Douglas J. Howe

Equality in lazy computation systems

The author introduces a general class of lazy computation systems and defines a natural program equivalence for them. He proves that if an extensionality condition holds of each of the operators of a computational system, then the equivalence relation is a congruence, so that the usual kinds of equality reasoning are valid for it. This condition is a simple syntactic one and is easy to verify for the various lazy computation systems considered so far. Conditions are given under which the equivalence coincides with observational congruence. These results have important consequences for type theories.<<ETX>>

The semantics of reflected proof

The authors lay the foundations for reasoning about proofs whose steps include both invocations of programs to build subproofs (tactics) and references to representations of proofs themselves (reflected proofs). The main result is the definition of a single type of proof which can mention itself, using a novel technique which finds a fixed point of a mapping between metalanguage and object language. This single type contrasts with hierarchies of types used in other approaches to accomplish the same classification. It is shown that these proofs are valid, and that every proof can be reduced to a proof involving only primitive inference rules. The extension of the results to proofs from which programs (such as tactics) can be derive and to proofs that can refer to a library of definitions and previously proven theorems is shown. It is believed that the mechanism of reflection is fundamental in building proof development systems, and its power is illustrated with applications to automating reasoning and describing modes of computation.<<ETX>>

Computational Metatheory in Nuprl

This paper describes an implementation within Nuprl of mechanisms that support the use of Nuprls type theory as a language for constructing theorem-proving procedures. The main component of the implementation is a large library of definitions, theorems and proofs. This library may be regarded as the beginning of a book of formal mathematics; it contains the formal development and explanation of a useful subset of Nuprls metatheory, and of a mechanism for translating results established about this embedded metatheory to the object level. Nuprls rich type theory, besides permitting the internal development of this partial reflection mechanism, allows us to make abstractions that drastically reduce the burden of establishing the correctness of new theorem-proving procedures. Our library includes a formally verified term-rewriting system.

On computational open-endedness in Martin-Lof's type theory

Computational open-endedness in a type theory is defined as the property that theorems remain true under extensions to the underlying programming language. Some properties related to open-endedness that are relevant to machine implementations of type theory are established. A class of computation systems, specified by a simple but fairly general kind of structural operational semantics, with respect to which P. Martin-Lofs (6th Int. Congress for Logic, Methodology, and Philosophy of Science, p.153-175, 1982) type theory (and most of its descendants) is open-ended is defined. It is shown that any such system validates a useful form of type free reasoning about program equivalence and that symbolic computation procedures can be automatically derived from these specifications. The main result is the definition of a particular computation system that includes a collection of oracles sufficient to provide a classical semantics for Martin-Lofs type theory in which the excluded middle law holds.<<ETX>>

Reasoning About Functional Programs in Nuprl

There are two ways of reasoning about functional programs in the constructive type theory of the Nuprl proof development system. Nuprl can be used in a conventional program-verification mode, in which functional programs are written in a familiar style and then proven to be correct. It can also be used in an extraction mode, where programs are not written explicitly, but instead are extracted from mathematical proofs. Nuprl is the only constructive type theory to support both of these approaches. These approaches are illustrated by applying Nuprl to Boyer and Moores “majority” algorithm.

Implementing Number Theory: An Experiment with Nuprl

We describe the results of an experiment in which the Nuprl proof development system was used in conjunction with a collection of simple proof-assisting programs to constructively prove a substantial theorem of number theory. We believe that these results indicate the promise of an approach to reasoning about computationally meaningful mathematics by which both proof construction and the results of formal reasoning are mathematically comprehensible.

Implementing Constructive Real Analysis: Preliminary Report

In this paper we present the results of an investigation into the use of the Nuprl proof development system to implement higher constructive mathematics. As a first step in exploring the issues involved, we have developed a basis for formalizing substantial parts of real analysis. More specifically, we have: developed type-theoretic representations of concepts from Bishops treatment of constructive mathematics that allow reasonably direct formalizations; used Nuprls facility for sound extension of its inference system to implement automated reasoners for analysis; and tested these ideas in a formalization of rational and real arithmetic and of a proof of the completeness theorem for the reals (every Cauchy sequence converges).

Reflecting the Open-Ended Computation System of Constructive Type Theory

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 the language in a clear way by adding simple primitive concepts that anticipate the reflection. This work provides a hook for developing methods 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.