Observationally-induced algebras in Domain Theory

TL;DR

This paper revises and simplifies the concept of observationally induced algebras within classical domain theory, applying it to powerdomains and revealing insights into the Plotkin powerdomain's structure and conditions for its characterization.

Contribution

It introduces a simplified framework for observationally induced algebras in domain theory and clarifies the structure of the Plotkin powerdomain, including conditions to recover its original form.

Findings

The simplified framework effectively models computational effects.

The Plotkin powerdomain exhibits an unexpected deviation under the new framework.

Additional conditions can restore the original Plotkin powerdomain structure.

Abstract

In this paper we revise and simplify the notion of observationally induced algebra introduced by Simpson and Schroeder for the purpose of modelling computational effects in the particular case where the ambient category is given by classical domain theory. As examples of the general framework we consider the various powerdomains. For the particular case of the Plotkin powerdomain the general recipe leads to a somewhat unexpected result which, however, makes sense from a Computer Science perspective. We analyze this "deviation" and show how to reobtain the original Plotkin powerdomain by imposing further conditions previously considered by R.~Heckmann and J.~Goubault-Larrecq.