Monads for Functional Programming
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Ambiguity and Incomplete Information in Categorical Models of Language
Ambiguity and Incomplete Information in Categorical Models of Language Dan Marsden University of Oxford [email protected] We investigate notions of ambiguity and partial information in categorical distributional models of natural language. Probabilistic ambiguity has previously been studied in [27, 26, 16] using Selinger’s CPM construction. This construction works well for models built upon vector spaces, as has been shown in quantum computational applications. Unfortunately, it doesn’t seem to provide a satis- factory method for introducing mixing in other compact closed categories such as the category of sets and binary relations. We therefore lack a uniform strategy for extending a category to model imprecise linguistic information. In this work we adopt a different approach. We analyze different forms of ambiguous and in- complete information, both with and without quantitative probabilistic data. Each scheme then cor- responds to a suitable enrichment of the category in which we model language. We view different monads as encapsulating the informational behaviour of interest, by analogy with their use in mod- elling side effects in computation. Previous results of Jacobs then allow us to systematically construct suitable bases for enrichment. We show that we can freely enrich arbitrary dagger compact closed categories in order to capture all the phenomena of interest, whilst retaining the important dagger compact closed structure. This allows us to construct a model with real convex combination of binary relations that makes non-trivial use of the scalars. Finally we relate our various different enrichments, showing that finite subconvex algebra enrichment covers all the effects under consideration. -
Comprehending Adts*
Comprehending ADTs* Martin Erwig School of EECS Oregon State University [email protected] Abstract We show how to generalize list comprehensions to work on abstract data types. First, we make comprehension notation automatically available for any data type that is specified as a constructor/destructor-pair (bialgebra). Second, we extend comprehensions to enable the use of dif- ferent types in one comprehension and to allow to map between different types. Third, we refine the translation of comprehensions to give a reasonable behavior even for types that do not obey the monad laws, which significantly extends the scope of comprehensions. Keywords: Functional Programming, Comprehension Syntax, Abstract Data Type, Generalized Fold, Monad 1 Introduction Set comprehensions are a well-known and appreciated notation from mathematics: an expression such as fE(x) j x 2 S; P(x)g denotes the set of values given by repeatedly evaluating the expression E for all x taken from the set S for which P(x) yields true. This notation has found its way into functional languages as list comprehensions [3, 17, 18]. Assuming that s denotes a list of numbers, the following Haskell expression computes squares of the odd numbers in s. [ x ∗ x j x s; odd x ] The main difference to the mathematical set comprehensions is the use of lists in the generator and as a result. In general, a comprehension consists of an expression defining the result values and of a list of qualifiers, which are either generators for supplying variable bindings or filters restricting these. Phil Wadler has shown that the comprehension syntax can be used not only for lists and sets, but more generally for arbitrary monads [19]. -
Structure Interpretation of Text Formats
1 1 Structure Interpretation of Text Formats 2 3 SUMIT GULWANI, Microsoft, USA 4 VU LE, Microsoft, USA 5 6 ARJUN RADHAKRISHNA, Microsoft, USA 7 IVAN RADIČEK, Microsoft, Austria 8 MOHAMMAD RAZA, Microsoft, USA 9 Data repositories often consist of text files in a wide variety of standard formats, ad-hoc formats, aswell 10 mixtures of formats where data in one format is embedded into a different format. It is therefore a significant 11 challenge to parse these files into a structured tabular form, which is important to enable any downstream 12 data processing. 13 We present Unravel, an extensible framework for structure interpretation of ad-hoc formats. Unravel 14 can automatically, with no user input, extract tabular data from a diverse range of standard, ad-hoc and 15 mixed format files. The framework is also easily extensible to add support for previously unseen formats, 16 and also supports interactivity from the user in terms of examples to guide the system when specialized data extraction is desired. Our key insight is to allow arbitrary combination of extraction and parsing techniques 17 through a concept called partial structures. Partial structures act as a common language through which the file 18 structure can be shared and refined by different techniques. This makes Unravel more powerful than applying 19 the individual techniques in parallel or sequentially. Further, with this rule-based extensible approach, we 20 introduce the novel notion of re-interpretation where the variety of techniques supported by our system can 21 be exploited to improve accuracy while optimizing for particular quality measures or restricted environments. -
