Type Universes as Kripke Worlds

What are mutable references; what do they mean? The answers to these questions have spawned lots of important theoretical work and form the foundation of many impactful tools. However, existing semantics collapse a key distinction: which allocations does a reference depend on? In this paper, we deco...

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Vydáno v:Proceedings of ACM on programming languages Ročník 9; číslo ICFP; s. 784 - 813
Hlavní autoři: Koronkevich, Paulette, Bowman, William J.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York, NY, USA ACM 05.08.2025
Témata:
Functional languages
Lambda calculus
Operational semantics
Procedures, functions and subroutines
Semantics
Software and its engineering
Theory of computation
Mutable References
Termination
Kripke models
Logical relations
ISSN:2475-1421, 2475-1421
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Shrnutí:What are mutable references; what do they mean? The answers to these questions have spawned lots of important theoretical work and form the foundation of many impactful tools. However, existing semantics collapse a key distinction: which allocations does a reference depend on? In this paper, we deconstruct the space of mutable higher-order references. We formalize a novel distinction—splitting the design space of references not only into higher-order vs (full-)ground references, but also dependency of an allocation on past vs future allocations. This distinction is fundamental to a thorny issue that arises in constructing semantic models of mutable references—the type-world circularity. The issue disappears for what we call predicative references, those that only quantify over past, not future, allocations, and for non-higher-order impredicative references. We design a syntax and semantics for each point in our newly described space. The syntax relies on a type universe hierarchy, à la dependent type theory, to kind the types of allocated terms, and stratify allocations. Each type universe corresponds to a semantic Kripke world, giving a lightweight syntactic mechanism to design and restrict heap shapes. The semantics bear a resemblance to work on regions, and suggest some connection between universe systems and regions, which we describe in some detail.
ISSN:2475-1421
2475-1421
DOI:10.1145/3747532