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@ -6500,18 +6500,27 @@ getting stuck on an unhandled operation. |
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\subsection{Effect sugar} |
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\label{sec:effect-sugar} |
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Every effect row which share the same effect variable must mention the |
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exact same operations to be complete. Their presence information may |
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differ. In higher-order effectful programming this can lead to |
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duplication of information, and thus verbose effect |
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signature. Verbosity can be a real nuisance in larger codebases. |
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The row polymorphism formalism underlying the effect system is rigid |
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with regard to presence information. Every effect row which share the |
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same effect variable must mention the exact same operations to be |
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complete, that is they must mention whether an operation is present, |
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absent, or polymorphic in their its presence. Consequently, in |
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higher-order effectful programming this can cause duplication of |
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information, which in turn can cause effect signatures to become |
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overly verbose. In most cases verbosity is undesirable if the extra |
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information is redundant, and in practice it can be real nuisance in |
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larger codebases. |
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% |
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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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To this end, I will take inspiration from the effect system of Frank, |
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which allows eliding redundant information in many |
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cases~\cite{LindleyMM17}. |
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% |
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I will describe an ad-hoc elaboration scheme, which is designed to |
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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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In the following I will describe an ad-hoc elaboration scheme for |
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effect rows, which is designed to guess the programmer's intent for |
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first-order and second-order functions, but it 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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@ -6521,44 +6530,50 @@ 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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$\map$ function for lists. |
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First, let us consider the familiar second-order function $\map$ for |
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lists which is completely parametric in its effects. |
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% |
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\[ |
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\map : \Record{\alpha \to \beta;\List~\alpha} \to \List~\beta |
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\ba{@{~}l@{~}l@{}l@{~}l} |
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&\map : \Record{\alpha \to \beta &;\List~\alpha} \to &\List~\beta\\ |
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\equiv&\map : \Record{\alpha \to \beta \eff \{\varepsilon\}&;\List~\alpha} |
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\to &\List~\beta \eff \{\varepsilon\} |
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\ea |
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\] |
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% |
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For this type to be correct in $\HCalc$ (and $\BCalc$ for that matter) |
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we must annotate the arrows with their effect signatures, e.g. |
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% |
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\[ |
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\map : \Record{\alpha \to \beta \eff \{\varepsilon\};\List~\alpha} |
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\to \List~\beta \eff \{\varepsilon\} |
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\] |
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we must annotate the computation type with their effect row. |
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% |
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The effect annotations do not convey any additional information, |
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These effect annotations do not convey any additional information, |
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because the function is entirely parametric in all effects, thus the |
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ink spent on the effect annotations is really wasted in this instance. |
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ink spent on the annotations is really wasted in this instance. |
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% |
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To fix this we simply need to instantiate each computation type with |
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the same effect variable $\varepsilon$. |
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A slight variation of this example is an effectful second-order |
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function that is parametric in its argument's effects. |
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A slightly more interesting example is a second-order function which |
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itself performs some operation, but is otherwise parametric in the |
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effects of its argument. |
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% |
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\[ |
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(A \to B_1 \eff \{\varepsilon\}) \to B_2 \eff \{\ell : A' \opto B';\varepsilon\} \simeq (A \to B_1 \eff \{\ell : A' \opto B';\varepsilon\}) \to B_2 \eff \{\ell : A' \opto B';\varepsilon\} |
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\ba{@{~}l@{~}l@{}l@{~}l} |
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&(A \to B_1 \eff \{ &\varepsilon\}) &\to B_2 \eff \{\ell : A' \opto B';\varepsilon\}\\ |
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\equiv&(A \to B_1 \eff \{\ell : A' \opto B';&\varepsilon\}) &\to B_2 \eff \{\ell : A' \opto B';\varepsilon\} |
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\ea |
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\] |
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% |
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For the effect row of the functional parameter to be correct it must |
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mention the $\ell$ operation. |
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To be type-correct both rows must mention the $\ell$ |
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operation. However, this information is redundant on the functional |
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parameter. The idea here is to push the information of the ambient |
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effect row $B_2$ inwards to $B_1$ such that the functional argument |
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can be granted the ability to perform $\ell$. |
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% |
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The idea here is to push the effect signature inwards. The following |
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infix function $\vartriangleleft$ copies operations and their presence |
