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Interdefinability WIP2

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Daniel Hillerström 5 years ago
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      thesis.tex

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thesis.tex
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@ -9,6 +9,10 @@
\usepackage{url}
\usepackage[sort&compress,square,numbers]{natbib} % Bibliography
\usepackage{bibentry} % Print bibliography entries inline.
\makeatletter % Redefine bibentry to omit hyperrefs
\renewcommand\bibentry[1]{\nocite{#1}{\frenchspacing
\@nameuse{BR@r@#1\@extra@b@citeb}}}
\makeatother
\nobibliography* % use the bibliographic data from the standard BibTeX setup.
\usepackage{breakurl}
\usepackage{amsmath} % Mathematics library
@ -11513,9 +11517,11 @@ environments.
%
\[
\bl
\sdtrans{-} : \TypeCat \to \TypeCat\\
\sdtrans{C \Rightarrow D} \defas
\sdtrans{C} \Rightarrow \Record{\UnitType \to \sdtrans{C}; \UnitType \to \sdtrans{D}} \medskip \medskip\\
\sdtrans{-} : \HandlerTypeCat \to \HandlerTypeCat\\
\sdtrans{A\eff E_1 \Rightarrow B\eff E_2} \defas
\sdtrans{A\eff E_1} \Rightarrow \Record{\UnitType \to \sdtrans{C \eff E_1}; \UnitType \to \sdtrans{B \eff E_2}} \eff \sdtrans{E_2} \medskip \medskip\\
% \sdtrans{C \Rightarrow D} \defas
% \sdtrans{C} \Rightarrow \Record{\UnitType \to \sdtrans{C}; \UnitType \to \sdtrans{D}} \medskip \medskip\\
\sdtrans{-} : \CompCat \to \CompCat\\
\sdtrans{\ShallowHandle \; M \; \With \; H} \defas
\ba[t]{@{}l}
@ -11545,7 +11551,67 @@ Each shallow handler is encoded as a deep handler that returns a pair
of thunks. The first forwards all operations, acting as the identity
on computations. The second interprets a single operation before
reverting to forwarding.
%
The following example illustrates an application of the translation on
the $\Pipe$ operator from Section~\ref{sec:pipes} applied to the
producer and consumer pair
$\Record{\Rec\;ones\,\Unit.\Do\;\Yield~1;ones\,\Unit;\lambda\Unit.\Do\;\Await\,\Unit
+ \Do\;\Await}$
%
\[
\ba{@{~}l@{~}l}
&\mathcal{D}\left\llbracket
\ba[m]{@{}l}
\ShallowHandle\;(\lambda\Unit.\Do\;\Await\,\Unit + \Do\;\Await\,\Unit)\,\Unit\;\With\\
\quad\ba[m]{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& \Return\;x\\
\OpCase{\Await}{\Unit}{r} &\mapsto& \Copipe\,\Record{r;\Rec\;ones\,\Unit.\Do\;\Yield~1;ones\,\Unit}
\ea
\ea
\right\rrbracket \medskip\\
=& \bl
\Let\;z \revto \bl
\Handle\;(\lambda\Unit.\Do\;\Await\,\Unit + \Do\;\Await\,\Unit)\,\Unit\;\With\\
\quad\ba[t]{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& \Return\;\Record{\lambda\Unit.\Return\;x;\lambda\Unit.\Return\;x}\\
\OpCase{\Await}{\Unit}{r} &\mapsto&\\
\multicolumn{3}{l}{
\quad\bl
\Let\;r \revto \lambda x.\Let\;z \revto r~x\;\In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit\;\In\\
\Return\;\Record{\bl
\lambda\Unit.\Let\;x \revto \Do\;\ell~p\;\In\;r~x;\\
\lambda\Unit. \sdtrans{\Copipe}\,\Record{r;\Rec\;ones\,\Unit.\Do\;\Yield~1;ones\,\Unit}}\el
\el}
\ea
\el\\
\In\;\Let\;\Record{f;g} = z\;\In\;g\,\Unit
\el
\ea
\]
%
Evaluation of both the left hand side and right hand side of the
equals sign yields the value $2 : \Int$. The $\Return$-case in the
image contains a redundant pair, because the $\Return$-case of $\Pipe$
is the identity. The translation of the $\Await$-case sets up the
forwarding component and handling component of the pair of thunks.
The translation commutes with substitution and preserves typability.
%
\begin{lemma}\label{lem:sdtrans-subst}
Let $\sigma$ denote a substitution. The translation $\sdtrans{-}$
commutes with substitution, i.e.
