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Pattern Matching
Functional Programming Steven Lau March 2015 before function programming... https://www.youtube.com/watch?v=92WHN-pAFCs Models of computation ● Turing machine ○ invented by Alan Turing in 1936 ● Lambda calculus ○ invented by Alonzo Church in 1930 ● more... Turing machine ● A machine operates on an infinite tape (memory) and execute a program stored ● It may ○ read a symbol ○ write a symbol ○ move to the left cell ○ move to the right cell ○ change the machine’s state ○ halt Turing machine Have some fun http://www.google.com/logos/2012/turing-doodle-static.html http://www.ioi2012.org/wp-content/uploads/2011/12/Odometer.pdf http://wcipeg.com/problems/desc/ioi1211 Turing machine incrementer state symbol action next_state _____ state 0 __1__ state 1 0 _ or 0 write 1 1 _10__ state 2 __1__ state 1 0 1 write 0 2 _10__ state 0 __1__ state 0 _00__ state 2 1 _ left 0 __0__ state 2 _00__ state 0 __0__ state 0 1 0 or 1 right 1 100__ state 1 _10__ state 1 2 0 left 0 100__ state 1 _10__ state 1 100__ state 1 _10__ state 1 100__ state 1 _10__ state 0 100__ state 0 _11__ state 1 101__ state 1 _11__ state 1 101__ state 1 _11__ state 0 101__ state 0 λ-calculus Beware! ● think mathematical, not C++/Pascal ● (parentheses) are for grouping ● variables cannot be mutated ○ x = 1 OK ○ x = 2 NO ○ x = x + 1 NO λ-calculus Simplification 1 of 2: ● Only anonymous functions are used ○ f(x) = x2+1 f(1) = 12+1 = 2 is written as ○ (λx.x2+1)(1) = 12+1 = 2 note that f = λx.x2+1 λ-calculus Simplification 2 of 2: ● Only unary functions are used ○ a binary function can be written as a unary function that return another unary function ○ (λ(x,y).x+y)(1,2) = 1+2 = 3 is written as [(λx.(λy.x+y))(1)](2) = [(λy.1+y)](2) = 1+2 = 3 ○ this technique is known as Currying Haskell Curry λ-calculus ● A lambda term has 3 forms: ○ x ○ λx.A ○ AB where x is a variable, A and B are lambda terms. -
What I Wish I Knew When Learning Haskell
What I Wish I Knew When Learning Haskell Stephen Diehl 2 Version This is the fifth major draft of this document since 2009. All versions of this text are freely available onmywebsite: 1. HTML Version http://dev.stephendiehl.com/hask/index.html 2. PDF Version http://dev.stephendiehl.com/hask/tutorial.pdf 3. EPUB Version http://dev.stephendiehl.com/hask/tutorial.epub 4. Kindle Version http://dev.stephendiehl.com/hask/tutorial.mobi Pull requests are always accepted for fixes and additional content. The only way this document will stayupto date and accurate through the kindness of readers like you and community patches and pull requests on Github. https://github.com/sdiehl/wiwinwlh Publish Date: March 3, 2020 Git Commit: 77482103ff953a8f189a050c4271919846a56612 Author This text is authored by Stephen Diehl. 1. Web: www.stephendiehl.com 2. Twitter: https://twitter.com/smdiehl 3. Github: https://github.com/sdiehl Special thanks to Erik Aker for copyediting assistance. Copyright © 20092020 Stephen Diehl This code included in the text is dedicated to the public domain. You can copy, modify, distribute and perform thecode, even for commercial purposes, all without asking permission. You may distribute this text in its full form freely, but may not reauthor or sublicense this work. Any reproductions of major portions of the text must include attribution. The software is provided ”as is”, without warranty of any kind, express or implied, including But not limitedtothe warranties of merchantability, fitness for a particular purpose and noninfringement. In no event shall the authorsor copyright holders be liable for any claim, damages or other liability, whether in an action of contract, tort or otherwise, Arising from, out of or in connection with the software or the use or other dealings in the software. -
Lambda Calculus and Functional Programming
