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Adjunction models for call-by-push-value with stacks

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Paul Blain Levy

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.

Keywords:
call-by-push-value, adjunction, CK-machine, monad,
denotational semantics, indexed category, continuations, possible worlds,
game semantics, call-by-name, call-by-value

2000 MSC:
18C50

*Theory and Applications of Categories,*
Vol. 14, 2005,
No. 5, pp 75-110.

http://www.tac.mta.ca/tac/volumes/14/5/14-05.dvi

http://www.tac.mta.ca/tac/volumes/14/5/14-05.ps

http://www.tac.mta.ca/tac/volumes/14/5/14-05.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/5/14-05.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/5/14-05.ps

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