8 Equivalent Models of Computation
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Confusion in the Church-Turing Thesis Has Obscured the Funda- Mental Limitations of Λ-Calculus As a Foundation for Programming Languages
Confusion in the Church-Turing Thesis Barry Jay and Jose Vergara University of Technology, Sydney {Barry.Jay,Jose.Vergara}@uts.edu.au August 20, 2018 Abstract The Church-Turing Thesis confuses numerical computations with sym- bolic computations. In particular, any model of computability in which equality is not definable, such as the λ-models underpinning higher-order programming languages, is not equivalent to the Turing model. However, a modern combinatory calculus, the SF -calculus, can define equality of its closed normal forms, and so yields a model of computability that is equivalent to the Turing model. This has profound implications for pro- gramming language design. 1 Introduction The λ-calculus [17, 4] does not define the limit of expressive power for higher- order programming languages, even when they are implemented as Turing ma- chines [69]. That this seems to be incompatible with the Church-Turing Thesis is due to confusion over what the thesis is, and confusion within the thesis itself. According to Soare [63, page 11], the Church-Turing Thesis is first mentioned in Steven Kleene’s book Introduction to Metamathematics [42]. However, the thesis is never actually stated there, so that each later writer has had to find their own definition. The main confusions within the thesis can be exposed by using the book to supply the premises for the argument in Figure 1 overleaf. The conclusion asserts the λ-definability of equality of closed λ-terms in normal form, i.e. syntactic equality of their deBruijn representations [20]. Since the conclusion is false [4, page 519] there must be a fault in the argument. -
Lambda Calculus Encodings
CMSC 330: Organization of Programming Languages Lambda Calculus Encodings CMSC330 Fall 2017 1 The Power of Lambdas Despite its simplicity, the lambda calculus is quite expressive: it is Turing complete! Means we can encode any computation we want • If we’re sufficiently clever... Examples • Booleans • Pairs • Natural numbers & arithmetic • Looping 2 Booleans Church’s encoding of mathematical logic • true = λx.λy.x • false = λx.λy.y • if a then b else c Ø Defined to be the expression: a b c Examples • if true then b else c = (λx.λy.x) b c → (λy.b) c → b • if false then b else c = (λx.λy.y) b c → (λy.y) c → c 3 Booleans (cont.) Other Boolean operations • not = λx.x false true Ø not x = x false true = if x then false else true Ø not true → (λx.x false true) true → (true false true) → false • and = λx.λy.x y false Ø and x y = if x then y else false • or = λx.λy.x true y Ø or x y = if x then true else y Given these operations • Can build up a logical inference system 4 Quiz #1 What is the lambda calculus encoding of xor x y? - xor true true = xor false false = false - xor true false = xor false true = true A. x x y true = λx.λy.x B. x (y true false) y false = λx.λy.y C. x (y false true) y if a then b else c = a b c not = λx.x false true D. y x y 5 Quiz #1 What is the lambda calculus encoding of xor x y? - xor true true = xor false false = false - xor true false = xor false true = true A. -
Programming Languages Shouldn't and Needn't Be Turing Complete
Programming languages shouldn't and needn't be Turing complete GABRIEL PICKARD Any algorithmic problem faced by application programmers in the wild1 can in principle be solved using a Turing incomplete programming language. [Rice 1953] suggests that Turing completeness bears a heavy price, fundamentally limiting the ability of automatic assistants to help programmers be more productive, create less bugs and write safer software. Nevertheless, no mainstream general purpose programming language is Turing incomplete, with the arguable exception of SQL. We identify historical causes for this discrepancy and argue that the current moment offers better conditions for introducing Turing incompleteness. We present an approach to eliminating Turing completeness during the language design process, with several examples suitable languages targeting application development. Designers of modern practical programming languages should strongly consider evading Turing completeness and enabling better verification, security, automated testing and distributed computing. 1 CEDING POWER TO GAIN CONTROL 1.1 Turing completeness considered harmful The history of programming language design2 has been a history of removing powers which were considered harmful (cf. [Dijkstra 1966]) and replacing them with tamer, more manageable and automated constructs, including: (1) Direct access to CPU registers and instructions ! higher level compilers (2) GOTO ! structured programming (3) Manual memory management ! garbage collection (and borrow checkers) (4) Arbitrary access ! information hiding (5) Side-effects and mutability ! pure functions and persistent data structures While not all application domains benefit from all the replacement steps above3, it appears as if removing some expressive power can often enable a new kind of programming, even a new kind of application altogether. -
Turing Machines
