Canonical Forms in Interactive Exercise Assistants

Interactive exercise assistants support students in practicing exercises, and acquiring procedural skills. Many mathematical topics can be practiced in such assistants. Ideally, an interactive exercise assistant not only validates final answers, but also comments on intermediate steps submitted by a student, provides hints on how to proceed, and presents worked-out examples. For these purposes, fine control over the symbolic simplification procedures of the underlying mathematical machinery is needed. In this paper, we introduce views for mathematical expressions. A view defines an equivalence relation by choosing a canonical form of mathematical expressions. We use views to track and recognize intermediate answers, to help in presenting expressions to a user, and to control the granularity of the steps in worked-out examples. We develop the concept of a view, discuss the laws it satisfies, and show how views are composed, which means that they can be used for multiple exercise classes.

Fusing Generic Functions

Contains fulltext : 60458.pdf (author's version ) (Open Access)Generic programming is accepted by the functional programming community as a valuable tool for program development. Several functional languages have adopted the generic scheme of type-indexed values. This scheme works by specialization of a generic function to a concrete type. However, the generated code is extremely inefficient compared to its hand-written counterpart. The performance penalty is so big that the practical usefulness of generic programming is compromised. In this paper we present a optimization algorithm that is able to completely eliminate the overhead intoduced by the specialization scheme for a large class of generic functions. The presented technique is based on consumer-producer elimination as exploited by fusion, a standard general purpose optimization method. We show that our algorithm is able to optimize many practical examples of generic functions

Efficient Generic Functional Programming

Contains fulltext : 60606.pdf (author's version ) (Open Access)Generic functions are defined by induction on the structural representation of types. As a consequence, by defining just a single generic operation, one acquires this operation over any particular data type. An instance on a specific type is generated by interpretation of the type's structure. A direct translation leads to extremely inefficient code that involves many conversions between types and their structural representations. In this paper we present an optimization technique based on compile-time symbolic evaluation. We prove that the optimization removes the overhead of the generated code for a considerable class of generic functions. The proof uses typing to identify intermediate data structures that should be eliminated. In essence, the output after optimization is similar to hand-written code

Dynamic Values Need Polytypic Functions - Draft -

Contains fulltext : 176175.pdf (preprint version ) (Open Access

Dynamic Construction of Generic Functions

This paper presents a library for the run-time construction and specialisation of generic or polytypic functions. This library utilises the type information that is available in dynamics to implement generic functions on their values. The library closely follows the static generic framework, both in its use and in its implementation. It can dynamically construct generic operations ranging from equality, map and parsers to pretty printers and generic graphical editors. A special feature of the library is that it can also be used to derive meaningful specialisations of generic functions that operate on the type representation of the dynamic

GAST: Generic Automated Software Testing

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There and Back Again: Arrows for Invertible Programming

Contains fulltext : 33151.pdf (publisher's version ) (Closed access) Contains fulltext : 33151.pdf (postprint version ) (Open Access)nul