Normalization for Cubical Type Theory

Sterling, Jonathan; Angiuli, Carlo

doi:10.1109/lics52264.2021.9470719

Cited by 14 publications

(20 citation statements)

References 31 publications

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“…Combining the application described in Section 6.3 with the internal perspective of Orton and Pitts [OP16], the realignment operation is included as an axiom of synthetic Tait computability [Ste21], the mathematical framework behind the recent normalization result for cubical type theory [SA21].…”

Section: Definitionmentioning

confidence: 99%

“…cit. to prove a generalized abstraction/parametricity theorem for a language of software packages ("modules") in the style of Standard ML; subsequently, Sterling and Angiuli [SA21] used STC to positively resolve the long-standing normalization conjecture for cubical type theory [Ang+21]. 5 Building on these results, Gratzer [Gra21] adapted STC to verify the analogous conjecture for multimodal type theory [Gra+20].…”

Section: Applications Of Realignmentmentioning

confidence: 99%

“…It is of practical interest to employ Martin-Löf type theory (MLTT) as an internal language for a variety of categories. In addition to the standard applications of internal methods to mathematics, the existence of models of MLTT in topoi is a critical ingredient for a number of recent results in type theory and programming languages, including the generalized abstraction theorem of Sterling and Harper [SH21] and the proofs of normalization for cubical type theory and multi-modal dependent type theory [Gra21;SA21].…”

Section: Introductionmentioning

confidence: 99%

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Strict universes for Grothendieck topoi

Gratzer¹,

Shulman²,

Sterling³

2022

Preprint

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Hofmann and Streicher famously showed how to lift Grothendieck universes into presheaf topoi, and Streicher has extended their result to the case of sheaf topoi by sheafification. In parallel, van den Berg and Moerdijk have shown in the context of algebraic set theory that similar constructions continue to apply even in weaker metatheories. Unfortunately, sheafification seems not to preserve an important realignment property enjoyed by the presheaf universes that plays a critical role in models of univalent type theory as well as synthetic Tait computability, a recent technique to establish syntactic properties of type theories and programming languages. In the context of multiple universes, the realignment property also implies a coherent choice of codes for connectives at each universe level, thereby interpreting the cumulativity laws present in popular formulations of Martin-Löf type theory.We observe that a slight adjustment to an argument of Shulman constructs a cumulative universe hierarchy satisfying the realignment property at every level in any Grothendieck topos. Hence one has direct-style interpretations of Martin-Löf type theory with cumulative universes into all Grothendieck topoi. A further implication is to extend the reach of recent synthetic methods in the semantics of cubical type theory and the syntactic metatheory of type theory and programming languages to all Grothendieck topoi.

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Section: Definitionmentioning

confidence: 99%

Section: Applications Of Realignmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Strict universes for Grothendieck topoi

Gratzer¹,

Shulman²,

Sterling³

2022

Preprint

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“…On the other hand we may view cost as a structure (proof-relevant property) over a program, and the projection of ordinary programs from cost-instrumented programs is implemented in our setting by the unit of the extensional modality 𝐴 → 𝐴. The perspective of cost as structure is an instance of a more general phenomenon pervading present-day work in type theory: notions that are ill-posed as properties of equivalence classes of typed terms can be recovered more objectively as structures defined over equivalence classes of typed terms, as in the work of Altenkirch and Kaposi [2016a]; Coquand [2019]; Sterling and Angiuli [2021]; .…”

Section: The Language Of Phase Distinctionsmentioning

confidence: 99%

A cost-aware logical framework

Niu

Sterling

Grodin

et al. 2022

Proc. ACM Program. Lang.

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We present calf , a c ost- a ware l ogical f ramework for studying quantitative aspects of functional programs. Taking inspiration from recent work that reconstructs traditional aspects of programming languages in terms of a modal account of phase distinctions , we argue that the cost structure of programs motivates a phase distinction between intension and extension . Armed with this technology, we contribute a synthetic account of cost structure as a computational effect in which cost-aware programs enjoy an internal noninterference property: input/output behavior cannot depend on cost. As a full-spectrum dependent type theory, calf presents a unified language for programming and specification of both cost and behavior that can be integrated smoothly with existing mathematical libraries available in type theoretic proof assistants. We evaluate calf as a general framework for cost analysis by implementing two fundamental techniques for algorithm analysis: the method of recurrence relations and physicist’s method for amortized analysis . We deploy these techniques on a variety of case studies: we prove a tight, closed bound for Euclid’s algorithm, verify the amortized complexity of batched queues, and derive tight, closed bounds for the sequential and parallel complexity of merge sort, all fully mechanized in the Agda proof assistant. Lastly we substantiate the soundness of quantitative reasoning in calf by means of a model construction.

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“…In this article we considered the phase distinction between syntactic and semantic, which allows one to prove parametricity results as well as other important metatheorems such as canonicity and normalization [55,125]. Example 6.2 (Type Refinements).…”

Section: Example 61 (Logical Relations)mentioning

confidence: 99%

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

Sterling

Harper

2021

J. ACM

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The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction , computational effects , and type abstraction . We contribute a fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects, placing projectibility of module expressions on a type-theoretic basis. Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a parametricity structure . Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof- relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs—simplifying the metatheory of strong sums over the collection of types, for although there can be no “relation classifying relations,” one easily accommodates a “family classifying small families.” Using the insight that logical relations/parametricity is itself a form of phase distinction between the syntactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types , by iterating our modal account of the phase distinction. We axiomatize a dependent type theory of parametricity structures using two pairs of complementary modalities (syntactic, semantic) and (static, dynamic), substantiated using the topos theoretic Artin gluing construction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant.

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Normalization for Cubical Type Theory

Cited by 14 publications

References 31 publications

Strict universes for Grothendieck topoi

Strict universes for Grothendieck topoi

A cost-aware logical framework

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

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