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@ -309,7 +309,7 @@ rather than $\rho$ and refer to it as an \emph{effect variable}). |
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The row variable in an open row type can be instantiated with |
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additional labels. Each label may occur at most once in each row (we |
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enforce this restriction at the level of kinds). We identify rows up |
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to the reordering of labels. |
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to the reordering of labels. We assume structural equality on labels. |
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% |
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% \begin{mathpar} |
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% \inferrule*[Lab=\rowlab{Closed}] |
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@ -612,7 +612,8 @@ Computations |
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\label{fig:base-language-type-rules} |
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\end{figure} |
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% |
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\dhil{There is some rendering issue with T-labels in the typing rules.} |
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% |
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The typing rules are given by |
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Figure~\ref{fig:base-language-type-rules}. In each typing rule, we |
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implicitly assume that each type is well-kinded with respect to the |
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@ -725,6 +726,84 @@ $N$, must have computation $C$ subject to the additional assumption |
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that the binder $x : A$. |
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\section{Dynamic semantics} |
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\label{sec:base-language-dynamic-semantics} |
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% |
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\begin{figure} |
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\begin{reductions} |
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\semlab{App} & (\lambda x^A . \, M) V &\reducesto& M[V/x] \\ |
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\semlab{TyApp} & (\Lambda \alpha^K . \, M) A &\reducesto& M[A/\alpha] \\ |
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\semlab{Split} & \Let \; \Record{\ell = x;y} = \Record{\ell = V;W} \; \In \; N &\reducesto& N[V/x,W/y] \\ |
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\semlab{Case$_1$} & |
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\Case \; (\ell~V)^R \{ \ell \; x \mapsto M; y \mapsto N\} &\reducesto& M[V/x] \\ |
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\semlab{Case$_2$} & |
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\Case \; (\ell~V)^R \{ \ell' \; x \mapsto M; y \mapsto N\} &\reducesto& N[(\ell~V)^R/y], \hfill\quad \text{if } \ell \neq \ell' \\ |
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\semlab{Let} & |
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\Let \; x \revto \Return \; V \; \In \; N &\reducesto& N[V/x] \\ |
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\semlab{Lift} & |
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\EC[M] &\reducesto& \EC[N], \hfill\quad \text{if } M \reducesto N \\ |
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\end{reductions} |
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\begin{syntax} |
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\slab{Evaluation contexts} & \mathcal{E} &::=& [\,] \mid \Let \; x \revto \mathcal{E} \; \In \; N |
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\end{syntax} |
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%%\[ |
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% Evaluation context lift |
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%% \inferrule*[Lab=\semlab{Lift}] |
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%% { M \reducesto N } |
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%% { \mathcal{E}[M] \reducesto \mathcal{E}[N]} |
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%% \] |
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\caption{Contextual small-step semantics} |
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\label{fig:base-language-small-step} |
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\end{figure} |
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% |
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In this section we present the dynamic semantics of \BCalc{}. We |
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choose to use a contextual small-step semantics, since in conjunction |
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with fine-grain call-by-value, it yields a considerably simpler |
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semantics than the traditional structural operational semantics |
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(SOS)~\cite{Plotkin04a}, because only the rule for let bindings admits |
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a continuation whereas in ordinary call-by-value SOS each congruence |
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rule admits a continuation. |
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% |
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Figure~\ref{fig:base-language-small-step} depicts the reduction |
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rules. The application rules \semlab{App} and \semlab{TyApp} |
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eliminates a lambda and type abstraction, respectively, by |
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substituting the argument for the parameter in their body computation |
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$M$. |
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% |
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Record splitting is handled by the \semlab{Split} rule: splitting on |
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some label $\ell$ binds the payload $V$ to $x$ and the remainder $W$ |
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to $y$ in the continuation $N$. |
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% |
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Disjunctive case splitting is handled by the two rules |
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\semlab{Case$_1$} and \semlab{Case$_2$}. The former rule handles the |
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success case, when the scrutinee's tag $\ell$ matches the tag of the |
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success clause, thus binds the payload $V$ to $x$ and proceeds to |
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evaluate the continuation $M$. The latter rule handles the |
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fall-through case, here the scrutinee gets bounds to $y$ and |
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evaluation proceeds with the continuation $N$. |
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% |
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The \semlab{Let} rule eliminates a trivial computation term |
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$\Return\;V$ by substituting $V$ for $x$ in the continuation $N$. |
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% |
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Recall from Section~\ref{sec:base-language-terms}, |
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Figure~\ref{fig:base-language-term-syntax} that the syntax of let |
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bindings allows a general computation term $M$ to occur on the right |
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hand side of the binding, i.e. $\Let\;x \revto M \;\In\;N$. Thus we |
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are seemingly stuck in the general case, as the \semlab{Let} rule only |
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applies if the right hand side is a trivial computation. |
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% |
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However, it is at this stage we make use of the notion of |
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\emph{evaluation contexts} due to \citet{Felleisen87}. An evaluation |
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context is syntactic construction which decompose the dynamic |
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semantics into a set of base rules (e.g. the rules presented thus far) |
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and an inductive rule, which enables us to focus on a particular |
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computation term, $M$, in some larger context, $\EC$, and reduce it in |
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the said context to another computation $N$ if $M$ reduces outside out |
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the context to that particular $N$. In our formalism, we call this |
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rule \semlab{Lift}. Evaluation contexts are generated from the empty |
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context $[~]$ and let expressions $\Let\;x \revto \EC \;\In\;N$. |
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\section{Row polymorphism} |
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\label{sec:row-polymorphism} |
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