Type Theory & Functional Programming
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Program #6: Word Count
CSc 227 — Program Design and Development Spring 2014 (McCann) http://www.cs.arizona.edu/classes/cs227/spring14/ Program #6: Word Count Due Date: March 11 th, 2014, at 9:00 p.m. MST Overview: The UNIX operating system (and its variants, of which Linux is one) includes quite a few useful utility programs. One of those is wc, which is short for Word Count. The purpose of wc is to give users an easy way to determine the size of a text file in terms of the number of lines, words, and bytes it contains. (It can do a bit more, but that’s all of the functionality that we are concerned with for this assignment.) Counting lines is done by looking for “end of line” characters (\n (ASCII 10) for UNIX text files, or the pair \r\n (ASCII 13 and 10) for Windows/DOS text files). Counting words is also straight–forward: Any sequence of characters not interrupted by “whitespace” (spaces, tabs, end–of–line characters) is a word. Of course, whitespace characters are characters, and need to be counted as such. A problem with wc is that it generates a very minimal output format. Here’s an example of what wc produces on a Linux system when asked to count the content of a pair of files; we can do better! $ wc prog6a.dat prog6b.dat 2 6 38 prog6a.dat 32 321 1883 prog6b.dat 34 327 1921 total Assignment: Write a Java program (completely documented according to the class documentation guidelines, of course) that counts lines, words, and bytes (characters) of text files. -
Pragmatic Quotient Types in Coq Cyril Cohen
Pragmatic Quotient Types in Coq Cyril Cohen To cite this version: Cyril Cohen. Pragmatic Quotient Types in Coq. International Conference on Interactive Theorem Proving, Jul 2013, Rennes, France. pp.16. hal-01966714 HAL Id: hal-01966714 https://hal.inria.fr/hal-01966714 Submitted on 29 Dec 2018 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. Pragmatic Quotient Types in Coq Cyril Cohen Department of Computer Science and Engineering University of Gothenburg [email protected] Abstract. In intensional type theory, it is not always possible to form the quotient of a type by an equivalence relation. However, quotients are extremely useful when formalizing mathematics, especially in algebra. We provide a Coq library with a pragmatic approach in two complemen- tary components. First, we provide a framework to work with quotient types in an axiomatic manner. Second, we program construction mecha- nisms for some specific cases where it is possible to build a quotient type. This library was helpful in implementing the types of rational fractions, multivariate polynomials, field extensions and real algebraic numbers. Keywords: Quotient types, Formalization of mathematics, Coq Introduction In set-based mathematics, given some base set S and an equivalence ≡, one may see the quotient (S/ ≡) as the partition {π(x) | x ∈ S} of S into the sets π(x) ˆ= {y ∈ S | x ≡ y}, which are called equivalence classes. -
Metaobject Protocols: Why We Want Them and What Else They Can Do
Metaobject protocols: Why we want them and what else they can do Gregor Kiczales, J.Michael Ashley, Luis Rodriguez, Amin Vahdat, and Daniel G. Bobrow Published in A. Paepcke, editor, Object-Oriented Programming: The CLOS Perspective, pages 101 ¾ 118. The MIT Press, Cambridge, MA, 1993. © Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. Metaob ject Proto cols WhyWeWant Them and What Else They Can Do App ears in Object OrientedProgramming: The CLOS Perspective c Copyright 1993 MIT Press Gregor Kiczales, J. Michael Ashley, Luis Ro driguez, Amin Vahdat and Daniel G. Bobrow Original ly conceivedasaneat idea that could help solve problems in the design and implementation of CLOS, the metaobject protocol framework now appears to have applicability to a wide range of problems that come up in high-level languages. This chapter sketches this wider potential, by drawing an analogy to ordinary language design, by presenting some early design principles, and by presenting an overview of three new metaobject protcols we have designed that, respectively, control the semantics of Scheme, the compilation of Scheme, and the static paral lelization of Scheme programs. Intro duction The CLOS Metaob ject Proto col MOP was motivated by the tension b etween, what at the time, seemed liketwo con icting desires. The rst was to have a relatively small but p owerful language for doing ob ject-oriented programming in Lisp. The second was to satisfy what seemed to b e a large numb er of user demands, including: compatibility with previous languages, p erformance compara- ble to or b etter than previous implementations and extensibility to allow further exp erimentation with ob ject-oriented concepts see Chapter 2 for examples of directions in which ob ject-oriented techniques might b e pushed. -
