A formal proof of the independence of the continuum hypothesis

Han, Jesse Michael; Doorn, Floris van

doi:10.1145/3372885.3373826

Cited by 16 publications

(12 citation statements)

References 36 publications

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“…Cost-aware logical framework. Decades of experience has shown the effectiveness of using dependent type theories to encode mathematics [Buzzard et al 2020;Gonthier 2008;Han and van Doorn 2020] and to verify behavioral properties of programs [Chlipala 2013;Lee et al 2007;Stump 2016; Ullrich 2016]. Our experience with calf suggests that dependent type theories are also an tool for analyzing intensional properties of programs including cost.…”

Section: Verification In Calfmentioning

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: Verification In Calfmentioning

confidence: 99%

A cost-aware logical framework

Niu

Sterling

Grodin

et al. 2022

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

show abstract

“…Cost-aware logical framework. Decades of experience has shown the effectiveness of using dependent type theories to encode mathematics [Buzzard et al 2020;Gonthier 2008;Han and van Doorn 2020] and to verify behavioral properties of programs [Chlipala 2013;Lee et al 2007;Stump 2016; Ullrich 2016]. Our experience with calf suggests that dependent type theories are also an appropriate tool for analyzing intensional properties of programs including cost.…”

Section: Verification In Calfmentioning

confidence: 99%

A cost-aware logical framework

Niu¹,

Sterling²,

Grodin³

et al. 2021

Preprint

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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.

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“…Lean Lean is an interactive theorem prover and functional programming language (de Moura et al, 2015). It has an extremely active community and is host to some of the most sophisticated formalized mathematics in the world, including scheme theory (Buzzard et al, 2019), forcing (Han & van Doorn, 2020), perfectoid spaces (Buzzard et al, 2020), and condensed mathematics (Scholze, 2020).…”

Section: Introductionmentioning

confidence: 99%

Proof Artifact Co-training for Theorem Proving with Language Models

Han¹,

Rute²,

Wu³

et al. 2021

Preprint

Self Cite

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Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT (Proof Artifact Co-Training), a general methodology for extracting abundant selfsupervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32% to 48%.

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A formal proof of the independence of the continuum hypothesis

Cited by 16 publications

References 36 publications

A cost-aware logical framework

A cost-aware logical framework

A cost-aware logical framework

Proof Artifact Co-training for Theorem Proving with Language Models

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