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information from its right argument to its left argument. |
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The following infix function $\vartriangleleft$ implements the inward |
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push of information by copying operations from its right parameter to |
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its left parameter. |
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% |
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\begin{equations} |
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\[ |
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\ba{@{~}r@{~}c@{~}l} |
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\vartriangleleft &:& \EffectCat \times \EffectCat \to \EffectCat\\ |
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E \vartriangleleft \{\cdot\} &\defas& E\\ |
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E \vartriangleleft \{\varepsilon\} &\defas& E\\ |
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@ -6567,30 +6582,36 @@ information from its right argument to its left argument. |
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\{\ell : P\} \uplus (E \vartriangleleft \{R\}) & \text{if } \ell \notin E\\ |
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\{R\} \vartriangleleft E & \text{otherwise} |
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\end{cases}\\ |
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\end{equations} |
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\ea |
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\] |
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% |
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This function essentially computes the union of the two effect rows. |
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Another example is an observably pure function whose input is |
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effectful, meaning the function must be handling the effects of its |
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argument internally (or not apply its argument at all). To capture the |
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intuition that operations have been handled, we would like to not |
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mention the handled operations in the function's effect signature, |
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however, the underlying row polymorphism formalism requires us to |
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explicitly mention handled operations as being polymorphic in their |
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presence, e.g. |
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The most frequent case to occur is a second-order function which |
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handles the effects of its argument. |
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% |
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\[ |
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(A \to B_1 \eff \{\ell : A' \opto B';\varepsilon\}) \to B_2 \eff \{\varepsilon\} \simeq (A \to B_1 \eff \{\ell : A' \opto B';\varepsilon\}) \to B_2 \eff \{\ell : \theta;\varepsilon\} |
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\ba{@{~}l@{~}l@{}l} |
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& (A \to B_1 \eff \{\ell : A' \opto B';\varepsilon\}) \to B_2 \eff \{ &\varepsilon\}\\ |
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\equiv& (A \to B_1 \eff \{\ell : A' \opto B';\varepsilon\}) \to B_2 \eff \{\ell : \theta;&\varepsilon\} |
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\ea |
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\] |
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% |
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The idea is to extract the effect signature of the domain $A$ and then |
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copy every operation from this signature into the function's signature |
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as polymorphic in its presence if it is not mentioned by the |
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function's signature. The following function implements the copy |
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aspect of this idea. |
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To capture the intuition that operations have been handled, we would |
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like to not mention the handled operations in the effect row attached |
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to $B_2$. |
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% |
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\begin{equations} |
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The idea is to propagate the information of the effect row attached to |
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$B_1$ outwards such that this information can be used to complete the |
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effect row on $B_2$. To complete the row we need to copy the |
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operations unique to the effect row of $B_1$ into the effect row of |
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$B_2$ and instantiate them with a fresh presence variable. |
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% |
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The following function $\vartriangleright$ propagates information from |
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its left parameter to its right parameter. |
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% |
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\[ |
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\ba{@{~}r@{~}c@{~}l} |
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\vartriangleright &:& \EffectCat \times \EffectCat \to \EffectCat\\ |
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\{\cdot\} \vartriangleright E &\defas& E\\ |
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\{\varepsilon\} \vartriangleright E &\defas& E\\ |
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@ -6604,62 +6625,144 @@ aspect of this idea. |
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\{\ell : P\} \uplus (\{R\} \vartriangleright E) & \text{if } \ell \notin E\\ |
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\{R\} \vartriangleright E & \text{otherwise} |
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\end{cases} |
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\end{equations} |
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\ea |
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\] |
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% |
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The only subtlety occur in the interesting case which is when an |
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operation is present in the left signature, but not in the right |
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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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operation is present in the left row, but not in the right row. In |
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this case we instantiate the operation with a fresh presence variable |
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in the output row. |
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Propagation of information in either direction should only happen if |
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the effect rows share the same effect variable. To avoid erroneous |
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propagation of information we implement and use the following guarded |
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variations of $\vartriangleleft$ and $\vartriangleright$. |
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% |
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\begin{equations} |
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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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\[ |
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\ba{@{}l@{\qquad}@{}l} |
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\ba[t]{@{~}r@{~}l@{~}c@{~}l} |
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\{R;\varepsilon\} \blacktriangleleft &\{R';\varepsilon\} &\defas& |
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\{R;\varepsilon\} \vartriangleleft \{R';\varepsilon\}\\ |
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\{R;\varepsilon\} \blacktriangleleft &\{R';\varepsilon'\} &\defas& |
