arXiv subjects

Robin Adams

Publications and source records attributed to Robin Adams.

9 recordsLinked to original sources

arXiv2025

A Graded Modal Type Theory for Pulse Schedules

The operations to be performed by a quantum computer are almost invariably given in the form of a quantum circuit. In the final stage of compilation, a quantum circuit must be translated into the input signals accepted by the quantum hardware itself. For a quantum computer based on superconducting qubits, this will be a sequence of microwave control pulses to be sent to the various input channels. A pulse schedule gives a full specification for which pulse should be applied to which channel at what time. There is as yet no language for these pulse schedules that is very amenable to formal semantics. In this paper, we propose such a language called GRAMPUS (GRAded Modal type theory for PUlse Schedules). It is a graded modal type theory, where the grades represent timing information: a variable $x :^{50} Q_1$ will represent a state of qubit $Q_1$ that will exist 50 nanoseconds in the future, and a variable $y :^{-75} Q_2$ will represent a state of qubit $Q_2$ that existed 75 nanoseconds in the past. We give the syntax for two type theories, one with grades (the annotated language) and one without (the plain language). We prove some metatheoretic properties, and describe the semantics in terms of category theory. We show that the input signals to a quantum chip forms a model of the annotated language. We also give a syntatic model, prove that it is initial, and hence prove soundness and completeness theorems.

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arXiv2023

Effect Algebras as Omega-categories

We show how an effect algebra $\mathcal{X}$ can be regarded as a category, where the morphisms $x \rightarrow y$ are the elements $f$ such that $x \leq f \leq y$. This gives an embedding $\mathbf{EA} \rightarrow \mathbf{Cat}$. The interval $[x,y]$ proves to be an effect algebra in its own right, so $\mathcal{X}$ is an $\mathbf{EA}$-enriched category. The construction can therefore be repeated, meaning that every effect algebra can be identified with a strict $\omega$-category. We describe explicitly the strict $\omega$-category structure for two classes of operators on a Hilbert space.

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arXiv2019

Constructing Independently Verifiable Privacy-Compliant Type Systems for Message Passing between Black-Box Components

Privacy by design (PbD) is the principle that privacy should be considered at every stage of the software engineering process. It is increasingly both viewed as best practice and required by law. It is therefore desirable to have formal methods that provide guarantees that certain privacy-relevant properties hold. We propose an approach that can be used to design a privacy-compliant architecture without needing to know the source code or internal structure of any individual component. We model an architecture as a set of agents or components that pass messages to each other. We present in this paper algorithms that take as input an architecture and a set of privacy constraints, and output an extension of the original architecture that satisfies the privacy constraints.

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arXiv2016

A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic

The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A computation becomes `stuck' when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over $\Omega$. There are proofs which inhabit propositions, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are paths which inhabit equations $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.

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arXiv2015

A Type Theory for Probabilistic and Bayesian Reasoning

This paper introduces a novel type theory and logic for probabilistic reasoning. Its logic is quantitative, with fuzzy predicates. It includes normalisation and conditioning of states. This conditioning uses a key aspect that distinguishes our probabilistic type theory from quantum type theory, namely the bijective correspondence between predicates and side-effect free actions (called instrument, or assert, maps). The paper shows how suitable computation rules can be derived from this predicate-action correspondence, and uses these rules for calculating conditional probabilities in two well-known examples of Bayesian reasoning in (graphical) models. Our type theory may thus form the basis for a mechanisation of Bayesian inference.

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arXiv2014

QPEL: Quantum Program and Effect Language

We present the syntax and rules of deduction of QPEL (Quantum Program and Effect Language), a language for describing both quantum programs, and properties of quantum programs - effects on the appropriate Hilbert space. We show how semantics may be given in terms of state-and-effect triangles, a categorical setting that allows semantics in terms of Hilbert spaces, C*-algebras, and other categories. We prove soundness and completeness results that show the derivable judgements are exactly those provable in all state-and-effect triangles.

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arXiv2009

Classical Predicative Logic-Enriched Type Theories

A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative systems of second order arithmetic ACAO and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logic-enriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACAO. The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.

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arXiv2008

Lambda-Free Logical Frameworks

We present the definition of the logical framework TF, the Type Framework. TF is a lambda-free logical framework; it does not include lambda-abstraction or product kinds. We give formal proofs of several results in the metatheory of TF, and show how it can be conservatively embedded in the logical framework LF: its judgements can be seen as the judgements of LF that are in beta-normal, eta-long normal form. We show how several properties, such as adequacy theorems for object theories and the injectivity of constants, can be proven more easily in TF, and then `lifted' to LF.

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arXiv2008

Weyl's Predicative Classical Mathematics as a Logic-Enriched Type Theory

We construct a logic-enriched type theory LTTW that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTW, including Weyl's definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics.

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