Recursion. Lecture 6: More Lambda Calculus Programming. Fixed Points. Recursion
|
|
- Reynard Fletcher
- 6 years ago
- Views:
Transcription
1 Recursion Lecture 6: More Lambda Calculus Programming CSC 131! Fall, 2014!! Kim Bruce Recursive definitions are handy! - fact = λn. cond (iszero n) 1 (Mult n (fact (Pred n)))! - Not a legal definition in lambda calculus because can t name functions!! Compute by expanding:! - fact 2! = cond (iszero 2) 1 (Mult 2 (fact (Pred 2)))! = Mult 2 (fact 1)! = Mult 2 (cond (iszero 1) 1 (Mult 1 (fact (Pred 1)))! = Mult 2 (Mult 1 (fact 0)) =... = Mult 2 (Mult 1 1) = 2 Recursion Fixed Points A different perspective: Start with! - fact = λn. cond (iszero n) 1 (Mult n (fact (Pred n)))! Let F stand for the closed term:! - λf. λn. cond (iszero n) 1 (Mult n (f (Pred n)))! - Notice F(fact) = fact.! - fact is a fixed point of F! - To find fact, need only find fixed point of F!! Easy w/ g(x) = x * x, but F???? Several fixed point operators:! - Ex: Y = λf. (λx. f (xx))(λx. f (xx))! Claim for all g, Y g = g (Y g)! Y g = (λf. (λx. f (xx))(λx. f (xx))) g! = (λx. g(xx))(λx. g(xx))! = g((λx. g(xx)) (λx. g(xx)))! = g (Y g)! If let x 0 = Y g, then g (x 0 ) = x 0. Invented by Haskell Curry
2 Factorial Computing Factorials Recursive definition:! - let F = λf. λn. cond (iszero n) 1 (Mult n (f (Pred n)))! - let fact = Y F! - then F(fact) = fact because Y always gives fixed points! Compute:! fact 0 = (F (fact)) 0 because fact is a fixed point of F! = cond (iszero 0) 1 (Mult 0 (fact (Pred 0)))! fact 1 = (F (fact)) 1 because fact is a fixed point of F! = (λn. cond (iszero n) 1 (Mult n (fact (Pred n)))) 1 expanding F! = cond (iszero 1) 1 (Mult 1 (fact (Pred 1))) applying it" = Mult 1 (fact (Pred 1)) by the definition of cond! = fact 0 by the definition of Mult and Pred! = 1 by the above calculation! = 1 by the definition of cond Lambda Calculus λ-calculus invented in 1928 by Church in Princeton & first published in 1932.! Goal to provide a foundation for logic! First to state explicit conversion rules.! Original version inconsistent, but corrected! - If this sentence is true then 1 = 2 problematic!!! 1933, definition of natural numbers Collaborators : Grad students:! - J. Barkley Rosser and Stephen Kleene! - Church-Rosser confluence theorem ensured consistency (earlier version inconsistent)! - Kleene showed λ-definable functions very rich! Equivalent to Herbrand-Gödel recursive functions! Equivalent to Turing-computable functions.! Founder of recursion theory, invented regular expressions! Church s thesis:! - λ-definability effectively computable O
3 Undecidability Alan Turing Turing! Convertibility problem for λ-calculus undecidable.! Validity in first-order predicate logic undecidable.! Proved independently year later by Turing.! - First showed halting problem undecidable , in Cambridge, England, definition of Turing machine! , in Princeton to get Ph.D. under Church.! , first published fixed point combinator! (λx. λy. (y (x x y))) (λx. λy. (y (x x y)))! - Kleene did not use fixed-point operator in defining functions on natural numbers!" - Broke German enigma code in WW2, Turing test AI! - Persecuted as homosexual, committed suicide in 1954 Combinatory Logic Schönfinkel - get rid of bound variables! Combinators K, S sufficient to write all computable functions:! - K x y = x! - S x y z = (x y)(x z)! Typed Lambda Calculus CL terms include K, S, variables and all terms created by function application.! All computable fcns, equivalent to λ-calculus
4 Types Definitions Start with base type e and build up types and terms:! - Type ::= e Type Type! - M ::= v (M M) λv : Type. M! Examples:! - Types: e, e e, e e e, (e e) e,...! - Terms: λx: e. x, λf: e e. λz: e. f(f(z)) Earlier definitions generalize over types t:! - true t = λx:t. λy:u. x! - n t = λs: t t. λz: t. s (n) (z)! Some untyped terms can t be typed:! - Ω = (λx. (x x))(λx. (x x))! - Y = λf. (λx. f (x x))(λx. f (x x)) Totally Awesome!! Theorem: If M is a term of the typed lambda calculus, then M has a unique normal form. I.e., every term of the typed lambda calculus is total.! Functional Languages Corollary: The typed lambda calculus does not include all computable functions.