Basics Computability Complexity Turing Machines Bruce Merry University of Cape Town 10 May 2012 Bruce Merry Turing Machines Basics Computability Complexity Outline 1 Basics Definition Building programs Turing Completeness 2 Computability Universal Machines Languages The Halting Problem 3 Complexity Non-determinism Complexity classes Satisfiability Bruce Merry Turing Machines Basics Definition Computability Building programs Complexity Turing Completeness Outline 1 Basics Definition Building programs Turing Completeness 2 Computability Universal Machines Languages The Halting Problem 3 Complexity Non-determinism Complexity classes Satisfiability Bruce Merry Turing Machines Basics Definition Computability Building programs Complexity Turing Completeness What Are Turing Machines? Invented by Alan Turing Hypothetical machines Formalise “computation” Alan Turing, 1912–1954 Bruce Merry Turing Machines If in state si and tape contains qj , write qk then move left/right and change to state sm A finite set of states, including a start state A tape that is infinite in both directions, containing finitely many non-blank symbols A head which points at one position on the tape A set of transitions Basics Definition Computability Building programs Complexity Turing Completeness What Are Turing Machines? Each Turing machine consists of A finite set of symbols, including a special blank symbol () Bruce Merry Turing Machines If in state si and tape contains qj , write qk then move left/right and change to state sm A tape that is infinite in both directions, containing -
Python C/C++ Java Perl Ruby
Programming Languages Big Ideas for CS 251 • What is a PL? Theory of Programming Languages • Why are new PLs created? Principles of Programming Languages – What are they used for? – Why are there so many? • Why are certain PLs popular? • What goes into the design of a PL? CS251 Programming Languages – What features must/should it contain? Spring 2018, Lyn Turbak – What are the design dimensions? – What are design decisions that must be made? Department of Computer Science Wellesley College • Why should you take this course? What will you learn? Big ideas 2 PL is my passion! General Purpose PLs • First PL project in 1982 as intern at Xerox PARC Java Perl • Created visual PL for 1986 MIT Python masters thesis • 1994 MIT PhD on PL feature Fortran (synchronized lazy aggregates) ML JavaScript • 1996 – 2006: worked on types Racket as member of Church project Haskell • 1988 – 2008: Design Concepts in Programming Languages C/C++ Ruby • 2011 – current: lead TinkerBlocks research team at Wellesley • 2012 – current: member of App Inventor development team CommonLisp Big ideas 3 Big ideas 4 Domain Specific PLs Programming Languages: Mechanical View HTML A computer is a machine. Our aim is to make Excel CSS the machine perform some specifieD acEons. With some machines we might express our intenEons by Depressing keys, pushing OpenGL R buIons, rotaEng knobs, etc. For a computer, Matlab we construct a sequence of instrucEons (this LaTeX is a ``program'') anD present this sequence to IDL the machine. Swift PostScript! – Laurence Atkinson, Pascal Programming Big ideas 5 Big ideas 6 Programming Languages: LinguisEc View Religious Views The use of COBOL cripples the minD; its teaching shoulD, therefore, be A computer language … is a novel formal regarDeD as a criminal offense. -
Unusable for Programming
UNUSABLE FOR PROGRAMMING DANIEL TEMKIN Independent, New York, USA [email protected] 2017.xCoAx.org Lisbon Computation Communication Aesthetics & X Abstract Esolangs are programming languages designed for non-practical purposes, often as experiments, par- odies, or experiential art pieces. A common goal of esolangs is to achieve Turing Completeness: the capability of implementing algorithms of any Keywords complexity. They just go about getting to Turing Esolangs Completeness with an unusual and impractical set Programming languages of commands, or unexpected waysDraft of representing Code art code. However, there is a much smaller class of Oulipo esolangs that are entirely “Unusable for Program- Tangible Computing ming.” These languages explore the very boundary of what a programming language is; producing uns- table programs, or for which no programs can be written at all. This is a look at such languages and how they function (or fail to). Final 1. INTRODUCTION Esolangs (for “esoteric programming languages”) include languages built as thought experiments, jokes, parodies critical of language practices, and arte- facts from imagined alternate computer histories. The esolang Unlambda, for example, is both an esolang masterpiece and typical of the genre. Written by David Madore in 1999, Unlambda gives us a version of functional programming taken to its most extreme: functions can only be applied to other functions, returning more functions, all of them unnamed. It’s a cyberlinguistic puzzle intended as a challenge and an experiment at the extremes of a certain type of language design. The following is a program that calculates the Fibonacci sequence. The code is nearly opaque, with little indication of what it’s doing or how it works. -
Modular Domain-Specific Language Components in Scala