DC Console Using DC Console Application Design Software
DC Console Using DC Console Application Design Software DC Console is easy-to-use, application design software developed specifically to work in conjunction with AML’s DC Suite. Create. Distribute. Collect. Every LDX10 handheld computer comes with DC Suite, which includes seven (7) pre-developed applications for common data collection tasks. Now LDX10 users can use DC Console to modify these applications, or create their own from scratch. AML 800.648.4452 Made in USA www.amltd.com Introduction This document briefly covers how to use DC Console and the features and settings. Be sure to read this document in its entirety before attempting to use AML’s DC Console with a DC Suite compatible device. What is the difference between an “App” and a “Suite”? “Apps” are single applications running on the device used to collect and store data. In most cases, multiple apps would be utilized to handle various operations. For example, the ‘Item_Quantity’ app is one of the most widely used apps and the most direct means to take a basic inventory count, it produces a data file showing what items are in stock, the relative quantities, and requires minimal input from the mobile worker(s). Other operations will require additional input, for example, if you also need to know the specific location for each item in inventory, the ‘Item_Lot_Quantity’ app would be a better fit. Apps can be used in a variety of ways and provide the LDX10 the flexibility to handle virtually any data collection operation. “Suite” files are simply collections of individual apps. Suite files allow you to easily manage and edit multiple apps from within a single ‘store-house’ file and provide an effortless means for device deployment. -
Countability of Inductive Types Formalized in the Object-Logic Level
Countability of Inductive Types Formalized in the Object-Logic Level QinxiangCao XiweiWu * Abstract: The set of integer number lists with finite length, and the set of binary trees with integer labels are both countably infinite. Many inductively defined types also have countably many ele- ments. In this paper, we formalize the syntax of first order inductive definitions in Coq and prove them countable, under some side conditions. Instead of writing a proof generator in a meta language, we develop an axiom-free proof in the Coq object logic. In other words, our proof is a dependently typed Coq function from the syntax of the inductive definition to the countability of the type. Based on this proof, we provide a Coq tactic to automatically prove the countability of concrete inductive types. We also developed Coq libraries for countability and for the syntax of inductive definitions, which have value on their own. Keywords: countable, Coq, dependent type, inductive type, object logic, meta logic 1 Introduction In type theory, a system supports inductive types if it allows users to define new types from constants and functions that create terms of objects of that type. The Calculus of Inductive Constructions (CIC) is a powerful language that aims to represent both functional programs in the style of the ML language and proofs in higher-order logic [16]. Its extensions are used as the kernel language of Coq [5] and Lean [14], both of which are widely used, and are widely considered to be a great success. In this paper, we focus on a common property, countability, of all first-order inductive types, and provide a general proof in Coq’s object-logic. -
Proving Termination of Evaluation for System F with Control Operators
Proving termination of evaluation for System F with control operators Małgorzata Biernacka Dariusz Biernacki Sergue¨ıLenglet Institute of Computer Science Institute of Computer Science LORIA University of Wrocław University of Wrocław Universit´ede Lorraine [email protected] [email protected] [email protected] Marek Materzok Institute of Computer Science University of Wrocław [email protected] We present new proofs of termination of evaluation in reduction semantics (i.e., a small-step opera- tional semantics with explicit representation of evaluation contexts) for System F with control oper- ators. We introduce a modified version of Girard’s proof method based on reducibility candidates, where the reducibility predicates are defined on values and on evaluation contexts as prescribed by the reduction semantics format. We address both abortive control operators (callcc) and delimited- control operators (shift and reset) for which we introduce novel polymorphic type systems, and we consider both the call-by-value and call-by-name evaluation strategies. 