Global Journal of Researches in Engineering Vol. 10 Issue 2 (Ver 1.0) June 2010 P a g e | 47 Lambda Calculus and Functional Programming Anahid Bassiri1Mohammad Reza. Malek2 GJRE Classification (FOR) 080299, 010199, 010203, Pouria Amirian3 010109 Abstract-The lambda calculus can be thought of as an idealized, Basis concept of a Turing machine is the present day Von minimalistic programming language. It is capable of expressing Neumann computers. Conceptually these are Turing any algorithm, and it is this fact that makes the model of machines with random access registers. Imperative functional programming an important one. This paper is programming languages such as FORTRAN, Pascal etcetera focused on introducing lambda calculus and its application. As as well as all the assembler languages are based on the way an application dikjestra algorithm is implemented using a Turing machine is instructed by a sequence of statements. lambda calculus. As program shows algorithm is more understandable using lambda calculus in comparison with In addition functional programming languages, like other imperative languages. Miranda, ML etcetera, are based on the lambda calculus. Functional programming is a programming paradigm that I. INTRODUCTION treats computation as the evaluation of mathematical ambda calculus (λ-calculus) is a useful device to make functions and avoids state and mutable data. It emphasizes L the theories realizable. Lambda calculus, introduced by the application of functions, in contrast with the imperative Alonzo Church and Stephen Cole Kleene in the 1930s is a programming style that emphasizes changes in state. formal system designed to investigate function definition, Lambda calculus provides a theoretical framework for function application and recursion in mathematical logic and describing functions and their evaluation. -
Purity in Erlang
Purity in Erlang Mihalis Pitidis1 and Konstantinos Sagonas1,2 1 School of Electrical and Computer Engineering, National Technical University of Athens, Greece 2 Department of Information Technology, Uppsala University, Sweden [email protected], [email protected] Abstract. Motivated by a concrete goal, namely to extend Erlang with the abil- ity to employ user-defined guards, we developed a parameterized static analysis tool called PURITY, that classifies functions as referentially transparent (i.e., side- effect free with no dependency on the execution environment and never raising an exception), side-effect free with no dependencies but possibly raising excep- tions, or side-effect free but with possible dependencies and possibly raising ex- ceptions. We have applied PURITY on a large corpus of Erlang code bases and report experimental results showing the percentage of functions that the analysis definitely classifies in each category. Moreover, we discuss how our analysis has been incorporated on a development branch of the Erlang/OTP compiler in order to allow extending the language with user-defined guards. 1 Introduction Purity plays an important role in functional programming languages as it is a corner- stone of referential transparency, namely that the same language expression produces the same value when evaluated twice. Referential transparency helps in writing easy to test, robust and comprehensible code, makes equational reasoning possible, and aids program analysis and optimisation. In pure functional languages like Clean or Haskell, any side-effect or dependency on the state is captured by the type system and is reflected in the types of functions. In a language like ERLANG, which has been developed pri- marily with concurrency in mind, pure functions are not the norm and impure functions can freely be used interchangeably with pure ones. -
Applicative Programming with Effects
Under consideration for publication in J. Functional Programming 1 FUNCTIONALPEARL Applicative programming with effects CONOR MCBRIDE University of Nottingham ROSS PATERSON City University, London Abstract In this paper, we introduce Applicative functors—an abstract characterisation of an ap- plicative style of effectful programming, weaker than Monads and hence more widespread. Indeed, it is the ubiquity of this programming pattern that drew us to the abstraction. We retrace our steps in this paper, introducing the applicative pattern by diverse exam- ples, then abstracting it to define the Applicative type class and introducing a bracket notation which interprets the normal application syntax in the idiom of an Applicative functor. Further, we develop the properties of applicative functors and the generic opera- tions they support. We close by identifying the categorical structure of applicative functors and examining their relationship both with Monads and with Arrows. 