Type-Inference Based Deforestation of Functional Programs

Total Page:16

File Type:pdf, Size:1020Kb

Type-Inference Based Deforestation of Functional Programs Type-Inference Based Deforestation of Functional Programs Von der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der Rheinisch-Westf¨alischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Informatiker Olaf Chitil aus S¨uchteln, jetzt Viersen Berichter: Universit¨atsprofessorDr. rer. nat. Klaus Indermark Universit¨atsprofessorDr. rer. nat. Michael Hanus Tag der m¨undlichen Pr¨ufung: 27. Oktober 2000 ii Abstract In lazy functional programming modularity is often achieved by using intermediate data structures to combine separate parts of a program. Each intermediate data structure is produced by one part and consumed by another one. Deforestation optimises a functional program by transformation into a program which does not produce such intermediate data structures. In this thesis we present a new method for deforestation, which combines a known method, short cut deforestation, with a new analysis that is based on type inference. Short cut deforestation eliminates an intermediate list by a single, local transformation. In return, short cut deforestation expects both producer and consumer of the intermediate list in a certain form. Whereas the required form of the consumer is generally considered desirable in a well-structured program anyway, the required form of the producer is only a crutch to enable deforestation. Hence only the list-producing functions of the standard libraries were defined in the desired form and short cut deforestation has been confined to compositions of these functions. Here, we present an algorithm which transforms an arbitrary producer into the re- quired form. Starting from the observation that short cut deforestation is based on a parametricity theorem of the second-order typed λ-calculus, we show how the construc- tion of the required form can be reduced to a type inference problem. Typability for the second-order typed λ-calculus is undecidable, but we only need to solve a partial type inference problem. For this problem we develop an algorithm based on the well-known Hindley-Milner type inference algorithm. The transformation of a producer often requires inlining of some function definitions. Type inference even indicates which function definitions need to be inlined. However, only limited inlining across module boundaries is practically feasible. Therefore, we extend the previously developed algorithm to split a function definition into a worker definition and a wrapper definition. We only need to inline the small wrapper definition, which transfers all information required for deforestation. The flexibility of type inference allows us to remove intermediate lists which original short cut deforestation cannot remove, even with hand-crafted producers. In contrast to most previous work on deforestation, we give a detailed proof of completeness and semantic correctness of our transformation. Acknowledgements My supervisor Prof. Klaus Indermark introduced me to the foundations of functional programming and taught me to precisely formulate my statements. He employed me at his chair, left me much freedom in selecting a subject for my thesis and showed trust in me all the time. Dr. Markus Mohnen introduced me to the area of optimisation through program transformation in general and deforestation in particular. The enthusiasm of Christian T¨uffersfor his Diplomarbeit on deforestation raised my interest in the area. Many people in and outside Aachen discussed the topic of my thesis with me. Of these Frank Huch has to be mentioned first. We talked so often about my deforestation method and he had non-enumerably many ideas for improvement. Prof. Michael Hanus was always helpful and agreed to be second examiner although my thesis has no logical elements. Prof. Simon Peyton Jones pointed out several shortcomings in the first ver- sions of my deforestation method and answered all my questions about the internals of the Glasgow Haskell compiler. Matthias H¨off,Dr. Armin K¨uhnemannand I compared several deforestation techniques, thus improving my understanding of these techniques. This was possible because Armin and Prof. Heiko Vogler had invited me to Dresden. Prof. Patricia Johann and Prof. Franklyn Turbak read versions of the thesis and made numerous suggestions for improvement. All the members of the Lehrstuhl f¨urInformatik II and the Lehr- und Forschungsgebiet f¨urInformatik II at the RWTH Aachen made working there such a pleasure, especially my office mates Dr. George Botorog and Thomas Richert. Finally, working all these years on this thesis would not have been possible without the support of my friends and my family. I thank all of you. Contents 1 Introduction 1 1.1 Intermediate Data Structures for Modularity . 