On the implementation of construction functions for non-free concrete data types

TL;DR

This paper explores how to implement construction functions for non-free concrete data types with invariants, proposing a method to ensure correctness and pattern matching, and introduces a prototype called Moca.

Contribution

It provides a detailed study of private data types, their construction functions, and presents Moca, a prototype addressing invariant preservation and pattern matching.

Findings

Private data types enable pattern matching and invariant preservation.

Construction functions can be systematically defined for non-free data types.

Moca prototype demonstrates practical implementation of these concepts.

Abstract

Many algorithms use concrete data types with some additional invariants. The set of values satisfying the invariants is often a set of representatives for the equivalence classes of some equational theory. For instance, a sorted list is a particular representative wrt commutativity. Theories like associativity, neutral element, idempotence, etc. are also very common. Now, when one wants to combine various invariants, it may be difficult to find the suitable representatives and to efficiently implement the invariants. The preservation of invariants throughout the whole program is even more difficult and error prone. Classically, the programmer solves this problem using a combination of two techniques: the definition of appropriate construction functions for the representatives and the consistent usage of these functions ensured via compiler verifications. The common way of ensuring consistency is to use an abstract data type for the representatives; unfortunately, pattern matching on representatives is lost. A more appealing alternative is to define a concrete data type with private constructors so that both compiler verification and pattern matching on representatives are granted. In this paper, we detail the notion of private data type and study the existence of construction functions. We also describe a prototype, called Moca, that addresses the entire problem of...