NOOP: A Domain-Theoretic Model of Nominally-Typed OOP

TL;DR

NOOP is a novel domain-theoretic model of object-oriented programming that incorporates nominal type information, demonstrating that type inheritance and subtyping are equivalent in nominally-typed languages, aligning with developer intuition.

Contribution

This paper introduces NOOP, the first domain-theoretic model that fully incorporates class/type names, clarifying the relationship between inheritance and subtyping in nominally-typed OOP.

Findings

Type inheritance and subtyping are fully identified in nominally-typed OOP.

NOOP provides a precise semantic foundation for analyzing nominal type systems.

Comparison with structural models highlights the benefits of nominal typing.

Abstract

The majority of industrial-strength object-oriented (OO) software is written using nominally-typed OO programming languages. Extant domain-theoretic models of OOP developed to analyze OO type systems miss, however, a crucial feature of these mainstream OO languages: nominality. This paper presents the construction of NOOP as the first domain-theoretic model of OOP that includes full class/type names information found in nominally-typed OOP. Inclusion of nominal information in objects of NOOP and asserting that type inheritance in statically-typed OO programming languages is an inherently nominal notion allow readily proving that type inheritance and subtyping are completely identified in these languages. This conclusion is in full agreement with intuitions of developers and language designers of these OO languages, and contrary to the belief that "inheritance is not subtyping," which came from assuming non-nominal (a.k.a., structural) models of OOP. To motivate the construction of NOOP, this paper briefly presents the benefits of nominal-typing to mainstream OO developers and OO language designers, as compared to structural-typing. After presenting NOOP, the paper further briefly compares NOOP to the most widely known domain-theoretic models of OOP. Leveraging the development of NOOP, the comparisons presented in this paper provide clear, brief and precise technical and mathematical accounts for the relation between nominal and structural OO type systems. NOOP, thus, provides a firmer semantic foundation for analyzing and progressing nominally-typed OO programming languages.