1 1.2 Deforestation . 2 1.3 Short Cut Deforestation . 3 1.4 Type Inference Identifies List Constructors . 5 1.5 Structure of the Thesis . 6 2 The Functional Language F 7 2.1 Syntax . 7 2.2 Substitutions and Renaming of Bound Variables . 10 2.2.1 Substitutions . 10 2.2.2 Renaming of Bound Variables . 13 2.3 Type System . 14 2.4 Evaluation of Programs . 18 2.5 Program Transformations . 22 2.6 Contextual Equivalence . 23 2.7 Kleene Equivalence . 26 2.8 The Short Cut Fusion Rule . 27 2.9 A Cost Model . 28 2.9.1 A Semantics for Lazy Evaluation . 29 2.9.2 Improvement . 33 3 Type-Inference Based Deforestation 35 3.1 List Abstraction through Type Inference . 35 3.1.1 Basic List Abstraction . 35 3.1.2 Polymorphic List Constructors . 37 3.2 Inlining of Definitions . 38 3.2.1 Determination of Variables that Need to be Inlined . 38 3.2.2 Instantiation of Polymorphism . 39 3.3 The Worker/Wrapper Scheme . 39 3.3.1 Functions that Produce Several Lists . 41 3.3.2 List Concatenation . 42 3.3.3 Recursively Defined Producers . 42 3.4 Functions that Consume their Own Result . 43 3.4.1 A larger example: inits ........................ 44 viii Contents 3.4.2 Deforestation Changes Complexity . 45 3.4.3 Inaccessible Recursive Arguments . 45 4 List Abstraction And Type Inference 47 4.1 The Instance Substitution . 47 4.2 List Replacement Phase . 49 4.2.1 Locally Bound Type Variables . 49 4.2.2 The List Replacement Algorithm . 49 4.2.3 Properties of the List Replacement Algorithm . 51 4.3 Type Inference Phase . 54 4.3.1 Idempotent Substitutions . 56 4.3.2 The Unification Algorithm ..................... 58 4.3.3 Consistency of Types of ConstructorU Variables . 65 4.3.4 Towards a Type Inference Algorithm . 68 4.3.5 The Type Inference Algorithm ................... 71 4.3.6 The Consistency Closure AlgorithmT . 80 4.3.7 Principal Consistent Typing . .C . 82 4.4 Instantiation of Variables and Abstraction Phase . 84 4.4.1 Superfluous Type Inference Variables . 84 4.4.2 Instantiation of Constructor Variables . 87 4.4.3 Merge of Constructor Variables . 88 4.4.4 Abstraction . 90 4.5 Properties and Implementation . 91 4.6 Alternative List Abstraction Algorithms . 92 5 The Deforestation Algorithm 95 5.1 The Worker/Wrapper Split Algorithm . 95 5.1.1 Non-Recursive Definitions . 95 5.1.2 Recursive Definitions . 96 5.1.3 Polymorphic Recursion . 98 5.1.4 Traversal Order . 100 5.2 The Complete Deforestation Algorithm . 101 5.2.1 Avoiding Inefficient Workers . 102 5.2.2 Consumers in foldr form . 102 5.2.3 Improvement by Short Cut Deforestation . 104 5.2.4 No Duplication of Evaluation | Linearity is Irrelevant . 107 5.3 Measuring an Example: Queens . 108 6 Extensions 115 6.1 A Worker/Wrapper Scheme for Consumers . 115 6.1.1 Scheme A . 116 6.1.2 Scheme B . 116 6.2 Deforestation Beyond Arguments of foldr . 117 6.3 More Efficient Algorithms . 118 6.4 Other Data Types than Lists . 119 7 Other Deforestation Methods 121 7.1 Short Cut Deforestation . 121 7.1.1 Gill's Worker/Wrapper Scheme . 121 7.1.2 Limitations of build . 122 7.1.3 Warm Fusion . 123 Contents ix 7.1.4 Short Cut Fusion in the Glasgow Haskell compiler . 124 7.2 Hylomorphism Fusion . 124 7.3 Unfold-Fold Transformations . 128 7.4 Attribute Grammar Composition . 130 7.5 Further Methods . 131 8 Conclusions 133 A Rules 149 B Standard List Functions Used in the Program Queens 151 Chapter 1 Introduction A primary feature of a good programming language for productive software development is that it supports writing of highly modular programs. A small program module which is weakly coupled with other modules can be both coded and debugged quickly. Fur- thermore, it can be reused in several different contexts. Even a complex task should be performed by a small module which uses other small modules. 1.1 Intermediate Data Structures for Modularity In functional programming functions are the basic modules from which programs are built. A powerful form of modularity is based on the use of intermediate data structures. Two separately defined functions can be glued together by an intermediate data structure that is produced by one function and consumed by the other. For example, the function any, which tests whether any element of a list xs satisfies a given predicate p, may be defined as follows in Haskell [PH+99]: any :: (a -> Bool) -> [a] -> Bool any p xs = or (map p xs) The function map applies p to all elements of xs yielding a list of boolean values. The function or combines these boolean values with the logical or operation (||). So we defined any by gluing together the producer map p xs and the consumer or by an intermediate list. We survey some more examples. A function anyTree shall test whether any element of a tree satisfies a given predicate. We can define this function by composing any with a general purpose function treeToList which simply produces a list of all elements stored in a tree.
Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Recursive Type Generativity
    JFP 17 (4 & 5): 433–471, 2007. c 2007 Cambridge University Press 433 doi:10.1017/S0956796807006429 Printed in the United Kingdom Recursive type generativity DEREK DREYER Toyota Technological Institute, Chicago, IL 60637, USA (e-mail: [email protected]) Abstract Existential types provide a simple and elegant foundation for understanding generative abstract data types of the kind supported by the Standard ML module system. However, in attempting to extend ML with support for recursive modules, we have found that the traditional existential account of type generativity does not work well in the presence of mutually recursive module definitions. The key problem is that, in recursive modules, one may wish to define an abstract type in a context where a name for the type already exists, but the existential type mechanism does not allow one to do so. We propose a novel account of recursive type generativity that resolves this problem. The basic idea is to separate the act of generating a name for an abstract type from the act of defining its underlying representation. To define several abstract types recursively, one may first “forward-declare” them by generating their names, and then supply each one’s identity secretly within its own defining expression. Intuitively, this can be viewed as a kind of backpatching semantics for recursion at the level of types. Care must be taken to ensure that a type name is not defined more than once, and that cycles do not arise among “transparent” type definitions. In contrast to the usual continuation-passing interpretation of existential types in terms of universal types, our account of type generativity suggests a destination-passing interpretation.
    [Show full text]
  • Compositional Data Types
    Compositional Data Types Patrick Bahr Tom Hvitved Department of Computer Science, University of Copenhagen Universitetsparken 1, 2100 Copenhagen, Denmark {paba,hvitved}@diku.dk Abstract require the duplication of functions which work both on general and Building on Wouter Swierstra’s Data types `ala carte, we present a non-empty lists. comprehensive Haskell library of compositional data types suitable The situation illustrated above is an ubiquitous issue in com- for practical applications. In this framework, data types and func- piler construction: In a compiler, an abstract syntax tree (AST) tions on them can be defined in a modular fashion. We extend the is produced from a source file, which then goes through different existing work by implementing a wide array of recursion schemes transformation and analysis phases, and is finally transformed into including monadic computations. Above all, we generalise recur- the target code. As functional programmers, we want to reflect the sive data types to contexts, which allow us to characterise a special changes of each transformation step in the type of the AST. For ex- yet frequent kind of catamorphisms. The thus established notion of ample, consider the desugaring phase of a compiler which reduces term homomorphisms allows for flexible reuse and enables short- syntactic sugar to the core syntax of the object language. To prop- cut fusion style deforestation which yields considerable speedups. erly reflect this structural change also in the types, we have to create We demonstrate our framework in the setting of compiler con- and maintain a variant of the data type defining the AST for the core struction, and moreover, we compare compositional data types with syntax.
