arXiv subjects

Todd Waugh Ambridge

Publications and source records attributed to Todd Waugh Ambridge.

3 recordsLinked to original sources

arXiv2025

A robust graph-based approach to observational equivalence

We propose a new step-wise approach to proving observational equivalence, and in particular reasoning about fragility of observational equivalence. Our approach is based on what we call local reasoning. The local reasoning exploits the graphical concept of neighbourhood, and it extracts a new, formal, concept of robustness as a key sufficient condition of observational equivalence. Moreover, our proof methodology is capable of proving a generalised notion of observational equivalence. The generalised notion can be quantified over syntactically restricted contexts instead of all contexts, and also quantitatively constrained in terms of the number of reduction steps. The operational machinery we use is given by a hypergraph-rewriting abstract machine inspired by Girard's Geometry of Interaction. The behaviour of language features, including function abstraction and application, is provided by hypergraph-rewriting rules. We demonstrate our proof methodology using the call-by-value lambda-calculus equipped with (higher-order) state.

cs.PL↗

arXiv2024

Exact Real Search: Formalised Optimisation and Regression in Constructive Univalent Mathematics

The real numbers are important in both mathematics and computation theory. Computationally, real numbers can be represented in several ways; most commonly using inexact floating-point data-types, but also using exact arbitrary-precision data-types which satisfy the expected mathematical properties of the reals. This thesis is concerned with formalising properties of certain types for exact real arithmetic, as well as utilising them computationally for the purposes of search, optimisation and regression. We develop, in a constructive and univalent type-theoretic foundation of mathematics, a formalised framework for performing search, optimisation and regression on a wide class of types. This framework utilises Martín Escardó's prior work on searchable types, along with a convenient version of ultrametric spaces -- which we call closeness spaces -- in order to consistently search certain infinite types using the functional programming language and proof assistant Agda. We formally define and prove the convergence properties of type-theoretic variants of global optimisation and parametric regression, problems related to search from the literature of analysis. As we work in a constructive setting, these convergence theorems yield computational algorithms for correct optimisation and regression on the types of our framework. Importantly, we can instantiate our framework on data-types from the literature of exact real arithmetic, allowing us to perform our variants of search, optimisation and regression on ternary signed-digit encodings of the real numbers, as well as a simplified version of Hans-J. Boehm's functional encodings of real numbers. Furthermore, we contribute to the extensive work on ternary signed-digits by formally verifying the definition of certain exact real arithmetic operations using the Escardó-Simpson interval object specification of compact intervals.

cs.LO↗

arXiv2020

A Constructive, Type-Theoretic Approach to Regression via Global Optimisation

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.

cs.LO↗