitchell Wand. Damien Sereni. Jean-Jacques Levy.´ Julia Lawall. -
Fundamentals of Type Inference Systems
Fundamentals of type inference systems Fritz Henglein DIKU, University of Copenhagen [email protected] February 1, 1991; revised January 31, 1994; updated August 9, 2009 1 Introduction These notes give a compact overview of established core type systems and of their fundamental properties. We emphasize the use and application of type systems in programming languages, but also mention their role in logic. Proofs are omitted, but references to relevant sources in the literature are usually given. 1.1 What is a \type"? There are many examples of types in programming languages: • Primitive types: int, float, bool • Compound types: products (records), sums (disjoint unions), lists, arrays • Recursively definable types, such as tree data types • Function types, e.g. int !int • Parametric polymorphic types • Abstract types, e.g. Java interfaces Loosely speaking a type is a description of a collection of related values. • A type has syntax (\description"): It denotes a collection. • A type has elements (\collection"): It makes sense to talk about ele- ments of a type; in particular, it may have zero, one or many elements 1 • A type's elements have common properties (\related"): Users of a type can rely on each element having some common properties|an interface| without having to know the identity of particular elements. Types incorporate multiple aspects: • Type as a set of values: This view focuses on how values are con- structed to be elements of a type; e.g. constructing the natural num- bers from 0 and the successor function; • Type as a an interface: This view focuses on how values can be used (\deconstructed") by a client; e.g. -
Introduction to Type Theory
Introduction to Type Theory Erik Barendsen version November 2005 selected from: Basic Course Type Theory Dutch Graduate School in Logic, June 1996 lecturers: Erik Barendsen (KUN/UU), Herman Geuvers (TUE) 1. Introduction In typing systems, types are assigned to objects (in some context). This leads to typing statements of the form a : τ, where a is an object and τ a type. Traditionally, there are two views on typing: the computational and the logical one. They differ in the interpretation of typing statements. In the computational setting, a is regarded as a program or algorithm, and τ as a specification of a (in some formalism). This gives a notion of partial correctness, namely well-typedness of programs. From the logical point of view, τ is a formula, and a is considered to be a proof of τ. Various logical systems have a type theoretic interpretation. This branch of type theory is related to proof theory because the constructive or algorithmic content of proofs plays an important role. The result is a realiz- ability interpretation: provability in the logical systems implies the existence of an ‘inhabitant’ in the corresponding type system. Church versus Curry Typing One can distinguish two different styles in typing, usually called after their inventors A. Church and H.B. Curry. The approach `ala Church assembles typed objects from typed components; i.e., the objects are constructed together with their types. We use type anno- tations to indicate the types of basic objects: if 0 denotes the type of natural numbers, then λx:0.x denotes the identity function on N, having type 0→0.