5 Commands/Statements Imperative Languages Characteristics! - Support for variables! represent memory locations for storing updatable values.! - Assignment operation! progress in computation depends on changes in values stored in variables.! - Repetition! flow of control guided by conditional and looping statements controlling order in which assignment statements are executed. Organized around notion of commands! Meaning of a statement is operation which, based on current contents of memory, and explicit values supplied to it, modifies the current contents of memory.! How are results of one command communicated to next? Problems Expressions... return a value, depending on the state of the computation.! Too low level.! Architecture dependent.! Does not generalize well to concurrent computation. Examples:! - Literals: 3, true, hello, 42.56! - Aggregates: arrays, records, sets, lists, etc. E.g. [1,3,5]! - Function calls: f(a,b), a + b * (c - d), (if x > 0 then sin else cos)(... )! - Conditional expressions: if x <> 0 then a/x else 1, case (historically only in functional languages)! - Named constants and variables: pi, x
6 Declarative Language Test Problems Within the scope of specific declarations of x 1,, x n, all occurrences of an expression e containing only variables x 1,, x n have the same value.! - Pure expressions are well behaved.! Fails w/imperative languages:! - x := x + y; y := 2 * x;! - Two occurrences of x have different values.! If DLT holds, then allows for optimizations Order of evaluation! - if (i >= 0 && a[i]!= 100)...! - What happens if i = -1?" - Is && commutative?" Side effects! - Destroy nice properties! - a[++x] = (y = x+1) + y + (++x)! - Does 2*(x++) = (x++) + (x++)? History of Functional Languages Functional Languages LISP 1957! Backus Turing award lecture (1977) - FP / FL! ML (1977) - Turing award for Robin Milner! - Meta language for theorem prover! SASL, KRC, and Miranda (Turner), Haskell (1990).! - Lazy languages! - Implemented using combinatory logic Pure LISP/Scheme (and Rex) are essentially sugared versions of untyped lambda calculus with constants.! ML & Haskell are typed lambda calculus w/ type inference and recursion definitions at each type.
Lecture 9: More Lambda Calculus / Types
Lecture 9: More Lambda Calculus / Types CSC 131 Spring, 2019 Kim Bruce Pure Lambda Calculus Terms of pure lambda calculus - M ::= v (M M) λv. M - Impure versions add constants, but not necessary! - Turing-complete
More information- M ::= v (M M) λv. M - Impure versions add constants, but not necessary! - Turing-complete. - true = λ u. λ v. u. - false = λ u. λ v.
Pure Lambda Calculus Lecture 9: More Lambda Calculus / Types CSC 131 Spring, 2019 Kim Bruce Terms of pure lambda calculus - M ::= v (M M) λv. M - Impure versions add constants, but not necessary! - Turing-complete
More informationUntyped Lambda Calculus
Advanced Topics in Programming Languages Untyped Lambda Calculus Oded Padon & Mooly Sagiv (original slides by Kathleen Fisher, John Mitchell, Shachar Itzhaky, S. Tanimoto ) Reference: Types and Programming
More informationCIS 500 Software Foundations Fall September 25
CIS 500 Software Foundations Fall 2006 September 25 The Lambda Calculus The lambda-calculus If our previous language of arithmetic expressions was the simplest nontrivial programming language, then the
More informationType Systems Winter Semester 2006
Type Systems Winter Semester 2006 Week 4 November 8 November 15, 2006 - version 1.1 The Lambda Calculus The lambda-calculus If our previous language of arithmetic expressions was the simplest nontrivial
More informationUntyped Lambda Calculus
Concepts in Programming Languages Recitation 5: Untyped Lambda Calculus Oded Padon & Mooly Sagiv (original slides by Kathleen Fisher, John Mitchell, Shachar Itzhaky, S. Tanimoto ) Reference: Types and
More informationChapter 5: The Untyped Lambda Calculus
Chapter 5: The Untyped Lambda Calculus What is lambda calculus for? Basics: syntax and operational semantics Programming in the Lambda Calculus Formalities (formal definitions) What is Lambda calculus
More information11/6/17. Outline. FP Foundations, Scheme. Imperative Languages. Functional Programming. Mathematical Foundations. Mathematical Foundations
Outline FP Foundations, Scheme In Text: Chapter 15 Mathematical foundations Functional programming λ-calculus LISP Scheme 2 Imperative Languages We have been discussing imperative languages C/C++, Java,
