Intro to Haskell Notes: Part 11
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Structured Recursion for Non-Uniform Data-Types
Structured recursion for non-uniform data-types by Paul Alexander Blampied, B.Sc. Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy, March 2000 Contents Acknowledgements .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1 Chapter 1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1. Non-uniform data-types .. .. .. .. .. .. .. .. .. .. .. .. 3 1.2. Program calculation .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.3. Maps and folds in Squiggol .. .. .. .. .. .. .. .. .. .. .. 11 1.4. Related work .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 14 1.5. Purpose and contributions of this thesis .. .. .. .. .. .. .. 15 Chapter 2. Categorical data-types .. .. .. .. .. .. .. .. .. .. .. .. 18 2.1. Notation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 19 2.2. Data-types as initial algebras .. .. .. .. .. .. .. .. .. .. 21 2.3. Parameterized data-types .. .. .. .. .. .. .. .. .. .. .. 29 2.4. Calculation properties of catamorphisms .. .. .. .. .. .. .. 33 2.5. Existence of initial algebras .. .. .. .. .. .. .. .. .. .. .. 37 2.6. Algebraic data-types in functional programming .. .. .. .. 57 2.7. Chapter summary .. .. .. .. .. .. .. .. .. .. .. .. .. 61 Chapter 3. Folding in functor categories .. .. .. .. .. .. .. .. .. .. 62 3.1. Natural transformations and functor categories .. .. .. .. .. 63 3.2. Initial algebras in functor categories .. .. .. .. .. .. .. .. 68 3.3. Examples and non-examples .. .. .. .. .. .. .. .. .. .. 77 3.4. Existence of initial algebras in functor categories .. .. .. .. 82 3.5. -
Binary Search Trees
Introduction Recursive data types Binary Trees Binary Search Trees Organizing information Sum-of-Product data types Theory of Programming Languages Computer Science Department Wellesley College Introduction Recursive data types Binary Trees Binary Search Trees Table of contents Introduction Recursive data types Binary Trees Binary Search Trees Introduction Recursive data types Binary Trees Binary Search Trees Sum-of-product data types Every general-purpose programming language must allow the • processing of values with different structure that are nevertheless considered to have the same “type”. For example, in the processing of simple geometric figures, we • want a notion of a “figure type” that includes circles with a radius, rectangles with a width and height, and triangles with three sides: The name in the oval is a tag that indicates which kind of • figure the value is, and the branches leading down from the oval indicate the components of the value. Such types are known as sum-of-product data types because they consist of a sum of tagged types, each of which holds on to a product of components. Introduction Recursive data types Binary Trees Binary Search Trees Declaring the new figure type in Ocaml In Ocaml we can declare a new figure type that represents these sorts of geometric figures as follows: type figure = Circ of float (* radius *) | Rect of float * float (* width, height *) | Tri of float * float * float (* side1, side2, side3 *) Such a declaration is known as a data type declaration. It consists of a series of |-separated clauses of the form constructor-name of component-types, where constructor-name must be capitalized. -
Full Functional Verification of Linked Data Structures
Full Functional Verification of Linked Data Structures Karen Zee Viktor Kuncak Martin C. Rinard MIT CSAIL, Cambridge, MA, USA EPFL, I&C, Lausanne, Switzerland MIT CSAIL, Cambridge, MA, USA ∗ [email protected] viktor.kuncak@epfl.ch [email protected] Abstract 1. Introduction We present the first verification of full functional correctness for Linked data structures such as lists, trees, graphs, and hash tables a range of linked data structure implementations, including muta- are pervasive in modern software systems. But because of phenom- ble lists, trees, graphs, and hash tables. Specifically, we present the ena such as aliasing and indirection, it has been a challenge to de- use of the Jahob verification system to verify formal specifications, velop automated reasoning systems that are capable of proving im- written in classical higher-order logic, that completely capture the portant correctness properties of such data structures. desired behavior of the Java data structure implementations (with the exception of properties involving execution time and/or mem- 1.1 Background ory consumption). Given that the desired correctness properties in- In principle, standard specification and verification approaches clude intractable constructs such as quantifiers, transitive closure, should work for linked data structure implementations. But in prac- and lambda abstraction, it is a challenge to successfully prove the tice, many of the desired correctness properties involve logical generated verification conditions. constructs such transitive closure and -
Recursive Type Generativity
