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      thesis.tex

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@ -7224,7 +7224,7 @@ operator as two mutually recursive handlers.
% %
\[ \[
\bl \bl
\Pipe : \Record{\UnitType \to \alpha \eff \{ \Yield : \beta \opto \UnitType \}, \UnitType \to \alpha\eff\{ \Await : \UnitType \opto \beta \}} \to \alpha \\
\Pipe : \Record{\UnitType \to \alpha \eff \{ \Yield : \beta \opto \UnitType \}; \UnitType \to \alpha\eff\{ \Await : \UnitType \opto \beta \}} \to \alpha \\
\Pipe\, \Record{p; c} \defas \Pipe\, \Record{p; c} \defas
\bl \bl
\ShallowHandle\; c\,\Unit \;\With\; \\ \ShallowHandle\; c\,\Unit \;\With\; \\
@ -7234,7 +7234,7 @@ operator as two mutually recursive handlers.
\ea \ea
\el\medskip\\ \el\medskip\\
\Copipe : \Record{\beta \to \alpha\eff\{ \Await : \UnitType \opto \beta\}, \UnitType \to \alpha\eff\{ \Yield : \beta \opto \UnitType\}} \to \alpha \\
\Copipe : \Record{\beta \to \alpha\eff\{ \Await : \UnitType \opto \beta\}; \UnitType \to \alpha\eff\{ \Yield : \beta \opto \UnitType\}} \to \alpha \\
\Copipe\, \Record{c; p} \defas \Copipe\, \Record{c; p} \defas
\bl \bl
\ShallowHandle\; p\,\Unit \;\With\; \\ \ShallowHandle\; p\,\Unit \;\With\; \\
@ -11200,167 +11200,269 @@ provides a way to extract the final value of some computation from a
configuration. configuration.
\subsection{Putting the machine into action} \subsection{Putting the machine into action}
\newcommand{\Chiid}{\ensuremath{\Chi_{\text{id}}}} \newcommand{\Chiid}{\ensuremath{\Chi_{\text{id}}}}
\newcommand{\kappaid}{\ensuremath{\kappa_{\text{id}}}} \newcommand{\kappaid}{\ensuremath{\kappa_{\text{id}}}}
\paragraph{Example} To make the transition rules in
Figure~\ref{fig:abstract-machine-semantics} concrete we give an
example of the abstract machine in action. We reuse the small producer
and consumer from Section~\ref{sec:shallow-handlers-tutorial}. We
reproduce their definitions here in ANF.
%
\[
\bl
\Ones \defas \Rec\;ones~\Unit. \Do\; \Yield~1; ones~\Unit \\
\AddTwo \defas
\lambda \Unit.
\Let\; x \revto \Do\;\Await~\Unit \;\In\;
\Let\; y \revto \Do\;\Await~\Unit \;\In\;
x + y \\
\el
\]%
%
Let $N_x$ denote the term $\Let\;x \revto \Do\;\Await~\Unit\;\In\;N_y$
and $N_y$ the term $\Let\;y \revto \Do\;\Await~\Unit\;\In\;x+y$.
%
%% For clarity, we annotate each bound variable $resume$ with a subscript
%% $\Await$ or $\Yield$ according to whether it was captured by a
%% consumer or a producer.
\newcommand{\incr}{\dec{incr}}
\newcommand{\Incr}{\dec{Incr}}
% %
Suppose $\Ones$, $\AddTwo$, $\Pipe$, and $\Copipe$ are bound in
$\env_\top$. Furthermore, let $H_\Pipe$ and $H_\Copipe$ denote the
pipe and copipe handler definitions. The machine begins by applying
$\Pipe$.
