9/23/2014. Why study? Lambda calculus. Church Rosser theorem Completeness of Lambda Calculus: Turing Complete

Size: px
Start display at page:

Download "9/23/2014. Why study? Lambda calculus. Church Rosser theorem Completeness of Lambda Calculus: Turing Complete"

Transcription

1 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 Complete 1 2 A framework developed in 1930s by Alonzo Church to study computations with functions Church wanted a minimal notation to expose only what tis essential Similar to BNF, Normal form in Rel Database Two operations with functions are essential: function creation function application In School: d/dx x n = nx n-1 [Power Rule] d/dx (f + g) = d/dx f + d/dx g [Sum Rule] Calculus is a branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables Basic syntactic notions Free and bound variables Functions, and Declarations λ Calculation rule Symbolic evaluation useful for discussing programs Used in optimization (inlining), macro expansion Correct macro processing requires variable renaming Illustrates some ideas about scope of binding Tremendous influence on design and analysis of programming languages Realistic languages are too large and complex to study from scratch as a whole Typical approach is to modularize the study into one feature at a time E.g., recursion, looping, exceptions, objects, etc. Then we assemble the features together slide 5 6 1

2 λ calculus is the standard testbed for studying programming language features Because of its minimality Despite its syntactic simplicity the λ calculus can easily encode: numbers, recursive data types, modules, imperative features, exceptions, etc. Certain language features necessitate more substantial extensions to λ calculus: Distributed & parallel languages: π calculus Object oriented languages: σ calculus 7 Whatever the next 700 languages turn out to be, they will surely be variants of. (Landin 1966) 8 John Backus: 1977 Turing Award Designer of Fortran, BNF, etc. Turing Award lecture Functional programming better than imperative i programming Easier to reason about functional programs More efficient due to parallelism Algebraic laws Reason about programs Optimizing compilers slide 9 To prove a program correct, must consider everything a program depends on In functional programs, dependence on any data structure is explicit fully and clearly expressed Therefore, it s easier to reason about functional programs slide 10 The only thing that we can do with a function is to apply it to an argument Church used the notation E 1 E 2 to denote the application of function E 1 to actual argument E 2 All functions are applied to a single argument Church introduced the notation λx. E to denote a function with formal argument xand with body E Functions do not have names names are not essential for the computation Functions have a single argument Once we understand how functions with one argument work we can generalize to multiple args

3 Only three kinds of expressions E ::= x variables E 1 E 2 function application λx. E function creation The form λx. Eis also called lambda abstraction, or simply abstraction Eare called λ terms or λ expressions As in all languages with variables, it is important to discuss the notion of scope Recall: the scope of an identifier is the portion of a program where the identifier is accessible An abstraction λx. E binds variable ibl x in E xis the newly introduced variable Eis the scope of x We say x is bound in λx. E Just like formal function arguments are bound in the function body Bound variable is a placeholder Variable x is bound in λx. (x+y) Function λx. (x+y) is same function as λz. (z+y) Compare to Caclulus x+y dx = z+y dz x P(x) = z P(z) slide 15 Expressions x + y Functions x + 2*y + z λx. (x+y) λz. (x + 2*y + z) Application (λx. (x+y)) 3 = 3 + y (λz. (x + 2*y + z)) 5 = x + 2*y + 5 Parsing: λx. f (f x) = λx.( f (f (x)) ) slide 16 Name of free (i.e., unbound) variable matters! Variable y is free in λx. (x+y) Function λx. (x+y) is not same as λx. (x+z) Occurrences y is free and bound in λx. ((λy. y+2) x) + y Just like in any language with static nested scoping, we have to worry about variable shadowing An occurrence of a variable might refer to differentthingsin things in differentcontext E.g., in CoolFun: let x E in x + (let x E in x) + x In λ calculus: λx. x (λx. x) x Renaming λx. x (λz.z) x slide

