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Chris Barrett (Aug 13 2023 at 09:57):I'd like to share my recently available Ph.D. thesis. Of particular interest to some here might be the easy corollary of my results that the first-order fragment of the language serves as a natural term language (and operational semantics) for string diagrams (see the introduction and chapter 4).
Abstract: The Functional Machine Calculus (FMC) was recently introduced as a generalization of the lambda-calculus to include higher-order global state, probabilistic and non-deterministic choice, and input and output, while retaining confluence. The calculus can encode both the call-by-name and call-by-value semantics of these effects. This is enabled by two independent generalizations, both natural from the perspective of the FMC’s operational semantics, which is given by a simple (multi-)stack machine.
The first generalization decomposes the syntax of the lambda-calculus in a way that allows for sequential composition of terms and the encoding of reduction strategies. Specifically, there exist translations of the call-by-name and call-by-value lambda-calculus which preserve operational semantics. The second parameterizes application and abstraction in terms of ‘locations’ (corresponding to the multiple stacks of the machine), which gives a unification of the operational semantics, syntax, and reduction rules of the given effects with those of the lambda-calculus. The FMC further comes equipped with a simple type system which restricts and captures the behaviour of effects.
This thesis makes two main contributions, showing that two fundamental properties of the lambda-calculus are preserved by the FMC. The first is to show that the categorical semantics of the FMC, modulo an appropriate equational theory, is given by the free Cartesian closed category. The equational theory is validated by a notion of observational equivalence. The second contribution is a proof that typed FMC-terms are strongly normalising. This is an extension (and small simplification) of Gandy’s proof for the lambda-calculus, which additionally emphasizes its latent operational intuition.