Tape diagrams provide a graphical representation for arrows of rig categories, namely categories equipped with two monoidal structures, ⊕ and ⊗, where ⊗ distributes over ⊕. However, their applicability is limited to categories where ⊕ is a biproduct, i.e., both a categorical product and a coproduct. In this work, we extend tape diagrams to deal with Kleisli categories of symmetric monoidal monads, presented by algebraic theories.

Effectful Mealy machines, which we introduce, are a generalization of Mealy machines with global effects determined by an effectful triple. We provide semantics of effectful Mealy machines in terms of both bisimilarity and traces: bisimilarity is characterized syntactically, via uniform feedback; traces are constructed coinductively in terms of streams. We prove that this framework characterizes standard causal processes and existing flavours of Mealy machine, bisimilarity, and trace equivalence. In the commutative case, we introduce a monoidal generalization of Raney's causal functions: monoidal causal processes.

Abstract Tape diagrams provide a convenient graphical notation for arrows of rig categories, i.e., categories equipped with two monoidal products, $$\oplus $$ ⊕ and $$\otimes $$ ⊗ . In this work, we introduce Kleene-Cartesian rig categories, namely rig categories where $$\otimes $$ ⊗ provides a Cartesian bicategory, while $$\oplus $$ ⊕ a Kleene bicategory.We show that the associated tape diagrams can conveniently deal with Hoare logic.

We introduce effectful Mealy machines - a general notion of Mealy machine with global effects - and give them semantics in terms of both bisimilarity and traces. Bisimilarity of effectful Mealy machines is characterized syntactically, via free uniform feedback. Traces of effectful Mealy machines are given a novel semantic coinductive universe in terms of effectful streams. We prove that this framework generalizes standard causal processes and captures existing flavours of Mealy machine, bisimilarity, and trace.

The categorical modeling of Petri nets has received much attention recently. The Dialectica construction has also had its fair share of attention. We revisit the use of the Dialectica construction as a categorical model for Petri nets generalising the original application to suggest that Petri nets with different kinds of transitions can be modelled in the same categorical framework. Transitions representing truth-values, probabilities, rates or multiplicities, evaluated in different algebraic structures called lineales are useful and are modelled here in the same category. We investigate (categorical instances of) this generalised model and its connections to more recent models of categorical nets. Final version for Fundamenta Informaticae

We introduce monoidal streams. Monoidal streams are a generalization of causal stream functions, which can be defined in cartesian monoidal categories, to arbitrary symmetric monoidal categories. In the same way that streams provide semantics to dataflow programming with pure functions, monoidal streams provide semantics to dataflow programming with theories of processes represented by a symmetric monoidal category. Monoidal streams also form a feedback monoidal category. In the same way that we can use a coinductive stream calculus to reason about signal flow graphs, we can use coinductive string diagrams to reason about feedback monoidal categories. As an example, we study syntax for a stochastic dataflow language, with semantics in stochastic monoidal streams. arXiv admin note: substantial text overlap with arXiv:2202.02061

We introduce partial Markov categories. In the same way that Markov categories encode stochastic processes, partial Markov categories encode stochastic processes with constraints, observations and updates. In particular, we prove a synthetic Bayes theorem; we apply it to define a syntactic partial theory of observations on any Markov category whose normalisations can be computed in the original Markov category. Finally, we formalise Evidential Decision Theory in terms of partial Markov categories, and provide examples.

We introduce monoidal width as a measure of complexity for morphisms in monoidal categories. Inspired by well-known structural width measures for graphs, like tree width and rank width, monoidal width is based on a notion of syntactic decomposition: a monoidal decomposition of a morphism is an expression in the language of monoidal categories, where operations are monoidal products and compositions, that specifies this morphism. Monoidal width penalises the composition operation along ``big'' objects, while it encourages the use of monoidal products. We show that, by choosing the correct categorical algebra for decomposing graphs, we can capture tree width and rank width. For matrices, monoidal width is related to the rank. These examples suggest monoidal width as a good measure for structural complexity of processes modelled as morphisms in monoidal categories.

We introduce monoidal streams: a generalization of causal stream functions to monoidal categories. In the same way that streams provide semantics to dataflow programming with pure functions, monoidal streams provide semantics to dataflow programming with theories of processes represented by a symmetric monoidal category. At the same time, monoidal streams form a feedback monoidal category, which can be used to interpret signal flow graphs. As an example, we study a stochastic dataflow language.