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another example

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Daniel Hillerström
2021-05-28 19:41:29 +01:00
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@@ -4374,18 +4374,18 @@ here, only the essential rules for the $\Cupto$ constructs.
The typing rule for $\Set$ uses the type embedded in the prompt to fix
the type of the whole computation $N$. Similarly, the typing rule for
$\Cupto$ uses the prompt type of its value argument to fix the answer
type for the continuation $k$. The type of the $\Cupto$ expression is
the same as the domain of the continuation, which at first glance may
seem strange. The intuition is that $\Cupto$ behaves as a let binding
for the continuation in the context of a $\Set$ expression, i.e.
%
\[
\bl
\Set\;p^{\prompttype~B}\;\In\;\EC[\Cupto\;p\;k^{A \to B}.M^B]^B\\
\reducesto \Let\;k \revto \lambda x^{A}.\EC[x]^B \;\In\;M^B
\el
\]
%
type for the continuation $k$. % The type of the $\Cupto$ expression is
% the same as the domain of the continuation, which at first glance may
% seem strange. The intuition is that $\Cupto$ behaves as a let binding
% for the continuation in the context of a $\Set$ expression, i.e.
% %
% \[
% \bl
% \Set\;p^{\prompttype~B}\;\In\;\EC[\Cupto\;p\;k^{A \to B}.M^B]^B\\
% \reducesto \Let\;k \revto \lambda x^{A}.\EC[x]^B \;\In\;M^B
% \el
% \]
% %
The dynamic semantics is generative to accommodate generation of fresh
prompts. Formally, the reduction relation is augmented with a store
@@ -4410,7 +4410,39 @@ active prompt $p$. After reification the prompt is removed and
evaluation continues as $M$. The $\slab{Resume}$ rule reinstalls the
captured context $\EC$ with the argument $V$ plugged in.
%
\dhil{Show some example of use\dots}
\citeauthor{GunterRR95}'s cupto provides similar behaviour to
\citeauthor{QueinnecS91}'s splitter in regards to being able to `jump
over prompt'. However, the separation of prompt creation from the
control reifier coupled with the ability to set prompts manually
provide a considerable amount of flexibility. For instance, consider
the following example which illustrates how control reifier $\Cupto$
may escape a matching control delimiter. Let us assume that two
distinct prompts $p$ and $p'$ have already been created.
%
\begin{derivation}
&2 + \Set\; p \;\In\; 3 + \Set\;p'\;\In\;(\Set\;p\;\In\;\lambda\Unit.\Cupto~p~k.k\,(k~1))\,\Unit,\{p,p'\}\\
\reducesto& \reason{\slab{Value}}\\
&2 + \Set\; p \;\In\; 3 + \Set\;p'\;\In\;(\lambda\Unit.\Cupto~p~k.k\,(k~1))\,\Unit,\{p,p'\}\\
\reducesto^+& \reason{\slab{Capture} $\EC = 3 + \Set\;p'\;\In\;[\,]$}\\
&2 + \qq{\cont_{\EC}}\,(\qq{\cont_{\EC}}\,1),\{p,p'\}\\
\reducesto& \reason{\slab{Resume} $\EC$ with $1$}\\
&2 + \qq{\cont_{\EC}}\,(3 + \Set\;p'\;\In\;1),\{p,p'\}\\
\reducesto^+& \reason{\slab{Value}}\\
&2 + \qq{\cont_{\EC}}\,4,\{p,p'\}\\
\reducesto& \reason{\slab{Resume} $\EC$ with $4$}\\
&2 + \Set\;p'\;In\;4,\{p,p'\}\\
\reducesto& \reason{\slab{Value}}\\
&2 + 4,\{p,p'\} \reducesto 6,\{p,p'\}
\end{derivation}
%
The prompt $p$ is used twice, and the dynamic scoping of $\Cupto$
means when it is evaluated it reifies the continuation up to the
nearest enclosing usage of the prompt $p$. Contrast this with the
morally equivalent example using splitter, which would get stuck on
the application of the control reifier, because it has escaped the
dynamic extent of its matching delimiter.
%
\paragraph{\citeauthor{PlotkinP09}'s effect handlers} In 2009,
\citet{PlotkinP09} introduced handlers for \citeauthor{PlotkinP01}'s