Stuart F. Allen

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>>

Innovations in computational type theory using Nuprl

Abstract For twenty years the Nuprl (“new pearl”) system has been used to develop software systems and formal theories of computational mathematics. It has also been used to explore and implement computational type theory (CTT)—a formal theory of computation closely related to Martin-Lofs intuitionistic type theory (ITT) and to the calculus of inductive constructions (CIC) implemented in the Coq prover. This article focuses on the theory and practice underpinning our use of Nuprl for much of the last decade. We discuss innovative elements of type theory, including new type constructors such as unions and dependent intersections, our theory of classes, and our theory of event structures. We also discuss the innovative architecture of Nuprl as a distributed system and as a transactional database of formal mathematics using the notion of abstract object identifiers. The database has led to an independent project called the Formal Digital Library, FDL, now used as a repository for Nuprl results as well as selected results from HOL, MetaPRL, and PVS. We discuss Howes set theoretic semantics that is used to relate such disparate theories and systems as those represented by these provers.

The Nuprl Open Logical Environment

The Nuprl system is a framework for reasoning about mathematics and programming. Over the years its design has been substantially improved to meet the demands of large-scale applications. Nuprl LPE, the newest release, features an open, distributed architecture centered around a flexible knowledge base and supports the cooperation of independent formal tools. This paper gives a brief overview of the system and the objectives that are addressed by its new architecture.

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.

Using Formal Reference to Enhance Authority and Integrity in Online Mathematical Texts

The amount and variety of digital information readily available to the public has become one of the defining features of the intellectual and scientific landscape. Digital information is bringing to the forefront new questions for computing and information science, e.g., how should this information be organized, searched, and evaluated. Universities, publishers, government, and other esteemed professionals bring unique and essential value to this enterprise that goes beyond their support of research – namely, intellectual authority. The imprimatur given to the information resources they own or sponsor is essential in helping individuals assess the validity of what they encounter on the Web.