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May. 30th, 2020
http://www.tac.mta.ca/tac/volumes/14/5/14-05.pdf
Call-by-push-value is a ”semantic machine code”, providing a set of simple primitives from which both the call-by-value and call-by-name paradigms are built. We present its operational semantics as a stack machine, suggesting a term judgement of stacks. We then see that CBPV, incorporating these stack terms, has a simple categorical semantics based on an adjunction between values and stacks. There are no coherence requirements.
We describe this semantics incrementally. First, we introduce locally indexed categories and the opGrothendieck construction, and use these to give the basic structure for interpreting the three judgements: values, stacks and computations. Then we look at the universal property required to interpret each type constructor. We define a model to be a strong adjunction with countable coproducts, countable products and exponentials.
We see a wide range of instances of this structure: we give examples for divergence, storage, erratic choice, continuations, possible worlds and games (with or without a bracketing condition), in each case resolving the strong monad from the literature into a strong adjunction. And we give ways of constructing models from other models.
Finally, we see that call-by-value and call-by-name are interpreted within the Kleisli and co-Kleisli parts, respectively, of a call-by-push-value adjunction.
Call-by-push-value is a ”semantic machine code”, providing a set of simple primitives from which both the call-by-value and call-by-name paradigms are built. We present its operational semantics as a stack machine, suggesting a term judgement of stacks. We then see that CBPV, incorporating these stack terms, has a simple categorical semantics based on an adjunction between values and stacks. There are no coherence requirements.
We describe this semantics incrementally. First, we introduce locally indexed categories and the opGrothendieck construction, and use these to give the basic structure for interpreting the three judgements: values, stacks and computations. Then we look at the universal property required to interpret each type constructor. We define a model to be a strong adjunction with countable coproducts, countable products and exponentials.
We see a wide range of instances of this structure: we give examples for divergence, storage, erratic choice, continuations, possible worlds and games (with or without a bracketing condition), in each case resolving the strong monad from the literature into a strong adjunction. And we give ways of constructing models from other models.
Finally, we see that call-by-value and call-by-name are interpreted within the Kleisli and co-Kleisli parts, respectively, of a call-by-push-value adjunction.
A Functional Functional Interpretation
May. 30th, 2020 04:47 pmhttps://www.pédrot.fr/articles/lics2014.pdf
In this paper, we present a modern reformulation of the Dialectica interpretation based on the linearized version of de Paiva. Contrarily to Go ̈del’s original translation which translated HA into system T, our presentation applies on untyped λ-terms and features nicer proof-theoretical properties.
In the Curry-Howard perspective, we show that the computational behaviour of this translation can be accurately described by the explicit stack manipulation of the Krivine abstract machine, thus giving it a direct-style operational explanation.
Finally, we give direct evidence that supports the fact our presentation is quite relevant, by showing that we can apply it to the dependently-typed calculus of constructions with universes CCω almost without any adaptation. This answers the question of the validity of Dialectica-like constructions in a dependent setting.
In this paper, we present a modern reformulation of the Dialectica interpretation based on the linearized version of de Paiva. Contrarily to Go ̈del’s original translation which translated HA into system T, our presentation applies on untyped λ-terms and features nicer proof-theoretical properties.
In the Curry-Howard perspective, we show that the computational behaviour of this translation can be accurately described by the explicit stack manipulation of the Krivine abstract machine, thus giving it a direct-style operational explanation.
Finally, we give direct evidence that supports the fact our presentation is quite relevant, by showing that we can apply it to the dependently-typed calculus of constructions with universes CCω almost without any adaptation. This answers the question of the validity of Dialectica-like constructions in a dependent setting.