    [Show full text]
  • 4. Types and Polymorphism
    4. Types and Polymorphism Oscar Nierstrasz Static types ok Semantics ok Syntax ok Program text Roadmap > Static and Dynamic Types > Type Completeness > Types in Haskell > Monomorphic and Polymorphic types > Hindley-Milner Type Inference > Overloading References > Paul Hudak, “Conception, Evolution, and Application of Functional Programming Languages,” ACM Computing Surveys 21/3, Sept. 1989, pp 359-411. > L. Cardelli and P. Wegner, “On Understanding Types, Data Abstraction, and Polymorphism,” ACM Computing Surveys, 17/4, Dec. 1985, pp. 471-522. > D. Watt, Programming Language Concepts and Paradigms, Prentice Hall, 1990 3 Conception, Evolution, and Application of Functional Programming Languages http://scgresources.unibe.ch/Literature/PL/Huda89a-p359-hudak.pdf On Understanding Types, Data Abstraction, and Polymorphism http://lucacardelli.name/Papers/OnUnderstanding.A4.pdf Roadmap > Static and Dynamic Types > Type Completeness > Types in Haskell > Monomorphic and Polymorphic types > Hindley-Milner Type Inference > Overloading What is a Type? Type errors: ? 5 + [ ] ERROR: Type error in application *** expression : 5 + [ ] *** term : 5 *** type : Int *** does not match : [a] A type is a set of values? > int = { ... -2, -1, 0, 1, 2, 3, ... } > bool = { True, False } > Point = { [x=0,y=0], [x=1,y=0], [x=0,y=1] ... } 5 The notion of a type as a set of values is very natural and intuitive: Integers are a set of values; the Java type JButton corresponds to all possible instance of the JButton class (or any of its possible subclasses). What is a Type? A type is a partial specification of behaviour? > n,m:int ⇒ n+m is valid, but not(n) is an error > n:int ⇒ n := 1 is valid, but n := “hello world” is an error What kinds of specifications are interesting? Useful? 6 A Java interface is a simple example of a partial specification of behaviour.
    [Show full text]
  • Computing Lectures
    Computing Lectures 1 R Basics Introduction to computing/programming using slides. Using R as a pocket calculator: • arithmetic operators (+, -, *, /, and ^) • elementary functions (exp and log, sin and cos) • Inf and NaN • pi and options(digits) • comparison operators (>, >=, <, <=, ==, !=): informally introduce logicals which will be discussed in more detail in unit 2. Using variables: • assignment using <- (\gets"): binding symbols to values) • inspect the binding (show the value of an object) by typing the symbol (name of the object): auto-prints. Show that explicitly calling print() has the same effect. • inspect available bindings (list objects) using ls() or objects() • remove bindings (\delete objects") using rm() Using sequences: • sequences, also known as vectors, can be combined using c() • functions seq() and rep() allow efficient creation of certain patterned sequences; shortcut : • subsequences can be selected by subscripting via [, using positive (numeric) indices (positions of elements to select) or negative indices (positions of elements to drop) • can compute with whole sequences: vectorization • summary functions: sum, prod, max, min, range • can also have sequences of character strings • sequences can also have names: use R> x <- c(A = 1, B = 2, C = 3) R> names(x) [1] "A" "B" "C" R> ## Change the names. R> names(x) <- c("d", "e", "f") R> x 1 d e f 1 2 3 R> ## Remove the names. R> names(x) <- NULL R> x [1] 1 2 3 for extracting and replacing the names, aka as getting and setting Using functions: • start with the simple standard example R> hell <- function() writeLines("Hello world.") to explain the constituents of functions: argument list (formals), body, and environment.
    [Show full text]
  • Table of Contents
    A Comprehensive Introduction to Vista Operating System Table of Contents Chapter 1 - Windows Vista Chapter 2 - Development of Windows Vista Chapter 3 - Features New to Windows Vista Chapter 4 - Technical Features New to Windows Vista Chapter 5 - Security and Safety Features New to Windows Vista Chapter 6 - Windows Vista Editions Chapter 7 - Criticism of Windows Vista Chapter 8 - Windows Vista Networking Technologies Chapter 9 -WT Vista Transformation Pack _____________________ WORLD TECHNOLOGIES _____________________ Abstraction and Closure in Computer Science Table of Contents Chapter 1 - Abstraction (Computer Science) Chapter 2 - Closure (Computer Science) Chapter 3 - Control Flow and Structured Programming Chapter 4 - Abstract Data Type and Object (Computer Science) Chapter 5 - Levels of Abstraction Chapter 6 - Anonymous Function WT _____________________ WORLD TECHNOLOGIES _____________________ Advanced Linux Operating Systems Table of Contents Chapter 1 - Introduction to Linux Chapter 2 - Linux Kernel Chapter 3 - History of Linux Chapter 4 - Linux Adoption Chapter 5 - Linux Distribution Chapter 6 - SCO-Linux Controversies Chapter 7 - GNU/Linux Naming Controversy Chapter 8 -WT Criticism of Desktop Linux _____________________ WORLD TECHNOLOGIES _____________________ Advanced Software Testing Table of Contents Chapter 1 - Software Testing Chapter 2 - Application Programming Interface and Code Coverage Chapter 3 - Fault Injection and Mutation Testing Chapter 4 - Exploratory Testing, Fuzz Testing and Equivalence Partitioning Chapter 5