4 Two λ terms that can be obtained from each other by a renaming of the bound variables are considered identical λx. xis identical to λy. yand to λz. z Intuition: by changing the name of a formal argument and of all its occurrences in the function body, the behavior of the function does not change fun (x)=x*x fun (y)=y*y fun(z) z*z in λ calculus such functions are considered identical 19 Convention: we will always rename bound variables so that they are all unique e.g., write λ x. x (λ y.y) xinstead of λ x. x (λ x.x) x Thismakesit it easyto seethe scope of bindings And also prevents serious confusion! 20 The identity function: I = def λx. x A function that given an argument ydiscards it andcomputes the identity function: λy. (λx. x) A function that given a function finvokes it on the identity function λf. f (λ x. x) Application associates to the left x y zparses as (x y) z Abstraction extends to the right as far as possible λx. x λy. x y zparses as λ x. (x (λy. ((x y) z))) And yields the the parse tree: λx app x λy app app x y z Syntactic sugar is syntax within a programming language that is designed to make things easier to read or to express. It makes the language "sweeter" for human use: things can be expressed more clearly, more concisely, or in an alternative style that some may prefer. conext.checking( new Expectations(){{ //Better one oneof(alarm).getattackalarm(null);}} ); Expectations exp = new Expectations(); exp.oneof(alarm).getattackalarm(null); conext.checking(exp); slide 23 function f(x) { return x+2; } f(5); block body (λf. f(5)) (λx. x+2) Syntactic sugar is syntax within a programming language that is designed to make things easier to read or to express. It makes the language "sweeter" for human use: things can be expressed more clearly, more concisely, or in an alternative style that some may prefer. declared function Same as λx. x+2 5 slide 24 4

5 The substitution of E for x in E (written [E /x]e ) Step 1. Rename bound variables in Eand E so they are unique Step 2. Perform the textual substitution of E for x in E Example: [y (λx. x) / x] λy. (λx. x) y x After renaming: [y (λv. v)/x] λz. (λu. u) z x After substitution: λz. (λu. u) z (y (λv. v)) 25 α reduction ( or renaming) λy. M α λv. (M [y av]) where v does not occur in M. Example : [y (λx. x) / x] λy. (λx. x) y x = [y (λv. v)/x] λz. (λu. u) z x β reduction (or substitution) (λx. M)N β M [ x an ] Note the syntax is different from Scheme: (λx.m)n ((lambda (x) M) N) 26 (λx. M)N M [ xan ] Replace all x s in M witn N The identity function: (λx. x) E [E / x] x= E Another example with the identity: (λf. f (λx. x)) (λx. x) [λx. x / f] f (λx. x)) = [(λx. x) / f] f (λy. y)) = (λx. x) (λy. y) [λy. y /x] x = λy. y A non terminating evaluation: (λx. xx)(λx. xx) [λx. xx / x]xx = [λy. yy / x] xx = (λy. yy)(λy. yy) 28 redex: Term of the form (λx. M)N Something that can be β reduced An expression is in normal form if it contains no redexes (redices). To evaluate a lambda expression, keep doing reductions until you get to normal form. The definition of substitution guarantees that evaluation respects static scoping: (λ x. (λy. y x)) (y (λx. x)) β λz. z (y (λv. v)) (y remains free, i.e., defined externally)

6 In a λ term, there could be more than one instance of (λ x. E) E (λ y. (λ x. x) y) E could reduce the inner or the outer lambda which one should we pick? inner (λ y. (λ x. x) y) E outer The Church Rosser theorem says that any order will compute the same result A result is a λ term that cannot be reduced further But we might want to fix the order of evaluation when we model a certain language (λy. [y/x] x) E = (λy. y) E [E/y] (λx. x) y =(λx. x) E E Given function f, return function f f λf. λx. f (f x) How does this work? (λf. λx. f (f x)) (λy. y+1) = λx. (λy. y+1) ((λy. y+1) x) = λx. (λy. y+1) (x+1) = λx. (x+1)+1 Given function f, return function f f fn f => fn x => f(f(x)) How does this work? (fn f => fn x => f(f(x))) (fn y => y + 1) = fn x => ((fn y => y + 1) ((fn y => y + 1) x)) = fn x => ((fn y => y + 1) (x + 1)) = fn x => ((x + 1) + 1) Same result if step 2 is altered slide 33 slide 34 Pure lambda calculus has only functions What if we want to compute with Booleans, numbers, lists, etc.? All these can be encoded in pure λ calculus The trick: do not encode what a value is but what we can do with it! For each data type, we have to describe how it can be used, as a function then we write that function in λ calculus The λ calculus can express data types (integers, Booleans, lists, trees, etc.) branching (using Booleans) recursion This is enough to encode Turing machines Encodings are fun But programming in pure λ calculus is painful we will add constants (0, 1, 2,, true, false, if thenelse, etc.) and we will add types

