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@ -91,6 +91,24 @@ |
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\newcommand{\XCancel}[1]{\Cancel[red,X,line width=1pt]{#1}} |
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% Structures |
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\tikzset{ |
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port/.style = {treenode, font=\Huge, draw=white, minimum width=0.5em, minimum height=0.5em}, |
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blackbox/.style = {rectangle, fill=black, draw=black, minimum width=2cm, minimum height=2cm}, |
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treenode/.style = {align=center, inner sep=3pt, text centered}, |
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opnode/.style = {treenode, rectangle, draw=black}, |
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leaf/.style = {treenode, draw=black, ellipse, thin}, |
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comptree/.style = {treenode, draw=black, regular polygon, regular polygon sides=3}, |
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highlight/.style = {draw=red,very thick}, |
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pencildraw/.style={ |
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black!75, |
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decorate, |
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decoration={random steps,segment length=0.8pt,amplitude=0.1pt} |
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}, |
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hbox/.style = {rectangle,draw=none, minimum width=6cm, minimum height=1cm}, |
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gbox/.style = {rectangle,draw=none,minimum width=2cm,minimum height=1cm} |
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} |
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%% |
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%% Theorem environments |
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%% |
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@ -4370,8 +4388,12 @@ of value types with a new arrow. |
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\slab{Computations} &M,N \in \CompCat &::=& \cdots \mid (\Do \; \ell~V)^E |
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\end{syntax} |
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% |
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The operation arrow, $\opto$, denotes the operation space. Contrary to |
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the function space constructor, $\to$, it does not have an associated |
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The operation arrow, $\opto$, denotes the operation space. The |
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operation space arrow is similar to the function space arrow in that |
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the type $A$ denotes the domain type of the operation, i.e. the type |
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of the operation payload, and the codomain type $B$ denotes the return |
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type of the operation. Contrary to the function space constructor, |
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$\to$, the operation space constructor does not have an associated |
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effect row. As we will see later, the reason that the operation space |
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constructor does not have an effect row is that the effects of an |
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operation is conferred by its handler. |
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@ -4380,34 +4402,62 @@ The intended behaviour of the new computation term $(\Do\; \ell~V)^E$ |
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is that it performs some operation $\ell$ with value argument |
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$V$. Thus the $\Do$-construct is similar to the typical |
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exception-signalling $\keyw{throw}$ or $\keyw{raise}$ constructs found |
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in programming languages with support for exceptions. The term is |
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annotated with an effect row $E$, providing a handle to obtain the |
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current effect context. We make use of this effect row during typing. |
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in programming languages with support for exceptions. In fact |
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operationally, an effectful operation may be thought of as an |
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exception which is resumable~\cite{Leijen17}. The term is annotated |
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with an effect row $E$, providing a way to make the current effect |
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context accessible during typing. |
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% |
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\begin{mathpar} |
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\inferrule*[Lab=\tylab{Do}] |
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{ E = \{\ell : A \opto B; R\} \\ |
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\typ{\Delta}{E} \\ |
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{ \typ{\Delta}{E} \\ |
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E = \{\ell : A \opto B; R\} \\ |
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\typ{\Delta;\Gamma}{V : A} |
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} |
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{\typc{\Delta;\Gamma}{(\Do \; \ell \; V)^E : B}{E}} |
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\end{mathpar} |
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% |
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An operation invocation is only well-typed if the effect row $E$ is |
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well-kinded and contains the said operation with a present type; in |
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other words, the current effect context permits the operation. The |
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argument type $A$ must be the same as the domain of the operation. |
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% |
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We slightly abuse notation by using the function space arrow, $\to$, |
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to also denote the operation space. Although, the function and |
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operation spaces are separate entities, we may think of the operation |
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space as a subspace of the function space in which every effect row is |
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empty, that is every operation has a type on the form |
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$A \to B \eff \emptyset$. The reason that the effect row is always |
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empty is that any effects an operation might have are ultimately |
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conferred by a handler. |
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\dhil{Introduce notation for computation trees.} |
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well-kinded and mentions the operation with a present type, or put |
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differently: the current effect context must permit an instance of the |
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operation to occur. The argument type $A$ must be the same as the |
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domain type of the operation. The type of the whole term is the |
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(value) return type of the operation paired with the current effect |
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context. |
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We have the basic machinery for writing effectful programs, albeit we |
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cannot evaluate those programs without handlers to ascribe a semantics |
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to the operations. |
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% \paragraph{Computation trees} We have the basic machinery for writing |
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% effectful programs, albeit we cannot evaluate those programs without |
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% handlers to ascribe a semantics to the operations. |
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% % |
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% For instance consider the signature of computations over some state at |
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% type $\beta$. |
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% % |
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% \[ |
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% \State~\beta \defas \{\Get : \UnitType \opto \beta; \Put : \beta \opto \UnitType\} |
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% \] |
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% % |
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% \[ |
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% \bl |
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% \dec{condupd} : \Record{\Bool;\beta} \to \beta \eff \State~\beta\\ |
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% \dec{condupd}~\Record{b;v} \defas |
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% \bl |
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% \Let\; x \revto \Do\;\Get\,\Unit\;\In\\ |
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% \If\;b\;\Then\;\;\Do\;\Put~v;\,x\; |
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% \Else\;x |
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% \el |
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% \el |
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% \] |
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% % |
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% This exact notation is due to \citet{Lindley14}, though the idea of |
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% using computation trees dates back to at least |
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% \citet{Kleene59,Kleene63}. |
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% \dhil{Introduce notation for computation trees.} |
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\subsection{Handling of effectful operations} |
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% |
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