More informationλ calculus Function application Untyped λ-calculus - Basic Idea Terms, Variables, Syntax β reduction Advanced Formal Methods
Course 2D1453, 2006-07 Advanced Formal Methods Lecture 2: Lambda calculus Mads Dam KTH/CSC Some material from B. Pierce: TAPL + some from G. Klein, NICTA Alonzo Church, 1903-1995 Church-Turing thesis First
More informationFunctional Programming and λ Calculus. Amey Karkare Dept of CSE, IIT Kanpur
Functional Programming and λ Calculus Amey Karkare Dept of CSE, IIT Kanpur 0 Software Development Challenges Growing size and complexity of modern computer programs Complicated architectures Massively
More informationOne of a number of approaches to a mathematical challenge at the time (1930): Constructibility
λ Calculus Church s λ Calculus: Brief History One of a number of approaches to a mathematical challenge at the time (1930): Constructibility (What does it mean for an object, e.g. a natural number, to
More informationThe Untyped Lambda Calculus
CS738: Advanced Compiler Optimizations The Untyped Lambda Calculus Amey Karkare karkare@cse.iitk.ac.in http://www.cse.iitk.ac.in/~karkare/cs738 Department of CSE, IIT Kanpur Reference Book Types and Programming
More information9/23/2014. Why study? Lambda calculus. Church Rosser theorem Completeness of Lambda Calculus: Turing Complete
Dr A Sahu Dept of Computer Science & Engineering IIT Guwahati Why study? Lambda calculus Syntax Evaluation Relationship to programming languages Church Rosser theorem Completeness of Lambda Calculus: Turing
More informationCS 4110 Programming Languages & Logics. Lecture 17 Programming in the λ-calculus
CS 4110 Programming Languages & Logics Lecture 17 Programming in the λ-calculus 10 October 2014 Announcements 2 Foster Office Hours 11-12 Enjoy fall break! Review: Church Booleans 3 We can encode TRUE,
More informationIntroduction to the Lambda Calculus
Introduction to the Lambda Calculus Overview: What is Computability? Church s Thesis The Lambda Calculus Scope and lexical address The Church-Rosser Property Recursion References: Daniel P. Friedman et
More informationChapter 11 :: Functional Languages
Chapter 11 :: Functional Languages Programming Language Pragmatics Michael L. Scott Copyright 2016 Elsevier 1 Chapter11_Functional_Languages_4e - Tue November 21, 2017 Historical Origins The imperative
More informationThe Lambda Calculus. 27 September. Fall Software Foundations CIS 500. The lambda-calculus. Announcements
CIS 500 Software Foundations Fall 2004 27 September IS 500, 27 September 1 The Lambda Calculus IS 500, 27 September 3 Announcements Homework 1 is graded. Pick it up from Cheryl Hickey (Levine 502). We
More informationPure Lambda Calculus. Lecture 17
Pure Lambda Calculus Lecture 17 Lambda Calculus Lambda Calculus (λ-calculus) is a functional notation introduced by Alonzo Church in the early 1930s to formalize the notion of computability. Pure λ-calculus
More informationProgramming Language Pragmatics
Chapter 10 :: Functional Languages Programming Language Pragmatics Michael L. Scott Historical Origins The imperative and functional models grew out of work undertaken Alan Turing, Alonzo Church, Stephen
More informationCOMP 1130 Lambda Calculus. based on slides by Jeff Foster, U Maryland
COMP 1130 Lambda Calculus based on slides by Jeff Foster, U Maryland Motivation Commonly-used programming languages are large and complex ANSI C99 standard: 538 pages ANSI C++ standard: 714 pages Java
More informationFunctional Programming. Big Picture. Design of Programming Languages
Functional Programming Big Picture What we ve learned so far: Imperative Programming Languages Variables, binding, scoping, reference environment, etc What s next: Functional Programming Languages Semantics
More informationCS 314 Principles of Programming Languages. Lecture 21
CS 314 Principles of Programming Languages Lecture 21 Zheng Zhang Department of Computer Science Rutgers University Wednesday 23 rd November, 2016 Zheng Zhang 1 CS@Rutgers University Class Information
More informationProgramming Languages. Programming with λ-calculus. Lecture 11: Type Systems. Special Hour to discuss HW? if-then-else int
CSE 230: Winter 2010 Principles of Programming Languages Lecture 11: Type Systems News New HW up soon Special Hour to discuss HW? Ranjit Jhala UC San Diego Programming with λ-calculus Encode: bool if-then-else
More informationIntroduction to the Lambda Calculus. Chris Lomont