%
\begin{derivation}
&\cek{\Pipe~\Record{\Ones, \AddTwo} \mid \env_\top \mid \kappaid}\\
\stepsto& \reason{apply $\Pipe$}\\
&\cek{\ShallowHandle\;c~\Unit\;\With\;H_\Pipe \mid \env_\top[c \mapsto (\emptyset, \AddTwo), p \mapsto (\emptyset, \Ones)] \mid \kappaid}\\
\stepsto& \reason{install $H_\Pipe$ with $\env_\Pipe = \env_\top[c \mapsto (\emptyset, \AddTwo), p \mapsto (\emptyset, \Ones)]$}\\
&\cek{c~\Unit \mid \env_\Pipe \mid (\nil, (\env_\Pipe, H_\Pipe)) \cons \kappaid}\\
\stepsto& \reason{apply $c$ and $\val{c}\env_\Pipe = (\emptyset, \AddTwo)$}\\
&\cek{N_x \mid \emptyset \mid (\nil, (\env_\Pipe, H_\Pipe)) \cons \kappaid}\\
\stepsto& \reason{focus on left operand}\\
&\cek{\Do\;\Await~\Unit \mid \emptyset \mid ([(\emptyset, x, N_y)], (\env_\Pipe, H_\Pipe)) \cons \kappaid}\\
\stepsto& \reason{shallow continuation capture $v_\Await = (\nil, [(\emptyset, x, N_y)])$}\\
&\cek{\Copipe~\Record{resume, p} \mid \env_\Pipe[resume \mapsto v_\Await] \mid \kappaid}\\
\end{derivation}
%
The invocation of $\Await$ begins a search through the machine
continuation to locate a matching handler. In this instance the
top-most handler $H_\Pipe$ handles $\Await$. The complete shallow
resumption consists of an empty continuation and a singleton pure
continuation. The former is empty as $H_\Pipe$ is a shallow handler,
meaning that it is discarded.
Evaluation continues by applying $\Copipe$.
%
\begin{derivation}
\stepsto& \reason{apply $\Copipe$}\\
&\cek{\ShallowHandle\; p~\Unit \;\With\;H_\Copipe \mid \env_\top[c \mapsto v_\Await, p \mapsto (\emptyset, \Ones)] \mid \kappaid}\\
\stepsto& \reason{install $H_\Copipe$ with $\env_\Copipe = \env_\top[c \mapsto v_\Await, p \mapsto (\emptyset, \Ones)]$}\\
&\cek{p~\Unit \mid \env_\Copipe \mid (\emptyset, (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
\stepsto^2& \reason{apply $p$, $\val{p}\env_\Copipe = (\emptyset, \Ones)$, and $\env_{\Ones} = \emptyset[ones \mapsto (\emptyset, \Ones)]$}\\
% &\cek{\Do\;\Yield~1;~ones~\Unit \mid \env_{ones} \mid (\nil, (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
% \stepsto^2& \reason{focus on $\Yield$}\\
&\cek{\Do\;\Yield~1 \mid \env_{\Ones} \mid ([(\env_{\Ones},\_,ones~\Unit)], (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
\stepsto& \reason{shallow continuation capture $v_\Yield = (\nil, [(\env_{\Ones}, \_, ones~\Unit)])$}\\
&\cek{\Pipe~\Record{resume, \lambda\Unit.c~s} \mid \env_\Copipe[s \mapsto 1,resume \mapsto v_\Yield] \mid \kappaid}
\end{derivation}
%
At this point the situation is similar as before: the invocation of
$\Yield$ causes the continuation to be unwound in order to find an
appropriate handler, which happens to be $H_\Copipe$. Next $\Pipe$ is
applied.
