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We generally welcome all relevant contributions. Personally I know nothing about probabilistic type theory, but would be happy to learn. Check out How to get started, writing in the nLab, etc., and let us know what you’re doing.
Yes, definitely! I’ve been occasionally looking out for work in this direction. No doubt there’s a range of ideas fitting this description.
Certainly there’s
But then there’s work by computational linguists, such as A Probabilistic Rich Type Theory for Semantic Interpretation.
Without using the name, ’probabilistic programming’ may involve a type theory: Semantics for probabilistic programming: higher-order functions, continuous distributions, and soft constraints
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