Recursive Type Generativity Derek Dreyer Toyota Technological Institute at Chicago [email protected] Abstract 1. Introduction Existential types provide a simple and elegant foundation for un- Recursive modules are one of the most frequently requested exten- derstanding generative abstract data types, of the kind supported by sions to the ML languages. After all, the ability to have cyclic de- the Standard ML module system. However, in attempting to extend pendencies between different files is a feature that is commonplace ML with support for recursive modules, we have found that the tra- in mainstream languages like C and Java. To the novice program- ditional existential account of type generativity does not work well mer especially, it seems very strange that the ML module system in the presence of mutually recursive module definitions. The key should provide such powerful mechanisms for data abstraction and problem is that, in recursive modules, one may wish to define an code reuse as functors and translucent signatures, and yet not allow abstract type in a context where a name for the type already exists, mutually recursive functions and data types to be broken into sepa- but the existential type mechanism does not allow one to do so. rate modules. Certainly, for simple examples of recursive modules, We propose a novel account of recursive type generativity that it is difficult to convincingly argue why ML could not be extended resolves this problem. The basic idea is to separate the act of gener- in some ad hoc way to allow them. However, when one considers ating a name for an abstract type from the act of defining its under- the semantics of a general recursive module mechanism, one runs lying representation. -
Programming the Capabilities of the PC Have Changed Greatly Since the Introduction of Electronic Computers
1 www.onlineeducation.bharatsevaksamaj.net www.bssskillmission.in INTRODUCTION TO PROGRAMMING LANGUAGE Topic Objective: At the end of this topic the student will be able to understand: History of Computer Programming C++ Definition/Overview: Overview: A personal computer (PC) is any general-purpose computer whose original sales price, size, and capabilities make it useful for individuals, and which is intended to be operated directly by an end user, with no intervening computer operator. Today a PC may be a desktop computer, a laptop computer or a tablet computer. The most common operating systems are Microsoft Windows, Mac OS X and Linux, while the most common microprocessors are x86-compatible CPUs, ARM architecture CPUs and PowerPC CPUs. Software applications for personal computers include word processing, spreadsheets, databases, games, and myriad of personal productivity and special-purpose software. Modern personal computers often have high-speed or dial-up connections to the Internet, allowing access to the World Wide Web and a wide range of other resources. Key Points: 1. History of ComputeWWW.BSSVE.INr Programming The capabilities of the PC have changed greatly since the introduction of electronic computers. By the early 1970s, people in academic or research institutions had the opportunity for single-person use of a computer system in interactive mode for extended durations, although these systems would still have been too expensive to be owned by a single person. The introduction of the microprocessor, a single chip with all the circuitry that formerly occupied large cabinets, led to the proliferation of personal computers after about 1975. Early personal computers - generally called microcomputers - were sold often in Electronic kit form and in limited volumes, and were of interest mostly to hobbyists and technicians. -
Certification of a Tool Chain for Deductive Program Verification Paolo Herms
Certification of a Tool Chain for Deductive Program Verification Paolo Herms To cite this version: Paolo Herms. Certification of a Tool Chain for Deductive Program Verification. Other [cs.OH]. Université Paris Sud - Paris XI, 2013. English. NNT : 2013PA112006. tel-00789543 HAL Id: tel-00789543 https://tel.archives-ouvertes.fr/tel-00789543 Submitted on 18 Feb 2013 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. UNIVERSITÉ DE PARIS-SUD École doctorale d’Informatique THÈSE présentée pour obtenir le Grade de Docteur en Sciences de l’Université Paris-Sud Discipline : Informatique PAR Paolo HERMS −! − SUJET : Certification of a Tool Chain for Deductive Program Verification soutenue le 14 janvier 2013 devant la commission d’examen MM. Roberto Di Cosmo Président du Jury Xavier Leroy Rapporteur Gilles Barthe Rapporteur Emmanuel Ledinot Examinateur Burkhart Wolff Examinateur Claude Marché Directeur de Thèse Benjamin Monate Co-directeur de Thèse Jean-François Monin Invité Résumé Cette thèse s’inscrit dans le domaine de la vérification du logiciel. Le but de la vérification du logiciel est d’assurer qu’une implémentation, un programme, répond aux exigences, satis- fait sa spécification. Cela est particulièrement important pour le logiciel critique, tel que des systèmes de contrôle d’avions, trains ou centrales électriques, où un mauvais fonctionnement pendant l’opération aurait des conséquences catastrophiques. -
Presentation on Ocaml Internals
