Christoph Kreitz

Theory of representations

Abstract An approach for a simple, general, and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented. A standard theory of effectivity on F = {f: N → N } has been developed in a previous paper. By representations δ: F →M this theory is extended to other sets M. Topological and recursion theoretical properties of representations are studied, where the final topology of a representation plays an essential role. It is shown that for any separable T0-space an (up to equivalence) unique admissible representation can be defined which reflects the topological properties correctly.

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 Horus and Ensemble projects: accomplishments and limitations

The Horus and Ensemble efforts culminated a multi-year Cornell research program in process group communication used for fault-tolerance, security and adaptation. Our intent was to understand the degree to which a single system could offer flexibility and yet maintain high performance, to explore the integration of fault tolerance with security and real-time mechanisms, and to increase trustworthiness of our solutions by applying formal methods. Here, we summarize the accomplishments of the effort and evaluate the successes and failures of the approach.

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.

MetaPRL – A Modular Logical Environment

MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive LCF-style tactic-based proof assistant, a logical framework, a logical programming environment, and a formal methods programming toolkit. MetaPRL is distributed under an open-source license and can be downloaded from http://metaprl.org/. This paper provides an overview of the system focusing on the features that did not exist in the previous generations of PRL systems.

The ILTP Problem Library for Intuitionistic Logic

The Intuitionistic Logic Theorem Proving (ILTP) library provides a platform for testing and benchmarking automated theorem proving (ATP) systems for intuitionistic propositional and first-order logic. It includes about 2,800 problems in a standardized syntax from 24 problem domains. For each problem an intuitionistic status and difficulty rating were obtained by running comprehensive tests of currently available intuitionistic ATP systems on all problems in the library. Thus, for the first time, the testing and evaluation of ATP systems for intuitionistic logic have been put on a firm basis.

Protocol switching: exploiting meta-properties

As we see a growing variety of network and application behaviors, it becomes more important that protocols adapt to their surroundings. Building adaptive protocols is complicated, and therefore we have considered building hybrid protocols that switch between specialized protocols. We show for which communication properties this is a correct solution, and classify these using a new concept called meta-properties. We also show how well these switches perform.

JProver: Integrating Connection-Based Theorem Proving into Interactive Proof Assistants

JProver is a first-order intuitionistic theorem prover that creates sequent-style proof objects and can serve as a proof engine in interactive proof assistants with expressive constructive logics. This paper gives a brief overview of JProvers proof technique, the generation of proof objects, and its integration into the Nuprl proof development system.

T-String Unification: Unifying Prefixes in Non-classical Proof Methods

For an efficient proof search in non-classical logics, particular in intuitionistic and modal logics, two similar approaches have been established: Wallens matrix characterization and Ohlbachs resolution calculus. Beside the usual term-unification both methods require a specialized string-unification to unify the so-called prefixes of atomic formulae (in Wallens notation) or world-paths (in Ohlbachs notation). For this purpose we present an efficient algorithm, called T-String-Unification, which computes a minimal set of most general unifiers. By transforming systems of equations we obtain an elegant unification procedure, which is applicable to the intuitionistic logic J and the modal logic S4. With some modifications we are also able to treat the modal logics D, K, D4, K4, S5, and T. We explain our method by an intuitive graphical presentation, prove correctness, completeness, minimality, and termination and investigate its complexity.

Representations of the real numbers and of the open subsets of the set of real numbers

Abstract In previous papers we have presented a unified Type 2 theory of computability and continuity and a theory of representations. In this paper the concepts developed so far are used for the foundation of a new kind of constructive analysis. Different standard representations of the real numbers are compared. It turns out that the crucial differences are of topological nature and that most of the representations (e.g., the decimal representation) are not reasonable for topological reasons. In the second part some effective representations of the open subsets of the real numbers are introduced and compared.