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@ -388,6 +388,11 @@ Evidently, control is a pervasive phenomenon in programming. However, |
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not every control phenomenon is equal in terms of programmability and |
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expressiveness. |
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\section{This thesis in a nutshell} |
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In this dissertation I am concerned only with |
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\citeauthor{PlotkinP09}'s deep handlers, their shallow variation, and |
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parameterised handlers which are a slight variation of deep handlers. |
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\section{Why first-class control matters} |
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\subsection{Flavours of control} |
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@ -399,22 +404,45 @@ expressiveness. |
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\section{State of effectful programming} |
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\subsection{Dark age of impurity} |
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The evolution of effectful programming has gone through several |
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characteristic phases. In this section I will provide a brief overview |
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of how effectful programming has evolved as well as providing a brief |
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introduction to the core concepts concerned with each phase. |
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\subsection{Early days of direct-style} |
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\subsection{Monadic enlightenment} |
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\subsection{Monadic epoch} |
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During the late 80s and early 90s monads rose to prominence as a |
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practical program structuring idiom for effectful |
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programming. Notably, they form the foundations for effectful |
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programming in Haskell, which adds special language-level support for |
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programming with monads~\cite{JonesABBBFHHHHJJLMPRRW99}. |
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practical programming idiom for structuring effectful |
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programming~\cite{Moggi89,Moggi91,Wadler92,Wadler92b,JonesW93,Wadler95,JonesABBBFHHHHJJLMPRRW99}. |
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% |
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The concept of monad has its origins in category theory and its |
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mathematical nature is well-understood~\cite{MacLane71,Borceux94}. The |
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emergence of monads as a programming abstraction began when |
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\citet{Moggi89,Moggi91} proposed to use monads as the basis for |
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denoting computational effects in denotational semantics such as |
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exceptions, state, nondeterminism, interactive input and |
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output. \citet{Wadler92,Wadler95} popularised monadic programming by |
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putting \citeauthor{Moggi89}'s proposal to use in functional |
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programming by demonstrating how monads increase the ease at which |
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programs may be retrofitted with computational effects. |
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% |
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In practical programming terms, monads may be thought of as |
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constituting a family of design patterns, where each pattern gives |
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rise to a distinct effect with its own operations. |
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% |
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% Notably, they form the foundations for effectful programming in |
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% Haskell, which adds special language-level support for programming |
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% with monads~\cite{JonesABBBFHHHHJJLMPRRW99}. |
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% |
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\begin{definition} |
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A monad is a triple $(T, \Return, \bind)$ where $T$ is some unary |
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type constructor, $\Return$ is an operation that lifts an arbitrary |
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value into the monad (sometimes this operation is called `the unit |
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operation'), and $\bind$ is an application operator the monad (this |
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operator is pronounced `bind'). Implementations of $\Return$ and |
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$\bind$ must conform to the following interface. |
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A monad is a triple $(T^{\TypeCat \to \TypeCat}, \Return, \bind)$ |
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where $T$ is some unary type constructor, $\Return$ is an operation |
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that lifts an arbitrary value into the monad (sometimes this |
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operation is called `the unit operation'), and $\bind$ is the |
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application operator of the monad (this operator is pronounced |
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`bind'). Adequate implementations of $\Return$ and $\bind$ must |
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conform to the following interface. |
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% |
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\[ |
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\bl |
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@ -431,43 +459,155 @@ programming with monads~\cite{JonesABBBFHHHHJJLMPRRW99}. |
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\end{reductions} |
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\end{definition} |
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% |
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Monads have given rise to various popular control-oriented programming |
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abstractions, e.g. the async/await idiom has its origins in monadic |
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Monads have a proven track record of successful applications. |
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Monads have also given rise to various popular control-oriented |
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programming abstractions, e.g. the asynchronous programming idiom |
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async/await has its origins in monadic |
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programming~\cite{Claessen99,LiZ07,SymePL11}. |
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\subsection{Option monad} |
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The $\Option$ type is a unary type constructor with two data |
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constructors, i.e. $\Option~A \defas [\Some:A|\None]$. |
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% \subsection{Option monad} |
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% The $\Option$ type is a unary type constructor with two data |
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% constructors, i.e. $\Option~A \defas [\Some:A|\None]$. |
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\begin{definition} The option monad is a monad equipped with a single |
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failure operation. |
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% \begin{definition} The option monad is a monad equipped with a single |
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% failure operation. |
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% % |
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% \[ |
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% \ba{@{~}l@{\qquad}@{~}r} |
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% \multicolumn{2}{l}{T~A \defas \Option~A} \smallskip\\ |
