Denotational Semantics; Lambda Calculus Basics Section and Practice Problems Week 4: Tue Feb 13 Fri Feb 16, 2018
|
|
- Jack Lee
- 5 years ago
- Views:
Transcription
1 Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Denotational Semantics; Lambda Calculus Basics Week 4: Tue Feb 13 Fri Feb 16, Denotational Semantics (a) Give the denotational semantic for each of the following IMP programs. That is, express the meaning of each of the following programs as a function from stores to stores. (i) a := b + 5; a := a b We can implement the rules and simplify to obtain: C [a := b + 5; a := a b]σ = σ[a σ[a σ(b) + 5](a) σ(b)] = σ[a (σ(b) + 5) σ(b)] (ii) if foo < 0 then bar := foo foo else bar := foo foo foo We take c to be the command above: { σ[bar σ(foo) σ(foo)] σ(foo) < 0 C [c]σ = σ[bar σ(foo) σ(foo) σ(foo)] otherwise (iii) bar := foo foo; if foo < 0 then skip else bar := bar foo (Hint: the answer to this question should be the same function as the answer to 1(b)ii above. You may have written the function down differently, but it should be the same mathematical function.) We have a very similar function to the above: { σ[bar σ(foo) σ(foo)] σ(foo) < 0 C [c]σ = σ[bar σ[bar σ(foo) σ(foo)](bar) σ(foo)] otherwise (iv) a := 0; b = 0; while a < 3 do b := b + c Note that the above term diverges. So C [a := 0; b = 0; while a < 3 do b := b + c] is the partial function with an empty domain.
2 (b) Consider the following loop. while foo < 5 do foo := foo + 1; bar := bar + 1 We will consider the denotational semantics of this loop. (i) What is the denotational semantics of the loop guard foo < 5? That is, what is the function B [foo < 5]? B [foo < 5] = {(σ, true) σ(foo) < 5} {(σ, false) σ(foo) 5} Equivalently: B [foo < 5] = { true if σ(foo) < 5 false if σ(foo) 5 (ii) What is the denotational semantics of the loop body foo := foo + 1; bar := bar + 1? That is, what is the function C [foo := foo + 1; bar := bar + 1]]? After some simplification: C [foo := foo + 1; bar := bar + 1]σ = σ[foo σ(foo) + 1, bar σ(bar) + 1] (iii) Recall that the semantics of the loop is the fixed point of the following higher-order function F. (This is from Section 1.2 of Lecture 6, where we have provided a specific loop guard b and loop body c for the higher-order function F b,c ). F : (Store Store) (Store Store) F (f) = {(σ, σ) (σ, false) B [foo < 5]} {(σ, σ ) (σ, true) B [foo < 5] σ. ((σ, σ ) C [foo := foo + 1; bar := bar + 1]] (σ, σ ) f)} That is, the semantics of the loop are: C [while foo < 5 do foo := foo + 1; bar := bar + 1] = F i ( ) i 0 = F ( ) F (F ( )) F (F (F ( )))... = fix(f ) Compute F ( ), F (F ( )), and F (F (F ( ))). In general, what is the domain of the partial function F i ( )? (Note that F i ( ) is F applied to the empty set i times, e.g., F 3 ( ) is F (F (F ( ))).) Page 2 of 6
3 For reference as to how we arrive at the below, please see lecture notes. F ( ) ={(σ, σ) σ(foo) 5} F 2 ( ) ={(σ, σ) σ(foo) 5} {(σ, σ[foo σ(foo) + 1, bar σ(bar) + 1]) σ(foo) = 4} ={(σ, σ) σ(foo) 5} {(σ, σ[foo 5, bar σ(bar) + 1]) σ(foo) = 4} F 3 ( ) ={(σ, σ) σ(foo) 5} {(σ, σ[foo σ(foo) + 1, bar σ(bar) + 1]) σ(foo) = 4} {(σ, σ[foo σ(foo) + 2, bar σ(bar) + 2]) σ(foo) = 3} ={(σ, σ) σ(foo) 5} {(σ, σ[foo 5, bar σ(bar) + 1]) σ(foo) = 4} {(σ, σ[foo 5, bar σ(bar) + 2]) σ(foo) = 3} In general, we have: F i ( ) ={(σ, σ) σ(foo) 5} {(σ, σ[foo 5, bar σ(bar) + 1]) σ(foo) + 1 = 5} {(σ, σ[foo 5, bar σ(bar) + 2]) σ(foo) + 2 = 5} {(σ, σ[foo 5, bar σ(bar) + (i 1)]) σ(foo) + (i 1) = 5} So we note that F i is defined for all σ such that σ(foo) i. 2 Lambda Calculus Basics (a) Variable Bindings Fully parenthesize each expression based on the standard parsing of λ-calculus expressions, i.e. you should parenthesize all applications and λ abstractions. Then, draw a box around all binding occurrences of variables, underline all usage occurrence of variables, and circle all free variables. For each bound usage occurrence, neatly draw an arrow to indicate its corresponding binding occurrence. (You may also use other methods to indicate binding occurrences of variables, usage occurrences of variables, free variables, and which uses correspond to which bindings.) λa. z λz. a y (λ a. z (λ z. a y )) (λz. z) λb. b λa. a a (λ z. z) (λ b. b (λ a. a a)) Page 3 of 6
4 λb. b λa. a b (λ b. b (λ a. a b)) λz. z λz. z z (λ z. z (λ z. z z)) λa. λb. (λa. a) λb. a (λ a. (λ b. (λ a. a) (λ b. a))) x λx. λx. x (λx. x) x (λ x. (λ x. x (λ x. x))) y (λy. y)(λy. z) y (λ y. y) (λ y. z ) (b) Alpha equivalence: Which of these three lambda-calculus expressions are alpha equivalent? i. λx. y λa. a x ii. λx. z λb. b x iii. λa. y λb. b a (Hint: to figure out whether two expressions are alpha equivalent, you need to know which variables are free and which variables are bound.) Expressions i. and iii. are alpha-equivalent. Note that alpha equivalence applies only to bound variables; free variables cannot be renamed. Page 4 of 6
