Haskell Programming with Nested Types: a Principled Approach
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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. -
GHC Reading Guide
GHC Reading Guide - Exploring entrances and mental models to the source code - Takenobu T. Rev. 0.01.1 WIP NOTE: - This is not an official document by the ghc development team. - Please refer to the official documents in detail. - Don't forget “semantics”. It's very important. - This is written for ghc 9.0. Contents Introduction 1. Compiler - Compilation pipeline - Each pipeline stages - Intermediate language syntax - Call graph 2. Runtime system 3. Core libraries Appendix References Introduction Introduction Official resources are here GHC source repository : The GHC Commentary (for developers) : https://gitlab.haskell.org/ghc/ghc https://gitlab.haskell.org/ghc/ghc/-/wikis/commentary GHC Documentation (for users) : * master HEAD https://ghc.gitlab.haskell.org/ghc/doc/ * latest major release https://downloads.haskell.org/~ghc/latest/docs/html/ * version specified https://downloads.haskell.org/~ghc/9.0.1/docs/html/ The User's Guide Core Libraries GHC API Introduction The GHC = Compiler + Runtime System (RTS) + Core Libraries Haskell source (.hs) GHC compiler RuntimeSystem Core Libraries object (.o) (libHsRts.o) (GHC.Base, ...) Linker Executable binary including the RTS (* static link case) Introduction Each division is located in the GHC source tree GHC source repository : https://gitlab.haskell.org/ghc/ghc compiler/ ... compiler sources rts/ ... runtime system sources libraries/ ... core library sources ghc/ ... compiler main includes/ ... include files testsuite/ ... test suites nofib/ ... performance tests mk/ ... build system hadrian/ ... hadrian build system docs/ ... documents : : 1. Compiler 1. Compiler Compilation pipeline 1. compiler The GHC compiler Haskell language GHC compiler Assembly language (native or llvm) 1. compiler GHC compiler comprises pipeline stages GHC compiler Haskell language Parser Renamer Type checker Desugarer Core to Core Core to STG STG to Cmm Assembly language Cmm to Asm (native or llvm) 1. -
Initial Algebra Semantics in Matching Logic
Technical Report: Initial Algebra Semantics in Matching Logic Xiaohong Chen1, Dorel Lucanu2, and Grigore Ro¸su1 1University of Illinois at Urbana-Champaign, Champaign, USA 2Alexandru Ioan Cuza University, Ia¸si,Romania xc3@illinois, [email protected], [email protected] July 24, 2020 Abstract Matching logic is a unifying foundational logic for defining formal programming language semantics, which adopts a minimalist design with few primitive constructs that are enough to express all properties within a variety of logical systems, including FOL, separation logic, (dependent) type systems, modal µ-logic, and more. In this paper, we consider initial algebra semantics and show how to capture it by matching logic specifications. Formally, given an algebraic specification E that defines a set of sorts (of data) and a set of operations whose behaviors are defined by a set of equational axioms, we define a corresponding matching logic specification, denoted INITIALALGEBRA(E), whose models are exactly the initial algebras of E. Thus, we reduce initial E-algebra semantics to the matching logic specifications INITIALALGEBRA(E), and reduce extrinsic initial E-algebra reasoning, which includes inductive reasoning, to generic, intrinsic matching logic reasoning. 1 Introduction Initial algebra semantics is a main approach to formal programming language semantics based on algebraic specifications and their initial models. It originated in the 1970s, when algebraic techniques started to be applied to specify basic data types such as lists, trees, stacks, etc. The original paper on initial algebra semantics [Goguen et al., 1977] reviewed various existing algebraic specification techniques and showed that they were all initial models, making the concept of initiality explicit for the first time in formal language semantics. -
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. -
MONADS and ALGEBRAS I I
\chap10" 2009/6/1 i i page 223 i i 10 MONADSANDALGEBRAS In the foregoing chapter, the adjoint functor theorem was seen to imply that the category of algebras for an equational theory T always has a \free T -algebra" functor, left adjoint to the forgetful functor into Sets. This adjunction describes the notion of a T -algebra in a way that is independent of the specific syntactic description given by the theory T , the operations and equations of which are rather like a particular presentation of that notion. In a certain sense that we are about to make precise, it turns out that every adjunction describes, in a \syntax invariant" way, a notion of an \algebra" for an abstract \equational theory." Toward this end, we begin with yet a third characterization of adjunctions. This one has the virtue of being entirely equational. 10.1 The triangle identities Suppose we are given an adjunction, - F : C D : U: with unit and counit, η : 1C ! UF : FU ! 1D: We can take any f : FC ! D to φ(f) = U(f) ◦ ηC : C ! UD; and for any g : C ! UD we have −1 φ (g) = D ◦ F (g): FC ! D: This we know gives the isomorphism ∼ HomD(F C; D) =φ HomC(C; UD): Now put 1UD : UD ! UD in place of g : C ! UD in the foregoing. We −1 know that φ (1UD) = D, and so 1UD = φ(D) = U(D) ◦ ηUD: i i i i \chap10" 2009/6/1 i i page 224 224 MONADS AND ALGEBRAS i i And similarly, φ(1FC ) = ηC , so −1 1FC = φ (ηC ) = FC ◦ F (ηC ): Thus we have shown that the two diagrams below commute. -
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. -
On Coalgebras Over Algebras