%
\begin{derivation}
\stepsto& \reason{apply $\Pipe$ and $\env_\Pipe' = \env_\top[c \mapsto (\env_\Copipe[c \mapsto v_\Await, s\mapsto 1]), p \mapsto v_\Yield])]$}\\
&\cek{\ShallowHandle\;c~\Unit\;\With\; H_\Pipe \mid \env_\Pipe' \mid \kappaid}\\
\stepsto& \reason{install $H_\Pipe$}\\
&\cek{c~\Unit \mid \env_\Pipe' \mid (\nil, (\env_\Pipe', H_\Pipe)) \cons \kappaid}\\
\stepsto& \reason{apply $c$ and $\val{c}\env_\Pipe' = (\env_\Copipe[c \mapsto v_\Await, s\mapsto 1])$}\\
&\cek{c~s \mid \env_\Copipe[c \mapsto v_\Await, s\mapsto 1] \mid (\nil, (\env_\Pipe', H_\Pipe)) \cons \kappaid}\\
\stepsto& \reason{shallow resume with $v_\Await = (\nil, [(\emptyset,x,N_y)])$}\\
&\cek{\Return\;1 \mid \env_\Pipe' \mid ([(\emptyset,x,N_y)], (\env_\Pipe', H_\Pipe)) \cons \kappaid}
\end{derivation}
%
Applying the resumption concatenates the first component (the
continuation) with the machine continuation. The second component
(pure continuation) gets concatenated with the pure continuation of
the top-most frame of the machine continuation. Thus in this
particular instance, the machine continuation is manipulated as
follows.
To better understand how the abstract machine concretely transitions
between configurations we will consider a small program consisting of
a deep, parameterised, and shallow handler.
% %
\[ \[
\ba{@{~}l@{~}l}
&\nil \concat ([(\emptyset,x,N_y)] \concat \nil, (\env_\Pipe', H_\Pipe)) \cons \kappaid\\
=& ([(\emptyset,x,N_y)], (\env_\Pipe', H_\Pipe)) \cons \kappaid
\bl
% H_\nondet : \alpha \eff \{\Choose : \UnitType \opto \Bool\} \Harrow \List~\alpha\\
% H_\nondet \defas
% \ba[t]{@{~}l@{~}c@{~}l}
% \Return\;x &\mapsto& [x]\\
% \OpCase{\Choose}{\Unit}{resume} &\mapsto& resume~\True \concat resume~\False
% \ea \smallskip\\
\nondet : (\UnitType \to \alpha \eff \{\Fork : \UnitType \opto \Bool\}) \to \List~\alpha\\
\nondet~m \defas \bl
\Handle\;m\,\Unit\;\With\\
~\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& [x]\\
\OpCase{\Fork}{\Unit}{resume} &\mapsto& resume~\True \concat resume~\False
\ea \ea
\el
\el
\] \]
% %
Because the control component contains the expression $\Return\;1$ and
the pure continuation is nonempty, the machine applies the pure
continuation.
\begin{derivation}
\stepsto& \reason{apply pure continuation, $\env_{\AddTwo} = \emptyset[x \mapsto 1]$}\\
&\cek{N_y \mid \env_{\AddTwo} \mid (\nil, (\env_\Pipe', H_\Pipe)) \cons \kappaid}\\
\stepsto& \reason{focus on right operand}\\
&\cek{\Do\;\Await~\Unit \mid \env_{\AddTwo} \mid ([(\env_{\AddTwo}, y, x + y)], (\env_\Pipe', H_\Pipe)) \cons \kappaid}\\
\stepsto^2& \reason{shallow continuation capture $w_\Await = (\nil, [(\env_{\AddTwo}, y, x + y)])$, apply $\Copipe$}\\
&\cek{\ShallowHandle\;p~\Unit \;\With\; \env_\Copipe \mid \env_\top[c \mapsto w_\Await, p \mapsto v_\Yield] \mid \kappaid}\\
\reducesto& \reason{install $H_\Copipe$ with $\env_\Copipe = \env_\top[c \mapsto w_\Await, p \mapsto v_\Yield]$}\\
&\cek{p~\Unit \mid \env_\Copipe \mid (\nil, (\env_\Copipe, H_\Copipe)) \cons \kappaid}
\end{derivation}
%
The variable $p$ is bound to the shallow resumption $v_\Yield$, thus
invoking it will transfer control back to the $\Ones$ computation.