7 What can we do with a Boolean? we can make a binary choice A Boolean is a function that given two choices selects ects one eof them true = def λx. λy. x false = def λx. λy. y if E 1 then E 2 else E 3 = def E 1 E 2 E 3 Example: if true then u else v is (λx. λy. x) u v β (λy. u) v β u What can we do with a pair? we can select one of its elements A pair is a function that given a boolean returns the left etor the right tee elemente mkpair x y = def λ b. x y fst p = def p true snd p = def p false Example: fst (mkpair x y) (mkpair x y) true true x y x What can we do with a natural number? we can iterate a number of times A natural number is a function that given an operation o f and da starting t value aues, applies f a number of times to s: 0 = def λf. λs. s 1 = def λf. λs. f s 2 = def λf. λs. f (f s) and so on The successor function succ n = def λf. λs. f (n f s) Addition add n 1 n 2 = def n 1 succ n 2 Multiplication mult n 1 n 2 = def n 1 (add n 2 ) 0 Testing equality with 0 iszero n = def n (λb. false) true mult (add 2) 0 (add 2) ((add 2) 0) 2 succ (add 2 0) 2 succ (2 succ 0) succ (succ (succ (succ 0))) succ (succ (succ (λf. λs. f (0 f s)))) succ (succ (succ (λf. λs. f s))) succ (succ (λg. λy. g ((λf. λs. f s) g y))) succ (succ (λg. λy. g (g y))) * λg. λy. g (g (g (g y))) =

Computer Science 203 Programming Languages Fall Lecture 10. Bindings, Procedures, Functions, Functional Programming, and the Lambda Calculus

Computer 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 information

Type Systems Winter Semester 2006

Type 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 information

CIS 500 Software Foundations Fall September 25

CIS 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 information

Foundations. Yu Zhang. Acknowledgement: modified from Stanford CS242

Foundations. 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 information

CS 4110 Programming Languages & Logics. Lecture 17 Programming in the λ-calculus

CS 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 information

Functional Programming and λ Calculus. Amey Karkare Dept of CSE, IIT Kanpur

Functional 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 information

CS 242. Fundamentals. Reading: See last slide

CS 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 information

Concepts of programming languages

Concepts 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 information

The Lambda Calculus. 27 September. Fall Software Foundations CIS 500. The lambda-calculus. Announcements

The 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 information

Harvard School of Engineering and Applied Sciences CS 152: Programming Languages. Lambda calculus

Harvard 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 information

1 Scope, Bound and Free Occurrences, Closed Terms

1 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 information

Pure Lambda Calculus. Lecture 17

Pure 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 information

Chapter 5: The Untyped Lambda Calculus

Chapter 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 information

More Lambda Calculus and Intro to Type Systems

More 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 information

Fundamentals and lambda calculus

Fundamentals 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 information

Introduction to the Lambda Calculus

Introduction 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 information

Lambda Calculus. Variables and Functions. cs3723 1

Lambda 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 information

λ calculus Function application Untyped λ-calculus - Basic Idea Terms, Variables, Syntax β reduction Advanced Formal Methods

λ 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 information

Programming Language Features. CMSC 330: Organization of Programming Languages. Turing Completeness. Turing Machine.