1 Introduction Termination of reductions is one of the crucial properties of typed λ-calculi. When considering a λ- calculus as a deterministic programming language, one is usually interested in termination of reductions according to a given evaluation strategy, such as call by value or call by name, rather than in more general normalization properties. A convenient format to specify such strategies is reduction semantics, i.e., a form of operational semantics with explicit representation of evaluation (reduction) contexts [14], where the evaluation contexts represent continuations [10]. Reduction semantics is particularly convenient for expressing non-local control effects and has been most successfully used to express the semantics of control operators such as callcc [14], or shift and reset [5]. -
Types Are Internal Infinity-Groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau
Types are internal infinity-groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau To cite this version: Antoine Allioux, Eric Finster, Matthieu Sozeau. Types are internal infinity-groupoids. 2021. hal- 03133144 HAL Id: hal-03133144 https://hal.inria.fr/hal-03133144 Preprint submitted on 5 Feb 2021 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. Types are Internal 1-groupoids Antoine Allioux∗, Eric Finstery, Matthieu Sozeauz ∗Inria & University of Paris, France [email protected] yUniversity of Birmingham, United Kingdom ericfi[email protected] zInria, France [email protected] Abstract—By extending type theory with a universe of defini- attempts to import these ideas into plain homotopy type theory tionally associative and unital polynomial monads, we show how have, so far, failed. This appears to be a result of a kind of to arrive at a definition of opetopic type which is able to encode circularity: all of the known classical techniques at some point a number of fully coherent algebraic structures. In particular, our approach leads to a definition of 1-groupoid internal to rely on set-level algebraic structures themselves (presheaves, type theory and we prove that the type of such 1-groupoids is operads, or something similar) as a means of presenting or equivalent to the universe of types. -
Acm Names Fellows for Innovations in Computing
CONTACT: Virginia Gold 212-626-0505 [email protected] ACM NAMES FELLOWS FOR INNOVATIONS IN COMPUTING 2014 Fellows Made Computing Contributions to Enterprise, Entertainment, and Education NEW YORK, January 8, 2015—ACM has recognized 47 of its members for their contributions to computing that are driving innovations across multiple domains and disciplines. The 2014 ACM Fellows, who hail from some of the world’s leading universities, corporations, and research labs, have achieved advances in computing research and development that are driving innovation and sustaining economic development around the world. ACM President Alexander L. Wolf acknowledged the advances made by this year’s ACM Fellows. “Our world has been immeasurably improved by the impact of their innovations. We recognize their contributions to the dynamic computing technologies that are making a difference to the study of computer science, the community of computing professionals, and the countless consumers and citizens who are benefiting from their creativity and commitment.” The 2014 ACM Fellows have been cited for contributions to key computing fields including database mining and design; artificial intelligence and machine learning; cryptography and verification; Internet security and privacy; computer vision and medical imaging; electronic design automation; and human-computer interaction. ACM will formally recognize the 2014 Fellows at its annual Awards Banquet in June in San Francisco. Additional information about the ACM 2014 Fellows, the awards event, as well as previous -
Journal of Applied Logic
JOURNAL OF APPLIED LOGIC AUTHOR INFORMATION PACK TABLE OF CONTENTS XXX . • Description p.1 • Impact Factor p.1 • Abstracting and Indexing p.1 • Editorial Board p.1 • Guide for Authors p.5 ISSN: 1570-8683 DESCRIPTION . This journal welcomes papers in the areas of logic which can be applied in other disciplines as well as application papers in those disciplines, the unifying theme being logics arising from modelling the human agent. For a list of areas covered see the Editorial Board. The editors keep close contact with the various application areas, with The International Federation of Compuational Logic and with the book series Studies in Logic and Practical Reasoning. Benefits to authors We also provide many author benefits, such as free PDFs, a liberal copyright policy, special discounts on Elsevier publications and much more. Please click here for more information on our author services. Please see our Guide for Authors for information on article submission. This journal has an Open Archive. All published items, including research articles, have unrestricted access and will remain permanently free to read and download 48 months after publication. All papers in the Archive are subject to Elsevier's user license. If you require any further information or help, please visit our Support Center IMPACT FACTOR . 