OCaml Internals Implementation of an ML descendant Theophile Ranquet Ecole Pour l’Informatique et les Techniques Avancées SRS 2014 [email protected] November 14, 2013 2 of 113 Table of Contents Variants and subtyping System F Variants Type oddities worth noting Polymorphic variants Cyclic types Subtyping Weak types Implementation details α ! β Compilers Functional programming Values Why functional programming ? Allocation and garbage Combinatory logic : SKI collection The Curry-Howard Compiling correspondence Type inference OCaml and recursion 3 of 113 Variants A tagged union (also called variant, disjoint union, sum type, or algebraic data type) holds a value which may be one of several types, but only one at a time. This is very similar to the logical disjunction, in intuitionistic logic (by the Curry-Howard correspondance). 4 of 113 Variants are very convenient to represent data structures, and implement algorithms on these : 1 d a t a t y p e tree= Leaf 2 | Node of(int ∗ t r e e ∗ t r e e) 3 4 Node(5, Node(1,Leaf,Leaf), Node(3, Leaf, Node(4, Leaf, Leaf))) 5 1 3 4 1 fun countNodes(Leaf)=0 2 | countNodes(Node(int,left,right)) = 3 1 + countNodes(left)+ countNodes(right) 5 of 113 1 t y p e basic_color= 2 | Black| Red| Green| Yellow 3 | Blue| Magenta| Cyan| White 4 t y p e weight= Regular| Bold 5 t y p e color= 6 | Basic of basic_color ∗ w e i g h t 7 | RGB of int ∗ i n t ∗ i n t 8 | Gray of int 9 1 l e t color_to_int= function 2 | Basic(basic_color,weight) −> 3 l e t base= match weight with Bold −> 8 | Regular −> 0 in 4 base+ basic_color_to_int basic_color 5 | RGB(r,g,b) −> 16 +b+g ∗ 6 +r ∗ 36 6 | Grayi −> 232 +i 7 6 of 113 The limit of variants Say we want to handle a color representation with an alpha channel, but just for color_to_int (this implies we do not want to redefine our color type, this would be a hassle elsewhere). -
Comparative Studies of Programming Languages; Course Lecture Notes
Comparative Studies of Programming Languages, COMP6411 Lecture Notes, Revision 1.9 Joey Paquet Serguei A. Mokhov (Eds.) August 5, 2010 arXiv:1007.2123v6 [cs.PL] 4 Aug 2010 2 Preface Lecture notes for the Comparative Studies of Programming Languages course, COMP6411, taught at the Department of Computer Science and Software Engineering, Faculty of Engineering and Computer Science, Concordia University, Montreal, QC, Canada. These notes include a compiled book of primarily related articles from the Wikipedia, the Free Encyclopedia [24], as well as Comparative Programming Languages book [7] and other resources, including our own. The original notes were compiled by Dr. Paquet [14] 3 4 Contents 1 Brief History and Genealogy of Programming Languages 7 1.1 Introduction . 7 1.1.1 Subreferences . 7 1.2 History . 7 1.2.1 Pre-computer era . 7 1.2.2 Subreferences . 8 1.2.3 Early computer era . 8 1.2.4 Subreferences . 8 1.2.5 Modern/Structured programming languages . 9 1.3 References . 19 2 Programming Paradigms 21 2.1 Introduction . 21 2.2 History . 21 2.2.1 Low-level: binary, assembly . 21 2.2.2 Procedural programming . 22 2.2.3 Object-oriented programming . 23 2.2.4 Declarative programming . 27 3 Program Evaluation 33 3.1 Program analysis and translation phases . 33 3.1.1 Front end . 33 3.1.2 Back end . 34 3.2 Compilation vs. interpretation . 34 3.2.1 Compilation . 34 3.2.2 Interpretation . 36 3.2.3 Subreferences . 37 3.3 Type System . 38 3.3.1 Type checking . 38 3.4 Memory management . -
“Scrap Your Boilerplate” Reloaded
“Scrap Your Boilerplate” Reloaded Ralf Hinze1, Andres L¨oh1, and Bruno C. d. S. Oliveira2 1 Institut f¨urInformatik III, Universit¨atBonn R¨omerstraße164, 53117 Bonn, Germany {ralf,loeh}@informatik.uni-bonn.de 2 Oxford University Computing Laboratory Wolfson Building, Parks Road, Oxford OX1 3QD, UK [email protected] Abstract. The paper “Scrap your boilerplate” (SYB) introduces a com- binator library for generic programming that offers generic traversals and queries. Classically, support for generic programming consists of two es- sential ingredients: a way to write (type-)overloaded functions, and in- dependently, a way to access the structure of data types. SYB seems to lack the second. As a consequence, it is difficult to compare with other approaches such as PolyP or Generic Haskell. In this paper we reveal the structural view that SYB builds upon. This allows us to define the combinators as generic functions in the classical sense. We explain the SYB approach in this changed setting from ground up, and use the un- derstanding gained to relate it to other generic programming approaches. Furthermore, we show that the SYB view is applicable to a very large class of data types, including generalized algebraic data types. 1 Introduction The paper “Scrap your boilerplate” (SYB) [1] introduces a combinator library for generic programming that offers generic traversals and queries. Classically, support for generic programming consists of two essential ingredients: a way to write (type-)overloaded functions, and independently, a way to access the structure of data types. SYB seems to lacks the second, because it is entirely based on combinators. -
CSI33 Data Structures
