Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets

We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way. We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers ‘black-and-white’ graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons

Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets

We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons

Tail Recursion Through Universal Invariants

Tail recursive constructions suggest a new semantics for datatypes, which allows a direct match between specifications and tail recursive programs. The semantics focusses on loops, their fixpoints, invariants and convergence. Convergent models of the natural numbers and lists are examined in detail, and, under very mild conditions, are shown to be equivalent to the corresponding initial algebra models. 1 Introduction Tail recursion is a central feature of program construction because of its efficiency, but is usually assigned a secondary place in semantics, which is dominated by primitive recursion as expressed through initial algebras. The success of this approach is testimony to the ease with which we can use initial algebras to specify functions, and their theoretical power. The difficulty is that whenever such a specification is to be translated into code there remains the need to optimise it, often by conversion into tail recursive form. Conversely, it is not at all easy to provi..