The Relational Machine Calculus
Chris Barrett, Daniel Castle, Willem Heijltjes
TL;DR
The paper introduces the Relational Machine Calculus (RMC) as a foundational model for first-order relational programming, derived from the FMC and enriched with duality, unification, Kleene algebra, and multi-location stacks. It provides a rigorous stack-machine semantics, a complete equational theory with confluent reduction, and both denotational and categorical semantics centered on Rel and Frobenius monoids. The RMC can encode a wide range of computational paradigms, from regular expressions and Prolog to Turing machines, interaction nets, and Petri nets, while preserving operational semantics. The work lays groundwork for a unified theory of relational programming with rigorous type discipline, duality-aware rewriting, and a pathway toward higher-order extensions and diagrammatic semantics, potentially impacting both language design and diagrammatic reasoning tools.
Abstract
This paper presents the Relational Machine Calculus (RMC): a simple, foundational model of first-order relational programming. The RMC originates from the Functional Machine Calculus (FMC), which generalizes the lambda-calculus and its standard call-by-name stack machine in two directions. One, "locations", introduces multiple stacks, which enable effect operators to be encoded into the abstraction and application constructs. The second, "sequencing", introduces the imperative notions of "skip" and "sequence", similar to kappa-calculus and concatenative programming languages. The key observation of the RMC is that the first-order fragment of the FMC exhibits a latent duality which, given a simple decomposition of the relevant constructors, can be concretely expressed as an involution on syntax. Semantically, this gives rise to a sound and complete calculus for string diagrams of Frobenius monoids. We consider unification as the corresponding symmetric generalization of beta-reduction. By further including standard operators of Kleene algebra, the RMC embeds a range of computational models: the kappa-calculus, logic programming, automata, Interaction Nets, and Petri Nets, among others. These embeddings preserve operational semantics, which for the RMC is again given by a generalization of the standard stack machine for the lambda-calculus. The equational theory of the RMC (which supports reasoning about its operational semantics) is conservative over both the first-order lambda-calculus and Kleene algebra, and can be oriented to give a confluent reduction relation.