Programming 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 information

CMSC 330: Organization of Programming Languages

CMSC 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 information

More Lambda Calculus and Intro to Type Systems

More 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 information

COMP80 Lambda Calculus Programming Languages Slides Courtesy of Prof. Sam Guyer Tufts University Computer Science History Big ideas Examples:

COMP80 Lambda Calculus Programming Languages Slides Courtesy of Prof. Sam Guyer Tufts University Computer Science History Big ideas Examples: COMP80 Programming Languages Slides Courtesy of Prof. Sam Guyer Lambda Calculus Formal system with three parts Notation for functions Proof system for equations Calculation rules called reduction Idea:

More information

CMPUT 325 : Lambda Calculus Basics. Lambda Calculus. Dr. B. Price and Dr. R. Greiner. 13th October 2004

CMPUT 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 information

CS152: Programming Languages. Lecture 7 Lambda Calculus. Dan Grossman Spring 2011

CS152: 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 information

Formal Semantics. Aspects to formalize. Lambda calculus. Approach

Formal 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 information

CSE 505: Concepts of Programming Languages

CSE 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 information

VU Semantik von Programmiersprachen

VU 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 information

Fundamentals 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) 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 information

Whereweare. 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. 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

5. Introduction to the Lambda Calculus. Oscar Nierstrasz

5. 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 information

CMSC 330: Organization of Programming Languages

CMSC 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 information

Functional Programming

Functional 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 information

Pure (Untyped) λ-calculus. Andrey Kruglyak, 2010

Pure (Untyped) λ-calculus. Andrey Kruglyak, 2010 Pure (Untyped) λ-calculus Andrey Kruglyak, 2010 1 Pure (untyped) λ-calculus An example of a simple formal language Invented by Alonzo Church (1936) Used as a core language (a calculus capturing the essential

More information

Activity. CSCI 334: Principles of Programming Languages. Lecture 4: Fundamentals II. What is computable? What is computable?

Activity. CSCI 334: Principles of Programming Languages. Lecture 4: Fundamentals II. What is computable? What is computable? Activity CSCI 334: Principles of Programming Languages Lecture 4: Fundamentals II Write a function firsts that, when given a list of cons cells, returns a list of the left element of each cons. ( (a. b)

More information

Lecture 9: More Lambda Calculus / Types

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.

- 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 information

Programming Languages. Programming with λ-calculus. Lecture 11: Type Systems. Special Hour to discuss HW? if-then-else int

Programming 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 information

Formal Systems and their Applications

Formal 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 information

Introduction to the λ-calculus

Introduction 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 information

INF 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. 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 information

Lambda Calculus. Lambda Calculus

Lambda 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 information

Introduction to Lambda Calculus. Lecture 5 CS 565 1/24/08

Introduction 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 information

11/6/17. Outline. FP Foundations, Scheme. Imperative Languages. Functional Programming. Mathematical Foundations. Mathematical Foundations

11/6/17. 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 information

CS 6110 S14 Lecture 1 Introduction 24 January 2014

CS 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 information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 4 MODULE, SPRING SEMESTER MATHEMATICAL FOUNDATIONS OF PROGRAMMING ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 4 MODULE, SPRING SEMESTER MATHEMATICAL FOUNDATIONS OF PROGRAMMING ANSWERS The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 4 MODULE, SPRING SEMESTER 2012 2013 MATHEMATICAL FOUNDATIONS OF PROGRAMMING ANSWERS Time allowed TWO hours Candidates may complete the front

More information

Constraint-based Analysis. Harry Xu CS 253/INF 212 Spring 2013

Constraint-based Analysis. Harry Xu CS 253/INF 212 Spring 2013 Constraint-based Analysis Harry Xu CS 253/INF 212 Spring 2013 Acknowledgements Many slides in this file were taken from Prof. Crista Lope s slides on functional programming as well as slides provided by

More information

Lambda Calculus-2. Church Rosser Property

Lambda 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 information

Introduction to Lambda Calculus. Lecture 7 CS /08/09

Introduction 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 information

Untyped Lambda Calculus

Untyped 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 information

CMSC 330: Organization of Programming Languages. Lambda Calculus

CMSC 330: Organization of Programming Languages. Lambda Calculus CMSC 330: Organization of Programming Languages Lambda Calculus 1 Turing Completeness Turing machines are the most powerful description of computation possible They define the Turing-computable functions