Introduction to the Lambda Calculus Chris Lomont 2010 2011 2012 www.lomont.org Leibniz (1646-1716) Create a universal language in which all possible problems can be stated Find a decision method to solve
More informationLecture 5: The Untyped λ-calculus
Lecture 5: The Untyped λ-calculus Syntax and basic examples Polyvios Pratikakis Computer Science Department, University of Crete Type Systems and Static Analysis Pratikakis (CSD) Untyped λ-calculus I CS49040,
More informationFundamentals and lambda calculus
Fundamentals and lambda calculus Again: JavaScript functions JavaScript functions are first-class Syntax is a bit ugly/terse when you want to use functions as values; recall block scoping: (function ()
More informationLecture #3: Lambda Calculus. Last modified: Sat Mar 25 04:05: CS198: Extra Lecture #3 1
Lecture #3: Lambda Calculus Last modified: Sat Mar 25 04:05:39 2017 CS198: Extra Lecture #3 1 Simplifying Python Python is full of features. Most are there to make programming concise and clear. Some are
More informationCSCE 314 TAMU Fall CSCE 314: Programming Languages Dr. Flemming Andersen. Haskell Basics
1 CSCE 314: Programming Languages Dr. Flemming Andersen Haskell Basics 2 Contents 1. Jump into Haskell: Using ghc and ghci (more detail) 2. Historical Background of Haskell 3. Lazy, Pure, and Functional
More informationFunctional Languages. Hwansoo Han
Functional Languages Hwansoo Han Historical Origins Imperative and functional models Alan Turing, Alonzo Church, Stephen Kleene, Emil Post, etc. ~1930s Different formalizations of the notion of an algorithm
More informationA Quick Overview. CAS 701 Class Presentation 18 November Department of Computing & Software McMaster University. Church s Lambda Calculus
A Quick Overview CAS 701 Class Presentation 18 November 2008 Lambda Department of Computing & Software McMaster University 1.1 Outline 1 2 3 Lambda 4 5 6 7 Type Problem Lambda 1.2 Lambda calculus is a
More informationCOP4020 Programming Languages. Functional Programming Prof. Robert van Engelen
COP4020 Programming Languages Functional Programming Prof. Robert van Engelen Overview What is functional programming? Historical origins of functional programming Functional programming today Concepts
More informationLambda Calculus. Lecture 4 CS /26/10
Lambda Calculus Lecture 4 CS 565 10/26/10 Pure (Untyped) Lambda Calculus The only value is a function Variables denote functions Functions always take functions as arguments Functions always return functions
More informationFunctional Languages and Higher-Order Functions
Functional Languages and Higher-Order Functions Leonidas Fegaras CSE 5317/4305 L12: Higher-Order Functions 1 First-Class Functions Values of some type are first-class if They can be assigned to local variables
More information301AA - Advanced Programming [AP-2017]
301AA - Advanced Programming [AP-2017] Lecturer: Andrea Corradini andrea@di.unipi.it Tutor: Lillo GalleBa galleba@di.unipi.it Department of Computer Science, Pisa Academic Year 2017/18 AP-2017-15: Func'onal
More informationMore Lambda Calculus and Intro to Type Systems
More Lambda Calculus and Intro to Type Systems Plan Heavy Class Participation Thus, wake up! Lambda Calculus How is it related to real life? Encodings Fixed points Type Systems Overview Static, Dyamic
More informationLambda Calculus. Lambda Calculus
Lambda Calculus Formalism to describe semantics of operations in functional PLs Variables are free or bound Function definition vs function abstraction Substitution rules for evaluating functions Normal
More informationComputer Science 203 Programming Languages Fall Lecture 10. Bindings, Procedures, Functions, Functional Programming, and the Lambda Calculus
1 Computer Science 203 Programming Languages Fall 2004 Lecture 10 Bindings, Procedures, Functions, Functional Programming, and the Lambda Calculus Plan Informal discussion of procedures and bindings Introduction
More information5. Introduction to the Lambda Calculus. Oscar Nierstrasz
5. Introduction to the Lambda Calculus Oscar Nierstrasz Roadmap > What is Computability? Church s Thesis > Lambda Calculus operational semantics > The Church-Rosser Property > Modelling basic programming
More information4/19/2018. Chapter 11 :: Functional Languages
Chapter 11 :: Functional Languages Programming Language Pragmatics Michael L. Scott Historical Origins The imperative and functional models grew out of work undertaken by Alan Turing, Alonzo Church, Stephen
More informationLambda Calculus.