5 (c) Evaluation For each of the following terms, do the following: (a) write the result of one step of the call-by-value reduction of the term; (b) write the result of one step of the call-by-name reduction of the term; and (c) write all possible results of one step under full β-reduction. If the term cannot take a step, please note that instead. (λz. λx. x x) (λy. y) The semantics for CBV and CBN are the same, we have: (λz. λx. x x) (λy. y) λx. x x There are no other possible reductions since only one application exists. λa. λb. (λc. c) (λd. d) There are no possible steps under CBV and CBN because neither allows reductions inside a lambda term. For full β-reduction, we have the following: λa. λb. (λc. c) (λd. d) λa. λb. λd. d (λx. x x x x) (λx. λy. x y) Under CBV and CBN we have: (λx. x x x x) (λx. λy. x y) (λx. λy. x y) (λx. λy. x y) (λx. λy. x y) (λx. λy. x y) and under full β=reduction we have no additional steps. (λx. λy. x y) ((λw. λz. w z) (λx. x)) Under CBV we have: (λx. λy. x y) ((λw. λz. w z) (λx. x)) (λx. λy. x y) (λz. (λx. x) z) and under CBN: (λx. λy. x y) ((λw. λz. w z) (λx. x)) λy. ((λw. λz. w z) (λx. x)) y There are no additional steps that we can take under full β-reduction. (λa. (λb. b a) a) (λz. z) (λw. w) First, note that the fully parenthetized expression is ((λa. (λb. b a) a) (λz. z)) (λw. w). Under both CBV and CBN we have (λa. (λb. b a) a) (λz. z) (λw. w) ((λb. b (λz. z)) (λz. z)) (λw. w). In addition, under full β-reduction we can also reduce the subexpression (λb. b a) a, giving us (λa. (λb. b a) a) (λz. z) (λw. w) (λa. a a) (λz. z) (λw. w). Page 5 of 6
6 (d) Suppose we have an applied lambda calculus with integers and addition. Write the sequence of expressions that the following lambda calculus term evaluates to under call-by-value semantics. Then do the same under call-by-name semantics. (λf. f (f 8)) (λx. x + 17) The term under CBN semantics evaluates to: (λf. f (f 8)) (λx. x + 17) (λx. x + 17) ((λx. x + 17) 8) ((λx. x + 17) 8) + 17 (8 + 17) and under CBV we have: (λf. f (f 8)) (λx. x + 17) (λx. x + 17) ((λx. x + 17) 8) (λx. x + 17) (8 + 17) (λx. x + 17) (25) Page 6 of 6
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 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 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 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 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 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 informationCMSC 330, Fall 2018 Midterm 2
CMSC 330, Fall 2018 Midterm 2 Name Teaching Assistant Kameron Aaron Danny Chris Michael P. Justin Cameron B. Derek Kyle Hasan Shriraj Cameron M. Alex Michael S. Pei-Jo Instructions Do not start this exam
More informationCMSC330 Fall 2016 Midterm #2 2:00pm/3:30pm
CMSC330 Fall 2016 Midterm #2 2:00pm/3:30pm Gradescope ID: (Gradescope ID is the First letter of your last name and last 5 digits of your UID) (If you write your name on the test, or your gradescope ID
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 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 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 informationCS 6110 S11 Lecture 25 Typed λ-calculus 6 April 2011
CS 6110 S11 Lecture 25 Typed λ-calculus 6 April 2011 1 Introduction Type checking is a lightweight technique for proving simple properties of programs. Unlike theorem-proving techniques based on axiomatic
More information1 Introduction. 3 Syntax
CS 6110 S18 Lecture 19 Typed λ-calculus 1 Introduction Type checking is a lightweight technique for proving simple properties of programs. Unlike theorem-proving techniques based on axiomatic semantics,
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 informationType Inference; Parametric Polymorphism; Records and Subtyping Section and Practice Problems Mar 20-23, 2018
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Type Inference; Parametric Polymorphism; Records and Subtyping Mar 20-23, 2018 1 Type Inference (a) Recall the constraint-based
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 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 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 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 informationDynamic Types, Concurrency, Type and effect system Section and Practice Problems Apr 24 27, 2018
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Apr 24 27, 2018 1 Dynamic types and contracts (a) To make sure you understand the operational semantics of dynamic types
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 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 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 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 informationCMSC 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 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 informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 21 Tuesday, April 15, 2014 1 Static program analyses For the last few weeks, we have been considering type systems.
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 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 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 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 informationCMSC 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 informationM. 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 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 informationCMSC 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 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 informationIf 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 informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 18 Thursday, April 3, 2014 1 Error-propagating semantics For the last few weeks, we have been studying type systems.
More informationOverview. 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 informationaxiomatic semantics involving logical rules for deriving relations between preconditions and postconditions.
CS 6110 S18 Lecture 18 Denotational Semantics 1 What is Denotational Semantics? So far we have looked at operational semantics involving rules for state transitions, definitional semantics involving translations
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 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 informationCSE-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 informationLambda 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 informationCSC173 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 informationExercise 1 (2+2+2 points)
1 Exercise 1 (2+2+2 points) The following data structure represents binary trees only containing values in the inner nodes: data Tree a = Leaf Node (Tree a) a (Tree a) 1 Consider the tree t of integers
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 informationCS-XXX: Graduate Programming Languages. Lecture 9 Simply Typed Lambda Calculus. Dan Grossman 2012
CS-XXX: Graduate Programming Languages Lecture 9 Simply Typed Lambda Calculus Dan Grossman 2012 Types Major new topic worthy of several lectures: Type systems Continue to use (CBV) Lambda Caluclus as our
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 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 informationFunctional Programming. Overview. Topics. Recall λ-terms. Examples
Topics Functional Programming Christian Sternagel Harald Zankl Evgeny Zuenko Department of Computer Science University of Innsbruck WS 2017/2018 abstract data types, algebraic data types, binary search
More informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 19 Tuesday, April 3, 2018 1 Introduction to axiomatic semantics The idea in axiomatic semantics is to give specifications
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 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 informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 14 Tuesday, March 24, 2015 1 Parametric polymorphism Polymorph means many forms. Polymorphism is the ability of
More informationExercise 1 ( = 22 points)
1 Exercise 1 (4 + 3 + 4 + 5 + 6 = 22 points) The following data structure represents polymorphic lists that can contain values of two types in arbitrary order: data DuoList a b = C a (DuoList a b) D b
More informationCSE-321 Programming Languages 2011 Final
Name: Hemos ID: CSE-321 Programming Languages 2011 Final Prob 1 Prob 2 Prob 3 Prob 4 Prob 5 Prob 6 Total Score Max 15 15 10 17 18 25 100 There are six problems on 18 pages in this exam, including one extracredit
More informationHarvard School of Engineering and Applied Sciences Computer Science 152
Harvard School of Engineering and Applied Sciences Computer Science 152 Lecture 17 Tuesday, March 30, 2010 1 Polymorph means many forms. Polymorphism is the ability of code to be used on values of different
More informationPure (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 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 informationCMSC330 Spring 2016 Midterm #2 9:30am/12:30pm/3:30pm
CMSC330 Spring 2016 Midterm #2 9:30am/12:30pm/3:30pm Name: Discussion Time: 10am 11am 12pm 1pm 2pm 3pm TA Name (Circle): Adam Anshul Austin Ayman Damien Daniel Jason Michael Patrick William Instructions
More informationCMSC330 Spring 2018 Midterm 2 9:30am/ 11:00am/ 3:30pm
CMSC330 Spring 2018 Midterm 2 9:30am/ 11:00am/ 3:30pm Name (PRINT YOUR NAME as it appears on gradescope ): SOLUTION Discussion Time (circle one) 10am 11am 12pm 1pm 2pm 3pm Instructions Do not start this
More informationLambda Calculus alpha-renaming, beta reduction, applicative and normal evaluation orders, Church-Rosser theorem, combinators
Lambda Calculus alpha-renaming, beta reduction, applicative and normal evaluation orders, Church-Rosser theorem, combinators Carlos Varela Rennselaer Polytechnic Institute February 11, 2010 C. Varela 1
More informationLast 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 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 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 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 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 informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 24 Thursday, April 19, 2018 1 Error-propagating semantics For the last few weeks, we have been studying type systems.
