The Essence of Functional Programming
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CSCI 2041: Lazy Evaluation
CSCI 2041: Lazy Evaluation Chris Kauffman Last Updated: Wed Dec 5 12:32:32 CST 2018 1 Logistics Reading Lab13: Lazy/Streams I Module Lazy on lazy Covers basics of delayed evaluation computation I Module Stream on streams A5: Calculon Lambdas/Closures I Arithmetic language Briefly discuss these as they interpreter pertain Calculon I 2X credit for assignment I 5 Required Problems 100pts Goals I 5 Option Problems 50pts I Eager Evaluation I Milestone due Wed 12/5 I Lazy Evaluation I Final submit Tue 12/11 I Streams 2 Evaluation Strategies Eager Evaluation Lazy Evaluation I Most languages employ I An alternative is lazy eager evaluation evaluation I Execute instructions as I Execute instructions only as control reaches associated expression results are needed code (call by need) I Corresponds closely to I Higher-level idea with actual machine execution advantages and disadvantages I In pure computations, evaluation strategy doesn’t matter: will produce the same results I With side-effects, when code is run matter, particular for I/O which may see different printing orders 3 Exercise: Side-Effects and Evaluation Strategy Most common place to see differences between Eager/Lazy eval is when functions are called I Eager eval: eval argument expressions, call functions with results I Lazy eval: call function with un-evaluated expressions, eval as results are needed Consider the following expression let print_it expr = printf "Printing it\n"; printf "%d\n" expr; ;; print_it (begin printf "Evaluating\n"; 5; end);; Predict results and output for both Eager and Lazy Eval strategies 4 Answers: Side-Effects and Evaluation Strategy let print_it expr = printf "Printing it\n"; printf "%d\n" expr; ;; print_it (begin printf "Evaluating\n"; 5; end);; Evaluation > ocamlc eager_v_lazy.ml > ./a.out Eager Eval # ocaml’s default Evaluating Printing it 5 Lazy Eval Printing it Evaluating 5 5 OCaml and explicit lazy Computations I OCaml’s default model is eager evaluation BUT. -
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 . -
Control Flow
Control Flow COMS W4115 Prof. Stephen A. Edwards Spring 2002 Columbia University Department of Computer Science Control Flow “Time is Nature’s way of preventing everything from happening at once.” Scott identifies seven manifestations of this: 1. Sequencing foo(); bar(); 2. Selection if (a) foo(); 3. Iteration while (i<10) foo(i); 4. Procedures foo(10,20); 5. Recursion foo(int i) { foo(i-1); } 6. Concurrency foo() jj bar() 7. Nondeterminism do a -> foo(); [] b -> bar(); Ordering Within Expressions What code does a compiler generate for a = b + c + d; Most likely something like tmp = b + c; a = tmp + d; (Assumes left-to-right evaluation of expressions.) Order of Evaluation Why would you care? Expression evaluation can have side-effects. Floating-point numbers don’t behave like numbers. Side-effects int x = 0; int foo() { x += 5; return x; } int a = foo() + x + foo(); What’s the final value of a? Side-effects int x = 0; int foo() { x += 5; return x; } int a = foo() + x + foo(); GCC sets a=25. Sun’s C compiler gave a=20. C says expression evaluation order is implementation-dependent. Side-effects Java prescribes left-to-right evaluation. class Foo { static int x; static int foo() { x += 5; return x; } public static void main(String args[]) { int a = foo() + x + foo(); System.out.println(a); } } Always prints 20. Number Behavior Basic number axioms: a + x = a if and only if x = 0 Additive identity (a + b) + c = a + (b + c) Associative a(b + c) = ab + ac Distributive Misbehaving Floating-Point Numbers 1e20 + 1e-20 = 1e20 1e-20 1e20 (1 + 9e-7) + 9e-7 6= 1 + (9e-7 + 9e-7) 9e-7 1, so it is discarded, however, 1.8e-6 is large enough 1:00001(1:000001 − 1) 6= 1:00001 · 1:000001 − 1:00001 · 1 1:00001 · 1:000001 = 1:00001100001 requires too much intermediate precision. -
210 CHAPTER 7. NAMES and BINDING Chapter 8
210 CHAPTER 7. NAMES AND BINDING Chapter 8 Expressions and Evaluation Overview This chapter introduces the concept of the programming environment and the role of expressions in a program. Programs are executed in an environment which is provided by the operating system or the translator. An editor, linker, file system, and compiler form the environment in which the programmer can enter and run programs. Interac- tive language systems, such as APL, FORTH, Prolog, and Smalltalk among others, are embedded in subsystems which replace the operating system in forming the program- development environment. The top-level control structure for these subsystems is the Read-Evaluate-Write cycle. The order of computation within a program is controlled in one of three basic ways: nesting, sequencing, or signaling other processes through the shared environment or streams which are managed by the operating system. Within a program, an expression is a nest of function calls or operators that returns a value. Binary operators written between their arguments are used in infix syntax. Unary and binary operators written before a single argument or a pair of arguments are used in prefix syntax. In postfix syntax, operators (unary or binary) follow their arguments. Parse trees can be developed from expressions that include infix, prefix and postfix operators. Rules for precedence, associativity, and parenthesization determine which operands belong to which operators. The rules that define order of evaluation for elements of a function call are as follows: • Inside-out: Evaluate every argument before beginning to evaluate the function. 211 212 CHAPTER 8. EXPRESSIONS AND EVALUATION • Outside-in: Start evaluating the function body. -
Exam 1 Review: Solutions Capture the Dragons
CSCI 1101B: Introduction to Computer Science Instructor: Prof. Harmon Exam 1 Review: Solutions Capture the Dragons Boolean Expressions 1. Of the following, which are Boolean expressions? (a)2+3 Not Boolean (b)2>3 Boolean (c) num >= 2 Boolean (d) num1 = num2 Not Boolean 2. If x = 3, and x <= y evaluates to True, what values can y have? y >= 3 3. Simplify each of the following expressions: (a) my_num or True Depends on what the value of my_num is. See rules as in #5. (b) False and "Mary" or True True (c) False and ("Mary" or True) False 4. Is x =< y the same as x <= y? No. Python only recognizes the latter. The former is invalid syntax. 5. If x = 1 and y = 2, explain the following results: (a) >>> x and y # = 2 (b) >>> x or y # = 1 Python uses lazy evaluation. • The expression x and y first evaluates x. If x is equivalent to False (or empty/zero), its value is returned. Otherwise, y is evaluated and the resulting value is returned. • The expression x or y first evaluates x. If x is equivalent to True (or non-empty/nonzero), its value is returned. Otherwise, y is evaluated and the resulting value is returned. 1 CSCI 1101B: Introduction to Computer Science Exam 1 Review: Solutions Conditionals 1. What is indentation, and how is it related to the concept of a code block? Indentation is the placement of text (to the left or right). Python uses indenta- tion to associate code into groups, or blocks. 2. What is the difference between a chained conditional and a nested conditional? Chained => if, elif, else Nested => conditionals exist (are "nested") inside each other 3. -
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.