arXiv subjects

Russell O'Connor

Publications and source records attributed to Russell O'Connor.

14 recordsLinked to original sources

arXiv2025

Formal Verification of the Safegcd Implementation

The modular inverse is an essential piece of computation required for elliptic curve operations used for digital signatures in Bitcoin and other applications. A novel approach to the extended Euclidean algorithm has been developed by Bernstein and Yang within the last few years and incorporated into the libsecp256k1 cryptographic library used by Bitcoin. However, novel algorithms introduce new risks of errors. To address this we have completed a computer verified proof of the correctness of (one of) libsecp256k1's modular inverse implementations with the Coq proof assistant using the Verifiable C's implementation of separation logic.

cs.CR↗

arXiv2019

Building on the Diamonds between Theories: Theory Presentation Combinators

To build a large library of mathematics, it seems more efficient to take advantage of the inherent structure of mathematical theories. Various theory presentation combinators have been proposed, and some have been implemented, in both legacy and current systems. Surprisingly, the ``standard library'' of most systems do not make pervasive use of these combinators. We present a set of combinators optimized for reuse, via the tiny theories approach. Our combinators draw their power from the inherent structure already present in the \emph{category of contexts} associated to a dependently typed language. The current work builds on ideas originating in CLEAR and Specware and their descendents (both direct and intellectual). Driven by some design criteria for user-centric library design, our library-building experience via the systematic use of combinators has fed back into the semantics of these combinators, and later into an updated syntax for them.

cs.LO↗

arXiv2017

Simplicity: A New Language for Blockchains

Simplicity is a typed, combinator-based, functional language without loops and recursion, designed to be used for crypto-currencies and blockchain applications. It aims to improve upon existing crypto-currency languages, such as Bitcoin Script and Ethereum's EVM, while avoiding some of the problems they face. Simplicity comes with formal denotational semantics defined in Coq, a popular, general purpose software proof assistant. Simplicity also includes operational semantics that are defined with an abstract machine that we call the Bit Machine. The Bit Machine is used as a tool for measuring the computational space and time resources needed to evaluate Simplicity programs. Owing to its Turing incompleteness, Simplicity is amenable to static analysis that can be used to derive upper bounds on the computational resources needed, prior to execution. While Turing incomplete, Simplicity can express any finitary function, which we believe is enough to build useful "smart contracts" for blockchain applications.

cs.PL↗

arXiv2015

A Representation Theorem for Second-Order Functionals

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of second-order functionals which are polymorphic over a class of functors. Types polymorphic over a class of functors are easily representable in languages such as Haskell, but are difficult to analyse and reason about. The concrete representation provided by the theorem is easier to analyse, but it might not be as convenient to implement. Therefore, depending on the task at hand, the change of representation may prove valuable in one direction or the other. We showcase the usefulness of the representation theorem with a range of examples. Concretely, we show how the representation theorem can be used to show that traversable functors are finitary containers, how parameterised coalgebras relate to very well-behaved lenses, and how algebraic effects might be implemented in a functional language.

cs.PL↗

arXiv2012

Theory Presentation Combinators

We motivate and give semantics to theory presentation combinators as the foundational building blocks for a scalable library of theories. The key observation is that the category of contexts and fibered categories are the ideal theoretical tools for this purpose.

cs.MS↗

arXiv2011

Functor is to Lens as Applicative is to Biplate: Introducing Multiplate

This paper gives two new categorical characterisations of lenses: one as a coalgebra of the store comonad, and the other as a monoidal natural transformation on a category of a certain class of coalgebras. The store comonad of the first characterisation can be generalized to a Cartesian store comonad, and the coalgebras of this Cartesian store comonad turn out to be exactly the Biplates of the Uniplate generic programming library. On the other hand, the monoidal natural transformations on functors can be generalized to work on a category of more specific coalgebras. This generalization turns out to be the type of compos from the Compos generic programming library. A theorem, originally conjectured by van Laarhoven, proves that these two generalizations are isomorphic, thus the core data types of the Uniplate and Compos libraries supporting generic program on single recursive types are the same. Both the Uniplate and Compos libraries generalize this core functionality to support mutually recursive types in different ways. This paper proposes a third extension to support mutually recursive data types that is as powerful as Compos and as easy to use as Uniplate. This proposal, called Multiplate, only requires rank 3 polymorphism in addition to the normal type class mechanism of Haskell.

cs.PL↗

arXiv2011

The MathScheme Library: Some Preliminary Experiments

We present some of the experiments we have performed to best test our design for a library for MathScheme, the mechanized mathematics software system we are building. We wish for our library design to use and reflect, as much as possible, the mathematical structure present in the objects which populate the library.

cs.MS↗

arXiv2010

Classical Mathematics for a Constructive World

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how classical results can be applied to constructive mathematics. Finally, we will see how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.

cs.LO↗

arXiv2008

Computing with Classical Real Numbers

There are two incompatible Coq libraries that have a theory of the real numbers; the Coq standard library gives an axiomatic treatment of classical real numbers, while the CoRN library from Nijmegen defines constructively valid real numbers. Unfortunately, this means results about one structure cannot easily be used in the other structure. We present a way interfacing these two libraries by showing that their real number structures are isomorphic assuming the classical axioms already present in the standard library reals. This allows us to use O'Connor's decision procedure for solving ground inequalities present in CoRN to solve inequalities about the reals from the Coq standard library, and it allows theorems from the Coq standard library to apply to problem about the CoRN reals.

cs.LO↗

arXiv2008

A computer verified, monadic, functional implementation of the integral

We provide a computer verified exact monadic functional implementation of the Riemann integral in type theory. Together with previous work by O'Connor, this may be seen as the beginning of the realization of Bishop's vision to use constructive mathematics as a programming language for exact analysis.

cs.LO↗

arXiv2008

A Computer Verified Theory of Compact Sets

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and displaying images with a computer. In this paper, I build upon existing work about complete metric spaces to define compact sets as the completion of the space of finite sets under the Hausdorff metric. This definition allowed me to quickly develop a computer verified theory of compact sets. I applied this theory to compute provably correct plots of uniformly continuous functions.

cs.LO↗

arXiv2008

Certified Exact Transcendental Real Number Computation in Coq

Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete metric spaces in the Coq proof assistant and used this library to build a constructive real number implementation including elementary real number functions and proofs of correctness. Using this library, I have created a tactic that automatically proves strict inequalities over closed elementary real number expressions by computation.

cs.LO↗

arXiv2006

A Monadic, Functional Implementation of Real Numbers

Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the correctness of these proofs. This paper develops a new implementation of the constructive real numbers and elementary functions for such proofs by using the monad properties of the completion operation on metric spaces. Bishop and Bridges's notion of regular sequences is generalized to, what I call, regular functions which form the completion of any metric space. Using the monad operations, continuous functions on length spaces (a common subclass of metric spaces) are created by lifting continuous functions on the original space. A prototype Haskell implementation has been created. I believe that this approach yields a real number library that is reasonably efficient for computation, and still simple enough to easily verify its correctness.

math.NA↗

arXiv2006

Essential Incompleteness of Arithmetic Verified by Coq

A constructive proof of the Goedel-Rosser incompleteness theorem has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive functions are proved to be representable in a weak axiom system. Formulas and proofs are encoded as natural numbers, and functions operating on these codes are proved to be primitive recursive. The weak axiom system is proved to be essentially incomplete. In particular, Peano arithmetic is proved to be consistent in Coq's type theory and therefore is incomplete.

cs.LO↗