More information

COMP 1130 Lambda Calculus. based on slides by Jeff Foster, U Maryland

COMP 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 information

Lambda Calculus. Lecture 4 CS /26/10

Lambda 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 information

Functional Programming

Functional 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 information

Lambda Calculus. Type Systems, Lectures 3. Jevgeni Kabanov Tartu,

Lambda Calculus. Type Systems, Lectures 3. Jevgeni Kabanov Tartu, Lambda Calculus Type Systems, Lectures 3 Jevgeni Kabanov Tartu, 13.02.2006 PREVIOUSLY ON TYPE SYSTEMS Arithmetical expressions and Booleans Evaluation semantics Normal forms & Values Getting stuck Safety

More information

Untyped Lambda Calculus

Untyped 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 information

Lecture 5: The Untyped λ-calculus

Lecture 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 information

Lexicografie computationala Feb., 2012

Lexicografie computationala Feb., 2012 Lexicografie computationala Feb., 2012 Anca Dinu University of Bucharest Introduction When we construct meaning representations systematically, we integrate information from two different sources: 1. The

More information

M. Snyder, George Mason University LAMBDA CALCULUS. (untyped)

M. Snyder, George Mason University LAMBDA CALCULUS. (untyped) 1 LAMBDA CALCULUS (untyped) 2 The Untyped Lambda Calculus (λ) Designed by Alonzo Church (1930s) Turing Complete (Turing was his doctoral student!) Models functions, always as 1-input Definition: terms,

More information

Recursion. Lecture 6: More Lambda Calculus Programming. Fixed Points. Recursion

Recursion. Lecture 6: More Lambda Calculus Programming. Fixed Points. Recursion 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

More information

The Untyped Lambda Calculus

The 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 information

One of a number of approaches to a mathematical challenge at the time (1930): Constructibility

One 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 information

λ-calculus Lecture 1 Venanzio Capretta MGS Nottingham

λ-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 information

Introduction to the Lambda Calculus. Chris Lomont

Introduction 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 information

CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages CMSC 330: Organization of Programming Languages Lambda Calculus Encodings CMSC 330 Spring 2017 1 Review A lambda calculus expression is defined as e ::= x variable λx.e abstraction (fun def) e e application

More information

Functions as data. Massimo Merro. 9 November Massimo Merro The Lambda language 1 / 21

Functions 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 information

Lambda Calculus. CS 550 Programming Languages Jeremy Johnson

Lambda Calculus. CS 550 Programming Languages Jeremy Johnson Lambda Calculus CS 550 Programming Languages Jeremy Johnson 1 Lambda Calculus The semantics of a pure functional programming language can be mathematically described by a substitution process that mimics

More information

CSE-321 Programming Languages 2010 Midterm

CSE-321 Programming Languages 2010 Midterm Name: Hemos ID: CSE-321 Programming Languages 2010 Midterm Score Prob 1 Prob 2 Prob 3 Prob 4 Total Max 15 30 35 20 100 1 1 SML Programming [15 pts] Question 1. [5 pts] Give a tail recursive implementation

More information

Organization of Programming Languages CS3200/5200N. Lecture 11

Organization of Programming Languages CS3200/5200N. Lecture 11 Organization of Programming Languages CS3200/5200N Razvan C. Bunescu School of Electrical Engineering and Computer Science bunescu@ohio.edu Functional vs. Imperative The design of the imperative languages

More information

CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages CMSC 330: Organization of Programming Languages Lambda Calculus CMSC 330 Summer 2017 1 100 years ago Albert Einstein proposed special theory of relativity in 1905 In the paper On the Electrodynamics of

More information

Programming Languages

Programming 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 information

Recursive Definitions, Fixed Points and the Combinator

Recursive 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 information

CMSC 330: Organization of Programming Languages. Lambda Calculus

CMSC 330: Organization of Programming Languages. Lambda Calculus CMSC 330: Organization of Programming Languages Lambda Calculus 1 100 years ago Albert Einstein proposed special theory of relativity in 1905 In the paper On the Electrodynamics of Moving Bodies 2 Prioritätsstreit,

More information

CITS3211 FUNCTIONAL PROGRAMMING

CITS3211 FUNCTIONAL PROGRAMMING CITS3211 FUNCTIONAL PROGRAMMING 9. The λ calculus Summary: This lecture introduces the λ calculus. The λ calculus is the theoretical model underlying the semantics and implementation of functional programming