Outline CSI33 Data Structures Department of Mathematics and Computer Science Bronx Community College December 2, 2015 CSI33 Data Structures Outline Outline 1 Chapter 13: Heaps, Balances Trees and Hash Tables Hash Tables CSI33 Data Structures Chapter 13: Heaps, Balances Trees and Hash TablesHash Tables Outline 1 Chapter 13: Heaps, Balances Trees and Hash Tables Hash Tables CSI33 Data Structures Chapter 13: Heaps, Balances Trees and Hash TablesHash Tables Python Dictionaries Various names are given to the abstract data type we know as a dictionary in Python: Hash (The languages Perl and Ruby use this terminology; implementation is a hash table). Map (Microsoft Foundation Classes C/C++ Library; because it maps keys to values). Dictionary (Python, Smalltalk; lets you "look up" a value for a key). Association List (LISP|everything in LISP is a list, but this type is implemented by hash table). Associative Array (This is the technical name for such a structure because it looks like an array whose 'indexes' in square brackets are key values). CSI33 Data Structures Chapter 13: Heaps, Balances Trees and Hash TablesHash Tables Python Dictionaries In Python, a dictionary associates a value (item of data) with a unique key (to identify and access the data). The implementation strategy is the same in any language that uses associative arrays. Representing the relation between key and value is a hash table. CSI33 Data Structures Chapter 13: Heaps, Balances Trees and Hash TablesHash Tables Python Dictionaries The Python implementation uses a hash table since it has the most efficient performance for insertion, deletion, and lookup operations. The running times of all these operations are better than any we have seen so far. -
Data Structures Are Ways to Organize Data (Informa- Tion). Examples
CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 1 ] What are Data Structures? Data structures are ways to organize data (informa- tion). Examples: simple variables — primitive types objects — collection of data items of various types arrays — collection of data items of the same type, stored contiguously linked lists — sequence of data items, each one points to the next one Typically, algorithms go with the data structures to manipulate the data (e.g., the methods of a class). This course will cover some more complicated data structures: how to implement them efficiently what they are good for CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 2 ] Abstract Data Types An abstract data type (ADT) defines a state of an object and operations that act on the object, possibly changing the state. Similar to a Java class. This course will cover specifications of several common ADTs pros and cons of different implementations of the ADTs (e.g., array or linked list? sorted or unsorted?) how the ADT can be used to solve other problems CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 3 ] Specific ADTs The ADTs to be studied (and some sample applica- tions) are: stack evaluate arithmetic expressions queue simulate complex systems, such as traffic general list AI systems, including the LISP language tree simple and fast sorting table database applications, with quick look-up CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 4 ] How Does C Fit In? Although data structures are universal (can be imple- mented in any programming language), this course will use Java and C: non-object-oriented parts of Java are based on C C is not object-oriented We will learn how to gain the advantages of data ab- straction and modularity in C, by using self-discipline to achieve what Java forces you to do. -
CIS 120 Midterm I October 13, 2017 Name (Printed): Pennkey (Penn
CIS 120 Midterm I October 13, 2017 Name (printed): PennKey (penn login id): My signature below certifies that I have complied with the University of Pennsylvania’s Code of Academic Integrity in completing this examination. Signature: Date: • Do not begin the exam until you are told it is time to do so. • Make sure that your username (a.k.a. PennKey, e.g. stevez) is written clearly at the bottom of every page. • There are 100 total points. The exam lasts 50 minutes. Do not spend too much time on any one question. • Be sure to recheck all of your answers. • The last page of the exam can be used as scratch space. By default, we will ignore anything you write on this page. If you write something that you want us to grade, make sure you mark it clearly as an answer to a problem and write a clear note on the page with that problem telling us to look at the scratch page. • Good luck! 1 1. Binary Search Trees (16 points total) This problem concerns buggy implementations of the insert and delete functions for bi- nary search trees, the correct versions of which are shown in Appendix A. First: At most one of the lines of code contains a compile-time (i.e. typechecking) error. If there is a compile-time error, explain what the error is and one way to fix it. If there is no compile-time error, say “No Error”. Second: even after the compile-time error (if any) is fixed, the code is still buggy—for some inputs the function works correctly and produces the correct BST, and for other inputs, the function produces an incorrect tree.