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@ -6510,16 +6510,16 @@ We can retrospectively fix this issue with some syntactic sugar rather |
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than redesigning the entire effect system to rectify this problem. |
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% |
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I will describe an ad-hoc elaboration scheme, which is designed to |
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work well for first-order and second-order functions, but might not |
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work so well for third-order and above. The reason for focusing on |
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first-order and second-order functions is that many familiar and |
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useful functions are either first-order or second-order, and in the |
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following sections we will mostly be working with first-order and |
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second-order functions (although, it should be noted that there exist |
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useful functions at higher order, e.g. in |
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Chapter~\ref{ch:handlers-efficiency} we shall use third-order |
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functions; for an example of a sixth order function see |
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\citet{Okasaki98}). |
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emulate Frank's effect polymorphic system~\cite{LindleyMM17} for |
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first-order and second-order functions, but might not work so well for |
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third-order and above. The reason for focusing on first-order and |
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second-order functions is that many familiar and useful functions are |
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either first-order or second-order, and in the following sections we |
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will mostly be working with first-order and second-order functions |
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(although, it should be noted that there exist useful functions at |
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higher order, e.g. in Chapter~\ref{ch:handlers-efficiency} we shall |
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use third-order functions; for an example of a sixth order function |
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see \citet{Okasaki98}). |
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First, let us consider some small instances of the kind of incomplete |
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effect rows we would like to be able to write. Consider the type of |
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@ -6612,46 +6612,54 @@ signature. In this case we instantiate the operation with a fresh |
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presence variable in the output signature. |
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% |
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\begin{equations} |
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\trval{-} &:& \ValTypeCat \to \ValTypeCat\\ |
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\trval{A \to B \eff E} &\defas& A' \to B' \eff E'\\ |
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\multicolumn{3}{l}{\quad\where~(A', E_A) \defas \trval{A}_{E}\;\keyw{and}\;(B', \_) = \trval{B}_{\{\varepsilon\}}\;\keyw{and}\; E' = E_A \vartriangleright E} \medskip\\ |
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\trval{-} &:& \ValTypeCat \times \EffectCat \to \ValTypeCat \times \EffectCat\\ |
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\trval{\UnitType}_{E_{amb}} &\defas& (\UnitType, \{\cdot\})\\ |
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\trval{A \to B \eff E}_{E_{amb}} &\defas& (A' \to B' \eff E', E_A \vartriangleright E)\\ |
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\multicolumn{3}{l}{\quad\where~E' = E \vartriangleleft E_{amb}\;\keyw{and}\;(A', E_A) = \trval{A}_{E'}\;\keyw{and}\;(B', \_) = \trval{B}_{\{\varepsilon\}}} |
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\xval{-} &:& \ValTypeCat \to \EffectCat\\ |
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\xval{\alpha} &\defas& \{\cdot\}\\ |
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\xval{\Record{R}} &\defas& \xval{[R]} \defas \xval{R}\\ |
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\xval{A \to C} &\defas& \xval{A} \vartriangleright \xcomp{C} \\ |
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\xval{\forall \alpha^K.C} &\defas& \xcomp{C} \smallskip\\ |
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\xcomp{-} &:& \CompTypeCat \to \EffectCat\\ |
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\xcomp{A \eff E} &\defas& \xval{A} \vartriangleright E \smallskip\\ |
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\xrow{-} &:& \RowCat \to \Effect\\ |
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\xrow{\cdot} &\defas& \xval{\rho} \defas \{\cdot\}\\ |
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\xrow{\ell : P;R} &\defas& \xval{P} \vartriangleright \xval{R} \smallskip\\ |
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\xpre{-} &:& \PresenceCat \to \EffectCat\\ |
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\xpre{\Pre{A}} &\defas& \xval{A}\\ |
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\xpre{\Abs} &\defas& \xpre{\theta} \defas \{\cdot\} |
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\end{equations} |
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% |
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\begin{equations} |
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\pval{-} &:& \ValTypeCat \times \EffectCat \to \ValTypeCat\\ |
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\pval{\alpha}_{\eamb} &\defas& \alpha\\ |
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\pval{\Record{R}}_{\eamb} &\defas& \Record{\prow{R}_{\eamb}}\\ |
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\pval{[R]}_{\eamb} &\defas& [\prow{R}_{\eamb}]\\ |
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\pval{A \to C}_{\eamb} &\defas& \pval{A}_{\eamb} \to \pcomp{C}_{\eamb} \\ |