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% \multirow{2}{*}{ |
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% \bl |
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% \Return : A \to T~A\\ |
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% \Return \defas \lambda x.\Some~x |
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% \el} & |
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% \multirow{2}{*}{ |
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% \bl |
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% \bind ~: T~A \to (A \to T~B) \to T~B\\ |
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% \bind ~\defas \lambda m.\lambda k.\Case\;m\;\{\None \mapsto \None;\Some~x \mapsto k~x\} |
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% \el}\\ & \smallskip\\ |
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% \multicolumn{2}{l}{ |
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% \bl |
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% \fail : A \to T~A\\ |
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% \fail \defas \lambda x.\None |
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% \el} |
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% \ea |
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% \] |
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% % |
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% \end{definition} |
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\subsubsection{State monad} |
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% |
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The state monad provides a way to encapsulate a state cell. |
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% |
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\begin{definition} |
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The state monad is defined over some fixed state type $S$. |
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% |
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\[ |
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\ba{@{~}l@{\qquad}@{~}r} |
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\multicolumn{2}{l}{T~A \defas \Option~A} \smallskip\\ |
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\ba{@{~}l@{\qquad\quad}@{~}r} |
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\multicolumn{2}{l}{T~A \defas S \to A \times S} \smallskip\\ |
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\multirow{2}{*}{ |
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\bl |
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\Return : A \to T~A\\ |
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\Return \defas \lambda x.\Some~x |
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\Return~x\defas \lambda st.\Record{x;st} |
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\el} & |
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\multirow{2}{*}{ |
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\bl |
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\bind ~: T~A \to (A \to T~B) \to T~B\\ |
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\bind ~\defas \lambda m.\lambda k.\Case\;m\;\{\None \mapsto \None;\Some~x \mapsto k~x\} |
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\el}\\ & \smallskip\\ |
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\multicolumn{2}{l}{ |
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\bind ~\defas \lambda m.\lambda k.\lambda st. \Let\;\Record{x;st'} = m~st\;\In\; (k~x)~st' |
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\el} \\ & % space hack to avoid the next paragraph from |
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% floating into the math environment. |
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\ea |
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\] |
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% |
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The $\Return$ of the monad is a state-accepting function of type |
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$S \to A \times S$ that returns its first argument paired with the |
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current state. The bind operator also produces a state-accepting |
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function of type $S \to A \times S$. The bind operator first |
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supplies the current state $st$ to the monad argument $m$. This |
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application yields a value result of type $A$ and an updated state |
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$st'$. The result is supplied to the continuation $k$, which |
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produces another state accepting function that gets applied to the |
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previously computed state value $st'$. |
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The state monad is equipped with two dual operations for accessing |
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and modifying the state encapsulated within the monad. |
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% |
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\[ |
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\ba{@{~}l@{\qquad\quad}@{~}r} |
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\multirow{2}{*}{ |
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\bl |
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\fail : A \to T~A\\ |
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\fail \defas \lambda x.\None |
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\el} |
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\getF : \UnitType \to T~S\\ |
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\getF~\Unit \defas \lambda st. \Record{st;st} |
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\el} & |
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\multirow{2}{*}{ |
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\bl |
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\putF : S \to T~\UnitType\\ |
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\putF~st \defas \lambda st'.\Record{\Unit;st} |
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\el} \\ & % space hack to avoid the next paragraph from |
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% floating into the math environment. |
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\ea |
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\] |
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% |
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Interactions between the two operations satisfy the following |
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equations~\cite{Gibbons12}. |
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% |
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\begin{reductions} |
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\slab{Get\textrm{-}put} & \getF\,\Unit \bind (\lambda st.\putF~st) &=& \Return\;\Unit\\ |
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\slab{Put\textrm{-}put} & \putF~st \bind (\lambda st.\putF~st') &=& \putF~st'\\ |
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\slab{Put\textrm{-}get} & \putF~st \bind (\lambda\Unit.\getF\,\Unit) &=& \putF~st \bind (\lambda \Unit.\Return\;st) |
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\end{reductions} |
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% |
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The first equation captures the intuition that getting a value and |
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then putting has no observable effect on the state cell. The second |
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equation states that only the latter of two consecutive puts is |
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observable. The third equation states that performing a get |
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immediately after putting a value is equivalent to returning that |
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value. |
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\end{definition} |
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\subsection{State monad} |
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The literature often presents the state monad with a fourth equation, |
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which states that $\getF$ is idempotent. However, this equation is |
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redundant as it is derivable from the first and third equations. |
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The following bit toggling function illustrates a use of the state |
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monad. |
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% |
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\[ |