%
\[
\sdtrans{V}\sdtrans{\sigma} = \sdtrans{V\sigma},\quad
\sdtrans{M}\sdtrans{\sigma} = \sdtrans{M\sigma},\quad
\sdtrans{H}\sdtrans{\sigma} = \sdtrans{H\sigma}.
\]
%
\end{lemma}
%
\begin{proof}
By induction on the structures of $V$, $M$, and $H$.
\end{proof}
% \paragraph{Example} We demonstrate the translation $\sdtrans{-}$ on
% the $\Pipe$ handler from Section~\ref{sec:shallow-handlers-tutorial}
% (recall that the variables $c$ and $p$ are bound to the consumer and
@ -11556,11 +11622,11 @@ reverting to forwarding.
% \ba{@{~}l@{~}l@{}l}
% &\mathcal{D}&\left\llbracket
% \ba[m]{@{~}l}
% \ShallowHandle\; c\,\Unit \;\With\; \\
% \lambda\Record{p;c}.\ShallowHandle\; c\,\Unit \;\With\; \\
% \quad
% \ba[m]{@{}l@{~}c@{~}l@{}}
% \Return~x &\mapsto& \Return\; x \\
% \Await~\Unit~resume &\mapsto& \Copipe\; \Record{resume; p} \\
% \OpCase{\Await}{\Unit}{r} &\mapsto& \Copipe\; \Record{r;p} \\
% \ea \\
% \ea\right\rrbracket = \\
% &
@ -11586,7 +11652,6 @@ reverting to forwarding.
% \ea
% \]
%
\begin{theorem}
If $\Delta; \Gamma \vdash M : C$ then $\sdtrans{\Delta};
\sdtrans{\Gamma} \vdash \sdtrans{M} : \sdtrans{C}$.
@ -11600,18 +11665,20 @@ If $\Delta; \Gamma \vdash M : C$ then $\sdtrans{\Delta};
\newcommand{\approxa}{\gtrsim}
As with the implementation of deep handlers as shallow handlers, the
implementation is again given by a local translation. However, this
time the administrative overhead is more significant. Reduction up to
congruence is insufficient and we require a more semantic notion of
administrative reduction.
implementation is again given by a typability-preserving local
translation. However, this time the administrative overhead is more
significant. Reduction up to congruence is insufficient and we require
a more semantic notion of administrative reduction.
\begin{definition}[Administrative evaluation contexts]
An evaluation context $\EC$ is administrative, $\admin(\EC)$, iff
An evaluation context $\EC \in \EvalCat$ is administrative,
$\admin(\EC)$, when the following two criteria hold.
\begin{enumerate}
\item For all values $V$, we have: $\EC[\Return\;V] \reducesto^\ast
\item For all values $V \in \ValCat$, we have: $\EC[\Return\;V] \reducesto^\ast
\Return\;V$
\item For all evaluation contexts $\EC'$, operations $\ell \in BL(\EC)
\backslash BL(\EC')$, values $V$:
\item For all evaluation contexts $\EC' \in \EvalCat$, operations
$\ell \in \BL(\EC) \backslash \BL(\EC')$, and values
$V \in \ValCat$:
%
\[
\EC[\EC'[\Do\;\ell\;V]] \reducesto^\ast \Let\; x \revto \Do\;\ell\;V \;\In\; \EC[\EC'[\Return\;x]].
@ -11658,16 +11725,19 @@ that administrative reduction may occur anywhere within a term.
%
The following lemma states that the forwarding component of the
translation is administrative.
%
\begin{lemma}\label{lem:sdtrans-admin}
For all shallow handlers $H$, the following context is administrative:
\begin{displaymath}
%
\[
\Let\; z \revto
\Handle\; [~] \;\With\; \sdtrans{H}\;
\In\;
\Let\; \Record{f;\_} = z\; \In\; f\,\Unit.
\end{displaymath}
\]
%
\end{lemma}
%
\begin{theorem}[Simulation up to administrative reduction]
If $M' \approxa \sdtrans{M}$ and $M \reducesto N$ then there exists
$N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
@ -11696,8 +11766,8 @@ homomorphic cases and show only the interesting cases.
%
\[
\bl
\PD{-} : \TypeCat \to \TypeCat\\
\PD{\Record{C, A} \Rightarrow^\param B \eff E} \defas \PD{C} \Rightarrow (\PD{A} \to \PD{B \eff E})\eff \PD{E} \medskip\\
\PD{-} : \HandlerTypeCat \to \HandlerTypeCat\\
\PD{\Record{C; A} \Rightarrow^\param B \eff E} \defas \PD{C} \Rightarrow (\PD{A} \to \PD{B \eff E})\eff \PD{E} \medskip\\
\PD{-} : \CompCat \to \CompCat\\
\PD{\ParamHandle\;M\;\With\;(q.\,H)(W)} \defas \left(\Handle\; \PD{M}\; \With\; \PD{H}_q\right)~\PD{W} \medskip\\
\PD{-} : \HandlerCat \times \ValCat \to \HandlerCat\\
@ -11705,20 +11775,74 @@ homomorphic cases and show only the interesting cases.
