Inductive-data-type Systems

TL;DR

This paper extends a combined algebraic and typed lambda-calculus language to include strictly positive inductive types, ensuring strong normalization and providing a robust computation model for data type systems.

Contribution

It reformulates and generalizes the General Schema to encompass strictly positive inductive types, simplifying the proof of strong normalization.

Findings

The extended system is strongly normalizing.

The reformulated schema captures a broader class of inductive types.

Provides a unified computation model for algebraic and typed languages.

Abstract

In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed lambda-calculus enriched by pattern-matching definitions following a certain format, called the "General Schema", which generalizes the usual recursor definitions for natural numbers and similar "basic inductive types". This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called "strictly positive", and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.