Rasmus Ejlers Møgelberg

Rasmus Ejlers Møgelberg

Type theories with multi-clocked guarded recursion provide a flexible framework for programming with coinductive types encoding productivity in types. Combining this with solutions to general guarded domain equations one can also construct relatively simple denotational models of programming languages with advanced features. These constructions have previously been explored in the setting of extensional type theory through a presheaf model, which proves correctness of encodings of W-types. That model has been adapted to presheaves of cubical sets (functors into the category of cubical sets), where the model verifies correctness of encodings also of coinductive types whose definitions involve quotient inductive types such as finite powersets or finite distributions. Likewise the cubical model also verifies correctness of coinductive predicates defined using existential quantification and allows the results to be related to the global world of cubical sets. This paper looks at how to extend the extensional presheaf model of multi-clocked guarded recursion to higher ordinals, so that correctness of encodings of coinductive types can be extended from W-types to those involving finite powersets and finite distributions, as well as coinductive predicates involving existential quantification. This extension will allow results previously proved in Clocked Cubical Type Theory to be interpreted in a model based on set-theory, proving the correctness of these results as understood in their usual set theoretic interpretation.

Giorgio Bacci, Rasmus Ejlers Møgelberg

Quantitative logic reasons about the degree to which formulas are satisfied. This paper studies the fundamental reasoning principles of higher-order quantitative logic and their application to reasoning about probabilistic programs and processes. We construct an affine calculus for $1$-bounded complete metric spaces and the monad for probability measures equipped with the Kantorovich distance. The calculus includes a form of guarded recursion interpreted via Banach's fixed point theorem, useful, e.g., for recursive programming with processes. We then define an affine higher-order quantitative logic for reasoning about terms of our calculus. The logic includes novel principles for guarded recursion, and induction over probability measures and natural numbers. We illustrate the expressivity of the logic by a sequence of case studies: Proving upper limits on bisimilarity distances of Markov processes, showing convergence of a temporal learning algorithm and of a random walk using a coupling argument.