%
\begin{derivation}
\stepsto & \reason{shallow resume with $v_\Yield = (\nil, [(\env_{\Ones}, \_, ones~\Unit)])$}\\
&\cek{\Return\;\Unit \mid \env_\Copipe \mid ([(\env_{\Ones}, \_, ones~\Unit)], (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
\stepsto^3& \reason{apply pure continuation, apply $ones$, focus on $\Yield$}\\
&\cek{\Do\;\Yield~1 \mid \env_{\Ones} \mid ([(\env_{\Ones}, \_, ones~\Unit)], (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
\end{derivation}
%
At this stage the machine repeats the transitions from before: the
shallow continuation of $\Do\;\Yield~1$ is captured, control passes to
the $\Yield$ clause in $H_\Copipe$, which again invokes $\Pipe$ and
subsequently installs the $H_\Pipe$ handler with an environment
$\env_\Pipe''$. The handler runs the computation $c~\Unit$, where $c$
is an abstraction over the resumption $w_\Await$ applied to the
yielded value $1$.
\[
\bl
% H_\incr : \Record{\Int;\alpha \eff \{\Incr : \UnitType \opto \Int\}} \Harrow^\param \alpha\\
% H_\incr \defas
% i.\,\ba[t]{@{~}l@{~}c@{~}l}
% \Return\;x &\mapsto& x\\
% \OpCase{\Incr}{\Unit}{resume} &\mapsto& resume\,\Record{i+1;i}
% \ea \smallskip\\
\incr : \Record{\Int;\UnitType \to \alpha\eff \{\Incr : \UnitType \opto \Int\}} \to \alpha\\
\incr\,\Record{i_0;m} \defas
\bl
\ParamHandle\;m\,\Unit\;\With\\
~\left(i.\,\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& x\\
\OpCase{\Incr}{\Unit}{resume} &\mapsto& resume\,\Record{i+1;i}
\ea\right)~i_0
\el
\el
\]
% %
\begin{derivation}
\stepsto^6 & \reason{by the above reasoning, shallow resume with $w_\Await = (\nil, [(\env_{\AddTwo}, y, x + y)])$}\\
&\cek{x + y \mid \env_{\AddTwo}[y \mapsto 1] \mid (\nil, (\env_\Pipe'', H_\Pipe)) \cons \kappaid}\\
\stepsto& \reason{$\val{x}\env_{\AddTwo}[y\mapsto 1] = 1$ and $\val{y}\env_{\AddTwo}[y\mapsto 1] = 1$}\\
&\cek{\Return\;2 \mid \env_{\AddTwo}[y \mapsto 1] \mid (\nil, (\env_\Pipe'', H_\Pipe)) \cons \kappaid}
\end{derivation}
\[
\bl
\Pipe : \Record{\UnitType \to \alpha \eff \{ \Yield : \beta \opto \UnitType \}; \UnitType \to \alpha\eff\{ \Await : \UnitType \opto \beta \}} \to \alpha \\
\Pipe\, \Record{p; c} \defas
\bl
\ShallowHandle\; c\,\Unit \;\With\; \\
~\ba[m]{@{}l@{~}c@{~}l@{}}
\Return~x &\mapsto& x \\
\OpCase{\Await}{\Unit}{resume} &\mapsto& \Copipe\,\Record{resume; p} \\
\ea
\el\medskip\\
\Copipe : \Record{\beta \to \alpha\eff\{ \Await : \UnitType \opto \beta\}; \UnitType \to \alpha\eff\{ \Yield : \beta \opto \UnitType\}} \to \alpha \\
\Copipe\, \Record{c; p} \defas
\bl
\ShallowHandle\; p\,\Unit \;\With\; \\
~\ba[m]{@{}l@{~}c@{~}l@{}}
\Return~x &\mapsto& x \\
\OpCase{\Yield}{y}{resume} &\mapsto& \Pipe\,\Record{resume; \lambda \Unit. c\, y} \\
\ea \\
\el \\
\el
\]
% %
Since the pure continuation is empty the $\Return$ clause of $H_\Pipe$
gets invoked with the value $2$. Afterwards the $\Return$ clause of the
identity continuation in $\kappaid$ is invoked, ultimately
transitioning to the following final configuration.