1 Introduction This is the story of a pattern that popped up time and again in our daily work, programming in Haskell (Peyton Jones, 2003), until the temptation to abstract it became irresistable. Let us illustrate with some examples. Sequencing commands One often wants to execute a sequence of commands and collect the sequence of their responses, and indeed there is such a function in the Haskell Prelude (here specialised to IO): sequence :: [IO a ] → IO [a ] sequence [ ] = return [] sequence (c : cs) = do x ← c xs ← sequence cs return (x : xs) In the (c : cs) case, we collect the values of some effectful computations, which we then use as the arguments to a pure function (:). We could avoid the need for names to wire these values through to their point of usage if we had a kind of ‘effectful application’. -
Learn You a Haskell for Great Good! © 2011 by Miran Lipovača 3 SYNTAX in FUNCTIONS 35 Pattern Matching
C ONTENTSINDETAIL INTRODUCTION xv So, What’s Haskell? .............................................................. xv What You Need to Dive In ........................................................ xvii Acknowledgments ................................................................ xviii 1 STARTING OUT 1 Calling Functions ................................................................. 3 Baby’s First Functions ............................................................. 5 An Intro to Lists ................................................................... 7 Concatenation ........................................................... 8 Accessing List Elements ................................................... 9 Lists Inside Lists .......................................................... 9 Comparing Lists ......................................................... 9 More List Operations ..................................................... 10 Texas Ranges .................................................................... 13 I’m a List Comprehension.......................................................... 15 Tuples ........................................................................... 18 Using Tuples............................................................. 19 Using Pairs .............................................................. 20 Finding the Right Triangle ................................................. 21 2 BELIEVE THE TYPE 23 Explicit Type Declaration .......................................................... 24 -
There Is No Fork: an Abstraction for Efficient, Concurrent, and Concise Data Access
There is no Fork: an Abstraction for Efficient, Concurrent, and Concise Data Access Simon Marlow Louis Brandy Jonathan Coens Jon Purdy Facebook Facebook Facebook Facebook [email protected] [email protected] [email protected] [email protected] Abstract concise business logic, uncluttered by performance-related details. We describe a new programming idiom for concurrency, based on In particular the programmer should not need to be concerned Applicative Functors, where concurrency is implicit in the Applica- with accessing external data efficiently. However, one particular tive <*> operator. The result is that concurrent programs can be problem often arises that creates a tension between conciseness and written in a natural applicative style, and they retain a high degree efficiency in this setting: accessing multiple remote data sources of clarity and modularity while executing with maximal concur- efficiently requires concurrency, and that normally requires the rency. This idiom is particularly useful for programming against programmer to intervene and program the concurrency explicitly. external data sources, where the application code is written without When the business logic is only concerned with reading data the use of explicit concurrency constructs, while the implementa- from external sources and not writing, the programmer doesn’t tion is able to batch together multiple requests for data from the care about the order in which data accesses happen, since there same source, and fetch data from multiple sources concurrently. are no side-effects that could make the result different when the Our abstraction uses a cache to ensure that multiple requests for order changes. So in this case the programmer would be entirely the same data return the same result, which frees the programmer happy with not having to specify either ordering or concurrency, from having to arrange to fetch data only once, which in turn leads and letting the system perform data access in the most efficient way to greater modularity. -
