Robin Adams

Pure type systems with judgemental equality

In a typing system, there are two approaches that may be taken to the notion of equality. One can use some external relation of convertibility defined on the terms of the grammar, such as

A Modular Hierarchy of Logical Frameworks

\beta

Structural subtyping for inductive types with functorial equality rules

-convertibility or

Weyl's predicative classical mathematics as a logic-enriched type theory

\beta \eta

A Type Theory for Probabilistic and Bayesian Reasoning

-convertibility; or one can introduce a judgement form for equality into the rules of the typing system itself. For quite some time, it has been an open problem whether the two systems produced by these two choices are equivalent. This problem is essentially the problem of proving that the Subject Reduction property holds in the system with judgemental equality. In this paper, we shall prove that the equivalence holds for all functional Pure Type Systems (PTSs). The proof essentially consists of proving the Church-Rosser Theorem for a typed version of parallel one-step reduction. This method should generalise easily to many typing systems which satisfy the Uniqueness of Types property.

QPEL: Quantum Program and Effect Language

We present a method for defining logical frameworks as a collection of features which are defined and behave independently of one another. Each feature is a set of grammar clauses and rules of deduction such that the result of adding the feature to a framework is a conservative extension of the framework itself. We show how several existing logical frameworks can be so built, and how several much weaker frameworks defined in this manner are adequate for expressing a wide variety of object logics.

Classical predicative logic-enriched type theories

In this paper we study subtyping for inductive types in dependent type theories in the framework of coercive subtyping. General structural subtyping rules for parameterised inductive types are formulated based on the notion of inductive schemata. Certain extensional equality rules play an important role in proving some of the crucial properties of the type system with these subtyping rules. In particular, it is shown that the structural subtyping rules are coherent and that transitivity is admissible in the presence of the functorial rules of computational equality.

A pluralist approach to the formalisation of mathematics

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 formalize many results from that book in LTTW, including Weyls 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 nonconstructive foundation for mathematics.

Coercive subtyping in lambda-free logical frameworks

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. 1998 ACM Subject Classification F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic — Lambda calculus and related systems; G.3 [Probability and Statistics]: Probabilistic algorithms; F.3.1 [Logics and Meanings of Programs]: Specifying and Verifying and Reasoning about Programs

Formalized metatheory with terms represented by an indexed family of types

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.