\[
\bl
\dec{prod}\,\Unit \defas \Do\;\Yield\,(\Do\;\Incr\,\Unit); \dec{prod}\,\Unit\\
\dec{cons}\,\Unit \defas \Let\;x \revto \Do\;\Await\,\Unit\;\In\;\Let\;y \revto \Do\;\Await\,\Unit\;\In\;x+y
\el
\]
% %
\begin{derivation} \begin{derivation}
\stepsto^2& \reason{by the above reasoning}\\
&\cek{\Return\;2 \mid \emptyset \mid \nil}
&\nondet\,(\lambda\Unit.\incr\,\Record{0;\lambda\Unit.\Record{\Do\;\Fork\,\Unit;\Pipe\,\Record{\dec{prod};\dec{cons}}}})\\
\stepsto& \reason{\mlab{Init} with $\env_0$}\\
&\cek{\nondet\,(\lambda\Unit.\incr\,\Record{0;\lambda\Unit.\Record{\Do\;\Fork\,\Unit;\Pipe\,\Record{\dec{prod};\dec{cons}}}}) \mid \env_0 \mid \sks_0}\\
\stepsto^+& \reason{\mlab{App}, \mlab{Handle}, \mlab{App}, \mlab{Handle^\param}, \mlab{App}, \mlab{Let}}\\
&\bl
\cek{\Do\;\Fork\,\Unit \mid \env_0 \mid ([(\env_0,x,\Record{x;\Pipe\,\Record{\dec{prod};\dec{cons}}})],\chi_\incr) \cons (\nil, \chi_\nondet) \cons \sks_0}\\
~\text{where } \chi_\incr \defas (\env_0[i \mapsto i],H_\incr)
\el\\
\stepsto^+& \reason{\mlab{Forward}, \mlab{Do^\param}}\\
&\bl
\cek{resume~\True \concat resume~\False \mid \env_\nondet \mid \sks_0}\\
~\text{where } \env_\nondet \defas \env_0[resume \mapsto [([(\env_0,x,\Record{x;\Pipe\,\Record{\dec{prod};\dec{cons}}})],\chi_\incr), (\nil, \chi_\nondet)]
\el
% &\bl
% \cek{c\,\Unit \mid \env_{\Pipe} \mid (\nil,\chi_\Pipe) \cons (\nil,\chi_\incr) \cons (\nil, \chi_\nondet) \cons \sks_0}\\
% \ba[t]{@{~}r@{~}c@{~}l}
% \text{where } \env_\Pipe &\defas& \env_0[pipe \mapsto (\env_0,\Pipe),p \mapsto (\env_0,\dec{prod}),c \mapsto (\env_0,\dec{cons})]\\
% \text{and } \chi_\Pipe &\defas& (\env_{\Pipe},H_{\Pipe})
% \ea
% \el\\
% \stepsto^+& \reason{\mlab{App}, \mlab{Let}}\\
% &\bl\cek{\Do\;\Await\,\Unit \mid \env_0 \mid ([(\env_0,x,x+\Do\;\Await\,\Unit)],\chi_\Pipe) \cons \sks_1}\\
% ~\text{where } \sks_1 \defas (\nil,\chi_\incr) \cons (\nil, \chi_\nondet) \cons \sks_0
% \el
\end{derivation} \end{derivation}
%
%\env_\top[c \mapsto (\env_\Copipe, \lambda\Unit.w_\Await~1),p \mapsto (\nil, [(\env_{\Ones}, \_, ones~\Unit)])]
\paragraph{Remark} If the main continuation is empty then the machine gets
stuck. This occurs when an operation is unhandled, and the forwarding
continuation describes the succession of handlers that have failed to
handle the operation along with any pure continuations that were
encountered along the way.
%
Assuming the input is a well-typed closed computation term $\typc{}{M
: A}{E}$, the machine will either not terminate, return a value of
type $A$, or get stuck failing to handle an operation appearing in
$E$. We now make the correspondence between the operational semantics
and the abstract machine more precise.
