Inductive and Functional Types in Ludics

TL;DR

This paper explores how inductive and functional types are represented in ludics using game semantics, providing explicit constructions for fixed points and analyzing their interactive properties and type safety issues.

Contribution

It introduces a novel interpretation of inductive and functional types in ludics through fixed points and offers insights into their interactive behaviors and safety limitations.

Findings

Inductive types are modeled as least fixed points in ludics.

An explicit construction for fixed points in ludics is provided.

Certain higher-order function types are shown to lack type safety.

Abstract

Ludics is a logical framework in which types/formulas are modelled by sets of terms with the same computational behaviour. This paper investigates the representation of inductive data types and functional types in ludics. We study their structure following a game semantics approach. Inductive types are interpreted as least fixed points, and we prove an internal completeness result giving an explicit construction for such fixed points. The interactive properties of the ludics interpretation of inductive and functional types are then studied. In particular, we identify which higher-order functions types fail to satisfy type safety, and we give a computational explanation.