A type-indexed function is a function that is defined for each member of some family of types. Haskell's type class mechanism provides collections of open type-indexed functions, in which the indexing family can be extended by defining a new type class instance but the collection of functions is fixed. The purpose of this paper is to present TypeCase: a design pattern that allows the definition of closed type-indexed functions, in which the index family is fixed but the collection of functions is extensible. It is inspired by Cheney and Hinze's work on lightweight approaches to generic programming. We generalise their techniques as a design pattern. Furthermore, we show that type-indexed functions with typeindexed types, and consequently generic functions with generic types, can also be encoded in a lightweight manner, thereby overcoming one of the main limitations of the lightweight approaches.

Datatype-generic programming is defining functions that depend on the structure, or “shape”, of datatypes. It has been around for more than 10 years, and a lot of progress has been made, in particular in the lazy functional programming language Haskell. There are more than 10 proposals for generic programming libraries or language extensions for Haskell. To compare and characterize the many generic programming libraries in a typed functional language, we introduce a set of criteria and develop a generic programming benchmark: a set of characteristic examples testing various facets of datatype-generic programming. We have implemented the benchmark for nine existing Haskell generic programming libraries and present the evaluation of the libraries. The comparison is useful for reaching a common standard for generic programming, but also for a programmer who has to choose a particular approach for datatype-generic programming.

The ITERATOR pattern gives a clean interface for element-by-element access to a collection. Imperative iterations using the pattern have two simultaneous aspects: mapping and accumulating. Various existing functional iterations model one or other of these, but not both simultaneously. We argue that McBride and Paterson’s idioms, and in particular the corresponding traverse operator, do exactly this, and therefore capture the essence of the ITERATOR pattern. We present some axioms for traversal, and illustrate with a simple example, the repmin problem.

Abstract. The last decade has seen a number of approaches to datatype-generic programming: PolyP, Functorial ML, ‘Scrap Your Boilerplate’, Generic Haskell, ‘Generics for the Masses’, etc. The approaches vary in sophistication and target audience: some propose full-blown programming languages, some suggest libraries, some can be seen as categorical programming methods. In these lecture notes we compare the various approaches to datatype-generic programming in Haskell. We introduce each approach by means of example, and we evaluate it along different dimensions (expressivity, ease of use, etc). 1

Introduction Arithmetic coding is a method for data compression. Although the idea was developed in the 1970's, it wasn't until the publication of an "accessible implementation " [14] that it achieved the popularity it has today. Over the past ten years arithmetic coding has been refined and its advantages and disadvantages over rival compression schemes, particularly Hu#man [9] and Shannon-Fano [5] coding, have been elucidated. Arithmetic coding produces a theoretically optimal compression under much weaker assumptions than Hu#man and Shannon-Fano, and can compress within one bit of the limit imposed by Shannon's Noiseless Coding Theorem [13]. Additionally, arithmetic coding is well suited to adaptive coding schemes, both character and word based. For recent perspectives on the subject, see [10, 12]. The "accessible implementation" of [14] consisted of a 300 line C program, and much of the paper was a blow-by-blow description of the workings of the code. There was little in the way

Unfolds generate data structures, and folds consume them.

Datatype-generic programs are programs that are parametrized by a datatype or type functor: whereas polymorphic programs abstract from the ‘integers ’ in ‘lists of integers’, datatype-generic programs abstract from the ‘lists of’. There are two main styles of datatype-generic programming: the Algebra of Programming approach, characterized by structured recursion operators arising from initial algebras and final coalgebras, and the Generic Haskell approach, characterized by case analysis over the structure of a datatype. We show that the former enjoys a kind of higherorder naturality, relating the behaviours of generic functions at different types; in contrast, the latter is ad hoc, with no coherence required or provided between the various clauses of a definition. Moreover, the naturality properties arise ‘for free’, simply from the parametrized types of the generic functions: we present a higherorder parametricity theorem for datatype-generic operators. Categories and Subject Descriptors D.3.3 [Programming languages]: Language constructs and features—Polymorphism, patterns, control structures, recursion; F.3.3 [Logics and meanings of programs]: Studies of program constructs—Program and recursion schemes, type structure; F.3.2 [Logics and meanings of programs]: Semantics of programming languages—Algebraic approaches to semantics; D.3.2 [Programming languages]: Language classifications—Functional languages.

Abstract. We argue that abstract datatypes — with public interfaces hiding private implementations — represent a form of codata rather than ordinary data, and hence that proof methods for corecursive programs are the appropriate techniques to use for reasoning with them. In particular, we show that the universal properties of unfold operators are perfectly suited for the task. We illustrate with the solution to a problem in the recent literature. 1

We show how the binary encoding and decoding of typed data and typed programs can be understood, programmed, and verified with the help of question-answer games. The encoding of a value is determined by the yes/no answers to a sequence of questions about that value; conversely, decoding is the interpretation of binary data as answers to the same question scheme. We introduce a general framework for writing and verifying gamebased codecs. We present games for structured, recursive, polymorphic, and indexed types, building up to a representation of well-typed terms in the simply-typed λ-calculus. The framework makes novel use of isomorphisms between types in the definition of games. The definition of isomorphisms together with additional simple properties make it easy to prove that codecs derived from games never encode two distinct values using the same code, never decode two codes to the same value, and interpret any bit sequence as a valid code for a value or as a prefix of a valid code.

The UNIX diff program finds the difference between two text files using a classic algorithm for determining the longest common subsequence; however, when working with structured input (e.g. program code), we often want to find the difference between tree-like data (e.g. the abstract syntax tree). In a functional programming language such as Haskell, we can represent this data with a family of (mutually recursive) datatypes. In this paper, we describe a functional, datatype-generic implementation of diff (and the associated program patch). Our approach requires advanced type system features to preserve type safety; therefore, we present the code in Agda, a dependently-typed language well-suited to datatypegeneric programming. In order to establish the usefulness of our work, we show that its efficiency can be improved with memoization and that it can also be defined in Haskell.