Lambda Calculus Oded Padon & Mooly Sagiv (original slides by Kathleen Fisher, John Mitchell, Shachar Itzhaky, S. Tanimoto, Stephen A. Edwards) Benjamin Pierce Types and Programming Languages http://www.cs.cornell.edu/courses/cs3110/2008fa/recitations/rec26.html
More informationCSE 505: Concepts of Programming Languages
CSE 505: Concepts of Programming Languages Dan Grossman Fall 2003 Lecture 6 Lambda Calculus Dan Grossman CSE505 Fall 2003, Lecture 6 1 Where we are Done: Modeling mutation and local control-flow Proving
More informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages. Lambda calculus
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Tuesday, February 19, 2013 The lambda calculus (or λ-calculus) was introduced by Alonzo Church and Stephen Cole Kleene in
More informationPart I. Historical Origins
Introduction to the λ-calculus Part I CS 209 - Functional Programming Dr. Greg Lavender Department of Computer Science Stanford University Historical Origins Foundations of Mathematics (1879-1936) Paradoxes
More informationVU Semantik von Programmiersprachen
VU Semantik von Programmiersprachen Agata Ciabattoni Institute für Computersprachen, Theory and Logic group (agata@logic.at) (A gentle) Introduction to λ calculus p. 1 Why shoud I studyλcalculus? p. 2
More informationRecursive Definitions, Fixed Points and the Combinator
Recursive Definitions, Fixed Points and the Combinator Dr. Greg Lavender Department of Computer Sciences University of Texas at Austin Recursive Self-Reference Recursive self-reference occurs regularly
More informationIntroduction to the l-calculus
Introduction to the l-calculus CS345 - Programming Languages Dr. Greg Lavender Department of Computer Sciences The University of Texas at Austin l-calculus in Computer Science a formal notation, theory,
More informationLambda Calculus. Variables and Functions. cs3723 1
Lambda Calculus Variables and Functions cs3723 1 Lambda Calculus Mathematical system for functions Computation with functions Captures essence of variable binding Function parameters and substitution Can
More informationHomework. Lecture 7: Parsers & Lambda Calculus. Rewrite Grammar. Problems
Homework Lecture 7: Parsers & Lambda Calculus CSC 131 Spring, 2019 Kim Bruce First line: - module Hmwk3 where - Next line should be name as comment - Name of program file should be Hmwk3.hs Problems How
More informationFoundations. Yu Zhang. Acknowledgement: modified from Stanford CS242
Spring 2013 Foundations Yu Zhang Acknowledgement: modified from Stanford CS242 https://courseware.stanford.edu/pg/courses/317431/ Course web site: http://staff.ustc.edu.cn/~yuzhang/fpl Reading Concepts
More informationJ. Barkley Rosser, 81, a professor emeritus of mathematics and computer science at the University of Wisconsin who had served in government, died
Church-Rosser J. Barkley Rosser, 81, a professor emeritus of mathematics and computer science at the University of Wisconsin who had served in government, died Sept. 5, 1989. Along with Alan Turing and
More informationFunctional Languages. CSE 307 Principles of Programming Languages Stony Brook University
Functional Languages CSE 307 Principles of Programming Languages Stony Brook University http://www.cs.stonybrook.edu/~cse307 1 Historical Origins 2 The imperative and functional models grew out of work
More informationPrinciples of Programming Languages COMP251: Functional Programming in Scheme (and LISP)
Principles of Programming Languages COMP251: Functional Programming in Scheme (and LISP) Prof. Dekai Wu Department of Computer Science and Engineering The Hong Kong University of Science and Technology
More informationMore Lambda Calculus and Intro to Type Systems
#1 More Lambda Calculus and Intro to Type Systems #2 Plan Heavy Class Participation Thus, wake up! (not actually kidding) Lambda Calculus How is it related to real life? Encodings Fixed points Type Systems
More informationFall 2013 Midterm Exam 10/22/13. This is a closed-book, closed-notes exam. Problem Points Score. Various definitions are provided in the exam.