\PD{\{\Return\;x \mapsto M\}}_q &\defas& \{\Return\;x \mapsto \lambda q. \PD{M}\}\\
\PD{\{\OpCase{\ell}{p}{r} \mapsto M\}}_q &\defas&
\{\OpCase{\ell}{p}{r} \mapsto \lambda q.
\Let\; r \revto \Return\;\lambda \Record{x,q'}. r~x~q'\;
\Let\; r \revto \Return\;\lambda \Record{x;q'}. r~x~q'\;
\In\; \PD{M}\}
\ea
\el
\]
%
The parameterised $\ParamHandle$ construct becomes an application of a
$\Handle$ construct to the translation of the parameter. The bodies of
return and operation clauses are each enclosed in a lambda abstraction
whose formal parameter is the handler parameter $q$. As a result the
ordinary deep resumption $r$ is a curried function. However, the uses
of $r$ in $M$ expects a binary function. To repair this discrepancy,
we construct an uncurried interface of $r$ via the function $r'$.
$\Handle$ construct to the translation of the parameter. The
translation of $\Return$ and operation clauses are parameterised by
the name of the handler parameter as each clause body is enclosed in a
$\lambda$-abstraction whose formal parameter is the handler parameter
$q$. As a result the ordinary deep resumption $r$ is a curried
function. However, the uses of $r$ in $M$ expects a binary
function. To repair this discrepancy, we construct an uncurried
interface of $r$ by embedding it under a binary $\lambda$-abstraction.
%
To illustrate the translation in action consider the following example
program that adds the results obtained by performing two invocations
of some stateful operation $\dec{Incr} : \UnitType \opto \Int$, which
increments some global counter and returns its prior value.
%
\[
\ba{@{~}l@{~}l}
&\mathcal{P}\left\llbracket
\ba[m]{@{}l}
\ParamHandle\;\Do\;\dec{Incr}\,\Unit + \Do\;\dec{Incr}\,\Unit\;\With\\
\left( q.\ba[m]{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& \Return\;\Record{x;q}\\
\OpCase{\dec{Incr}}{\Unit}{r} &\mapsto& r\,\Record{q;q+1}
\ea\right)~40
\ea
\right\rrbracket \medskip\\
=& \left(
\ba[m]{@{}l}
\Handle\;\Do\;\dec{Incr}\,\Unit + \Do\;\dec{Incr}\,\Unit\;\With\\
\quad\ba[t]{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& \lambda q.\Return\;\Record{x;q}\\
\OpCase{\dec{Incr}}{\Unit}{r} &\mapsto& \lambda q. \Let\;r \revto \Return\;\lambda\Record{x;q}.r~x~q\;\In\;r\,\Record{q;q+1}
\ea
\ea\right)~40
\ea
\]
%
Evaluation of the program on either side of the equals sign yields
$\Record{81;42} : \Int$. The translation desugars the parameterised
handler into an ordinary deep handler that makes use of the
parameter-passing idiom to maintain the state of the handled
computation~\cite{Pretnar15}.
The translation commutes with substitution and preserves typability.
%
\begin{lemma}\label{lem:pd-subst}
Let $\sigma$ denote a substitution. The translation $\sdtrans{-}$
commutes with substitution, i.e.
%
\[
\sdtrans{V}\sdtrans{\sigma} = \sdtrans{V\sigma},\quad
\sdtrans{M}\sdtrans{\sigma} = \sdtrans{M\sigma},\quad
\sdtrans{H}\sdtrans{\sigma} = \sdtrans{(q.\,H)\sigma}.
\]
%
\end{lemma}
%
\begin{proof}
By induction on the structures of $V$, $M$, and $q.H$.
\end{proof}
%
\begin{theorem}
If $\Delta; \Gamma \vdash M : C$ then $\PD{\Delta};
\sdtrans{\Gamma} \vdash \PD{M} : \PD{C}$.
@ -11727,20 +11851,71 @@ If $\Delta; \Gamma \vdash M : C$ then $\PD{\Delta};
\begin{proof}
By induction on typing derivations.
\end{proof}
%
This translation of parameterised handlers simulates the native
semantics. As with the simulation of deep handlers via shallow
handlers in Section~\ref{sec:deep-as-shallow}, this simulation is only up
to congruence due to the need for an application of a pure function to
a variable to be reduced. The interesting cases of the proof appear in
Appendix~\ref{sec:param-proof}.
handlers in Section~\ref{sec:deep-as-shallow}, this simulation is only
up to congruence due to the need for an application of a pure function
to a variable to be reduced.