% \paragraph{Example} To make the transition rules in
% Figure~\ref{fig:abstract-machine-semantics} concrete we give an
% example of the abstract machine in action. We reuse the small producer
% and consumer from Section~\ref{sec:shallow-handlers-tutorial}. We
% reproduce their definitions here in ANF.
% %
% \[
% \bl
% \Ones \defas \Rec\;ones~\Unit. \Do\; \Yield~1; ones~\Unit \\
% \AddTwo \defas
% \lambda \Unit.
% \Let\; x \revto \Do\;\Await~\Unit \;\In\;
% \Let\; y \revto \Do\;\Await~\Unit \;\In\;
% x + y \\
% \el
% \]%
% %
% Let $N_x$ denote the term $\Let\;x \revto \Do\;\Await~\Unit\;\In\;N_y$
% and $N_y$ the term $\Let\;y \revto \Do\;\Await~\Unit\;\In\;x+y$.
% %
% %% For clarity, we annotate each bound variable $resume$ with a subscript
% %% $\Await$ or $\Yield$ according to whether it was captured by a
% %% consumer or a producer.
% %
% Suppose $\Ones$, $\AddTwo$, $\Pipe$, and $\Copipe$ are bound in
% $\env_\top$. Furthermore, let $H_\Pipe$ and $H_\Copipe$ denote the
% pipe and copipe handler definitions. The machine begins by applying
% $\Pipe$.
% %
% \begin{derivation}
% &\cek{\Pipe~\Record{\Ones, \AddTwo} \mid \env_\top \mid \kappaid}\\
% \stepsto& \reason{apply $\Pipe$}\\
% &\cek{\ShallowHandle\;c~\Unit\;\With\;H_\Pipe \mid \env_\top[c \mapsto (\emptyset, \AddTwo), p \mapsto (\emptyset, \Ones)] \mid \kappaid}\\
% \stepsto& \reason{install $H_\Pipe$ with $\env_\Pipe = \env_\top[c \mapsto (\emptyset, \AddTwo), p \mapsto (\emptyset, \Ones)]$}\\
% &\cek{c~\Unit \mid \env_\Pipe \mid (\nil, (\env_\Pipe, H_\Pipe)) \cons \kappaid}\\
% \stepsto& \reason{apply $c$ and $\val{c}\env_\Pipe = (\emptyset, \AddTwo)$}\\
% &\cek{N_x \mid \emptyset \mid (\nil, (\env_\Pipe, H_\Pipe)) \cons \kappaid}\\
% \stepsto& \reason{focus on left operand}\\
% &\cek{\Do\;\Await~\Unit \mid \emptyset \mid ([(\emptyset, x, N_y)], (\env_\Pipe, H_\Pipe)) \cons \kappaid}\\
% \stepsto& \reason{shallow continuation capture $v_\Await = (\nil, [(\emptyset, x, N_y)])$}\\
% &\cek{\Copipe~\Record{resume, p} \mid \env_\Pipe[resume \mapsto v_\Await] \mid \kappaid}\\
% \end{derivation}
% %
% The invocation of $\Await$ begins a search through the machine
% continuation to locate a matching handler. In this instance the
% top-most handler $H_\Pipe$ handles $\Await$. The complete shallow
% resumption consists of an empty continuation and a singleton pure
% continuation. The former is empty as $H_\Pipe$ is a shallow handler,
% meaning that it is discarded.
% Evaluation continues by applying $\Copipe$.