More information

Lecture #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: 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 information

Last class. CS Principles of Programming Languages. Introduction. Outline

Last class. CS Principles of Programming Languages. Introduction. Outline Last class CS6848 - Principles of Programming Languages Principles of Programming Languages V. Krishna Nandivada IIT Madras Interpreters A Environment B Cells C Closures D Recursive environments E Interpreting

More information

CS152: Programming Languages. Lecture 11 STLC Extensions and Related Topics. Dan Grossman Spring 2011

CS152: Programming Languages. Lecture 11 STLC Extensions and Related Topics. Dan Grossman Spring 2011 CS152: Programming Languages Lecture 11 STLC Extensions and Related Topics Dan Grossman Spring 2011 Review e ::= λx. e x e e c v ::= λx. e c τ ::= int τ τ Γ ::= Γ, x : τ (λx. e) v e[v/x] e 1 e 1 e 1 e

More information

Fall 2013 Midterm Exam 10/22/13. This is a closed-book, closed-notes exam. Problem Points Score. Various definitions are provided in the exam.

Fall 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 information

CSE 341: Programming Languages

CSE 341: Programming Languages CSE 341: Programming Languages Autumn 2005 Lecture 10 Mutual Recursion, Equivalence, and Syntactic Sugar CSE 341 Autumn 2005, Lecture 10 1 Mutual Recursion You ve already seen how multiple functions can

More information

CIS 500 Software Foundations Midterm I

CIS 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 information

Overview. A normal-order language. Strictness. Recursion. Infinite data structures. Direct denotational semantics. Transition semantics

Overview. A normal-order language. Strictness. Recursion. Infinite data structures. Direct denotational semantics. Transition semantics Overview A normal-order language Strictness Recursion Infinite data structures Direct denotational semantics Transition semantics Lazy (call-by-need) evaluation and its semantics A Normal-Order Language

More information

Denotational Semantics; Lambda Calculus Basics Section and Practice Problems Week 4: Tue Feb 13 Fri Feb 16, 2018

Denotational Semantics; Lambda Calculus Basics Section and Practice Problems Week 4: Tue Feb 13 Fri Feb 16, 2018 Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Denotational Semantics; Lambda Calculus Basics Week 4: Tue Feb 13 Fri Feb 16, 2018 1 Denotational Semantics (a) Give the

More information

More Lambda Calculus and Intro to Type Systems

More Lambda Calculus and Intro to Type Systems More Lambda Calculus and Intro to Type Systems #1 One Slide Summary The lambda calculus is a model of computation or a programming language that is as expressive as a Turing machine. The lambda calculus

More information

Lambda Calculus. Adrian Groza. Department of Computer Science Technical University of Cluj-Napoca

Lambda Calculus. Adrian Groza. Department of Computer Science Technical University of Cluj-Napoca Lambda Calculus Adrian Groza Department of Computer Science Technical University of Cluj-Napoca Outline 1 λ-calculus 2 Operational Semantics Syntax Conversions Normal Form 3 Lambda Calculus as a Functional

More information

Polymorphic lambda calculus Princ. of Progr. Languages (and Extended ) The University of Birmingham. c Uday Reddy

Polymorphic lambda calculus Princ. of Progr. Languages (and Extended ) The University of Birmingham. c Uday Reddy 06-02552 Princ. of Progr. Languages (and Extended ) The University of Birmingham Spring Semester 2016-17 School of Computer Science c Uday Reddy2016-17 Handout 6: Polymorphic Type Systems 1. Polymorphic

More information

CITS3211 FUNCTIONAL PROGRAMMING. 10. Programming in the pure λ calculus

CITS3211 FUNCTIONAL PROGRAMMING. 10. Programming in the pure λ calculus CITS3211 FUNCTIONAL PROGRAMMING 10. Programming in the pure λ calculus Summary: This lecture demonstates the power of the pure λ calculus by showing that there are λ expressions that can be used to emulate

More information

CSC173 Lambda Calculus Lambda Calculus Evaluation (10 min) Evaluate this expression (β-reduce with normal-order evaluation):

CSC173 Lambda Calculus Lambda Calculus Evaluation (10 min) Evaluate this expression (β-reduce with normal-order evaluation): CSC173 Lambda Calculus 014 Please write your name on the bluebook. You may use two sides of handwritten notes. There are 90 possible points and 75 is a perfect score. Stay cool and please write neatly.