Submitted to GPCE ’10 Modular Domain-Specific Language Components in Scala Christian Hofer Klaus Ostermann Aarhus University, Denmark University of Marburg, Germany [email protected] [email protected] Abstract simply compose these languages, if the concrete representations of Programs in domain-specific embedded languages (DSELs) can be their types in the host language match. Assuming Scala as our host represented in the host language in different ways, for instance im- language, we can then write a term like: plicitly as libraries, or explicitly in the form of abstract syntax trees. app(lam((x: Region) )union(vr(x),univ)), Each of these representations has its own strengths and weaknesses. scale(circle(3), The implicit approach has good composability properties, whereas add(vec(1,2),vec(3,4)))) the explicit approach allows more freedom in making syntactic pro- This term applies a function which maps a region to its union with gram transformations. the universal region to a circle that is scaled by a vector. Traditional designs for DSELs fix the form of representation, However, the main advantage of this method is also its main which means that it is not possible to choose the best representation disadvantage. It restricts the implementation to a fixed interpreta- for a particular interpretation or transformation. We propose a new tion which has to be compositional, i. e., the meaning of an expres- design for implementing DSELs in Scala which makes it easy to sion may only depend only on the meaning of its sub-expressions, use different program representations at the same time. -
Encoding Data in Lambda Calculus: an Introduction
Encoding Data in Lambda Calculus: An Introduction Frank(Peng) Fu September 26, 2017 Abstract Lambda calculus is a formalism introduced by Alonzo Church in the 1930s for his research on the foundations of mathematics. It is now widely used as a theoretical foundation for the functional program- ming languages (e.g. Haskell, OCaml, Lisp). I will first give a short introduction to lambda calculus, then I will discuss how to encode natural numbers using the encoding schemes invented by Alonzo Church, Dana Scott and Michel Parigot. Although we will mostly focus on numbers, these encoding schemes also works for more general data structures such as lists and trees. If time permits, I will talk about the type theoretical aspects of these encodings. 1 Introduction to Lambda Calculus Lambda calculus was invented by Alonzo Church, a lot of early results are due to him and his students. Currently, the definitive reference for lambda calculus is the book by Henk Barendregt [1]. Definition 1 (Lambda Calculus) The set of lambda term Λ is defined inductively as following. • x 2 Λ for any variable x. • If e 2 Λ, then λx.e 2 Λ. • If e1; e2 2 Λ, then e1 e2 2 Λ. Some computer scientists express lambda terms as: e; n ::= x j e1 e2 j λx.e. Lambda terms are almost symbolic, except we only consider lambda terms modulo alpha-equivalence, i.e. we view λx.e as the same term as λy:[y=x]e, where y does not occurs in e. Definition 2 Beta-reduction: (λx.e1) e2 [e2=x]e1, where [e2=x]e1 means the result of replacing all the variable x in e1 by e2. -
Typed -Calculi and Superclasses of Regular Transductions
Typed λ-calculi and superclasses of regular transductions Lê Thành Dung˜ Nguy˜ên LIPN, UMR 7030 CNRS, Université Paris 13, France https://nguyentito.eu/ [email protected] Abstract We propose to use Church encodings in typed λ-calculi as the basis for an automata-theoretic counterpart of implicit computational complexity, in the same way that monadic second-order logic provides a counterpart to descriptive complexity. Specifically, we look at transductions i.e. string-to-string (or tree-to-tree) functions – in particular those with superlinear growth, such as polyregular functions, HDT0L transductions and Sénizergues’s “k-computable mappings”. Our first results towards this aim consist showing the inclusion of some transduction classes in some classes defined by λ-calculi. In particular, this sheds light on a basic open question on the expressivity of the simply typed λ-calculus. We also encode regular functions (and, by changing the type of programs considered, we get a larger subclass of polyregular functions) in the elementary affine λ-calculus, a variant of linear logic originally designed for implicit computational complexity. 2012 ACM Subject Classification Theory of computation → Lambda calculus; Theory of computa- tion → Transducers; Theory of computation → Linear logic Keywords and phrases streaming string transducers, simply typed λ-calculus, linear logic Acknowledgements Thanks to Pierre Pradic for bringing to my attention the question (asked by Mikołaj Bojańczyk) of relating transductions and linear logic and for the proof of Theorem 2.9. This work also benefited from discussions on automata theory with Célia Borlido, Marie Fortin and Jérémy Ledent, and on the simply typed λ-calculus with Damiano Mazza. -
Exotic Programming Languages