    [Show full text]
  • Scrap Your Boilerplate: a Practical Design Pattern for Generic Programming
    Scrap Your Boilerplate: A Practical Design Pattern for Generic Programming Ralf Lammel¨ Simon Peyton Jones Vrije Universiteit, Amsterdam Microsoft Research, Cambridge Abstract specified department as spelled out in Section 2. This is not an un- We describe a design pattern for writing programs that traverse data usual situation. On the contrary, performing queries or transforma- structures built from rich mutually-recursive data types. Such pro- tions over rich data structures, nowadays often arising from XML grams often have a great deal of “boilerplate” code that simply schemata, is becoming increasingly important. walks the structure, hiding a small amount of “real” code that con- Boilerplate code is tiresome to write, and easy to get wrong. More- stitutes the reason for the traversal. over, it is vulnerable to change. If the schema describing the com- Our technique allows most of this boilerplate to be written once and pany’s organisation changes, then so does every algorithm that re- for all, or even generated mechanically, leaving the programmer curses over that structure. In small programs which walk over one free to concentrate on the important part of the algorithm. These or two data types, each with half a dozen constructors, this is not generic programs are much more adaptive when faced with data much of a problem. In large programs, with dozens of mutually structure evolution because they contain many fewer lines of type- recursive data types, some with dozens of constructors, the mainte- specific code. nance burden can become heavy. Our approach is simple to understand, reasonably efficient, and it handles all the data types found in conventional functional program- Generic programming techniques aim to eliminate boilerplate code.
    [Show full text]
  • Lecture 5. Data Types and Type Classes Functional Programming
    Lecture 5. Data types and type classes Functional Programming [Faculty of Science Information and Computing Sciences] 0 I function call and return as only control-flow primitive I no loops, break, continue, goto I instead: higher-order functions! Goal of typed purely functional programming: programs that are easy to reason about So far: I data-flow only through function arguments and return values I no hidden data-flow through mutable variables/state I instead: tuples! [Faculty of Science Information and Computing Sciences] 1 Goal of typed purely functional programming: programs that are easy to reason about So far: I data-flow only through function arguments and return values I no hidden data-flow through mutable variables/state I instead: tuples! I function call and return as only control-flow primitive I no loops, break, continue, goto I instead: higher-order functions! [Faculty of Science Information and Computing Sciences] 1 I high-level declarative data structures I no explicit reference-based data structures I instead: (immutable) algebraic data types! Goal of typed purely functional programming: programs that are easy to reason about Today: I (almost) unique types I no inheritance hell I instead of classes + inheritance: variant types! I (almost): type classes [Faculty of Science Information and Computing Sciences] 2 Goal of typed purely functional programming: programs that are easy to reason about Today: I (almost) unique types I no inheritance hell I instead of classes + inheritance: variant types! I (almost): type classes
    [Show full text]
  • Implementing Reflection in Nuprl
    IMPLEMENTING REFLECTION IN NUPRL A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Eli Barzilay January 2006 c 2006 Eli Barzilay ALL RIGHTS RESERVED IMPLEMENTING REFLECTION IN NUPRL Eli Barzilay, Ph.D. Cornell University 2006 Reflection is the ability of some entity to describe itself. In a logical context, it is the ability of a logic to reason about itself. Reflection is, therefore, placed at the core of meta-mathematics, making it an important part of formal reasoning; where it revolves mainly around syntax and semantics — the main challenge is in making the syntax of the logic become part of its semantic domain. Given its importance, it is surprising that logical computer systems tend to avoid the subject, or provide poor tools for reflective work. This is in sharp contrast to the area of programming languages, where reflection is well researched and used in a variety of ways where it plays an central role. One factor in making reflection inaccessible in logical systems is the relative difficulty that is immediately encountered when formalizing syntax: dealing with formal syntax means dealing with structures that involve bindings, and in a logical context it seems natural to use the same formal tools to describe syntax — often limiting the usability of such formalizations to specific theories and toy examples. G¨odelnumbers are an example for a reflective formalism that serves its purpose, yet is impractical as a basis for syntactical reasoning in applied systems. In programming languages, there is a simple yet elegant strategy for imple- menting reflection: instead of making a system that describes itself, the system is made available to itself.
    [Show full text]