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\pval{\forall \alpha^K.C}_{\eamb} &\defas& \forall\alpha^K.\pcomp{C} \smallskip\\ |
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\pcomp{-} &:& \CompTypeCat \times \EffectCat \to \CompTypeCat\\ |
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\pcomp{A \eff E}_{\eamb} &\defas& \pval{A}_{\eamb} \eff E \vartriangleleft \eamb \smallskip\\ |
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\prow{-} &:& \RowCat \times \EffectCat \to \RowCat\\ |
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\prow{\cdot}_{\eamb} &\defas& \{\cdot\}\\ |
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\prow{\rho}_{\eamb} &\defas& \{\rho\}\\ |
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\prow{\ell : P;R}_{\eamb} &\defas& \ppre{P};\prow{R} \smallskip\\ |
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\ppre{-} &:& \PresenceCat \times \EffectCat \to \EffectCat\\ |
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\ppre{\Pre{A}}_{\eamb} &\defas& \pval{A}_{\eamb}\\ |
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\ppre{\Abs}_{\eamb} &\defas& \Abs\\ |
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\ppre{\theta} &\defas& \theta |
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\end{equations} |
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% which can be a real nuisance for programming as it scales poorly to large codebases. |
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% The idea is to push the effects inwards. |
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% |
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\begin{equations} |
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\trval{-} &:& \ValTypeCat \to \ValTypeCat\\ |
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\trval{A \to B \eff E} &\defas& \pval{A}_{E'} \to \pval{B}_{E'} \eff E' \vartriangleright E\\ |
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\multicolumn{3}{l}{\quad\where~E' = \xval{A} \vartriangleright \xval{B}} |
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\end{equations} |
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% \begin{equations} |
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% \pp &:& \EffectCat \to \EffectCat\\ |
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% \pp(\{\cdot\}) &\defas& \{\cdot\}\\ |
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% \pp(\{\varepsilon\}) &\defas& \{\varepsilon\}\\ |
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% \pp(\{\ell : A \opto B;R\}) &\defas& \{\ell : \theta \} \cup \pp(\{R\}),\quad \text{fresh}~\theta |
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% \trval{-} &:& \ValTypeCat \to \ValTypeCat\\ |
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% \trval{A \to B \eff E} &\defas& A' \to B' \eff E'\\ |
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% \multicolumn{3}{l}{\quad\where~(A', E_A) \defas \trval{A}_{E}\;\keyw{and}\;(B', \_) = \trval{B}_{\{\varepsilon\}}\;\keyw{and}\; E' = E_A \vartriangleright E} \medskip\\ |
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% \trval{-} &:& \ValTypeCat \times \EffectCat \to \ValTypeCat \times \EffectCat\\ |
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% \trval{\UnitType}_{E_{amb}} &\defas& (\UnitType, \{\cdot\})\\ |
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% \trval{A \to B \eff E}_{E_{amb}} &\defas& (A' \to B' \eff E', E_A \vartriangleright E)\\ |
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% \multicolumn{3}{l}{\quad\where~E' = (E_A \vartriangleright E) \vartriangleleft E_{amb}\;\keyw{and}\;(A', E_A) = \trval{A}_{E'}\;\keyw{and}\;(B', \_) = \trval{B}_{\{\varepsilon\}}} |
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% \end{equations} |
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% % |
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% \[ |
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% \bl |
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% \trcomp{-} : \ValTypeCat \to \ValTypeCat\\ |
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% \trcomp{A} \defas \trcomp{A}_{\{\varepsilon\}} \smallskip\\ |
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% \ba{@{}r@{~}c@{~}l} |
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% \trcomp{-} &:& \ValTypeCat \times \EffectCat \to \ValTypeCat \times \EffectCat\\ |
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% \trcomp{A \to B}_E &\defas& (A' \to B' \eff E', E')\\ |
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% \multicolumn{3}{l}{\quad\where~(A', E_A) = \trval{A}_E\;\keyw{and}\;(B', \_) = \trval{B}_{\{\varepsilon\}}\;\keyw{and}\;E' = E \cup \pp(E_A)}\\ |
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% \trcomp{A \to B \eff E}_{E'} &\defas& \trval{A}_{E \cup E'} \to \trval{B}_{\{\varepsilon\}} \eff E \cup E' |
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% \ea\smallskip\\ |
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% % |
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% \ba{@{}r@{~}c@{~}l} |
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% \trcomp{-} &:& \HandlerTypeCat \to \HandlerTypeCat\\ |
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% \trcomp{A \Harrow B} &\defas& \trcomp{B}_{\{\varepsilon\}} \eff \{\varepsilon\} \Harrow \trcomp{B}_{\{\varepsilon\}} \eff \{\varepsilon\}\\ |
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% \trcomp{A \eff E_A \Harrow B} &\defas& \trcomp{A}_{E_A} \Harrow \trcomp{B}_{\{\varepsilon\}} \eff \pp(E),\quad \text{fresh}~\varepsilon\\ |
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% \trcomp{A \Harrow B \eff E_B} &\defas& \trcomp{A}_{E_B} \Harrow \trcomp{B}_{\{\varepsilon\}} \eff E_B\\ |
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% \trcomp{A \eff E_A \Harrow B \eff E_B} &\defas& \trcomp{A}_{E} \eff E \Harrow \trcomp{B}_{\{\varepsilon\}} \eff E'\\ |
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% \multicolumn{3}{l}{\quad \where~E = E_A \cup E_B\;\keyw{and}\;E' = \pp(E_A) \cup E_B} |
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% \ea\\ |
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% \el |
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% \] |
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\section{Composing \UNIX{} with effect handlers} |
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\label{sec:deep-handlers-in-action} |
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