Overlap Between Type Theory and Semantics in Programming Languages and Linguistics

The field of programming languages and linguistics share several interesting areas of overlap, particularly in the domain of Categorial Grammar. This type of grammatical structure provides a framework where parts of speech are defined not by arbitrary labels like verb, adjective, or noun, but by function type signatures. These signatures indicate what a particular linguistic element can be combined with and the type of constituent produced by such a combination.

Understanding Parts of Speech in Type Theory

In this approach, parts of speech are functions with specific type signatures. For instance, a verb can be viewed as a function that takes a subject and an object and returns a sentence. Therefore, the categorial grammar provides a systematic way to understand and parse sentences by defining the roles and relationships between linguistic elements based on their function types.

Using type-directed parsing, you can take a sequence of tokens with assigned types/parts of speech and combine them progressively in the order allowed by their types to build up a parse tree. This method can eventually lead to a sentence structure or a higher-level constituent. This approach aligns with the natural language processing techniques where each part of speech corresponds to a function in lambda calculus.

Type- Directed Parsing and its Application

Type-directed parsing is generally considered a bad idea in the design of programming languages due to its complexity and potential for conflicts. However, it has been applied in certain contexts. For example, some parts of the Perl programming language are notoriously difficult to parse statically, necessitating runtime type information to disambiguate the syntax. In such cases, a combination of type definitions and runtime information is required to make parsing possible.

Concatenative Languages and Natural Languages: An Analogy

A special class of programming languages known as concatenative languages, such as FORTH, Factor, and Joy, exhibit characteristics that parallel those of natural languages. These languages do not require explicit bracketing for function calls and parse trees are essentially the same as function call trees. This parallels the way natural languages work, where parameter definitions are often implicit and resolved through context rather than explicit declarations. For instance, anaphora (referring to a previously mentioned noun) in natural languages becomes significantly more complex than the simple type-based parameter binding in concatenative programming languages or categorial grammar.

Conclusion: The Intersection Between Programming and Linguistics

The intersection between type theory in programming languages and semantics in linguistics is a rich and complex area of study. Categorial grammar, with its focus on function type signatures, provides a powerful tool for parsing and understanding natural language structures. Similarly, the design of concatenative programming languages shows how such grammatical principles can be applied beyond natural language processing to construct and understand complex computational systems.

While type-directed parsing might not be the norm in mainstream programming languages, its application in specific contexts and its parallel in natural languages suggest a deeper exploration into the fundamental principles that govern both human language and computational systems.