Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in a discrete-time setting. The core of this work is to generalise the discrete-time picture to continuous time by providing a notion of behavioural equivalence for continuous-time Markov processes. In Chen et al. [(2019). Electronic Notes in Theoretical Computer Science 347 45–63.], we proposed two equivalent definitions of bisimulation for continuous-time stochastic processes where the evolution is a flow through time: the first one as an equivalence relation and the second one as a cospan of morphisms. In Chen et al. [(2020). Electronic Notes in Theoretical Computer Science.], we developed the theory further: we introduced different concepts that correspond to different behavioural equivalences and compared them to bisimulation. In particular, we studied the relation between bisimulation and symmetry groups of the dynamics. We also provided a game interpretation for two of the behavioural equivalences. The present work unifies the cited conference presentations and gives detailed proofs.

Robinson’s unification algorithm can be identified as the underlying machinery of both C. Meredith’s rule D (condensed detachment) in propositional logic and of the construction of principal types in lambda calculus and combinatory logic. In combinatory logic, it also plays a crucial role in the construction of Meyer, Bunder & Powers’ Fool’s model. This paper now considers pattern matching, the unidirectional variant of unification, as a basis for logical inference, typing, and a very simple and natural model for untyped combinatory logic. An analysis of the new typing scheme will enable us to characterize a large class of terms of combinatory logic which do not change their principal type when being weakly reduced. We also consider the question whether the major or the minor premisse should be used as the fixed pattern.

We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level n can be constructed in HoTT for any externally fixed natural number n. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where n will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky’s Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be assumed by postulating that the inner and outer natural numbers types are isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of $(\infty,1)$-category and give some examples.

The categorical models of differential linear logic (LL) are additive categories and those of the differential lambda-calculus are left-additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential LL are concerned, these models feature finite nondeterminism and indeed these languages are essentially non-deterministic. We introduce a categorical framework for differentiation which does not require additivity and is compatible with deterministic models such as coherence spaces and probabilistic models such as probabilistic coherence spaces.

We show how to apply forward and reverse mode Combinatory Homomorphic Automatic Differentiation (CHAD) (Vákár (2021). ESOP, 607–634; Vákár and Smeding (2022). ACM Transactions on Programming Languages and Systems 44 (3) 20:1–20:49.) to total functional programming languages with expressive type systems featuring the combination of

• • tuple types;

• • sum types;

• • inductive types;

• • coinductive types;

• • function types.

We achieve this by analyzing the categorical semantics of such types in $\Sigma$-types (Grothendieck constructions) of suitable categories. Using a novel categorical logical relations technique for such expressive type systems, we give a correctness proof of CHAD in this setting by showing that it computes the usual mathematical derivative of the function that the original program implements. The result is a principled, purely functional and provably correct method for performing forward- and reverse-mode automatic differentiation (AD) on total functional programming languages with expressive type systems.

Weak $\omega$-categories are notoriously difficult to define because of the very intricate nature of their axioms. Various approaches have been explored based on different shapes given to the cells. Interestingly, homotopy type theory encompasses a definition of weak $\omega$-groupoid in a globular setting, since every type carries such a structure. Starting from this remark, Brunerie could extract this definition of globular weak $\omega$-groupoids, formulated as a type theory. By refining its rules, Finster and Mimram have then defined a type theory called $\mathsf{CaTT}$, whose models are weak $\omega$-categories. Here, we generalize this approach to monoidal weak $\omega$-categories. Based on the principle that they should be equivalent to weak $\omega$-categories with only one 0-cell, we are able to derive a type theory $\mathsf{MCaTT}$ whose models are monoidal weak $\omega$-categories. This requires changing the rules of the theory in order to encode the information carried by the unique 0-cell. The correctness of the resulting type theory is shown by defining a pair of translations between our type theory $\mathsf{MCaTT}$ and the type theory $\mathsf{CaTT}$. Our main contribution is to show that these translations relate the models of our type theory to the models of the type theory $\mathsf{CaTT}$ consisting of $\omega$-categories with only one 0-cell by analyzing in details how the notion of models interact with the structural rules of both type theories.

Several relational program logics have been introduced for integrating reasoning about relational properties of programs and measurement of quantitative difference between computational effects. Toward a general framework for such logics, in this paper, we formalize the concept of quantitative difference between computational effects as divergences on monads, then develop a relational program logic called approximate computational relational logic (acRL for short). It supports generic computational effects and divergences on them. The semantics of the acRL is given by graded strong relational liftings constructed from divergences on monads. We derive two instantiations of the acRL: (1) for the verification of various kinds of differential privacy of higher-order functional probabilistic programs and (2) the other for measuring difference of distributions of cost between higher-order functional probabilistic programs with a cost counting operator.