More information

The Untyped Lambda Calculus

The Untyped Lambda Calculus Resources: The slides of this lecture were derived from [Järvi], with permission of the original author, by copy & x = 1 let x = 1 in... paste or by selection, annotation, or rewording. [Järvi] is in turn

More information

Advanced Topics in Programming Languages Lecture 3 - Models of Computation

Advanced 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 information

Elixir, functional programming and the Lambda calculus.

Elixir, functional programming and the Lambda calculus. Elixir, functional programming and the Lambda calculus. Programming II - Elixir Version Johan Montelius Spring Term 2018 Introduction In this tutorial you re going to explore lambda calculus and how it

More information

dynamically typed dynamically scoped

dynamically typed dynamically scoped Reference Dynamically Typed Programming Languages Part 1: The Untyped λ-calculus Jim Royer CIS 352 April 19, 2018 Practical Foundations for Programming Languages, 2/e, Part VI: Dynamic Types, by Robert

More information

Programming Languages Lecture 14: Sum, Product, Recursive Types

Programming Languages Lecture 14: Sum, Product, Recursive Types CSE 230: Winter 200 Principles of Programming Languages Lecture 4: Sum, Product, Recursive Types The end is nigh HW 3 No HW 4 (= Final) Project (Meeting + Talk) Ranjit Jhala UC San Diego Recap Goal: Relate

More information

If a program is well-typed, then a type can be inferred. For example, consider the program

If a program is well-typed, then a type can be inferred. For example, consider the program CS 6110 S18 Lecture 24 Type Inference and Unification 1 Type Inference Type inference refers to the process of determining the appropriate types for expressions based on how they are used. For example,

More information

Lambda Calculus and Type Inference

Lambda Calculus and Type Inference Lambda Calculus and Type Inference Björn Lisper Dept. of Computer Science and Engineering Mälardalen University bjorn.lisper@mdh.se http://www.idt.mdh.se/ blr/ October 13, 2004 Lambda Calculus and Type

More information

Homework. Lecture 7: Parsers & Lambda Calculus. Rewrite Grammar. Problems

Homework. 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 information

CIS 500 Software Foundations Midterm I Answer key October 8, 2003

CIS 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 information

Graphical Untyped Lambda Calculus Interactive Interpreter

Graphical Untyped Lambda Calculus Interactive Interpreter Graphical Untyped Lambda Calculus Interactive Interpreter (GULCII) Claude Heiland-Allen https://mathr.co.uk mailto:claude@mathr.co.uk Edinburgh, 2017 Outline Lambda calculus encodings How to perform lambda

More information

CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages CMSC 330: Organization of Programming Languages Lambda Calculus Encodings CMSC 330 Summer 2017 1 The Power of Lambdas Despite its simplicity, the lambda calculus is quite expressive: it is Turing complete!

More information

CMSC 336: Type Systems for Programming Languages Lecture 4: Programming in the Lambda Calculus Acar & Ahmed 22 January 2008.

CMSC 336: Type Systems for Programming Languages Lecture 4: Programming in the Lambda Calculus Acar & Ahmed 22 January 2008. CMSC 336: Type Systems for Programming Languages Lecture 4: Programming in the Lambda Calculus Acar & Ahmed 22 January 2008 Contents 1 Announcements 1 2 Solution to the Eercise 1 3 Introduction 1 4 Multiple

More information

From the λ-calculus to Functional Programming Drew McDermott Posted

From the λ-calculus to Functional Programming Drew McDermott Posted From the λ-calculus to Functional Programming Drew McDermott drew.mcdermott@yale.edu 2015-09-28 Posted 2015-10-24 The λ-calculus was intended from its inception as a model of computation. It was used by

More information

CSE-321 Programming Languages 2012 Midterm

CSE-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 information