Reflecting the computation system of constructive type theory in itself∗

The computation system of constructive type theory is open-ended so that theorems about computation will hold for a broad class of extensions to the system. We show that despite this openness it is possible to completely reflect the computation system into itself in a clear way by adding simple primitive concepts that anticipate the reflection. This work provides a method to modify the built-in evaluator and to treat the issues of intensionality and computational complexity in programming logics and provides a basis for reflecting the deductive apparatus of type theory. In this abstract we use the term “reflection” to refer to grammatical constructions which allow a language to talk about itself. This capability is important in natural language, and in fact was used in the first sentence of this abstract (as well as in this sentence). Reflection is also an important mechanism in formal languages. In Lisp it is used to provide an extensible syntax. Formal logical calculi also use it to provide an extensible inference system [DS79], by allowing users to state new rules of inference and prove that they are sound. Reflection need not be explicitly provided because it can sometimes be achieved be a technique known as gödelization, used by Gödel to prove his incompleteness theorem by reflecting the relation “p is a proof of P” inside the pure language of arithmetic. This mechanism not only provides a basis for reasoning about computation, but also a means of modifying evaluation, say to make it more efficient. For instance it is possible to provide other function evaluation procedures such as “call by value” in addition to the basic lazy evaluation. Reflection also provides a basis for resoning about syntax. We can define basic operators on terms at the reflected level, e.g. substitution, renaming, pattern matching, unification, etc. These can be given just as at the metaleval, providing an internal account of the basic system operators. Moreover, because the term structure is so general, its reflection provides a way ∗Supported in part by NSF grant CCR-8616552 and ONR grant N00014-88-K-0409.