Foundations for structured programming with GADTs

GADTs are at the cutting edge of functional programming and become more widely used every day. Nevertheless, the semantic foundations underlying GADTs are not well understood. In this paper we solve this problem by showing that the standard theory of data types as carriers of initial algebras of functors can be extended from algebraic and nested data types to GADTs. We then use this observation to derive an initial algebra semantics for GADTs, thus ensuring that all of the accumulated knowledge about initial algebras can be brought to bear on them. Next, we use our initial algebra semantics for GADTs to derive expressive and principled tools — analogous to the well-known and widely-used ones for algebraic and nested data types — for reasoning about, programming with, and improving the performance of programs involving, GADTs; we christen such a collection of tools for a GADT an initial algebra package. Along the way, we give a constructive demonstration that every GADT can be reduced to one which uses only the equality GADT and existential quantification. Although other such reductions exist in the literature, ours is entirely local, is independent of any particular syntactic presentation of GADTs, and can be implemented in the host language, rather than existing solely as a metatheoretical artifact. The main technical ideas underlying our approach are (i) to modify the notion of a higher-order functor so that GADTs can be seen as carriers of initial algebras of higher-order functors, and (ii) to use left Kan extensions to trade arbitrary GADTs for simpler-but-equivalent ones for which initial algebra semantics can be derive

A principled approach to programming with nested types in Haskell

Initial algebra semantics is one of the cornerstones of the theory of modern functional programming languages. For each inductive data type, it provides a Church encoding for that type, a build combinator which constructs data of that type, a fold combinator which encapsulates structured recursion over data of that type, and a fold/build rule which optimises modular programs by eliminating from them data constructed using the buildcombinator, and immediately consumed using the foldcombinator, for that type. It has long been thought that initial algebra semantics is not expressive enough to provide a similar foundation for programming with nested types in Haskell. Specifically, the standard folds derived from initial algebra semantics have been considered too weak to capture commonly occurring patterns of recursion over data of nested types in Haskell, and no build combinators or fold/build rules have until now been defined for nested types. This paper shows that standard folds are, in fact, sufficiently expressive for programming with nested types in Haskell. It also defines buildcombinators and fold/build fusion rules for nested types. It thus shows how initial algebra semantics provides a principled, expressive, and elegant foundation for programming with nested types in Haskell

Fold-based fusion as a library: a generative programming pearl

Fusion is a program optimisation technique commonly implemented using special-purpose compiler support. In this paper, we present an alternative approach, implementing fold-based fusion as a standalone library. We use staging to compose operations on folds; the operations are partially evaluated away, yielding code that does not construct unnecessary intermediate data structures. The technique extends to partitioning and grouping of collections

A Family Of Syntactic Logical Relations For The Semantics Of Haskell-Like Languages

Logical relations are a fundamental and powerful tool for reasoning about programs in languages with parametric polymorphism. Logical relations suitable for reasoning about observational behavior in polymorphic calculi supporting various programming language features have been introduced in recent years. Unfortunately, the calculi studied are typically idealized, and the results obtained for them over only partial insight into the impact of such features on observational behavior in implemented languages. In this paper we show how to bring reasoning via logical relations closer to bear on real languages by deriving results that are more pertinent to an intermediate language for the (mostly) lazy functional language Haskell like GHC Core. To provide a more ?ne-grained analysis of program behavior than is possible by reasoning about program equivalence alone, we work with an abstract notion of relating observational behavior of computations which has among its specializations both observational equivalence and observational approximation. We take selective strictness into account, and we consider the impact of different kinds of computational failure, e.g., divergence versus failed pattern matching, because such distinctions are significant in practice. Once distinguished, the relative de?nedness of different failure causes needs to be considered, because different orders here induce different observational relations on programs (including the choice between equivalence and approximation). Our main contribution is the construction of an entire family of logical relations, parameterized over a definedness order on failure causes, each member of which characterizes the corresponding observational relation. Although we deal with properties very much tied to types, we base our results on a type-erasing semantics since this is more faithful to actual implementations