An Introduction to Applicative Functors
An Introduction to Applicative Functors Bocheng Zhou What Is an Applicative Functor? ● An Applicative functor is a Monoid in the category of endofunctors, what's the problem? ● WAT?! Functions in Haskell ● Functions in Haskell are first-order citizens ● Functions in Haskell are curried by default ○ f :: a -> b -> c is the curried form of g :: (a, b) -> c ○ f = curry g, g = uncurry f ● One type declaration, multiple interpretations ○ f :: a->b->c ○ f :: a->(b->c) ○ f :: (a->b)->c ○ Use parentheses when necessary: ■ >>= :: Monad m => m a -> (a -> m b) -> m b Functors ● A functor is a type of mapping between categories, which is applied in category theory. ● What the heck is category theory? Category Theory 101 ● A category is, in essence, a simple collection. It has three components: ○ A collection of objects ○ A collection of morphisms ○ A notion of composition of these morphisms ● Objects: X, Y, Z ● Morphisms: f :: X->Y, g :: Y->Z ● Composition: g . f :: X->Z Category Theory 101 ● Category laws: Functors Revisited ● Recall that a functor is a type of mapping between categories. ● Given categories C and D, a functor F :: C -> D ○ Maps any object A in C to F(A) in D ○ Maps morphisms f :: A -> B in C to F(f) :: F(A) -> F(B) in D Functors in Haskell class Functor f where fmap :: (a -> b) -> f a -> f b ● Recall that a functor maps morphisms f :: A -> B in C to F(f) :: F(A) -> F(B) in D ● morphisms ~ functions ● C ~ category of primitive data types like Integer, Char, etc. -
It's All About Morphisms
It’s All About Morphisms Uberto Barbini @ramtop https://medium.com/@ramtop About me OOP programmer Agile TDD Functional PRogramming Finance Kotlin Industry Blog: https://medium.com/@ ramtop Twitter: @ramtop #morphisms#VoxxedVienna @ramtop Map of this presentation Monoid Category Monad Functor Natural Transformation Yoneda Applicative Morphisms all the way down... #morphisms#VoxxedVienna @ramtop I don’t care about Monads, why should I? Neither do I what I care about is y el is to define system behaviour ec Pr #morphisms#VoxxedVienna @ramtop This presentation will be a success if most of you will not fall asleep #morphisms#VoxxedVienna @ramtop This presentation will be a success if You will consider that Functional Programming is about transformations and preserving properties. Not (only) lambdas and flatmap #morphisms#VoxxedVienna @ramtop What is this Category thingy? Invented in 1940s “with the goal of understanding the processes that preserve mathematical structure.” “Category Theory is about relation between things” “General abstract nonsense” #morphisms#VoxxedVienna @ramtop Once upon a time there was a Category of Stuffed Toys and a Category of Tigers... #morphisms#VoxxedVienna @ramtop A Category is defined in 5 steps: 1) A collection of Objects #morphisms#VoxxedVienna @ramtop A Category is defined in 5 steps: 2) A collection of Arrows #morphisms#VoxxedVienna @ramtop A Category is defined in 5 steps: 3) Each Arrow works on 2 Objects #morphisms#VoxxedVienna @ramtop A Category is defined in 5 steps: 4) Arrows can be combined #morphisms#VoxxedVienna -
Deep Dive Into Programming with CAPS HMPP Dev-Deck
Introduction to HMPP Hybrid Manycore Parallel Programming Московский государственный университет, Лоран Морен, 30 август 2011 Hybrid Software for Hybrid Hardware • The application should stay hardware resilient o New architectures / languages to master o Hybrid solutions evolve: we don‟t want to redo the work each time • Kernel optimizations are the key to get performance o An hybrid application must get the best of its hardware resources 30 август 2011 Москва - Лоран Морен 2 Methodology to Port Applications Methodology to Port Applications Hotspots Parallelization •Optimize CPU code •Performance goal •Exhibit Parallelism •Validation process •Move kernels to GPU •Hotspots selection •Continuous integration Define your Port your Hours to Days parallel application Days to Weeks project on GPU GPGPU operational Phase 1 application