[Paper Review] Domain Theory: An Introduction

Robert Cartwright, Rebecca J. Parsons+1 more|arXiv (Cornell University)|May 19, 2016

semigroups and automata theory4 references3 citations

TL;DR

This monograph presents a revised, accessible formulation of domain theory using finitary bases—closed under least upper bounds on finite consistent subsets—offering a mathematically rigorous framework for modeling computational data values, operations, and recursion in programming languages. The key contribution is a unified, effective presentation of domains that supports higher-order computation, fixed points, and the semantics of typed lambda calculus via projections into a universal domain.

ABSTRACT

This monograph is an ongoing revision of "Lectures On A Mathematical Theory of Computation" by Dana Scott. Scott's monograph uses a formulation of domains called neighborhood systems in which finite elements are selected subsets of a master set of objects called "tokens". Since tokens have little intuitive significance, Scott has discarded neighborhood systems in favor of an equivalent formulation of domains called information systems. Unfortunately, he has not rewritten his monograph to reflect this change. We have rewritten Scott's monograph in terms of finitary bases instead of information systems. A finitary basis is an information system that is closed under least upper bounds on finite consistent subsets. This convention ensures that every finite answer is represented by a single basis object instead of a set of objects.

Motivation & Objective

• To provide a modern, accessible formulation of domain theory that replaces outdated neighborhood systems with finitary bases.
• To establish a foundation for modeling infinite, partially defined, and finite maximal data values in a unified computational framework.
• To demonstrate how domain constructors (products, function spaces) and recursive definitions can be formalized using approximable mappings and projections.
• To show that all computable functions and constructions in the typed lambda calculus can be embedded within a universal domain via effective presentations.
• To support the semantics of higher-order programming languages by ensuring computability and fixed-point properties through effective domain constructors.

Proposed method

• Reformulates Dana Scott’s original domain theory using finitary bases instead of information systems or neighborhood systems, ensuring every finite data value is represented by a single basis element.
• Defines domains as complete partial orders (CPOs) with countable, directed-complete bases, where finite elements represent approximations to infinite values.
• Introduces approximable mappings and continuous functions as the core morphisms between domains, preserving approximation order and least upper bounds.
• Constructs domain constructors (products, function spaces) via categorical constructions and proves their computability using retractions and projections.
• Uses the universal domain 𝒰 as a carrier for all computable elements, with all domains embedded as subdomains via finitary projections.
• Translates typed lambda calculus into the universal domain using type embeddings and projection-based isomorphisms, preserving semantics and computability.

Experimental results

Research questions

• RQ1 How can domain theory be reformulated to eliminate reliance on abstract, non-intuitive constructs like neighborhood systems?
• RQ2 What is the role of finitary bases in ensuring effective, canonical representations of finite data values and their approximations?
• RQ3 How can domain constructors such as products and function spaces be defined and proven computable within a unified framework?
• RQ4 Can all computable functions and recursive definitions be captured within a single universal domain via effective presentations?
• RQ5 How can the semantics of the typed lambda calculus be embedded into a universal domain while preserving type structure and computability?

Key findings

• Finitary bases provide a more intuitive and effective alternative to information systems and neighborhood systems, ensuring every finite data value is represented by a single basis object.
• The universal domain 𝒰 supports all computable elements and serves as a carrier for all effectively presented domains, with every domain embedded as a subdomain via finitary projections.
• All domain constructors—products, function spaces, and multiary compositions—are shown to be computable combinators on projections over the universal domain.
• Recursive definitions and fixed points are captured via the standard fixed-point method, with all computable maps guaranteed to have computable fixed points.
• The typed lambda calculus can be fully embedded into the universal domain through projection-based isomorphisms, preserving types and semantics.
• Effective presentations of domains ensure that computable elements correspond exactly to effectively presented subdomains of the universal domain.