Quotients, inductive types, and quotient inductive types

Fiore, Marcelo; Pitts, Andrew M.; Steenkamp, S. C.

doi:10.48550/arxiv.2101.02994

Cited by 8 publications

(13 citation statements)

References 22 publications

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“…In Fig. 2 we define the intensional modality as a quotient inductive type [Altenkirch and Kaposi 2016b;Fiore et al 2021]. In categorical language the intensional modality is the pushout 𝐴 ⊔ 𝐴× ¶ E ¶ E of the projection maps of 𝐴 × ¶ E .…”

Section: Closed/intensional Modalitymentioning

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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Section: Closed/intensional Modalitymentioning

confidence: 99%

A cost-aware logical framework

Niu

Sterling

Grodin

et al. 2022

Proc. ACM Program. Lang.

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show abstract

“…While Cubical Agda [39] accepts all three HIT definitions, the status of defining HITs in HoTT and Cubical type theories in general is an open research question. Various schemas for defining HITs have been proposed [35,14,11,22,23]. HITs have also been used in other computer science applications [5,24,6,4].…”

Section: Multiset Equalitymentioning

confidence: 99%

Free Commutative Monoids in Homotopy Type Theory

Choudhury¹,

Fiore²

2021

Preprint

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We develop a constructive theory of finite multisets, defining them as free commutative monoids in Homotopy Type Theory. We formalise two algebraic presentations of this construction using 1-HITs, establishing the categorical universal property for each and thereby their equivalence. These presentations correspond to equational theories including a commutation axiom. In this setting, we prove important structural combinatorial properties of singleton multisets arising from concatenations and projections of multisets. This is done in generality, without assuming decidable equality on the carrier set. Further, as applications, we present a constructive formalisation of the relational model of differential linear logic and use it to characterise the equality type of multisets. This leads us to a novel conditional equational presentation of the finite-multiset construction and thereby to a sound and complete deduction system for multiset equality.

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“…In Fig. 2 we define the intensional modality as a quotient inductive type [Altenkirch and Kaposi 2016b;Fiore et al 2021]. In categorical language the intensional modality is the pushout…”

Section: Closed/intensional Modalitymentioning

confidence: 99%

A cost-aware logical framework

Niu¹,

Sterling²,

Grodin³

et al. 2021

Preprint

View full text Add to dashboard Cite

We present calf, a cost-aware logical framework 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.

show abstract

Quotients, inductive types, and quotient inductive types

Cited by 8 publications

References 22 publications

A cost-aware logical framework

A cost-aware logical framework

Free Commutative Monoids in Homotopy Type Theory

A cost-aware logical framework

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