with known potential Phase 2 Tuning Optimize your GPGPU •Exploit CPU and GPU application •Optimize kernel execution •Reduce CPU-GPU data transfers Weeks to Months 30 август 2011 Москва - Лоран Морен 4 Take a decision • Go • Dense hotspot • Fast GPU kernels • Low CPU-GPU data transfers • Midterm perspective: Prepare to manycore parallelism • No Go o Flat profile o GPU results not accurate enough • cannot validate the computation o Slow GPU kernels • i.e. no speedup to be expected o Complex data structures o Big memory space requirement 30 август 2011 Москва - Лоран Морен 5 Tools in the Methodology Hotspots Parallelization • Profiling tools Gprof, Kachegrind, Vampir, Acumem, ThreadSpotter •HMPP Workbench •Debugging -
The Quantum IO Monad Thorsten Altenkirch and Alexander S
1 The Quantum IO Monad Thorsten Altenkirch and Alexander S. Green School of Computer Science, The University of Nottingham Abstract The Quantum IO monad is a purely functional interface to quantum programming implemented as a Haskell library. At the same time it provides a constructive semantics of quantum programming. The QIO monad separates reversible (i.e. unitary) and irreversible (i.e. prob- abilistic) computations and provides a reversible let operation (ulet), allowing us to use ancillas (auxiliary qubits) in a modular fashion. QIO programs can be simulated either by calculating a probability distribu- tion or by embedding it into the IO monad using the random number generator. As an example we present a complete implementation of Shor’s algorithm. 1.1 Introduction We present an interface from a pure functional programming language, Haskell, to quantum programming: the Quantum IO monad, and use it to implement Shor’s factorisation algorithm. The implementation of the QIO monad provides a constructive semantics for quantum programming, i.e. a functional program which can also be understood as a mathematical model of quantum computing. Actually, the Haskell QIO library is only a first approximation of such a semantics, we would like to move to a more expressive language which is also logically sound. Here we are thinking of a language like Agda (Norell (2007)), which is based on Martin L¨of’s Type Theory. We have already investigated this approach of functional specifications of effects in a classical context (Swierstra and Altenkirch (2007, 2008); Swierstra (2008)). At the same 1 2 Thorsten Altenkirch and Alexander S. -
Why Functional Programming Is Good … …When You Like Math - Examples with Haskell
Why functional programming is good … …when you like math - examples with Haskell . Jochen Schulz Georg-August Universität Göttingen 1/18 Table of contents 1. Introduction 2. Functional programming with Haskell 3. Summary 1/18 Programming paradigm imperative (e.g. C) object-oriented (e.g. C++) functional (e.g. Haskell) (logic) (symbolc) some languages have multiple paradigm 2/18 Side effects/pure functions . side effect . Besides the return value of a function it has one or more of the following modifies state. has observable interaction with outside world. pure function . A pure function always returns the same results on the same input. has no side-effect. .also refered to as referential transparency pure functions resemble mathematical functions. 3/18 Functional programming emphasizes pure functions higher order functions (partial function evaluation, currying) avoids state and mutable data (Haskell uses Monads) recursion is mostly used for loops algebraic type system strict/lazy evaluation (often lazy, as Haskell) describes more what is instead what you have to do 4/18 Table of contents 1. Introduction 2. Functional programming with Haskell 3. Summary 5/18 List comprehensions Some Math: S = fx2jx 2 N; x2 < 20g > [ x^2 | x <- [1..10] , x^2 < 20 ] [1,4,9,16] Ranges (and infinite ranges (don’t do this now) ) > a = [1..5], [1,3..8], ['a'..'z'], [1..] [1,2,3,4,5], [1,3,5,7], "abcdefghijklmnopqrstuvwxyz" usually no direct indexing (needed) > (head a, tail a, take 2 a, a !! 2) (1,[2,3,4,5],[1,2],3) 6/18 Functions: Types and Typeclasses Types removeNonUppercase :: [Char] -> [Char] removeNonUppercase st = [ c | c <- st, c `elem ` ['A'..'Z']] Typeclasses factorial :: (Integral a) => a -> a factorial n = product [1..n] We also can define types and typeclasses and such form spaces.