2016: 0.838 © Clarivate Analytics Journal Citation Reports 2017 ABSTRACTING AND INDEXING . Zentralblatt MATH Scopus EDITORIAL BOARD . Executive Editors Dov M. Gabbay, King's College London, London, UK Sarit Kraus, Bar-llan University, -
Sequent Calculus and Equational Programming (Work in Progress)
Sequent Calculus and Equational Programming (work in progress) Nicolas Guenot and Daniel Gustafsson IT University of Copenhagen {ngue,dagu}@itu.dk Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums. 1 Programming with Equations Functional programming has proved extremely useful in making the task of writing correct software more abstract and thus less tied to the specific, and complex, architecture of modern computers. This, is in a large part, due to its extensive use of types as an abstraction mechanism, specifying in a crisp way the intended behaviour of a program, but it also relies on its declarative style, as a mathematical approach to functions and data structures. However, the vast gain in expressivity obtained through the development of dependent types makes the programming task more challenging, as it amounts to the question of proving complex theorems — as illustrated by the double nature of proof assistants such as Coq [11] and Agda [18]. Keeping this task as simple as possible is then of the highest importance, and it requires the use of a clear declarative style. -
On Modeling Homotopy Type Theory in Higher Toposes
Review: model categories for type theory Left exact localizations Injective fibrations On modeling homotopy type theory in higher toposes Mike Shulman1 1(University of San Diego) Midwest homotopy type theory seminar Indiana University Bloomington March 9, 2019 Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Some caveats 1 Classical metatheory: ZFC with inaccessible cardinals. 2 Classical homotopy theory: simplicial sets. (It's not clear which cubical sets can even model the (1; 1)-topos of 1-groupoids.) 3 Will not mention \elementary (1; 1)-toposes" (though we can deduce partial results about them by Yoneda embedding). 4 Not the full \internal language hypothesis" that some \homotopy theory of type theories" is equivalent to the homotopy theory of some kind of (1; 1)-category. Only a unidirectional interpretation | in the useful direction! 5 We assume the initiality hypothesis: a \model of type theory" means a CwF. -
Multi-Level Constraints
Multi-Level Constraints Tony Clark1 and Ulrich Frank2 1 Aston University, UK, [email protected] 2 University of Duisburg-Essen, DE, [email protected] Abstract. Meta-modelling and domain-specific modelling languages are supported by multi-level modelling which liberates model-based engi- neering from the traditional two-level type-instance language architec- ture. Proponents of this approach claim that multi-level modelling in- creases the quality of the resulting systems by introducing a second ab- straction dimension and thereby allowing both intra-level abstraction via sub-typing and inter-level abstraction via meta-types. Modelling ap- proaches include constraint languages that are used to express model semantics. Traditional languages, such as OCL, support intra-level con- straints, but not inter-level constraints. This paper motivates the need for multi-level constraints, shows how to implement such a language in a reflexive language architecture and applies multi-level constraints to an example multi-level model. 1 Introduction Conceptual models aim to bridge the gap between natural languages that are required to design and use a system and implementation languages. To this end, general-purpose modelling languages (GPML) like the UML consist of concepts that represent semantic primitives such as class, attribute, etc., that, on the one hand correspond to concepts of foundational ontologies, e.g., [4], and on the other hand can be nicely mapped to corresponding elements of object-oriented programming languages. Since GPML can be used to model a wide range of systems, they promise at- tractive economies of scale. At the same time, their use suffers from the fact that they offer generic concepts only.