Fabio Zanasi

Interacting Hopf algebras

We introduce the theory IHR of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IHR are derived using Lacks approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid–comonoid pairs. This construction is instrumental in showing that IHR is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R.

A Categorical Semantics of Signal Flow Graphs.

We introduce \(\mathbb{IH}\), a sound and complete graphical theory of vector subspaces over the field of polynomial fractions, with relational composition. The theory is constructed in modular fashion, using Lack’s approach to composing PROPs with distributive laws.

Interacting Bialgebras are Frobenius

Bialgebras and Frobenius algebras are different ways in which monoids and comonoids interact as part of the same theory. Such theories feature in many fields: e.g. quantum computing, compositional semantics of concurrency, network algebra and component-based programming. In this paper we study an important sub-theory of Coecke and Duncans ZX-calculus, related to strongly-complementary observables, where two Frobenius algebras interact. We characterize its free model as a category of ℤ2-vector subspaces. Moreover, we use the framework of PROPs to exhibit the modular structure of its algebra via a universal construction involving span and cospan categories of ℤ2-matrices and distributive laws between PROPs. Our approach demonstrates that the Frobenius structures result from the interaction of bialgebras.

Rewriting modulo symmetric monoidal structure

String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory.An important role in many such approaches is played by equational theories of diagrams, typically oriented and applied as rewrite rules. This paper lays a comprehensive foundation for this form of rewriting. We interpret diagrams combinatorially as typed hypergraphs and establish the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hypergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure.We illustrate our approach with a proof of termination for the theory of non-commutative bimonoids.

A Predicate/State Transformer Semantics for Bayesian Learning

This paper establishes a link between Bayesian inference (learning) and predicate and state transformer operations from programming semantics and logic. Specifically, a very general definition of backward inference is given via first applying a predicate transformer and then conditioning. Analogously, forward inference involves first conditioning and then applying a state transformer. These definitions are illustrated in many examples in discrete and continuous probability theory and also in quantum theory.

Bialgebraic Semantics for Logic Programming

We study different behavioral metrics, such as those arising from both branching and linear-time semantics, in a coalgebraic setting. Given a coalgebra

A Universal Construction for (Co)Relations.

\alpha\colon X \to HX

The Algebra of Partial Equivalence Relations

for a functor

Lawvere Categories as Composed PROPs

H \colon \mathrm{Set}\to \mathrm{Set}

Brzozowski goes concurrent:A Kleene Theorem for pomset languages

, we define a framework for deriving pseudometrics on