Probabilistic Boolean circuits have recently been proposed as a string-diagrammatic foundation for finite probabilistic programming. In this paper, we present a complete set of axioms for their semantics in terms of Markov kernels. Our approach is based on two intermediate results: completeness for partial Boolean circuits and completeness for probabilistic Boolean tapes, a diagrammatic language for rig categories.

The study of categories that abstract the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. This paper strives to provide a modern presentation of these ``categories for relations'', including their enriched version, further showing how they arise as Kleisli categories of symmetric monoidal monads. The resulting taxonomy aims at bringing clarity and organisation to the many related concepts and frameworks occurring in the literature.

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.