We show that, when the actions of a Mazurkiewicz trace are considered not merely as atomic but as transformations from a specified type of inputs to a specified type of outputs, we obtain a novel notion of presentation for effectful categories (also known as generalized Freyd categories), a well-known algebraic structure in the semantics of side-effecting computation. Like the usual representation of traces as graphs, our notion of presentation gives rise to a graphical representation of morphisms in effectful categories. We use our presentations to give a construction of the commuting tensor product of free effectful categories, capturing the combination of systems in which the actions of each must commute with one another, while still permitting exchange of resources.

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 We introduce context-free languages of morphisms in monoidal categories, extending recent work on the categorification of context-free languages, and regular languages of string diagrams. Context-free languages of string diagrams include classical context-free languages of words, trees, and hypergraphs, when instantiated over appropriate monoidal categories. We prove a representation theorem for context-free languages of string diagrams: every such language arises as the image under a monoidal functor of a regular language of string diagrams.

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.

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

Premonoidal categories are monoidal categories without the interchange law while effectful categories are premonoidal categories with a chosen monoidal subcategory of interchanging morphisms. In the same sense that string diagrams, pioneered by Joyal and Street, are an internal language for monoidal categories, we show that string diagrams with an added "runtime object", pioneered by Alan Jeffrey, are an internal language for effectful categories and can be used as string diagrams for effectful, premonoidal, and Freyd categories.

We introduce the normal produoidal category of monoidal contexts over an arbitrary monoidal category. In the same sense that a monoidal morphism represents a process, a monoidal context represents an incomplete process: a piece of a decomposition, possibly containing missing parts. We characterize monoidal contexts in terms of universal properties. In particular, symmetric monoidal contexts coincide with monoidal lenses, endowing them with a novel universal property. We apply this algebraic structure to the analysis of multi-party interaction protocols in arbitrary theories of processes.

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 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.

It is not usual practice in the evolutionary algorithms area to benchmark different operations in order to choose the best language for a single or multilanguage implementation. Researchers rely instead on common practice or frameworks using mainstream languages. That is why it is usual practice to choose compiled languages (namely Java or C/C++) when implementing evolutionary algorithms, without considering other languages or rejecting them outright on the basis of performance. Since there is a myriad of languages nowadays, we considered it an interesting challenge to measure their speed when performing frequent operations in evolutionary algorithms. In this paper we have tested three basic evolutionary algorithm operations over binary chromosomes: bitflip mutation, crossover and the OneMax fitness function. As a performance measure, the speed for both popular and not so popular computer languages have been used. In general, the results confirm that compiled languages scale and perform better, but also in some cases have a behaviour that is independent of the size of the chromosome. Additionally, results show that other languages, such as Go (compiled) or Python (interpreted) are fast enough for most purposes. Besides, these experiments show which of these operations are, in fact, the best for choosing an implementation language based on its performance.