%
\begin{theorem}[Simulation of parameterised handlers by deep
handlers]
\label{thm:param-simulation}
If $M \reducesto N$ then $\PD{M} \reducesto^+_{\mathrm{cong}} \PD{N}$.
\end{theorem}
%
\begin{proof}
By induction on $M$. There are only two interesting cases to
consider.
\begin{description}
\item[Case]
$M = \ParamHandle\; \Return\;V\;\With\;(q.~H)(W) \reducesto
N[V/x,W/q]$, where $\hret = \{ \Return\; x \mapsto N \}$.
%
\begin{derivation}
&\PD{\ParamHandle\;\Return\;V\;\With\;(q.~H)(W)}\\
=& \reason{definition of $\PD{-}$}\\
&(\Handle\;\Return\;\PD{V}\;\With\;\PD{H}_q)~\PD{W}\\
\reducesto& \reason{$\semlab{Ret}$ with $\PD{\hret}_q = \{\Return~x \mapsto \lambda q. \PD{N}\}$}\\
&(\lambda q. \PD{N}[\PD{V}/x])~\PD{W}\\
\reducesto& \reason{$\semlab{App}$}\\
&\PD{N}[\PD{V}/x,\PD{W}/q]\\
=& \reason{lemma~\ref{lem:pd-subst}}\\
&\PD{N[V/x,W/q]}
\end{derivation}
\item[Case] $M = \ParamHandle\; \EC[\Do\;\ell~V]\;\With\;(q.~H)(W) \reducesto \\
\qquad
N[V/p,W/q,\lambda \Record{x,q'}. \ParamHandle\;\EC[\Return\;x]\;\With\;(q.~H)(q')/r]$, where $\hell = \{ \OpCase{\ell}{p}{r} \mapsto N \}$.
\begin{derivation}
&\PD{\ParamHandle\;\EC[\Do\;\ell~V]\;\With\;(q.~H)(W)}\\
=& \reason{definition of $\PD{-}$}\\
&(\Handle\;\EC[\Do\;\ell~\PD{V}]\;\With\;\PD{H}_q)~\PD{W}\\
\reducesto& \reason{$\semlab{Op}$ with $\PD{\hell}_q = \{\ell~p~r \mapsto \lambda q. \Let\,r' \revto \lambda \Record{x,q'}. r~x~q\,\In\, \PD{N}[r'/r]\}$}\\
&((\lambda q. \bl
\Let\;r' \revto \lambda \Record{x,q'}. r~x~q\; \In \\
\PD{N}[r'/r])[\PD{V}/p,\lambda x.\Handle\;\EC[\Return\;x]\;\With\;\PD{H}_q)~\PD{W}\\
\el \\
=& \reason{definition of $[-]$}\\
&(\lambda q. \bl
\Let\; r' \revto \lambda \Record{x,q'}. (\lambda x. \Handle\;\EC[\Return\;x]\;\With\; \PD{H}_q)~x~q'\;\In \\
\PD{N}[\PD{V}/p,r'/r])\\
\el \\
\reducesto& \reason{$\semlab{App}$}\\
&\Let\;r' \revto \lambda \Record{x,q'}. r~x~q'\; \In\; \PD{N}[r'/r,\PD{V}/p,\PD{V}/q]\\
\reducesto& \reason{$\semlab{Let}$}\\
&\PD{N}[\bl
\PD{V}/p,\PD{W}/q, \\
\lambda \Record{x,q'}. (\lambda x. \Handle\;\EC[\Return\;x]\;\With\; \PD{H}_q)~x~q'/r]\\
\el \\
\reducesto_{\mathrm{cong}}& \reason{$\semlab{App}$}\\
&\PD{N}[\PD{V}/p,\PD{W}/q,\lambda \Record{x,q'}. (\Handle\;\EC[\Return\;x]\;\With\; \PD{H}_q)~q'/r]\\
=& \reason{definition of $\PD{-}$}\\
&\PD{N}[\PD{V}/p,\PD{W}/q,\lambda \Record{x,q'}. \PD{\ParamHandle\;\EC[\Return\;x]\;\With\;(q.~H)(q')}/r]\\
=& \reason{lemma~\ref{lem:pd-subst}}\\
&\PD{N[V/p,W/q,\lambda \Record{x,q'}. \ParamHandle\;\EC[\Return\;x]\;\With\; (q.~H)(q')/r]}
\end{derivation}
\end{description}
\end{proof}
\section{Related work}
\citet{ForsterKLP17,ForsterKLP19} \citet{PirogPS19}, \citet{Shan04}

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