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\bl |
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T~A \defas \Int \to (A \times \Int)\smallskip\\ |
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\incrEven : \UnitType \to T~\Bool\\ |
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\incrEven~\Unit \defas \getF |
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\bind (\lambda st. |
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\putF\,(1+st) |
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\bind \lambda\Unit. \Return\;(\even~st))) |
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\el |
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\] |
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% |
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We fix the state type of our monad to be the integer type. The type |
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signature of the function $\incrEven$ may be read as describing a |
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thunk that returns a boolean value, and whilst computing this boolean |
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value the function may perform any effectful operations given by the |
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monad $T$~\cite{Moggi91,Wadler92}, i.e. $\getF$ and |
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$\putF$. Operationally, the function retrieves the current value of |
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the state cell via the invocation of $\getF$. The bind operator passes |
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this value onto the continuation, which increments the value and |
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invokes $\putF$. The continuation applies a predicate |
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$\even : \Int \to \Bool$ to the original state value to test whether |
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its parity is even. The structure of the monad means that the result |
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of running this computation gives us a pair consisting of boolean |
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value indicating whether the initial state was even and the final |
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state value. For instance, if the initial state value is $\True$, then |
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the computation yields |
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% |
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\[ |
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\incrEven~\Unit~4 \reducesto^+ \Record{\True;5}, |
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\] |
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% |
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where the first component contains the initial state value and the |
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second component contains the final state value. |
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\subsection{Continuation monad} |
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\subsubsection{Continuation monad} |
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As in J.R.R. Tolkien's fictitious Middle-earth~\cite{Tolkien54} there |
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exists one monad to rule them all, one monad to realise them, one |
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monad to subsume them all, and in the term language bind them. This |
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@ -483,7 +623,7 @@ powerful monad is the \emph{continuation monad}. |
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\multirow{2}{*}{ |
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\bl |
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\Return : A \to T~A\\ |
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\Return \defas \lambda x.\lambda k.k~x |
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\Return~x\defas \lambda k.k~x |
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\el} & |
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\multirow{2}{*}{ |
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\bl |
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@ -502,26 +642,78 @@ $m$ to an anonymous function that accepts a value $x$ of type |
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$A$. This value $x$ is supplied to the immediate continuation $k$ |
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alongside the current continuation $f$. |
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Scheme's undelimited control operator $\Callcc$ is definable as a |
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monadic operation on the continuation monad~\cite{Wadler92}. |
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If we instantiate $R = S \to A \times S$ for some type $S$ then we |
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can implement the state monad inside the continuation monad. |
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% |
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\[ |
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\ba{@{~}l@{\qquad\quad}@{~}r} |
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\multirow{2}{*}{ |
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\bl |
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\Callcc : ((A \to T~B) \to T~A) \to T~A\\ |
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\Callcc \defas \lambda f.\lambda k. f\,(\lambda x.\lambda.k'.k~x)~k |
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\el |
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\getF : \UnitType \to T~S\\ |
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\getF~\Unit \defas \lambda k.\lambda st.k~st~st |
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\el} & |
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\multirow{2}{*}{ |
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\bl |
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\putF : S \to T~\UnitType\\ |
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\putF~st \defas \lambda k.\lambda st'.k~\Unit~st |
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\el} \\ & % space hack to avoid the next paragraph from |
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% floating into the math environment. |
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\ea |
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\] |
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% |
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The $\getF$ operation takes as input a (binary) continuation $k$ of |
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type $S \to S \to A \times S$ and produces a state-accepting function |
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that applies the continuation to the given state $st$. The first |
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occurrence of $st$ is accessible to the caller of $\getF$, whilst the |
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second occurrence passes the value $st$ onto the next operation |
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invocation on the monad. The operation $\putF$ works in the same |
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way. The primary difference is that $\putF$ does not return the value |
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of the state cell; instead it returns simply the unit value $\Unit$. |
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% |
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Using this continuation monad we can interpret $\incrEven$. |
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% |
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\[ |
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\incrEven~\Unit~(\lambda x.\lambda st. \Record{x;st})~4 \reducesto^+ \Record{\True;5} |
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\] |
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% |
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The initial continuation $(\lambda x.\lambda st. \Record{x;st})$ |
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corresponds to the $\Return$ of the state monad. |
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% Scheme's undelimited control operator $\Callcc$ is definable as a |
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% monadic operation on the continuation monad~\cite{Wadler92}. |
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% % |
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% \[ |
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% \bl |
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% \Callcc : ((A \to T~B) \to T~A) \to T~A\\ |
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% \Callcc \defas \lambda f.\lambda k. f\,(\lambda x.\lambda.k'.k~x)~k |
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% \el |