Programming Languages Fall 2013 Midterm Exam 10/22/13 Time Limit: 100 Minutes Name (Print): Graduate Center I.D. This is a closed-book, closed-notes exam. Various definitions are provided in the exam.
More informationIntroduction. chapter Functions
chapter 1 Introduction In this chapter we set the stage for the rest of the book. We start by reviewing the notion of a function, then introduce the concept of functional programming, summarise the main
More informationFunctional Programming
Functional Programming CS331 Chapter 14 Functional Programming Original functional language is LISP LISt Processing The list is the fundamental data structure Developed by John McCarthy in the 60 s Used
More informationLECTURE 16. Functional Programming
LECTURE 16 Functional Programming WHAT IS FUNCTIONAL PROGRAMMING? Functional programming defines the outputs of a program as a mathematical function of the inputs. Functional programming is a declarative
More informationIntroduction to Lambda Calculus. Lecture 7 CS /08/09
Introduction to Lambda Calculus Lecture 7 CS 565 02/08/09 Lambda Calculus So far, we ve explored some simple but non-interesting languages language of arithmetic expressions IMP (arithmetic + while loops)
More informationCostly software bugs that could have been averted with type checking
Type Checking Class Notes from Lectures 6 and 7 Lahav Yeffet and Ori Folger The material in these lectures is based on the textbook Types and Programming Languages by Benjamin Pierce. Here is a list of
More informationFormal Systems and their Applications
Formal Systems and their Applications Dave Clarke (Dave.Clarke@cs.kuleuven.be) Acknowledgment: these slides are based in part on slides from Benjamin Pierce and Frank Piessens 1 Course Overview Introduction
More informationCMSC 330: Organization of Programming Languages
CMSC 330: Organization of Programming Languages Lambda Calculus CMSC 330 1 Programming Language Features Many features exist simply for convenience Multi-argument functions foo ( a, b, c ) Use currying
More informationCMPUT 325 : Lambda Calculus Basics. Lambda Calculus. Dr. B. Price and Dr. R. Greiner. 13th October 2004
CMPUT 325 : Lambda Calculus Basics Dr. B. Price and Dr. R. Greiner 13th October 2004 Dr. B. Price and Dr. R. Greiner CMPUT 325 : Lambda Calculus Basics 1 Lambda Calculus Lambda calculus serves as a formal
More informationLambda Calculus-2. Church Rosser Property
Lambda Calculus-2 Church-Rosser theorem Supports referential transparency of function application Says if it exists, normal form of a term is UNIQUE 1 Church Rosser Property Fundamental result of λ-calculus:
More informationIntroduction to Lambda Calculus. Lecture 5 CS 565 1/24/08
Introduction to Lambda Calculus Lecture 5 CS 565 1/24/08 Lambda Calculus So far, we ve explored some simple but non-interesting languages language of arithmetic expressions IMP (arithmetic + while loops)
More informationCIS 500 Software Foundations Midterm I Answer key October 8, 2003
CIS 500 Software Foundations Midterm I Answer key October 8, 2003 Inductive Definitions Review: Recall that the function size, which calculates the total number of nodes in the abstract syntax tree of
More informationCIS 500 Software Foundations Midterm I
CIS 500 Software Foundations Midterm I October 11, 2006 Name: Student ID: Email: Status: Section: registered for the course not registered: sitting in to improve a previous grade not registered: just taking
More informationChapter 5: The Untyped Lambda Calculus
Chapter 5: The Untyped Lambda Calculus What is lambda calculus for? Basics: syntax and operational semantics Programming in the Lambda Calculus Formalities (formal definitions) What is Lambda calculus
More informationProgramming Languages
Programming Languages Lambda Calculus and Scheme CSCI-GA.2110-003 Fall 2011 λ-calculus invented by Alonzo Church in 1932 as a model of computation basis for functional languages (e.g., Lisp, Scheme, ML,
More information1 Scope, Bound and Free Occurrences, Closed Terms
CS 6110 S18 Lecture 2 The λ-calculus Last time we introduced the λ-calculus, a mathematical system for studying the interaction of functional abstraction and functional application. We discussed the syntax
More information301AA - Advanced Programming