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\{R;\varepsilon\} |
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\ea & |
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\ba[t]{@{~}r@{~}l@{~}c@{~}l} |
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\{R;\varepsilon\} \blacktriangleright &\{R';\varepsilon\} &\defas& |
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\{R;\varepsilon\} \vartriangleright \{R';\varepsilon\}\\ |
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\{R;\varepsilon\} \blacktriangleright &\{R';\varepsilon'\} &\defas& |
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\{R';\varepsilon'\} |
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\ea |
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\ea |
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\] |
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% |
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The following function $\inward{-}$ pushes the ambient effect row |
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$\eamb$ inward a given type. I will omit the homomorphic cases as |
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there is only one interesting case. |
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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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\pcomp{A \eff E}_{\eamb} &\defas& \pval{A}_{\eamb} \eff E \blacktriangleleft \eamb |
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\end{equations} |
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% |
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The following function $\outward{-}$ combines and propagates the |
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effect rows of a type outward. Again, I omit the homomorphic cases. |
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% |
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\[ |
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\ba{@{}l@{\qquad}@{}l} |
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\ba[t]{@{~}r@{~}c@{~}l} |
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\xval{-} &:& \ValTypeCat \to \EffectCat\\ |
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\xval{\alpha} &\defas& \{\cdot\}\\ |
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\xval{A \to C} &\defas& \xval{A} \blacktriangleright \xcomp{C} \\ |
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\ea & |
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\ba[t]{@{~}r@{~}c@{~}l} |
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\xcomp{-} &:& \CompTypeCat \to \EffectCat\\ |
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\xcomp{A \eff E} &\defas& \xval{A} \blacktriangleright E \smallskip\\ |
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\ea \smallskip\\ |
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\ba[t]{@{~}r@{~}c@{~}l} |
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\xpre{-} &:& \PresenceCat \to \EffectCat\\ |
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% \xpre{\Pre{A}} &\defas& \xval{A}\\ |
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\xpre{\Abs} &\defas& \xpre{\theta} \defas \{\cdot\} |
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\ea & |
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\ba[t]{@{~}r@{~}c@{~}l} |
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\xrow{-} &:& \RowCat \to \Effect\\ |
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\xrow{\cdot} &\defas& \xval{\rho} \defas \{\cdot\}\\ |
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\xrow{\ell : P;R} &\defas& \xval{P} \blacktriangleright \xval{R} |
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\ea |
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\ea |
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\] |
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% \[ |
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% \ba{@{}l@{\qquad}@{}l} |
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% \ba[t]{@{~}r@{~}c@{~}l} |
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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} \blacktriangleright \xcomp{C} \\ |
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% \xval{\forall \alpha^K.C} &\defas& \xcomp{C} |
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% \ea & |
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% \ba[t]{@{~}r@{~}c@{~}l} |
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% \xcomp{-} &:& \CompTypeCat \to \EffectCat\\ |
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% \xcomp{A \eff E} &\defas& \xval{A} \blacktriangleright 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} \blacktriangleright \xval{R} |
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% \ea\\ |
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% \multicolumn{2}{c}{ |
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% \ba{@{~}r@{~}c@{~}l} |
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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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% \ea} |
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% \ea |
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% \] |
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% |
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% \[ |
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% \ba{@{}l@{\qquad}@{}l} |
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% \ba[t]{@{~}r@{~}c@{~}l} |
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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} |
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% \ea & |
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% \ba[t]{@{~}r@{~}c@{~}l} |
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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 \blacktriangleleft \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} |
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% \ea\\ |
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% \multicolumn{2}{c}{ |
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% \ba[t]{@{~}r@{~}c@{~}l} |
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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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% \ea} |
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% \ea |
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% \] |
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% |
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We combine all of the above functions to implement the effect row |
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elaboration for top-level function types. |
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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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\trval{A \to B \eff E} &\defas& \pval{A}_{E'} \to \pval{B}_{E'} \eff E'\\ |
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\multicolumn{3}{l}{\quad\where~E' = (\xval{A} \blacktriangleright E) \blacktriangleright (\xval{B} \blacktriangleright E)} |
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\end{equations} |
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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_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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The function $\trval{-}$ traverses the abstract syntax of its argument |
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twice. The first traversal propagates effect information outwards to |
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the ambient effect row $E$. The second traversal pushes the full |
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ambient information $E'$ inwards. |
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% |
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As a remark, note that the function $\trval{-}$ do not have to |
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consider handler types, because they cannot appear at the top-level in |
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$\HCalc$. With this syntactic sugar in place we can program with |
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second-order effectful functions without having to write down |
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redundant information. |
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\section{Composing \UNIX{} with effect handlers} |
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\label{sec:deep-handlers-in-action} |
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