Monadic augment and generalised short cut fusion

Monads are commonplace programming devices that are used to uniformly structure computations with effects such as state, exceptions, and I/O. This paper further develops the monadic programming paradigm by investigating the extent to which monadic computations can be optimised by using generalisations of short cut fusion to eliminate monadic structures whose sole purpose is to “glue together ” monadic program components. We make several contributions. First, we show that every inductive type has an associated build combinator and an associated short cut fusion rule. Second, we introduce the notion of an inductive monad to describe those monads that give rise to inductive types, and we give examples of such monads which are widely used in functional programming. Third, we generalise the standard augment combinators and cata/augment fusion rules for algebraic data types to types induced by inductive monads. This allows us to give the first cata/augment rules for some common data types, such as rose trees. Fourth, we demonstrate the practical applicability of our generalisations by providing Haskell implementations for all concepts and examples in the paper. Finally, we offer deep theoretical insights by showing that the augment combinators are monadic in nature, and thus that our cata/build and cata/augment rules are arguably the best generally applicable fusion rules obtainable

Functional Query Languages with Categorical Types

We study three category-theoretic types in the context of functional query languages (typed lambda-calculi extended with additional operations for bulk data processing). The types we study are:Engineering and Applied Science

Tools for Reasoning about Effectful Declarative Programs

In the pure functional language Haskell, nearly all side-effects that a function can produce have to be noted in its type. This includes input/output, propagation of a state, and nondeterminism. If no side-effects are noted, such a function acts like a mathematical function, i.e., mapping arguments to unique results. In that case, expressions in a program can be reasoned about like mathematical expressions. In addition to this socalled equational reasoning, the type system also enables type based reasoning. One example are free theorems - equations between expressions that are true only due to the types of the expressions involved. Some such statements serve as formal justification for optimization strategies in compilers. The thesis at hand investigates two generalizations of such methods for programs not free of side-effects, i.e., effectful programs. First, effectful traversals of data structures are being studied. The most important contribution in this part is that a data structure can be lawfully traversed if, and only if, it is isomorphic to a polynomial functor. This result links the widespread interface of traversing to a clear intuition regarding the structure and behavior of the data type. Furthermore, tools are presented facilitating convenient proofs about effectful traversals. Second, free theorems for the functional-logic language Curry are derived. Due to the close relationship between both languages, Curry can be understood as Haskell with built-in nondeterminism, i.e., a built-in side-effect. Equational and type based reasoning can both be adapted to Curry to a certain degree. In particular, short cut fusion - a very fertile runtime optimization - is enabled for Curry

Specialising Parsers for Queries

Many software systems consist of data processing components that analyse large datasets to gather information and learn from these. Often, only part of the data is relevant for analysis. Data processing systems contain an initial preprocessing step that filters out the unwanted information. While efficient data analysis techniques and methodologies are accessible to non-expert programmers, data preprocessing seems to be forgotten, or worse, ignored. This despite real performance gains being possible by efficiently preprocessing data. Implementations of the data preprocessing step traditionally have to trade modularity for performance: to achieve the former, one separates the parsing of raw data and filtering it, and leads to slow programs because of the creation of intermediate objects during execution. The efficient version is a low-level implementation that interleaves parsing and querying. In this dissertation we demonstrate a principled and practical technique to convert the modular, maintainable program into its interleaved efficient counterpart. Key to achieving this objective is the removal, or deforestation, of intermediate objects in a program execution. We first show that by encoding data types using Böhm-Berarducci encodings (often referred to as Church encodings), and combining these with partial evaluation for function composition we achieve deforestation. This allows us to implement optimisations themselves as libraries, with minimal dependence on an underlying optimising compiler. Next we illustrate the applicability of this approach to parsing and preprocessing queries. The approach is general enough to cover top-down and bottom-up parsing techniques, and deforestation of pipelines of operations on lists and streams. We finally present a set of transformation rules that for a parser on a nested data format and a query on the structure, produces a parser specialised for the query. As a result we preserve the modularity of writing parsers and queries separately while also minimising resource usage. These transformation rules combine deforested implementations of both libraries to yield an efficient, interleaved result