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Extensional equivalence and singleton types

Content Provider	ACM Digital Library
Author	Stone, Christopher A. Harper, Robert
Copyright Year	2006
Abstract	We study the $&lamda;^{ΠΣS}_{≤}$ calculus, which contains singleton types $\textit{S}(\textit{M})$ classifying terms of base type provably equivalent to the term $\textit{M}.$ The system includes dependent types for pairs and functions (Σ and Π) and a subtyping relation induced by regarding singletons as subtypes of the base type. The decidability of type checking for this language is non-obvious, since to type check we must be able to determine equivalence of well-formed terms. But in the presence of singleton types, the provability of an equivalence judgment Γ ⊢ $M_{1}$ $&eqiv;M_{2}$ : $\textit{A}$ can depend both on the typing context Γ and on the particular type $\textit{A}$ at which $M_{1}$ and $M_{2}$ are compared.We show how to prove decidability of term equivalence, hence of type checking, in $&lamda;^{ΠΣS}_{≤}$ by exhibiting a type-directed algorithm for directly computing normal forms. The correctness of normalization is shown using an unusual variant of Kripke logical relations organized around sets; rather than defining a logical equivalence relation, we work directly with (subsets of) the corresponding equivalence classes.We then provide a more efficient algorithm for checking type equivalence without constructing normal forms. We also show that type checking, subtyping, and all other judgments of the system are decidable.The $&lamda;^{ΠΣS}_{≤}$ calculus models type constructors and kinds in the intermediate language used by the TILT compiler for Standard ML to implement the SML module system. The decidability of $&lamda;^{ΠΣS}_{≤}$ term equivalence allows us to show decidability of type checking for TILT's intermediate language. We also obtain a consistency result that allows us to prove type safety for the intermediate language. The algorithms derived here form the core of the type checker used for internal type checking in TILT.
Starting Page	676
Ending Page	722
Page Count	47
File Format	PDF
ISSN	15293785
e-ISSN	1557945X
DOI	10.1145/1183278.1183281
Volume Number	7
Issue Number	4
Journal	ACM Transactions on Computational Logic (TOCL)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2006-10-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Singleton types Equivalence algorithms Logical relations
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Logic Computational Mathematics Computer Science

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