Patricia Johann

Foundations for structured programming with GADTs

GADTs are at the cutting edge of functional programming and becomemore 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 derivean 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 bederived.

Initial algebra semantics is enough

Initial algebra semantics is a cornerstone of the theory of modern functional programming languages. For each inductive data type, it provides a fold combinator encapsulating structured recursion over data of that type, a Church encoding, a build combinator which constructs data of that type, and a fold/build rule which optimises modular programs by eliminating intermediate data of 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. Specifically, the folds have been considered too weak to capture commonly occurring patterns of recursion, and no Church encodings, build combinators, or fold/build rules have been given for nested types. This paper overturns this conventional wisdom by solving all of these problems.

Warm fusion in Stratego: A case study in generation of program transformation systems

Stratego is a domain-specific language for the specification of program transformation systems. The design of Stratego is based on the paradigm of rewriting strategies: user-definable programs in a little language of strategy operators determine where and in what order transformation rules are (automatically) applied to a program. The separation of rules and strategies supports modularity of specifications. Stratego also provides generic features for specification of program traversals.In this paper we present a case study of Stratego as applied to a non-trivial problem in program transformation. We demonstrate the use of Stratego in eliminating intermediate data structures from (also known as deforesting) functional programs via the warm fusion algorithm of Launchbury and Sheard. This algorithm has been specified in Stratego and embedded in a fully automatic transformation system for kernel Haskell. The entire system consists of about 2600 lines of specification code, which breaks down into 1850 lines for a general framework for Haskell transformation and 750 lines devoted to a highly modular, easily extensible specification of the warm fusion transformer itself. Its successful design and construction provides further evidence that programs generated from Stratego specifications are suitable for integration into real systems, and that rewriting strategies are a good paradigm for the implementation of such systems.

An improved general E -unification method

A method for E-unification in arbitrary equational theories E is presented which directly generalizes the standard technique of Narrowing. The method is defined in terms of transformations on systems, building upon and refining results of Gallier and Snyder.

Staged notational definitions

Recent work proposed defining type-safe macros via interpretation into a multi-stage language. The utility of this approach was illustrated with a language called MacroML, in which all type checking is carried out before macro expansion. Building on this work, the goal of this paper is to develop a macro language that makes it easy for programmers to reason about terms locally. We show that defining the semantics of macros in this manner helps in developing and verifying not only type systems for macro languages but also equational reasoning principles. Because the MacroML calculus is sensetive to renaming of (what appear locally to be) bound variables, we present a calculus of staged notational definitions (SND) that eliminates the renaming problem but retains MacroMLs phase distinction. Additionally, SND incorporates the generality of Griffins account of notational definitions. We exhibit a formal equational theory for SND and prove its soundness.

Fusing Logic and Control with Local Transformations: An Example Optimization

Abstract programming supports the separation of logical concerns from issues of control in program construction. While this separation of concerns leads to reduced code size and increased reusability of code, its main disadvantage is the computa- tional overhead it incurs. Fusion techniques can be used to combine the reusability of abstract programs with the eciency of specialized programs. In this paper we illustrate some of the ways in which rewriting strategies can be used to separate the denition of program transformation rules from the strategies under which they are applied. Doing so supports the generic denition of program transformation components. Fusion techniques for strategies can then be used to specialize such generic components. We show how the generic innermost rewriting strategy can be optimized by fusing it with the rules to which it is applied. Both the optimization and the programs to which the optimization applies are specied in the strategy language Stratego. The optimization is based on small transformation rules that are applied locally under the control of strategies, using special knowledge about the contexts in which the rules are applied.

A combinatory logic approach to higher-order E -unification

Let E be a first-order equational theory. A translation of higher-order E-unification problems into a combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modification of ordinary narrowing is shown to be a complete method for enumerating higher-order E-unifiers. In fact, we treat a more general problem, in which the types of terms contain type variables.

When is a type refinement an inductive type

Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the N-indexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purposespecific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations often store redundant information about data and their refinements. This paper shows how to generically derive inductive characterisations of refinements of inductive types, and argues that these characterisations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. These characterisations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined.

Generic fibrational induction

This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs’ elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, a sound induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of a particular syntactic form. We establish the soundness of our generic induction rule by reducing induction to iteration. We then show how our generic induction rule can be instantiated to give induction rules for the data types of rose trees, definite hereditary sets, and hyperfunctions. The first of these lies outside the scope of Hermida and Jacobs’ work because it is not polynomial, and as far as we are aware, no induction rules have been known to exist for the second and third in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set.

Refining Inductive Types

Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the N-indexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purpose-specific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations must often store redundant information about data and their refinements. In this paper we show how to generically derive inductive characterizations of refinements of inductive types, and argue that these characterizations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. Our characterizations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined.