An investigation of the laws of traversals

Traversals of data structures are ubiquitous in programming. Consequently, it is important to be able to characterise those structures that are traversable and understand their algebraic properties. Traversable functors have been characterised by McBride and Paterson as those equipped with a distributive law over arbitrary applicative functors; however, laws that fully capture the intuition behind traversals are missing. This article is an attempt to remedy this situation by proposing laws for characterising traversals that capture the intuition behind them. To support our claims, we prove that finitary containers are traversable in our sense and argue that elements in a traversable structure are visited exactly once.Fil: Jaskelioff, Mauro Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentina. Universidad Nacional de Rosario; ArgentinaFil: Rypacek, Ondrej. King's College London; Reino Unid

Traversal Query Language For Scala.Meta

With the rise of metaprogramming in Scala, manipulating ASTs has become a daily job. Yet the standard API provides only low-level mechanisms to transform or to collect information on those data structures. Moreover, those mechanisms often force the programmer to manipulate state in order to retrieve information on these ASTs. In this report, we try to solve those problems by introducing TQL, a high-level combinator Scala library to transform and query data structures in a purely functional way. Parser combinators allow to combine several small parsers to build a bigger one in an expressive way. In this report, we argue that we can apply the same concept to data structure manipulation and construct complicated traversers on top of smaller ones. Yet combinators may feel unnatural or too complicated for certain usage. We therefore built a library on top of TQL to manipulate data structures as a collection. We then put TQL in practice to scala.meta ASTs, and describe the challenges we face when traversing a real-word data structure, especially performance-wise

Enhancing Semantic Bidirectionalization via Shape Bidirectionalizer Plug-ins

Matsuda et al. (2007) and Voigtlander (2009) have introduced two techniques that given a source-to-view function provide an update propagation function mapping an original source and an updated view back to an updated source, subject to standard consistency conditions. Previously, we developed a synthesis of the two techniques, based on a separation of shape and content aspects (Voigtlander et al. 2010). Here, we carry that idea further, reworking the technique of Voigtlander such that any shape bidirectionalizer (based on the work of Matsuda et al. or not) can be used as a plug-in, to good effect. We also provide a data-type-generic account, enabling wider reuse, including the use of pluggable bidirectionalization itself as a plug-in

Paraiso : An Automated Tuning Framework for Explicit Solvers of Partial Differential Equations

We propose Paraiso, a domain specific language embedded in functional programming language Haskell, for automated tuning of explicit solvers of partial differential equations (PDEs) on GPUs as well as multicore CPUs. In Paraiso, one can describe PDE solving algorithms succinctly using tensor equations notation. Hydrodynamic properties, interpolation methods and other building blocks are described in abstract, modular, re-usable and combinable forms, which lets us generate versatile solvers from little set of Paraiso source codes. We demonstrate Paraiso by implementing a compressive hydrodynamics solver. A single source code less than 500 lines can be used to generate solvers of arbitrary dimensions, for both multicore CPUs and GPUs. We demonstrate both manual annotation based tuning and evolutionary computing based automated tuning of the program.Comment: 52 pages, 14 figures, accepted for publications in Computational Science and Discover

Maximum Segment Sum, Monadically (distilled tutorial, with solutions)

The maximum segment sum problem is to compute, given a list of integers, the largest of the sums of the contiguous segments of that list. This problem specification maps directly onto a cubic-time algorithm; however, there is a very elegant linear-time solution too. The problem is a classic exercise in the mathematics of program construction, illustrating important principles such as calculational development, pointfree reasoning, algebraic structure, and datatype-genericity. Here, we take a sideways look at the datatype-generic version of the problem in terms of monadic functional programming, instead of the traditional relational approach; the presentation is tutorial in style, and leavened with exercises for the reader.Comment: Revision of the article in Proceedings DSL 2011, EPTCS 66, arXiv:1109.0323, to provide solutions to the exercise

Tealeaves: Structured Monads for Generic First-Order Abstract Syntax Infrastructure

Verifying the metatheory of a formal system in Coq involves a lot of tedious "infrastructural" reasoning about variable binders. We present Tealeaves, a generic framework for first-order representations of variable binding that can be used to develop this sort of infrastructure once and for all. Given a particular strategy for representing binders concretely, such as locally nameless or de Bruijn indices, Tealeaves allows developers to implement modules of generic infrastructure called backends that end users can simply instantiate to their own syntax. Our framework rests on a novel abstraction of first-order abstract syntax called a decorated traversable monad (DTM) whose equational theory provides reasoning principles that replace tedious induction on terms. To evaluate Tealeaves, we have implemented a multisorted locally nameless backend providing generic versions of the lemmas generated by LNgen. We discuss case studies where we instantiate this generic infrastructure to simply-typed and polymorphic lambda calculi, comparing our approach to other utilities

chemf : a purely functional chemistry toolkit

Background: Although programming in a type-safe and referentially transparent style offers several advantages over working with mutable data structures and side effects, this style of programming has not seen much use in chemistry-related software. Since functional programming languages were designed with referential transparency in mind, these languages offer a lot of support when writing immutable data structures and side-effects free code. We therefore started implementing our own toolkit based on the above programming paradigms in a modern, versatile programming language. Results: We present our initial results with functional programming in chemistry by first describing an immutable data structure for molecular graphs together with a couple of simple algorithms to calculate basic molecular properties before writing a complete SMILES parser in accordance with the OpenSMILES specification. Along the way we show how to deal with input validation, error handling, bulk operations, and parallelization in a purely functional way. At the end we also analyze and improve our algorithms and data structures in terms of performance and compare it to existing toolkits both object-oriented and purely functional. All code was written in Scala, a modern multi-paradigm programming language with a strong support for functional programming and a highly sophisticated type system. Conclusions: We have successfully made the first important steps towards a purely functional chemistry toolkit. The data structures and algorithms presented in this article perform well while at the same time they can be safely used in parallelized applications, such as computer aided drug design experiments, without further adjustments. This stands in contrast to existing object-oriented toolkits where thread safety of data structures and algorithms is a deliberate design decision that can be hard to implement. Finally, the level of type-safety achieved by Scala highly increased the reliability of our code as well as the productivity of the programmers involved in this project

A Representation Theorem for Second-Order Functionals

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of second-order functionals which are polymorphic over a class of functors. Types polymorphic over a class of functors are easily representable in languages such as Haskell, but are difficult to analyse and reason about. The concrete representation provided by the theorem is easier to analyse, but it might not be as convenient to implement. Therefore, depending on the task at hand, the change of representation may prove valuable in one direction or the other. We showcase the usefulness of the representation theorem with a range of examples. Concretely, we show how the representation theorem can be used to show that traversable functors are finitary containers, how parameterised coalgebras relate to very well-behaved lenses, and how algebraic effects might be implemented in a functional language

‘do’ unchained: embracing local imperativity in a purely functional language (functional pearl)

Purely functional programming languages pride themselves with reifying effects that are implicit in imperative languages into reusable and composable abstractions such as monads. This reification allows for more exact control over effects as well as the introduction of new or derived effects. However, despite libraries of more and more powerful abstractions over effectful operations being developed, syntactically the common \u27do\u27 notation still lags behind equivalent imperative code it is supposed to mimic regarding verbosity and code duplication. In this paper, we explore extending \u27do\u27 notation with other imperative language features that can be added to simplify monadic code: local mutation, early return, and iteration. We present formal translation rules that compile these features back down to purely functional code, show that the generated code can still be reasoned over using an implementation of the translation in the Lean 4 theorem prover, and formally prove the correctness of the translation rules relative to a simple static and dynamic semantics in Lean