The Marriage of Effects and Monads
The Marriage of Effects and Monads PHILIP WADLER Avaya Labs and PETER THIEMANN Universit¨at Freiburg Gifford and others proposed an effect typing discipline to delimit the scope of computational ef- fects within a program, while Moggi and others proposed monads for much the same purpose. Here we marry effects to monads, uniting two previously separate lines of research. In partic- ular, we show that the type, region, and effect system of Talpin and Jouvelot carries over di- rectly to an analogous system for monads, including a type and effect reconstruction algorithm. The same technique should allow one to transpose any effect system into a corresponding monad system. Categories and Subject Descriptors: D.3.1 [Programming Languages]: Formal Definitions and Theory; F.3.2 [Logics and Meanings of Programs]: Semantics of Programming Languages— Operational semantics General Terms: Languages, Theory Additional Key Words and Phrases: Monad, effect, type, region, type reconstruction 1. INTRODUCTION Computational effects, such as state or continuations, are powerful medicine. If taken as directed they may cure a nasty bug, but one must be wary of the side effects. For this reason, many researchers in computing seek to exploit the benefits of computational effects while delimiting their scope. Two such lines of research are the effect typing discipline, proposed by Gifford and Lucassen [Gifford and Lucassen 1986; Lucassen 1987], and pursued by Talpin and Jouvelot [Talpin and Jouvelot 1992, 1994; Talpin 1993] among others, and the use of monads, proposed by Moggi [1989, 1990], and pursued by Wadler [1990, 1992, 1993, 1995] among others. Effect systems are typically found in strict languages, such as FX [Gifford et al. -
CSCI 2041: Lazy Evaluation
CSCI 2041: Lazy Evaluation Chris Kauffman Last Updated: Wed Dec 5 12:32:32 CST 2018 1 Logistics Reading Lab13: Lazy/Streams I Module Lazy on lazy Covers basics of delayed evaluation computation I Module Stream on streams A5: Calculon Lambdas/Closures I Arithmetic language Briefly discuss these as they interpreter pertain Calculon I 2X credit for assignment I 5 Required Problems 100pts Goals I 5 Option Problems 50pts I Eager Evaluation I Milestone due Wed 12/5 I Lazy Evaluation I Final submit Tue 12/11 I Streams 2 Evaluation Strategies Eager Evaluation Lazy Evaluation I Most languages employ I An alternative is lazy eager evaluation evaluation I Execute instructions as I Execute instructions only as control reaches associated expression results are needed code (call by need) I Corresponds closely to I Higher-level idea with actual machine execution advantages and disadvantages I In pure computations, evaluation strategy doesn’t matter: will produce the same results I With side-effects, when code is run matter, particular for I/O which may see different printing orders 3 Exercise: Side-Effects and Evaluation Strategy Most common place to see differences between Eager/Lazy eval is when functions are called I Eager eval: eval argument expressions, call functions with results I Lazy eval: call function with un-evaluated expressions, eval as results are needed Consider the following expression let print_it expr = printf "Printing it\n"; printf "%d\n" expr; ;; print_it (begin printf "Evaluating\n"; 5; end);; Predict results and output for both Eager and Lazy Eval strategies 4 Answers: Side-Effects and Evaluation Strategy let print_it expr = printf "Printing it\n"; printf "%d\n" expr; ;; print_it (begin printf "Evaluating\n"; 5; end);; Evaluation > ocamlc eager_v_lazy.ml > ./a.out Eager Eval # ocaml’s default Evaluating Printing it 5 Lazy Eval Printing it Evaluating 5 5 OCaml and explicit lazy Computations I OCaml’s default model is eager evaluation BUT. -
Lecture Notes on the Lambda Calculus 2.4 Introduction to Β-Reduction
2.2 Free and bound variables, α-equivalence. 8 2.3 Substitution ............................. 10 Lecture Notes on the Lambda Calculus 2.4 Introduction to β-reduction..................... 12 2.5 Formal definitions of β-reduction and β-equivalence . 13 Peter Selinger 3 Programming in the untyped lambda calculus 14 3.1 Booleans .............................. 14 Department of Mathematics and Statistics 3.2 Naturalnumbers........................... 15 Dalhousie University, Halifax, Canada 3.3 Fixedpointsandrecursivefunctions . 17 3.4 Other data types: pairs, tuples, lists, trees, etc. ....... 20 Abstract This is a set of lecture notes that developed out of courses on the lambda 4 The Church-Rosser Theorem 22 calculus that I taught at the University of Ottawa in 2001 and at Dalhousie 4.1 Extensionality, η-equivalence, and η-reduction. 22 University in 2007 and 2013. Topics covered in these notes include the un- typed lambda calculus, the Church-Rosser theorem, combinatory algebras, 4.2 Statement of the Church-Rosser Theorem, and some consequences 23 the simply-typed lambda calculus, the Curry-Howard isomorphism, weak 4.3 Preliminary remarks on the proof of the Church-Rosser Theorem . 25 and strong normalization, polymorphism, type inference, denotational se- mantics, complete partial orders, and the language PCF. 4.4 ProofoftheChurch-RosserTheorem. 27 4.5 Exercises .............................. 32 Contents 5 Combinatory algebras 34 5.1 Applicativestructures. 34 1 Introduction 1 5.2 Combinatorycompleteness . 35 1.1 Extensionalvs. intensionalviewoffunctions . ... 1 5.3 Combinatoryalgebras. 37 1.2 Thelambdacalculus ........................ 2 5.4 The failure of soundnessforcombinatoryalgebras . .... 38 1.3 Untypedvs.typedlambda-calculi . 3 5.5 Lambdaalgebras .......................... 40 1.4 Lambdacalculusandcomputability . 4 5.6 Extensionalcombinatoryalgebras . 44 1.5 Connectionstocomputerscience . -
