Intensional polymorphism, the ability to dispatch to di#erent routines based on types at run time, enables a variety of advanced implementation techniques for polymorphic languages, including tag-free garbage collection, unboxed function arguments, polymorphic marshalling, and flattened data structures. To date, languages that support intensional polymorphism have required a type-passing (as opposed to type-erasure) interpretation where types are constructed and passed to polymorphic functions at run time. Unfortunately, type-passing su#ers from a number of drawbacks: it requires duplication of run-time constructs at the term and type levels, it prevents abstraction, and it severely complicates polymorphic closure conversion.

Interpretation is an implicit part of today’s programming; it has great power but is overused and has significant costs. For example, interpreters are typically significantly hard to understand and hard to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned attempt to redress the problem of interpretation. It incorporates an awareness of the undesirability of interpretation with observations that definitions and a certain style of programming appear to offer alternatives to it. Application of TFP is expected to lead to a number of significant outcomes, theoretical as well as practical. Primary among these are novel programming languages to lessen or eliminate the use of interpretation in programming, leading to better-quality software. However, TFP contains a number of lacunae in its current formulation, which hinder development of these outcomes. Among others, formal semantics and type-systems for TFP languages are yet to be discovered, the means to reduce interpretation in programs is to be determined, and a detailed explication is needed of interpretation, definition, and the differences between the two. Most important of all however is the need to develop a complete understanding of the nature of interpretation. In this work, suitable type-systems for TFP languages are identified, and guidance given regarding the construction of appropriate formal semantics. Techniques, based around the ‘fold’ operator, are identified and developed for modifying programs so as to reduce the amount of interpretation they contain. Interpretation as a means of language-extension is also investigated. v Finally, the nature of interpretation is considered. Numerous hypotheses relating to it considered in detail. Combining the results of those analyses with discoveries from elsewhere in this work leads to the proposal that interpretation is not, in fact, symbol-based computation, but is in fact something more fundamental: computation that varies with input. We discuss in detail various implications of this characterisation, including its practical application. An often more-useful property, ‘inherent interpretiveness’, is also motivated and discussed in depth. Overall, our inquiries act to give conceptual and theoretical foundations for practical TFP.