Formalising Mathematics in Simple Type Theory

Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics. This point is seen by examining formal proofs of a theorem about stereographic projections. A formalisation using the HOL Light proof assistant is contrasted with one using Isabelle/HOL. Harrison's technique for formalising Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic type classes. However, every formal system can be outgrown, and mathematics should be formalised with a view that it will eventually migrate to a new formalism

Parallel Verification of Natural Deduction Proof Graphs

Graph-based interactive theorem provers offer a visual representation of proofs, explicitly representing the dependencies and inferences between each of the proof steps in a graph or hypergraph format. The number and complexity of these dependency links can determine how long it takes to verify the validity of the entire proof. Towards this end, we present a set of parallel algorithms for the formal verification of graph-based natural-deduction (ND) style proofs. We introduce a definition of layering that captures dependencies between the proof steps (nodes). Nodes in each layer can then be verified in parallel as long as prior layers have been verified. To evaluate the performance of our algorithms on proof graphs, we propose a framework for finding the performance bounds and patterns using directed acyclic network topologies (DANTs). This framework allows us to create concrete instances of DANTs for empirical evaluation of our algorithms. With this, we compare our set of parallel algorithms against a serial implementation with two experiments: one scaling both the problem size and the other scaling the number of threads. Our findings show that parallelization results in improved verification performance for certain DANT instances. We also show that our algorithms scale for certain DANT instances with respect to the number of threads.Comment: In Proceedings LFMTP 2023, arXiv:2311.0991

Computational effects and operations: an overview

We overview a programme to provide a unified semantics for computational effects based upon the notion of a countable enriched Lawvere theory. We define the notion of countable enriched Lawvere theory, show how the various leading examples of computational effects, except for continuations, give rise to them, and we compare the definition with that of a strong monad. We outline how one may use the notion to model three natural ways in which to combine computational effects: by their sum, by their commutative combination, and by distributivity. We also outline a unified account of operational semantics. We present results we have already shown, some partial results, and our plans for further development of the programme