More informationCITS3211 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 informationIntro to semantics; Small-step semantics Lecture 1 Tuesday, January 29, 2013
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 1 Tuesday, January 29, 2013 1 Intro to semantics What is the meaning of a program? When we write a program, we use
More informationLecture 5: Lazy Evaluation and Infinite Data Structures
Lecture 5: Lazy Evaluation and Infinite Data Structures Søren Haagerup Department of Mathematics and Computer Science University of Southern Denmark, Odense October 3, 2017 How does Haskell evaluate a
More informationAbstract register machines Lecture 23 Tuesday, April 19, 2016
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 23 Tuesday, April 19, 2016 1 Why abstract machines? So far in the class, we have seen a variety of language features.
More informationExercise 1 ( = 24 points)
1 Exercise 1 (4 + 5 + 4 + 6 + 5 = 24 points) The following data structure represents polymorphic binary trees that contain values only in special Value nodes that have a single successor: data Tree a =
More informationConstrained Optimization and Lagrange Multipliers
Constrained Optimization and Lagrange Multipliers MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Constrained Optimization In the previous section we found the local or absolute
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 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 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 informationn Closed book n You are allowed 5 cheat pages n Practice problems available in Course Materials n Check grades in Rainbow grades
Announcements Announcements n HW12: Transaction server n Due today at 2pm n You can use up to 2 late days, as always n Final Exam, December 17, 3-6pm, in DCC 308 and 318 n Closed book n You are allowed
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 informationVariables. Substitution
Variables Elements of Programming Languages Lecture 4: Variables, binding and substitution James Cheney University of Edinburgh October 6, 2015 A variable is a symbol that can stand for another expression.
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 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 informationLambdaLab: An Interactive λ-calculus Reducer for Learning
LambdaLab: An Interactive λ-calculus Reducer for Learning Abstract Daniel Sainati Cornell University dhs253@cornell.edu In advanced programming languages curricula, the λ-calculus often serves as the foundation
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 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 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 informationORIE 6300 Mathematical Programming I September 2, Lecture 3
ORIE 6300 Mathematical Programming I September 2, 2014 Lecturer: David P. Williamson Lecture 3 Scribe: Divya Singhvi Last time we discussed how to take dual of an LP in two different ways. Today we will
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 informationExercise 1 ( = 18 points)
1 Exercise 1 (4 + 5 + 4 + 5 = 18 points) The following data structure represents polymorphic binary trees that contain values only in special Value nodes that have a single successor: data Tree a = Leaf
More informationCS 6110 S11 Lecture 12 Naming and Scope 21 February 2011
CS 6110 S11 Lecture 12 Naming and Scope 21 February 2011 In this lecture we introduce the topic of scope in the context of the λ-calculus and define translations from λ-cbv to FL for the two most common
More informationCSE505, Fall 2012, Midterm Examination October 30, 2012
CSE505, Fall 2012, Midterm Examination October 30, 2012 Rules: The exam is closed-book, closed-notes, except for one side of one 8.5x11in piece of paper. Please stop promptly at Noon. You can rip apart
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 informationLecture 9: Typed Lambda Calculus
Advanced Topics in Programming Languages Spring Semester, 2012 Lecture 9: Typed Lambda Calculus May 8, 2012 Lecturer: Mooly Sagiv Scribe: Guy Golan Gueta and Shachar Itzhaky 1 Defining a Type System for
More informationNote that in this definition, n + m denotes the syntactic expression with three symbols n, +, and m, not to the number that is the sum of n and m.
CS 6110 S18 Lecture 8 Structural Operational Semantics and IMP Today we introduce a very simple imperative language, IMP, along with two systems of rules for evaluation called small-step and big-step semantics.
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 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 informationTyped Lambda Calculus and Exception Handling
Typed Lambda Calculus and Exception Handling Dan Zingaro zingard@mcmaster.ca McMaster University Typed Lambda Calculus and Exception Handling p. 1/2 Untyped Lambda Calculus Goal is to introduce typing
More informationCSCI B522 Lecture 11 Naming and Scope 8 Oct, 2009
CSCI B522 Lecture 11 Naming and Scope 8 Oct, 2009 Lecture notes for CS 6110 (Spring 09) taught by Andrew Myers at Cornell; edited by Amal Ahmed, Fall 09. 1 Static vs. dynamic scoping The scope of a variable
More information