A C++11 implementation of arbitrary-rank tensors for high-performance computing

This article discusses an efficient implementation of tensors of arbitrary rank by using some of the idioms introduced by the recently published C++ ISO Standard (C++11). With the aims at providing a basic building block for high-performance computing, a single Array class template is carefully crafted, from which vectors, matrices, and even higher-order tensors can be created. An expression template facility is also built around the array class template to provide convenient mathematical syntax. As a result, by using templates, an extra high-level layer is added to the C++ language when dealing with algebraic objects and their operations, without compromising performance. The implementation is tested running on both CPU and GPU.Comment: 21 pages, 6 figures, 1 tabl

Canonical Abstract Syntax Trees

This paper presents Gom, a language for describing abstract syntax trees and generating a Java implementation for those trees. Gom includes features allowing the user to specify and modify the interface of the data structure. These features provide in particular the capability to maintain the internal representation of data in canonical form with respect to a rewrite system. This explicitly guarantees that the client program only manipulates normal forms for this rewrite system, a feature which is only implicitly used in many implementations

The Use of the Binding Argument in the Debate about Location

In this paper I inquire into the methodological status of one of the arguments that have figured prominently in contemporary debates about the semantics of a variety of expressions, the so-called “Binding Argument”. My inquiry is limited to the case of meteorological sentences like “It is raining”, but my conclusion can be extended to other types of sentences as well. Following Jason Stanley, I distinguish between three interpretations of the argument. My focus is on the third, weakest interpretation, according to which postulating variables for locations in the logical form of meteorological sentences is the best available explanation of the binding phenomena that such sentences give rise to. My aim is to show that even in this weak interpretation, the argument cannot be reasonably taken to hold. I accomplish this by showcasing several alternative ways to account for the binding data that have not been, as of yet, ruled out as flawed

MatchPy: A Pattern Matching Library

Pattern matching is a powerful tool for symbolic computations, based on the well-defined theory of term rewriting systems. Application domains include algebraic expressions, abstract syntax trees, and XML and JSON data. Unfortunately, no lightweight implementation of pattern matching as general and flexible as Mathematica exists for Python Mathics,MacroPy,patterns,PyPatt. Therefore, we created the open source module MatchPy which offers similar pattern matching functionality in Python using a novel algorithm which finds matches for large pattern sets more efficiently by exploiting similarities between patterns.Comment: arXiv admin note: substantial text overlap with arXiv:1710.0007

Array operators using multiple dispatch: a design methodology for array implementations in dynamic languages

Arrays are such a rich and fundamental data type that they tend to be built into a language, either in the compiler or in a large low-level library. Defining this functionality at the user level instead provides greater flexibility for application domains not envisioned by the language designer. Only a few languages, such as C++ and Haskell, provide the necessary power to define $n$-dimensional arrays, but these systems rely on compile-time abstraction, sacrificing some flexibility. In contrast, dynamic languages make it straightforward for the user to define any behavior they might want, but at the possible expense of performance. As part of the Julia language project, we have developed an approach that yields a novel trade-off between flexibility and compile-time analysis. The core abstraction we use is multiple dispatch. We have come to believe that while multiple dispatch has not been especially popular in most kinds of programming, technical computing is its killer application. By expressing key functions such as array indexing using multi-method signatures, a surprising range of behaviors can be obtained, in a way that is both relatively easy to write and amenable to compiler analysis. The compact factoring of concerns provided by these methods makes it easier for user-defined types to behave consistently with types in the standard library.Comment: 6 pages, 2 figures, workshop paper for the ARRAY '14 workshop, June 11, 2014, Edinburgh, United Kingdo

Mechanising syntax with binders in Coq

Mechanising binders in general-purpose proof assistants such as Coq is cumbersome and difficult. Yet binders, substitutions, and instantiation of terms with substitutions are a critical ingredient of many programming languages. Any practicable mechanisation of the meta-theory of the latter hence requires a lean formalisation of the former. We investigate the topic from three angles: First, we realise formal systems with binders based on both pure and scoped de Bruijn algebras together with basic syntactic rewriting lemmas and automation. We automate this process in a compiler called Autosubst; our final tool supports many-sorted, variadic, and modular syntax. Second, we justify our choice of realisation and mechanise a proof of convergence of the sigma calculus, a calculus of explicit substitutions that is complete for equality of the de Bruijn algebra corresponding to the lambda calculus. Third, to demonstrate the practical usefulness of our approach, we provide concise, transparent, and accessible mechanised proofs for a variety of case studies refined to de Bruijn substitutions.Die Mechanisierung von Bindern in universellen Beweisassistenten wie Coq ist arbeitsaufwändig und schwierig. Binder, Substitutionen und die Instantiierung von Substitutionen sind jedoch kritischer Bestandteil vieler Programmiersprachen. Deshalb setzt eine praktikable Mechanisierung der Metatheorie von Programmiersprachen eine elegante Formalisierung von Bindern voraus. Wir nähern uns dem Thema aus drei Richtungen an: Zuerst realisieren wir formale Systeme mit Bindern mit Hilfe von reinen und indizierten de Bruijn Algebren, zusammen mit grundlegenden syntaktischen Gleichungen und Automatisierung. Wir automatisieren diesen Prozess in einem Kompilierer namens Autosubst. Unser finaler Kompilierer unterstützt Sortenlogik, variadische Syntax und modulare Syntax. Zweitens rechtfertigen wir unsere Repräsentation und mechanisieren einen Beweis der Konvergenz des SP-Kalküls, einem Kalkül expliziter Substitutionen der bezüglich der Gleichheit der puren de Bruijn Algebra des -Kalküls vollständig ist. Drittens entwickeln wir kurze, transparente und leicht zugängliche mechanisierte Beweise für diverse Fallstudien, die wir an de Bruijn Substitutionen angepasst haben. Wir weisen so die praktische Anwendbarkeit unseres Ansatzes nach

Efficient Pattern Matching in Python

Pattern matching is a powerful tool for symbolic computations. Applications include term rewriting systems, as well as the manipulation of symbolic expressions, abstract syntax trees, and XML and JSON data. It also allows for an intuitive description of algorithms in the form of rewrite rules. We present the open source Python module MatchPy, which offers functionality and expressiveness similar to the pattern matching in Mathematica. In particular, it includes syntactic pattern matching, as well as matching for commutative and/or associative functions, sequence variables, and matching with constraints. MatchPy uses new and improved algorithms to efficiently find matches for large pattern sets by exploiting similarities between patterns. The performance of MatchPy is investigated on several real-world problems