I. Babich, Master student V. Spivachuk, PHD in Phil., As. Prof., research advisor Khmelnytskyi National University EXOTIC PROGRAMMING LANGUAGES The word ―exotic‖ is defined by the Oxford dictionary as ―of a kind not ordinarily encountered‖. If that is so, what is meant by exotic programming languages? These are languages that are not used for commercial purposes or even for anything ―useful‖. This definition can be applied a little loosely and, hence, many different types of programming languages can be classified as ―exotic‖. So, after some consideration, I have included three types of languages as belonging to this category. Exotic programming languages include languages that are intentionally designed to be difficult to learn and program with. Such languages are often used to analyse the power of programming languages. They are known as esoteric programming languages. Some exotic languages, known as joke languages, are created for the sake of fun. And the third type includes non – English – based programming languages. The definition clearly mentions that these languages do not serve any practical purpose. Then what’s all the fuss about exotic programming languages? To understand their importance, we need to first understand what makes a programming language a programming language. The “Turing completeness” of programming languages A language qualifies as a programming language in the true sense only if it is ―Turing complete‖ [1, c. 4]. A language that is Turing complete can be used to represent any computable algorithm. Many of the so called languages are, in fact, not languages, in the true sense. Examples of tools that are not Turing complete and hence not languages in the strict sense include HTML, XML, JSON, etc. -
Proof Pearl: Proving a Simple Von Neumann Machine Turing Complete
Proof Pearl: Proving a Simple Von Neumann Machine Turing Complete J Strother Moore Dept. of Computer Science, University of Texas, Austin, TX, USA [email protected] http://www.cs.utexas.edu Abstract. In this paper we sketch an ACL2-checked proof that a simple but unbounded Von Neumann machine model is Turing Complete, i.e., can do anything a Turing machine can do. The project formally revisits the roots of computer science. It requires re-familiarizing oneself with the definitive model of computation from the 1930s, dealing with a simple “modern” machine model, thinking carefully about the formal statement of an important theorem and the specification of both total and partial programs, writing a verifying compiler, including implementing an X86- like call/return protocol and implementing computed jumps, codifying a code proof strategy, and a little “creative” reasoning about the non- termination of two machines. Keywords: ACL2, Turing machine, Java Virtual Machine (JVM), ver- ifying compiler 1 Prelude I have often taught an undergraduate course at the University of Texas at Austin entitled A Formal Model of the Java Virtual Machine. In the course, students are taught how to model sophisticated computing engines and, to a lesser extent, how to prove theorems about such engines and their programs with the ACL2 theorem prover [5]. The course starts with a pedagogical (“toy”) JVM-like model which the students elaborate over the semester towards a more realistic model, which is then compared to an accurate JVM model[9]. The pedagogical model is called M1: a stack based machine providing a fixed number of registers (JVM’s “locals”), an unbounded operand stack, and an execute-only program providing the following bytecode instructions ILOAD, ISTORE, ICONST, IADD, ISUB, IMUL, IFEQ, GOTO, and HALT, with unbounded arithmetic. -
NCCU Programming Languages Concepts 程式語言
NCCU Programming Languages: Concepts & Principles 程式語言 Instructor: 資科系 陳恭副教授 Spring 2006 Lecture 1 Contacts • Instructor: Dr. Kung Chen – E-mail: [email protected] –Office: 大仁樓200210 – Office Hours: Tuesday: 11am-12am • Teaching assistants: 資科系研究生 – E-mail: [email protected] [email protected] –Office: 資科系程式語言與軟體方法實驗室 • Class web page: – http://www.cs.nccu.edu/~chenk/Courses/PL What shall we study? Don’t Get Confused by the Course Name 教哪個程式語言? C++? Java? C#? … No! A “principles” course aims to teach the underlying concepts behind all programming languages (Concept-driven). Course Pre-requisite • Required course for CS major (junior) – Implies that it’s not an easy course • Prerequisite – Experience in C programming – Experience in C++ or Java programming • Non-CS major students – Better talk to the instructor or be well-motivated Programming languages Programming languages you have used you’ve heard of • C, Java, C++, C#, • Lisp, … •Basic, … •… Programming Languages’ Tower of Babel • Why are there so many programming languages? "I speak Spanish to God, Italian to women, French to men, and German to my horse." — Emperor Charles V (1500-1558) Why So Many Languages? • Application domains have distinctive (and conflicting) needs • Examples: – Scientific Computing: high performance – Business: report generation – Artificial intelligence: symbolic computation – Systems programming: low-level access – Real-time systems: timing constraints – Special purpose languages –… Why So Many Languages? (Cont’d) • Evolution: Our understanding of programming evolves, so do languages grow! • The advance of programming methods usually lead to new programming languages: – Structured programming • Pascal – Object-Oriented Programming • Simula 67, Smalltalk, C++ – Functional programming • Socio-economic factors – Proprietary interests, commercial advantage • Personal preference Different Programming Language Design Philosophies C Other languages If all you have is a hammer, then everything looks like a nail.