% %
% \begin{derivation}
% \stepsto& \reason{apply $\Copipe$}\\
% &\cek{\ShallowHandle\; p~\Unit \;\With\;H_\Copipe \mid \env_\top[c \mapsto v_\Await, p \mapsto (\emptyset, \Ones)] \mid \kappaid}\\
% \stepsto& \reason{install $H_\Copipe$ with $\env_\Copipe = \env_\top[c \mapsto v_\Await, p \mapsto (\emptyset, \Ones)]$}\\
% &\cek{p~\Unit \mid \env_\Copipe \mid (\emptyset, (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
% \stepsto^2& \reason{apply $p$, $\val{p}\env_\Copipe = (\emptyset, \Ones)$, and $\env_{\Ones} = \emptyset[ones \mapsto (\emptyset, \Ones)]$}\\
% % &\cek{\Do\;\Yield~1;~ones~\Unit \mid \env_{ones} \mid (\nil, (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
% % \stepsto^2& \reason{focus on $\Yield$}\\
% &\cek{\Do\;\Yield~1 \mid \env_{\Ones} \mid ([(\env_{\Ones},\_,ones~\Unit)], (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
% \stepsto& \reason{shallow continuation capture $v_\Yield = (\nil, [(\env_{\Ones}, \_, ones~\Unit)])$}\\
% &\cek{\Pipe~\Record{resume, \lambda\Unit.c~s} \mid \env_\Copipe[s \mapsto 1,resume \mapsto v_\Yield] \mid \kappaid}
% \end{derivation}
% %
% At this point the situation is similar as before: the invocation of
% $\Yield$ causes the continuation to be unwound in order to find an
% appropriate handler, which happens to be $H_\Copipe$. Next $\Pipe$ is
% applied.
% %
% \begin{derivation}
% \stepsto& \reason{apply $\Pipe$ and $\env_\Pipe' = \env_\top[c \mapsto (\env_\Copipe[c \mapsto v_\Await, s\mapsto 1]), p \mapsto v_\Yield])]$}\\
% &\cek{\ShallowHandle\;c~\Unit\;\With\; H_\Pipe \mid \env_\Pipe' \mid \kappaid}\\
% \stepsto& \reason{install $H_\Pipe$}\\
% &\cek{c~\Unit \mid \env_\Pipe' \mid (\nil, (\env_\Pipe', H_\Pipe)) \cons \kappaid}\\
% \stepsto& \reason{apply $c$ and $\val{c}\env_\Pipe' = (\env_\Copipe[c \mapsto v_\Await, s\mapsto 1])$}\\
% &\cek{c~s \mid \env_\Copipe[c \mapsto v_\Await, s\mapsto 1] \mid (\nil, (\env_\Pipe', H_\Pipe)) \cons \kappaid}\\
% \stepsto& \reason{shallow resume with $v_\Await = (\nil, [(\emptyset,x,N_y)])$}\\
% &\cek{\Return\;1 \mid \env_\Pipe' \mid ([(\emptyset,x,N_y)], (\env_\Pipe', H_\Pipe)) \cons \kappaid}
% \end{derivation}
% %
% Applying the resumption concatenates the first component (the
% continuation) with the machine continuation. The second component
% (pure continuation) gets concatenated with the pure continuation of
% the top-most frame of the machine continuation. Thus in this
% particular instance, the machine continuation is manipulated as
% follows.
% %
% \[
% \ba{@{~}l@{~}l}
% &\nil \concat ([(\emptyset,x,N_y)] \concat \nil, (\env_\Pipe', H_\Pipe)) \cons \kappaid\\
% =& ([(\emptyset,x,N_y)], (\env_\Pipe', H_\Pipe)) \cons \kappaid
% \ea
% \]
% %
% Because the control component contains the expression $\Return\;1$ and
% the pure continuation is nonempty, the machine applies the pure
% continuation.
% \begin{derivation}
% \stepsto& \reason{apply pure continuation, $\env_{\AddTwo} = \emptyset[x \mapsto 1]$}\\
% &\cek{N_y \mid \env_{\AddTwo} \mid (\nil, (\env_\Pipe', H_\Pipe)) \cons \kappaid}\\
% \stepsto& \reason{focus on right operand}\\
% &\cek{\Do\;\Await~\Unit \mid \env_{\AddTwo} \mid ([(\env_{\AddTwo}, y, x + y)], (\env_\Pipe', H_\Pipe)) \cons \kappaid}\\
% \stepsto^2& \reason{shallow continuation capture $w_\Await = (\nil, [(\env_{\AddTwo}, y, x + y)])$, apply $\Copipe$}\\
% &\cek{\ShallowHandle\;p~\Unit \;\With\; \env_\Copipe \mid \env_\top[c \mapsto w_\Await, p \mapsto v_\Yield] \mid \kappaid}\\
% \reducesto& \reason{install $H_\Copipe$ with $\env_\Copipe = \env_\top[c \mapsto w_\Await, p \mapsto v_\Yield]$}\\
% &\cek{p~\Unit \mid \env_\Copipe \mid (\nil, (\env_\Copipe, H_\Copipe)) \cons \kappaid}
% \end{derivation}
% %
% The variable $p$ is bound to the shallow resumption $v_\Yield$, thus
% invoking it will transfer control back to the $\Ones$ computation.