301AA - Advanced Programming Lecturer: Andrea Corradini andrea@di.unipi.it h;p://pages.di.unipi.it/corradini/ Course pages: h;p://pages.di.unipi.it/corradini/dida@ca/ap-18/ AP-2018-18: Lambda Calculus,
More informationFormal Semantics. Aspects to formalize. Lambda calculus. Approach
Formal Semantics Aspects to formalize Why formalize? some language features are tricky, e.g. generalizable type variables, nested functions some features have subtle interactions, e.g. polymorphism and
More informationCS152: Programming Languages. Lecture 7 Lambda Calculus. Dan Grossman Spring 2011
CS152: Programming Languages Lecture 7 Lambda Calculus Dan Grossman Spring 2011 Where we are Done: Syntax, semantics, and equivalence For a language with little more than loops and global variables Now:
More informationFundamentals of Artificial Intelligence COMP221: Functional Programming in Scheme (and LISP)
Fundamentals of Artificial Intelligence COMP221: Functional Programming in Scheme (and LISP) Prof. Dekai Wu Department of Computer Science and Engineering The Hong Kong University of Science and Technology
More informationCS 6110 S14 Lecture 1 Introduction 24 January 2014
CS 6110 S14 Lecture 1 Introduction 24 January 2014 1 Introduction What is a program? Is it just something that tells the computer what to do? Yes, but there is much more to it than that. The basic expressions
More informationConcepts of programming languages
Concepts of programming languages Lecture 5 Wouter Swierstra 1 Announcements Submit your project proposal to me by email on Friday; The presentation schedule in now online Exercise session after the lecture.
More informationFunctions as data. Massimo Merro. 9 November Massimo Merro The Lambda language 1 / 21
Functions as data Massimo Merro 9 November 2011 Massimo Merro The Lambda language 1 / 21 The core of sequential programming languages In the mid 1960s, Peter Landin observed that a complex programming
More informationMATH Iris Loeb.
MATH 134 http://www.math.canterbury.ac.nz/math134/09/su1/c Iris Loeb I.Loeb@math.canterbury.ac.nz Office Hours: Thur 10.00-11.00, Room 703 (MSCS Building) The Limits of Formal Logic We now turn our attention
More informationITT8060 Advanced Programming
ITT8060 Advanced Programming In F# Juhan Ernits Welcome to Advanced Programming (in F#)! Teachers: Juhan Ernits Hendrik Maarand Course web page http://courses.cs.ttu.ee/pages/itt8060 Contact: juhan.ernits@ttu.ee
More informationCOS 326 Functional programming: an elegant weapon for the modern age
COS 326 Functional programming: an elegant weapon for the modern age In 1936, Alonzo Church invented the lambda calculus. He called it a logic, but it was a language of pure functions -- the world's first
More informationFunctional Programming
Functional Programming COMS W4115 Prof. Stephen A. Edwards Spring 2003 Columbia University Department of Computer Science Original version by Prof. Simon Parsons Functional vs. Imperative Imperative programming
More informationINF 212 ANALYSIS OF PROG. LANGS LAMBDA CALCULUS. Instructors: Crista Lopes Copyright Instructors.
INF 212 ANALYSIS OF PROG. LANGS LAMBDA CALCULUS Instructors: Crista Lopes Copyright Instructors. History Formal mathematical system Simplest programming language Intended for studying functions, recursion
More informationFundamentals and lambda calculus. Deian Stefan (adopted from my & Edward Yang s CSE242 slides)
Fundamentals and lambda calculus Deian Stefan (adopted from my & Edward Yang s CSE242 slides) Logistics Assignments: Programming assignment 1 is out Homework 1 will be released tomorrow night Podcasting:
More informationAdvanced Topics in Programming Languages Lecture 3 - Models of Computation
Advanced Topics in Programming Languages Lecture 3 - Models of Computation Andrey Leshenko Maxim Borin 16/11/2017 1 Lambda Calculus At the beginning of the last century, several attempts were made to theoretically
More informationLecture Notes on Data Representation
Lecture Notes on Data Representation 15-814: Types and Programming Languages Frank Pfenning Lecture 9 Tuesday, October 2, 2018 1 Introduction In this lecture we ll see our type system in action. In particular
More informationlambda calculus History 11/28/13 Jianguo Lu A formal system designed to inves:gate func:on defini:on, func:on applica:on, and recursion.
