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@ -3552,6 +3552,8 @@ macro-expressible in terms of the existing constructs. |
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I choose here to treat them as primitives in order to keep the |
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presentation relatively concise. |
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\dhil{Remark that a `generalised continuation' is a defunctionalised continuation.} |
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\subsection{Dynamic terms: the target calculus revisited} |
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\begin{figure}[t] |
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@ -3663,7 +3665,7 @@ for the CPS translation with generalised continuations |
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functions that resumptions get converted to have the same shape as the |
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translation of source level functions -- this is required because the |
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operational semantics does not treat resumptions as distinct |
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first-class objects, but rather as a special kinds of functions |
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first-class objects, but rather as a special kinds of functions. |
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\subsection{Translation with generalised continuations} |
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\label{sec:cps-gen-conts} |
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@ -3771,7 +3773,201 @@ now uniform: only a single continuation frame is inspected at any |
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time. |
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\subsubsection{Correctness} |
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\dhil{Todo} |
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\label{sec:cps-gen-cont-correctness} |
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% |
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The correctness of this CPS translation |
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(Theorem~\ref{thm:ho-simulation-gen-cont}) follows closely the |
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correctness result for the higher-order uncurried CPS translation for |
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deep handlers (Theorem~\ref{thm:ho-simulation}). Save for the |
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syntactic difference, the most notable difference is the extension of |
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the operation handling lemma (Lemma~\ref{lem:handle-op-gen-cont}) to |
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cover shallow handling in addition to deep handling. Each lemma is |
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stated in terms of static continuations, where the shape of the top |
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element is always known statically, i.e., it is of the form |
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$\sRecord{\sV_{fs}, \sRecord{\sV_{\mret},\sV_{\mops}}} \scons |
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\sW$. Moreover, the static values $\sV_{fs}$, $\sV_{\mret}$, and |
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$\sV_{\mops}$ are all reflected dynamic terms (i.e., of the form |
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$\reflect V$). This fact is used implicitly in the proofs. |
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% |
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\begin{lemma}[Substitution]\label{lem:subst-gen-cont} |
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% |
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The CPS translation commutes with substitution in value terms |
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% |
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\[ |
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\cps{W}[\cps{V}/x] = \cps{W[V/x]}, |
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\] |
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% |
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and with substitution in computation terms |
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\[ |
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\ba{@{}l@{~}l} |
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&(\cps{M} \sapp (\sRecord{\sV_{fs},\sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW))[\cps{V}/x]\\ |
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= &\cps{M[V/x]} \sapp (\sRecord{\sV_{fs},\sRecord{\sV_{\mret},\sV_{\mops}}} \scons\sW)[\cps{V}/x], |
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\ea |
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\] |
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% |
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and with substitution in handler definitions |
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% |
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\begin{equations} |
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\cps{\hret}[\cps{V}/x] |
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&=& \cps{\hret[V/x]},\\ |
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\cps{\hops}[\cps{V}/x] |
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&=& \cps{\hops[V/x]}. |
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\end{equations} |
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\end{lemma} |
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% |
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In order to reason about the behaviour of the \semlab{Op} and |
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\semlab{Op^\dagger} rules, which are defined in terms of evaluation |
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contexts, we extend the CPS translation to evaluation contexts, using |
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the same translations as for the corresponding constructs in $\SCalc$. |
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% |
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\begin{equations} |
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\cps{[~]} |
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&=& \slam \shk. \shk \\ |
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\cps{\Let\; x \revto \EC \;\In\; N} |
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&=& |
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\begin{array}[t]{@{}l} |
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\slam \sRecord{\shf, \sRecord{\svhret, \svhops}} \scons \shk.\\ |
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\quad \cps{\EC} \sapp (\bl\sRecord{\reflect((\dlam x\,\dhk.\bl\Let\;\dRecord{fs,\dRecord{\vhret,\vhops}}\dcons \dhk'=\dhk\;\In\\ |
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\cps{N} \sapp (\sRecord{\reflect fs, \sRecord{\reflect \vhret, \reflect \vhops}} \scons \reflect \dhk')) \dcons \reify\shf),\el\\ |
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\sRecord{\svhret,\svhops}} \scons \shk)\el |
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\end{array} |
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\\ |
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\cps{\Handle^\depth\; \EC \;\With\; H} |
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&=& \slam \shk.\cps{\EC} \sapp (\sRecord{[], \cps{H}^\depth} \scons \shk) |
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\end{equations} |
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% |
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The following lemma is the characteristic property of the CPS |
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translation on evaluation contexts. |
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% |
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This allows us to focus on the computation within an evaluation |
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context. |
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% |
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\begin{lemma}[Evaluation context decomposition] |
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\label{lem:decomposition-gen-cont} |
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\[ |
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\cps{\EC[M]} \sapp (\sRecord{\sV_{fs}, \sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW) |
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= |
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\cps{M} \sapp (\cps{\EC} \sapp (\sRecord{\sV_{fs}, \sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW)) |
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\] |
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\end{lemma} |
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% |
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By definition, reifying a reflected dynamic value is the identity |
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($\reify \reflect V = V$), but we also need to reason about the |
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inverse composition. As a result of the invariant that the translation |
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always has static access to the top of the handler stack, the |
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translated terms are insensitive to whether the remainder of the stack |
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is statically known or is a reflected version of a reified stack. This |
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is captured by the following lemma. The proof is by induction on the |