% %
% \begin{derivation}
% \stepsto & \reason{shallow resume with $v_\Yield = (\nil, [(\env_{\Ones}, \_, ones~\Unit)])$}\\
% &\cek{\Return\;\Unit \mid \env_\Copipe \mid ([(\env_{\Ones}, \_, ones~\Unit)], (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
% \stepsto^3& \reason{apply pure continuation, apply $ones$, focus on $\Yield$}\\
% &\cek{\Do\;\Yield~1 \mid \env_{\Ones} \mid ([(\env_{\Ones}, \_, ones~\Unit)], (\env_\Copipe, H_\Copipe)) \cons \kappaid}\\
% \end{derivation}
% %
% At this stage the machine repeats the transitions from before: the
% shallow continuation of $\Do\;\Yield~1$ is captured, control passes to
% the $\Yield$ clause in $H_\Copipe$, which again invokes $\Pipe$ and
% subsequently installs the $H_\Pipe$ handler with an environment
% $\env_\Pipe''$. The handler runs the computation $c~\Unit$, where $c$
% is an abstraction over the resumption $w_\Await$ applied to the
% yielded value $1$.
% %
% \begin{derivation}
% \stepsto^6 & \reason{by the above reasoning, shallow resume with $w_\Await = (\nil, [(\env_{\AddTwo}, y, x + y)])$}\\
% &\cek{x + y \mid \env_{\AddTwo}[y \mapsto 1] \mid (\nil, (\env_\Pipe'', H_\Pipe)) \cons \kappaid}\\
% \stepsto& \reason{$\val{x}\env_{\AddTwo}[y\mapsto 1] = 1$ and $\val{y}\env_{\AddTwo}[y\mapsto 1] = 1$}\\
% &\cek{\Return\;2 \mid \env_{\AddTwo}[y \mapsto 1] \mid (\nil, (\env_\Pipe'', H_\Pipe)) \cons \kappaid}
% \end{derivation}
% %
% Since the pure continuation is empty the $\Return$ clause of $H_\Pipe$
% gets invoked with the value $2$. Afterwards the $\Return$ clause of the
% identity continuation in $\kappaid$ is invoked, ultimately
% transitioning to the following final configuration.
% %
% \begin{derivation}
% \stepsto^2& \reason{by the above reasoning}\\
% &\cek{\Return\;2 \mid \emptyset \mid \nil}
% \end{derivation}
% %
% %\env_\top[c \mapsto (\env_\Copipe, \lambda\Unit.w_\Await~1),p \mapsto (\nil, [(\env_{\Ones}, \_, ones~\Unit)])]
% \paragraph{Remark} If the main continuation is empty then the machine gets
% stuck. This occurs when an operation is unhandled, and the forwarding
% continuation describes the succession of handlers that have failed to
% handle the operation along with any pure continuations that were
% encountered along the way.
% %
% Assuming the input is a well-typed closed computation term $\typc{}{M
% : A}{E}$, the machine will either not terminate, return a value of
% type $A$, or get stuck failing to handle an operation appearing in
% $E$. We now make the correspondence between the operational semantics
% and the abstract machine more precise.
\section{Realisability and efficiency implications} \section{Realisability and efficiency implications}
\label{subsec:machine-realisability} \label{subsec:machine-realisability}

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