Lecture 10. Functors and Monads Functional Programming
Lecture 10. Functors and monads Functional Programming [Faculty of Science Information and Computing Sciences] 0 Goals I Understand the concept of higher-kinded abstraction I Introduce two common patterns: functors and monads I Simplify code with monads Chapter 12 from Hutton’s book, except 12.2 [Faculty of Science Information and Computing Sciences] 1 Functors [Faculty of Science Information and Computing Sciences] 2 Map over lists map f xs applies f over all the elements of the list xs map :: (a -> b) -> [a] -> [b] map _ [] = [] map f (x:xs) = f x : map f xs > map (+1)[1,2,3] [2,3,4] > map even [1,2,3] [False,True,False] [Faculty of Science Information and Computing Sciences] 3 mapTree _ Leaf = Leaf mapTree f (Node l x r) = Node (mapTree f l) (f x) (mapTree f r) Map over binary trees Remember binary trees with data in the inner nodes: data Tree a = Leaf | Node (Tree a) a (Tree a) deriving Show They admit a similar map operation: mapTree :: (a -> b) -> Tree a -> Tree b [Faculty of Science Information and Computing Sciences] 4 Map over binary trees Remember binary trees with data in the inner nodes: data Tree a = Leaf | Node (Tree a) a (Tree a) deriving Show They admit a similar map operation: mapTree :: (a -> b) -> Tree a -> Tree b mapTree _ Leaf = Leaf mapTree f (Node l x r) = Node (mapTree f l) (f x) (mapTree f r) [Faculty of Science Information and Computing Sciences] 4 Map over binary trees mapTree also applies a function over all elements, but now contained in a binary tree > t = Node (Node Leaf 1 Leaf) 2 Leaf > mapTree (+1) t Node -
Coffeescript Accelerated Javascript Development.Pdf
Download from Wow! eBook <www.wowebook.com> What readers are saying about CoffeeScript: Accelerated JavaScript Development It’s hard to imagine a new web application today that doesn’t make heavy use of JavaScript, but if you’re used to something like Ruby, it feels like a significant step down to deal with JavaScript, more of a chore than a joy. Enter CoffeeScript: a pre-compiler that removes all the unnecessary verbosity of JavaScript and simply makes it a pleasure to write and read. Go, go, Coffee! This book is a great introduction to the world of CoffeeScript. ➤ David Heinemeier Hansson Creator, Rails Just like CoffeeScript itself, Trevor gets straight to the point and shows you the benefits of CoffeeScript and how to write concise, clear CoffeeScript code. ➤ Scott Leberknight Chief Architect, Near Infinity Though CoffeeScript is a new language, you can already find it almost everywhere. This book will show you just how powerful and fun CoffeeScript can be. ➤ Stan Angeloff Managing Director, PSP WebTech Bulgaria Download from Wow! eBook <www.wowebook.com> This book helps readers become better JavaScripters in the process of learning CoffeeScript. What’s more, it’s a blast to read, especially if you are new to Coffee- Script and ready to learn. ➤ Brendan Eich Creator, JavaScript CoffeeScript may turn out to be one of the great innovations in web application development; since I first discovered it, I’ve never had to write a line of pure JavaScript. I hope the readers of this wonderful book will be able to say the same. ➤ Dr. Nic Williams CEO/Founder, Mocra CoffeeScript: Accelerated JavaScript Development is an excellent guide to Coffee- Script from one of the community’s most esteemed members. -
1. Language of Operads 2. Operads As Monads
OPERADS, APPROXIMATION, AND RECOGNITION MAXIMILIEN PEROUX´ All missing details can be found in [May72]. 1. Language of operads Let S = (S; ⊗; I) be a (closed) symmetric monoidal category. Definition 1.1. An operad O in S is a collection of object fO(j)gj≥0 in S endowed with : • a right-action of the symmetric group Σj on O(j) for each j, such that O(0) = I; • a unit map I ! O(1) in S; • composition operations that are morphisms in S : γ : O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) −! O(j1 + ··· + jk); defined for each k ≥ 0, j1; : : : ; jk ≥ 0, satisfying natural equivariance, unit and associativity relations. 0 0 A morphism of operads : O ! O is a sequence j : O(j) ! O (j) of Σj-equivariant morphisms in S compatible with the unit map and γ. Example 1.2. ⊗j Let X be an object in S. The endomorphism operad EndX is defined to be EndX (j) = HomS(X ;X), ⊗j with unit idX , and the Σj-right action is induced by permuting on X . Example 1.3. Define Assoc(j) = ` , the associative operad, where the maps γ are defined by equivariance. Let σ2Σj I Com(j) = I, the commutative operad, where γ are the canonical isomorphisms. Definition 1.4. Let O be an operad in S. An O-algebra (X; θ) in S is an object X together with a morphism of operads ⊗j θ : O ! EndX . Using adjoints, this is equivalent to a sequence of maps θj : O(j) ⊗ X ! X such that they are associative, unital and equivariant. -
Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl
Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl To cite this version: Wolfram Kahl. Categories of Coalgebras with Monadic Homomorphisms. 12th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2014, Grenoble, France. pp.151-167, 10.1007/978-3-662-44124-4_9. hal-01408758 HAL Id: hal-01408758 https://hal.inria.fr/hal-01408758 Submitted on 5 Dec 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl McMaster University, Hamilton, Ontario, Canada, [email protected] Abstract. Abstract graph transformation approaches traditionally con- sider graph structures as algebras over signatures where all function sym- bols are unary. Attributed graphs, with attributes taken from (term) algebras over ar- bitrary signatures do not fit directly into this kind of transformation ap- proach, since algebras containing function symbols taking two or more arguments do not allow component-wise construction of pushouts. We show how shifting from the algebraic view to a coalgebraic view of graph structures opens up additional flexibility, and enables treat- ing term algebras over arbitrary signatures in essentially the same way as unstructured label sets. -
A Critique of Abelson and Sussman Why Calculating Is Better Than
A critique of Abelson and Sussman - or - Why calculating is better than scheming Philip Wadler Programming Research Group 11 Keble Road Oxford, OX1 3QD Abelson and Sussman is taught [Abelson and Sussman 1985a, b]. Instead of emphasizing a particular programming language, they emphasize standard engineering techniques as they apply to programming. Still, their textbook is intimately tied to the Scheme dialect of Lisp. I believe that the same approach used in their text, if applied to a language such as KRC or Miranda, would result in an even better introduction to programming as an engineering discipline. My belief has strengthened as my experience in teaching with Scheme and with KRC has increased. - This paper contrasts teaching in Scheme to teaching in KRC and Miranda, particularly with reference to Abelson and Sussman's text. Scheme is a "~dly-scoped dialect of Lisp [Steele and Sussman 19781; languages in a similar style are T [Rees and Adams 19821 and Common Lisp [Steele 19821. KRC is a functional language in an equational style [Turner 19811; its successor is Miranda [Turner 1985k languages in a similar style are SASL [Turner 1976, Richards 19841 LML [Augustsson 19841, and Orwell [Wadler 1984bl. (Only readers who know that KRC stands for "Kent Recursive Calculator" will have understood the title of this There are four language features absent in Scheme and present in KRC/Miranda that are important: 1. Pattern-matching. 2. A syntax close to traditional mathematical notation. 3. A static type discipline and user-defined types. 4. Lazy evaluation. KRC and SASL do not have a type discipline, so point 3 applies only to Miranda, Philip Wadhr Why Calculating is Better than Scheming 2 b LML,and Orwell. -
Richard S. Uhler
Smten and the Art of Satisfiability-Based Search by Richard S. Uhler B.S., University of California, Los Angeles (2008) S.M., Massachusetts Institute of Technology (2010) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2014 c Massachusetts Institute of Technology 2014. All rights reserved. Author.............................................................. Department of Electrical Engineering and Computer Science August 22, 2014 Certified by. Jack B. Dennis Professor of Computer Science and Engineering Emeritus Thesis Supervisor Accepted by . Leslie A. Kolodziejski Chairman, Committee for Graduate Students 2 Smten and the Art of Satisfiability-Based Search by Richard S. Uhler Submitted to the Department of Electrical Engineering and Computer Science on August 22, 2014, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science Abstract Satisfiability (SAT) and Satisfiability Modulo Theories (SMT) have been leveraged in solving a wide variety of important and challenging combinatorial search prob- lems, including automatic test generation, logic synthesis, model checking, program synthesis, and software verification. Though in principle SAT and SMT solvers sim- plify the task of developing practical solutions to these hard combinatorial search problems, in practice developing an application that effectively uses a SAT or SMT solver can be a daunting task. SAT and SMT solvers have limited support, if any, for queries described using the wide array of features developers rely on for describing their programs: procedure calls, loops, recursion, user-defined data types, recursively defined data types, and other mechanisms for abstraction, encapsulation, and modu- larity.