lambda calculus Jianguo Lu 1 History Lambda calculus A formal system designed to inves:gate func:on defini:on, func:on applica:on, and recursion. Introduced by Alonzo Church in 1930s. The calculus can
More informationLambda calculus. Wouter Swierstra and Alejandro Serrano. Advanced functional programming - Lecture 6
Lambda calculus Advanced functional programming - Lecture 6 Wouter Swierstra and Alejandro Serrano 1 Today Lambda calculus the foundation of functional programming What makes lambda calculus such a universal
More informationIntroduction to the λ-calculus
Announcements Prelim #2 issues: o Problem 5 grading guide out shortly o Problem 3 (hashing) issues Will be on final! Friday RDZ office hours are 11-12 not 1:30-2:30 1 Introduction to the λ-calculus Today
More informationCSE-321 Programming Languages 2012 Midterm
Name: Hemos ID: CSE-321 Programming Languages 2012 Midterm Prob 1 Prob 2 Prob 3 Prob 4 Prob 5 Prob 6 Total Score Max 14 15 29 20 7 15 100 There are six problems on 24 pages in this exam. The maximum score
More informationCSE 505. Lecture #9. October 1, Lambda Calculus. Recursion and Fixed-points. Typed Lambda Calculi. Least Fixed Point
Lambda Calculus CSE 505 Lecture #9 October 1, 2012 Expr ::= Var λ Var. Expr (Expr Expr) Key Concepts: Bound and Free Occurrences Substitution, Reduction Rules: α, β, η Confluence and Unique Normal Form
More informationThe Eval/Apply Cycle Eval. Evaluation and universal machines. Examining the role of Eval. Eval from perspective of language designer
Evaluation and universal machines What is the role of evaluation in defining a language? How can we use evaluation to design a language? The Eval/Apply Cycle Eval Exp & env Apply Proc & args Eval and Apply
More informationCS 242. Fundamentals. Reading: See last slide
CS 242 Fundamentals Reading: See last slide Syntax and Semantics of Programs Syntax The symbols used to write a program Semantics The actions that occur when a program is executed Programming language
More informationCSCI-GA Scripting Languages
CSCI-GA.3033.003 Scripting Languages 12/02/2013 OCaml 1 Acknowledgement The material on these slides is based on notes provided by Dexter Kozen. 2 About OCaml A functional programming language All computation
More informationλ-calculus Lecture 1 Venanzio Capretta MGS Nottingham
λ-calculus Lecture 1 Venanzio Capretta MGS 2018 - Nottingham Table of contents 1. History of λ-calculus 2. Definition of λ-calculus 3. Data Structures 1 History of λ-calculus Hilbert s Program David Hilbert
More informationThe Lambda Calculus. notes by Don Blaheta. October 12, A little bondage is always a good thing. sk
The Lambda Calculus notes by Don Blaheta October 2, 2000 A little bondage is always a good thing. sk We ve seen that Scheme is, as languages go, pretty small. There are just a few keywords, and most of
More informationProgramming Language Features. CMSC 330: Organization of Programming Languages. Turing Completeness. Turing Machine.
CMSC 330: Organization of Programming Languages Lambda Calculus Programming Language Features Many features exist simply for convenience Multi-argument functions foo ( a, b, c ) Ø Use currying or tuples
More informationCMSC 330: Organization of Programming Languages
CMSC 330: Organization of Programming Languages Lambda Calculus CMSC 330 1 Programming Language Features Many features exist simply for convenience Multi-argument functions foo ( a, b, c ) Ø Use currying
More informationThe Untyped Lambda Calculus
Resources: The slides of this lecture were derived from x = 1 [Järvi], with permission of the original author, by copy & let x = 1 in... paste or by selection, annotation, or rewording. [Järvi] is in turn
More informationCOMP 4161 NICTA Advanced Course. Advanced Topics in Software Verification. Toby Murray, June Andronick, Gerwin Klein
COMP 4161 NICTA Advanced Course Advanced Topics in Software Verification Toby Murray, June Andronick, Gerwin Klein λ 1 Last time... λ calculus syntax free variables, substitution β reduction α and η conversion
More informationλ calculus is inconsistent
Content Rough timeline COMP 4161 NICTA Advanced Course Advanced Topics in Software Verification Gerwin Klein, June Andronick, Toby Murray λ Intro & motivation, getting started [1] Foundations & Principles
More informationWhereweare. CS-XXX: Graduate Programming Languages. Lecture 7 Lambda Calculus. Adding data structures. Data + Code. What about functions
Whereweare CS-XXX: Graduate Programming Languages Lecture 7 Lambda Calculus Done: Syntax, semantics, and equivalence For a language with little more than loops and global variables Now: Didn t IMP leave
More information