On coalgebras over algebras Adriana Balan1 Alexander Kurz2 1University Politehnica of Bucharest, Romania 2University of Leicester, UK 10th International Workshop on Coalgebraic Methods in Computer Science A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 1 / 31 Outline 1 Motivation 2 The final coalgebra of a continuous functor 3 Final coalgebra and lifting 4 Commuting pair of endofunctors and their fixed points A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 2 / 31 Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+) Locally finitely presentable categories: Hom(B; L) completion of Hom(B; I ) for all finitely presentable objects B [Adamek 2003] Motivation Starting data: category C, endofunctor H : C −! C Among fixed points: final coalgebra, initial algebra Categories enriched over complete metric spaces: unique fixed point [Adamek, Reiterman 1994] Categories enriched over cpo: final coalgebra L coincides with initial algebra I [Plotkin, Smyth 1983] A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31 Locally finitely presentable categories: Hom(B; L) completion of Hom(B; I ) for all finitely presentable objects B [Adamek 2003] Motivation Starting data: category C, endofunctor H : C −! C Among fixed points: final coalgebra, initial algebra Categories enriched over complete metric spaces: unique fixed point [Adamek, Reiterman 1994] Categories enriched over cpo: final coalgebra L coincides with initial algebra I [Plotkin, Smyth 1983] Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+) A. -
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. -
A Note on the Under-Appreciated For-Loop
A Note on the Under-Appreciated For-Loop Jos´eN. Oliveira Techn. Report TR-HASLab:01:2020 Oct 2020 HASLab - High-Assurance Software Laboratory Universidade do Minho Campus de Gualtar – Braga – Portugal http://haslab.di.uminho.pt TR-HASLab:01:2020 A Note on the Under-Appreciated For-Loop by Jose´ N. Oliveira Abstract This short research report records some thoughts concerning a simple algebraic theory for for-loops arising from my teaching of the Algebra of Programming to 2nd year courses at the University of Minho. Interest in this so neglected recursion- algebraic combinator grew recently after reading Olivier Danvy’s paper on folding over the natural numbers. The report casts Danvy’s results as special cases of the powerful adjoint-catamorphism theorem of the Algebra of Programming. A Note on the Under-Appreciated For-Loop Jos´eN. Oliveira Oct 2020 Abstract This short research report records some thoughts concerning a sim- ple algebraic theory for for-loops arising from my teaching of the Al- gebra of Programming to 2nd year courses at the University of Minho. Interest in this so neglected recursion-algebraic combinator grew re- cently after reading Olivier Danvy’s paper on folding over the natural numbers. The report casts Danvy’s results as special cases of the pow- erful adjoint-catamorphism theorem of the Algebra of Programming. 1 Context I have been teaching Algebra of Programming to 2nd year courses at Minho Uni- versity since academic year 1998/99, starting just a few days after AFP’98 took place in Braga, where my department is located. -
Haskell-Like S-Expression-Based Language Designed for an IDE
Department of Computing Imperial College London MEng Individual Project Haskell-Like S-Expression-Based Language Designed for an IDE Author: Supervisor: Michal Srb Prof. Susan Eisenbach June 2015 Abstract The state of the programmers’ toolbox is abysmal. Although substantial effort is put into the development of powerful integrated development environments (IDEs), their features often lack capabilities desired by programmers and target primarily classical object oriented languages. This report documents the results of designing a modern programming language with its IDE in mind. We introduce a new statically typed functional language with strong metaprogramming capabilities, targeting JavaScript, the most popular runtime of today; and its accompanying browser-based IDE. We demonstrate the advantages resulting from designing both the language and its IDE at the same time and evaluate the resulting environment by employing it to solve a variety of nontrivial programming tasks. Our results demonstrate that programmers can greatly benefit from the combined application of modern approaches to programming tools. I would like to express my sincere gratitude to Susan, Sophia and Tristan for their invaluable feedback on this project, my family, my parents Michal and Jana and my grandmothers Hana and Jaroslava for supporting me throughout my studies, and to all my friends, especially to Harry whom I met at the interview day and seem to not be able to get rid of ever since. ii Contents Abstract i Contents iii 1 Introduction 1 1.1 Objectives ........................................ 2 1.2 Challenges ........................................ 3 1.3 Contributions ...................................... 4 2 State of the Art 6 2.1 Languages ........................................ 6 2.1.1 Haskell .................................... -
Dualising Initial Algebras
Math. Struct. in Comp. Science (2003), vol. 13, pp. 349–370. c 2003 Cambridge University Press DOI: 10.1017/S0960129502003912 Printed in the United Kingdom Dualising initial algebras NEIL GHANI†,CHRISTOPHLUTH¨ ‡, FEDERICO DE MARCHI†¶ and J O H NPOWER§ †Department of Mathematics and Computer Science, University of Leicester ‡FB 3 – Mathematik und Informatik, Universitat¨ Bremen §Laboratory for Foundations of Computer Science, University of Edinburgh Received 30 August 2001; revised 18 March 2002 Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonads, and, dually, initial algebras form both monads and comonads. In developing these theories we strive to provide them with an associated notion of syntax. In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras of the representing monad. We attempt to emulate this result for the coalgebraic case by first defining a notion of cosignature and coequation and then proving that the models of such coalgebraic presentations are precisely the coalgebras of the representing comonad. 1. Introduction While the theory of coalgebras for an endofunctor is well developed, less attention has been given to comonads.Wefeel this is a pity since the corresponding theory of monads on Set explains the key concepts of universal algebra such as signature, variables, terms, substitution, equations,and so on.