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Deep handlers static semantics.

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Daniel Hillerström
2020-12-05 18:53:54 +00:00
parent a4e504e24b
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thesis.tex
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@@ -1800,7 +1800,7 @@ second application of $\Control$ captures the remainder of the
computation of to $\Prompt$. However, the captured context gets
discarded, because the continuation $k'$ is never invoked.
%
\dhil{Multi-prompts: more liberal typing, no interference}
\dhil{Mention control0/prompt0 and the control hierarchy}
\paragraph{\citeauthor{DanvyF90}'s shift and reset} Shift and reset
first appeared in a technical report by \citeauthor{DanvyF89} in
@@ -2222,8 +2222,14 @@ Effect handler definitions occupy their own syntactic category.
\mid \{ \OpCase{\ell}{p}{k} \mapsto N \} \uplus H\\
\end{syntax}
%
A handler consists of a $\Return$-clause and zero or more operation
clauses.
An effect handler consists of a $\Return$-clause and zero or more
operation clauses. Each operation clause binds the payload of the
matching operation $\ell$ to $p$ and the continuation of the operation
invocation to $k$ in $N$.
Effect handlers introduces a new syntactic category of signatures, and
extends the value types with operation types. Operation and handler
application both appear as computation forms.
%
\begin{syntax}
&\Sigma \in \mathsf{Sig} &::=& \emptyset \mid \{ \ell : A \opto B \} \uplus \Sigma\\
@@ -2231,29 +2237,64 @@ clauses.
&M,N \in \CompCat &::=& \cdots \mid \Do\;\ell\,V \mid \Handle \; M \; \With \; H\\[1ex]
\end{syntax}
%
A signature is a collection of labels with operation types. An
operation type $A \opto B$ is similar to the function type in that $A$
denotes the domain (type of the argument) of the operation, and $B$
denotes the codomain (return type). For simplicity, we will just
assume a global fixed signature. The form $\Do\;\ell\,V$ is the
application form for operations. It applies an operation $\ell$ with
payload $V$. The construct $\Handle\;M\;\With\;H$ handles a
computation $M$ with handler $H$.
%
\begin{mathpar}
\inferrule*
{{\bl
% C = A \eff \{(\ell_i : A_i \opto B_i)_i; R\} \\
% D = B \eff \{(\ell_i : P_i)_i; R\}\\
\{\ell_i : A_i \opto B_i\}_i \in \Sigma \\
H = \{\Return\;x \mapsto M\} \uplus \{ \OpCase{\ell_i}{p_i}{k_i} \mapsto N_i \}_i \\
\el}\\\\
\typ{\Gamma, x : A;\Sigma}{M : D}\\\\
[\typ{\Gamma,p_i : A_i, k_i : B_i \to D;\Sigma}{N_i : D}]_i
}
{\typ{\Gamma;\Sigma}{H : C \Harrow D}}
\end{mathpar}
\begin{mathpar}
\inferrule*
{\{ \ell : A \opto B \} \in \Sigma \\ \typ{\Gamma;\Sigma}{V : A}}
{\typ{\Gamma;\Sigma}{\Do\;\ell\,V : B}}
\inferrule*
{
\typ{\Gamma}{M : C} \\
\typ{\Gamma}{H : C \Harrow D}
}
{\typ{\Gamma;\Sigma}{\Handle \; M \; \With\; H : D}}
%\mprset{flushleft}
\inferrule*
{{\bl
% C = A \eff \{(\ell_i : A_i \opto B_i)_i; R\} \\
% D = B \eff \{(\ell_i : P_i)_i; R\}\\
\{\ell_i : A_i \opto B_i\}_i \in \Sigma \\
H = \{\Return\;x \mapsto M\} \uplus \{ \OpCase{\ell_i}{p_i}{r_i} \mapsto N_i \}_i \\
\el}\\\\
\typ{\Gamma, x : A;\Sigma}{M : D}\\\\
[\typ{\Gamma,p_i : A_i, r_i : B_i \to D;\Sigma}{N_i : D}]_i
}
{\typ{\Gamma;\Sigma}{H : C \Harrow D}}
\end{mathpar}
%
The first typing rule checks that the operation label of each
operation clause is declared in the signature $\Sigma$. The signature
provides the necessary information to construct the type of the
payload parameters $p_i$ and the continuations $k_i$. Note that the
domain of each continuation $k_i$ is compatible with the codomain of
$\ell_i$, and the codomain of $k_i$ is compatible with the codomain of
the handler.
%
The second and third typing rules are application of operations and
handlers, respectively. The rule for operation application simply
inspects the signature to check that the operation is declared, and
that the type of the payload is compatible with the declared type.
This particular presentation is nominal, because operations are
declared up front. Nominal typing is the only sound option in the
absence of an effect system (unless we restrict operations to work
over a fixed type, say, an integer). In
Chapter~\ref{ch:unary-handlers} we see a different presentation based
on structural typing.
The dynamic semantics of effect handlers are similar to that of
$\fcontrol$, though, the $\slab{Value}$ rule is more interesting.
%
\begin{reductions}
\slab{Value} & \Handle\; V \;\With\;H &\reducesto& M[V/x], \text{ where } \{\Return\;x \mapsto M\} \in H\\
\slab{Capture} & \Handle\;\EC[\Do\;\ell~V] \;\With\; H &\reducesto& M[V/p,\qq{\cont_{\Record{\EC;H}}}/k],\\
@@ -2319,8 +2360,12 @@ $\ShallowHandle\; - \;\With\;-$.
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]\\
\end{reductions}
%
Chapter~\ref{ch:unary-handlers} contains further examples of deep and
shallow handlers in action.
Deep handlers can be used to simulate shift0 and reset0.
Deep handlers can be used to simulate shift0 and
reset0~\cite{KammarLO13}.
%
\begin{equations}
\sembr{\shiftz~k.M} &\defas& \Do\;\dec{Shift0}~(\lambda k.M)\\
@@ -2328,13 +2373,14 @@ Deep handlers can be used to simulate shift0 and reset0.
\ba[t]{@{~}l}\Handle\;m\,\Unit\;\With\\
~\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& x\\
\OpCase{\dec{Shift0}}{f}{k} &\mapsto& f\,(\lambda x. \Continue~k~x)
\OpCase{\dec{Shift0}}{f}{k} &\mapsto& f\,k
\ea
\ea
\end{equations}
%
Shallow handlers can be used to simulate control0 and prompt0.
Shallow handlers can be used to simulate control0 and
prompt0~\cite{KammarLO13}.
%
\begin{equations}
\sembr{\Controlz~k.M} &\defas& \Do\;\dec{Control0}~(\lambda k.M)\\
@@ -2347,7 +2393,7 @@ Shallow handlers can be used to simulate control0 and prompt0.
\ba[t]{@{~}l}\ShallowHandle\;m\,\Unit\;\With\\
~\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& x\\
\OpCase{\dec{Control0}}{f}{k} &\mapsto& prompt0\,(\lambda\Unit.f\,(\lambda x. \Continue~k~x))
\OpCase{\dec{Control0}}{f}{k} &\mapsto& prompt0\,(\lambda\Unit.f\,k)
\ea
\ea
\el