Indexed Data Types Research Articles

The essence of ornaments

AbstractFunctional programmers from all horizons strive to use, and sometimes abuse, their favorite type system in order to capture the invariants of their programs. A widely used tool in that trade consists in defining finely indexed datatypes. Operationally, these typesclassifythe programmer's data, following the ML tradition. Logically, these typesenforcethe program invariants in a novel manner. This new programming pattern, by which one programsoverinductive definitions to account for some invariants, lead to the development of a theory of ornaments (McBride, 2011Ornamental Algebras, Algebraic Ornaments. Unpublished). However, ornaments originate as a dependently-typed object and may thus appear rather daunting to a functional programmer of the non-dependent kind. This article aims at presenting ornamentsfrom first-principlesand, in particular, to declutter their presentation from syntactic considerations. To do so, we shall give a sufficiently abstract model of indexed datatypes by means of many-sorted signatures. In this process, we formalize our intuition that an indexed datatype is the combination of a data-structureand a data-logic. Over this abstraction of datatypes, we shall recast the definition of ornaments, effectively giving a model of ornaments. Benefiting both from the operationalandabstract nature of many-sorted signatures, ornaments should appear applicable and, one hopes, of interest beyond the type-theoretic circles, case in point being languages with generalized abstract datatypes or refinement types.

Indexed codata types

Indexed data types allow us to specify and verify many interesting invariants about finite data in a general purpose programming language. In this paper we investigate the dual idea: indexed codata types, which allow us to describe data-dependencies about infinite data structures. Unlike finite data which is defined by constructors, we define infinite data by observations. Dual to pattern matching on indexed data which may refine the type indices, we define copattern matching on indexed codata where type indices guard observations we can make. Our key technical contributions are three-fold: first, we extend Levy's call-by-push value language with support for indexed (co)data and deep (co)pattern matching; second, we provide a clean foundation for dependent (co)pattern matching using equality constraints; third, we describe a small-step semantics using a continuation-based abstract machine, define coverage for indexed (co)patterns, and prove type safety. This is an important step towards building a foundation where (co)data type definitions and dependent types can coexist.

Indexed Induction and Coinduction, Fibrationally

This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a dual to the sound induction rule for inductive types that we developed previously. That is, we present a sound coinduction rule for any data type arising as the carrier of the final coalgebra of a functor, thus relaxing Hermida and Jacobs' restriction to polynomial functors. To achieve this we introduce the notion of a quotient category with equality (QCE) that i) abstracts the standard notion of a fibration of relations constructed from a given fibration; and ii) plays a role in the theory of coinduction dual to that played by a comprehension category with unit (CCU) in the theory of induction. Secondly, we show that inductive and coinductive indexed types also admit sound induction and coinduction rules. Indexed data types often arise as carriers of initial algebras and final coalgebras of functors on slice categories, so we give sufficient conditions under which we can construct, from a CCU (QCE) U:E \rightarrow B, a fibration with base B/I that models indexing by I and is also a CCU (resp., QCE). We finish the paper by considering the more general case of sound induction and coinduction rules for indexed data types when the indexing is itself given by a fibration.