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structure of $M$ (after generalising the statement to stacks of |
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arbitrary depth), and relies on the observation that translated terms |
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either access the top of the handler stack, or reify the stack to use |
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dynamically, whereupon the distinction between reflected and reified |
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becomes void. Again, this lemma holds when the top of the static |
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continuation is known. |
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% |
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\begin{lemma}[Reflect after reify] |
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\label{lem:reflect-after-reify-gen-cont} |
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\[ |
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\cps{M} \sapp (\sRecord{\sV_{fs}, \sRecord{\sV_{\mret},\sV_{\mops}}} \scons \reflect \reify \sW) |
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= |
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\cps{M} \sapp (\sRecord{\sV_{fs}, \sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW). |
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\] |
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\end{lemma} |
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The next lemma states that the CPS translation correctly simulates |
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forwarding. The proof is by inspection of how the translation of |
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operation clauses treats non-handled operations. |
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% |
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\begin{lemma}[Forwarding]\label{lem:forwarding-gen-cont} |
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If $\ell \notin dom(H_1)$ then: |
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% |
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\[ |
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\bl |
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\cps{\hops_1}^\delta \dapp \dRecord{\ell, \dRecord{V_p, V_{\dhkr}}} \dapp (\dRecord{V_{fs}, \cps{H_2}^\delta} \dcons W) |
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\reducesto^+ \qquad \\ |
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\hfill |
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\cps{\hops_2}^\delta \dapp \dRecord{\ell, \dRecord{V_p, \dRecord{V_{fs}, \cps{H_2}^\delta} \dcons V_{\dhkr}}} \dapp W. \\ |
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\el |
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\] |
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% |
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\end{lemma} |
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The following lemma is central to our simulation theorem. It |
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characterises the sense in which the translation respects the handling |
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of operations. Note how the values substituted for the resumption |
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variable $r$ in both cases are in the image of the translation of |
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$\lambda$-terms in the CPS translation. This is thanks to the precise |
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way that the reductions rules for resumption construction works in our |
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dynamic language, as described above. |
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% |
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\begin{lemma}[Handling]\label{lem:handle-op-gen-cont} |
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Suppose $\ell \notin BL(\EC)$ and $\hell = \{\ell\,p\,r \mapsto N_\ell\}$. If $H$ is deep then |
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% |
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\[ |
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\bl |
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\cps{\Do\;\ell\;V} \sapp (\cps{\EC} \sapp (\sRecord{\snil, \cps{H}} \scons \sRecord{\sV_{fs},\sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW)) \reducesto^+ \\ |
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\quad (\cps{N_\ell} \sapp \sRecord{\sV_{fs},\sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW)\\ |
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\qquad \quad [\cps{V}/p, |
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\dlam x\,\dhk.\bl |
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\Let\;\dRecord{fs, \dRecord{\vhret, \vhops}} \dcons \dhk' = \dhk\;\In\;\\ |
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\cps{\Return\;x} \sapp (\cps{\EC} \sapp (\sRecord{\snil, \cps{H}} \scons \sRecord{\reflect \dlk, \sRecord{\reflect \vhret, \reflect \vhops}} \scons \reflect\dhk'))/r]. \\ |
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\el\\ |
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\el |
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\] |
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% |
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Otherwise if $H$ is shallow then |
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% |
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\[ |
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\bl |
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\cps{\Do\;\ell\;V} \sapp (\cps{\EC} \sapp (\sRecord{\snil, \cps{H}^\dagger} \scons \sRecord{\sV_{fs},\sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW)) \reducesto^+ \\ |
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\quad (\cps{N_\ell} \sapp \sRecord{\sV_{fs},\sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW)\\ |
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\qquad [\cps{V}/p, \dlam x\,\dhk. \bl |
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\Let\;\dRecord{\dlk, \dRecord{\vhret, \vhops}} \dcons \dhk' = \dhk \;\In \\ |
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\cps{\Return\;x} \sapp (\cps{\EC} \sapp (\sRecord{\reflect \dlk, \sRecord{\reflect \vhret, \reflect \vhops}} \scons \reflect\dhk'))/r]. \\ |
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\el \\ |
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\el |
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\] |
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% |
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\end{lemma} |
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\medskip |
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Now the main result for the translation: a simulation result in the |
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style of \citet{Plotkin75}. |
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% |
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\begin{theorem}[Simulation] |
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\label{thm:ho-simulation-gen-cont} |
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If $M \reducesto N$ then |
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\[ |
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\cps{M} \sapp (\sRecord{\sV_{fs}, \sRecord{\sV_{\mret},\sV_{\mops}}} |
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\scons \sW) \reducesto^+ \cps{N} \sapp (\sRecord{\sV_{fs}, |
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\sRecord{\sV_{\mret},\sV_{\mops}}} \scons \sW). |
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\] |
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\end{theorem} |
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\begin{proof} |
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The proof is by case analysis on the reduction relation using Lemmas |
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\ref{lem:decomposition-gen-cont}--\ref{lem:handle-op-gen-cont}. In |
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particular, the \semlab{Op} and \semlab{Op^\dagger} cases follow |
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from Lemma~\ref{lem:handle-op-gen-cont}. |
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\end{proof} |
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In common with most CPS translations, full abstraction does not hold |
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(a function could count the number of handlers it is invoked within by |
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examining the continuation, for example). However, as the semantics is |
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deterministic it is straightforward to show a backward simulation |
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result. |
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% |
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\begin{lemma}[Backwards simulation] |
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If $\pcps{M} \reducesto^+ V$ then there exists $W$ |
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such that $M \reducesto^* W$ and $\pcps{W} = V$. |
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\end{lemma} |
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% |
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\begin{corollary} |
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$M \reducesto^\ast V$ if and only if $\pcps{M} \reducesto^\ast \pcps{V}$. |
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\end{corollary} |
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\section{Related work} |
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\label{sec:cps-related-work} |
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