The syntax and semantics of quantitative type theory

Atkey, Robert; (2018) The syntax and semantics of quantitative type theory. In: LICS '18. ACM, New York, pp. 56-65. ISBN 978-1-4503-5583-4 (

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We present Quantitative Type Theory, a Type Theory that records usage information for each variable in a judgement, based on a previous system by McBride. The usage information is used to give a realizability semantics using a variant of Linear Combinatory Algebras, refining the usual realizability semantics of Type Theory by accurately tracking resource behaviour. We define the semantics in terms of Quantitative Categories with Families, a novel extension of Categories with Families for modelling resource sensitive type theories.