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% \] |
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\subsection{Free monad} |
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\subsubsection{Free monad} |
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% |
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\begin{figure} |
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\centering |
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\compTreeEx |
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\caption{Computation tree for $\incrEven$.}\label{fig:comptree} |
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\end{figure} |
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% |
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Just like other monads the free monad satisfies the monad laws, |
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however, unlike other monads the free monad does not perform any |
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computation \emph{per se}. Instead the free monad builds an abstract |
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representation of the computation in form of a computation tree, whose |
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interior nodes correspond to an invocation of some operation on the |
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monad, where each outgoing edge correspond to a possible continuation |
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of the operation; the leaves correspond to return values. The meaning |
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of a free monadic computation is ascribed by a separate function, or |
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interpreter, that traverses the computation tree. |
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of the operation; the leaves correspond to return |
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values. Figure~\ref{fig:comptree} depicts the computation tree for the |
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$\incrEven$ function. This particular computation tree has infinite |
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width, because the operation $\getF$ has infinitely many possible |
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continuations (we take the denotation of $\Int$ to be |
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$\mathbb{Z}$). Contrary, each $\putF$ node has only one outgoing edge, |
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because $\putF$ has only a single possible continuation, namely, the |
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trivial continuation $\Unit$. |
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The meaning of a free monadic computation is ascribed by a separate |
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function, or interpreter, that traverses the computation tree. |
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The shape of computation trees is captured by the following generic |
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type definition. |
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@ -549,11 +741,23 @@ operation is another computation tree node. |
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% |
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\[ |
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\bl |
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T~A \defas \Free~F~A \smallskip\\ |
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\ba{@{~}l@{\qquad}@{~}r} |
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\multicolumn{2}{l}{T~A \defas \Free~F~A} \smallskip\\ |
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\multirow{2}{*}{ |
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\bl |
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\Return : A \to T~A\\ |
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\Return \defas \lambda x.\return~x\\ |
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\Return~x \defas \return~x |
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\el} & |
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\multirow{2}{*}{ |
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\bl |
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\bind ~: T~A \to (A \to T~B) \to T~B\\ |
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\bind ~\defas \lambda m.\lambda k.\Case\;m\;\{\return~x \mapsto k~x;\OpF~y \mapsto \OpF\,(\fmap\,(\lambda m'. m' \bind k)\,y)\} |
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\bind ~\defas \lambda m.\lambda k.\Case\;m\;\{ |
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\bl |
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\return~x \mapsto k~x;\\ |
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\OpF~y \mapsto \OpF\,(\fmap\,(\lambda m'. m' \bind k)\,y)\} |
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\el |
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\el}\\ & \\ & |
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\ea |
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\el |
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\] |
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% |
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@ -563,7 +767,7 @@ operation is another computation tree node. |
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\[ |
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\bl |
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\DoF : F~A \to \Free~F~A\\ |
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\DoF \defas \lambda op.\OpF\,(\fmap\,(\lambda x.\return~x)\,op) |
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\DoF~op \defas \OpF\,(\fmap\,(\lambda x.\return~x)\,op) |
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\el |
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\] |
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% |
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@ -571,21 +775,52 @@ operation is another computation tree node. |
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\[ |
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\bl |
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\dec{FreeState}~S~R \defas [\Get:S \to R|\Put:\Record{S;R}|\Done] \smallskip\\ |
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\dec{FreeState}~S~R \defas [\Get:S \to R|\Put:\Record{S;R}] \smallskip\\ |
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\fmap~f~x \defas \Case\;x\;\{ |
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\bl |
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\Get~k \mapsto \Get\,(\lambda st.f\,(k~st));\\ |
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\Put~st'~k \mapsto \Put\,\Record{st';f~k};\\ |
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\Done \mapsto \Done\} |
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\el \smallskip\\ |
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\Put~st'~k \mapsto \Put\,\Record{st';f~k}\} |
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\el |
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\el |
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\] |
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% |
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\[ |
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\ba{@{~}l@{\qquad\quad}@{~}r} |
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\multirow{2}{*}{ |
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\bl |
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\getF : \UnitType \to T~S\\ |
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\getF~\Unit \defas \DoF~(\Get\,(\lambda x.x)))\\ |
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\getF~\Unit \defas \DoF~(\Get\,(\lambda x.x)) |
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\el} & |
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\multirow{2}{*}{ |
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\bl |
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\putF : S \to T~\UnitType\\ |
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\putF~st \defas \DoF~(\Put\,\Record{st;\Unit}) |
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\el} \\ & % space hack to avoid the next paragraph from |
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% floating into the math environment. |
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\ea |
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\] |
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% |
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\[ |
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\bl |
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\runState : S \to \Free\,(\dec{FreeState}~S)\,A \to A \times S\\ |
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\runState~st~m \defas |
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\Case\;m\;\{ |
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\ba[t]{@{~}l@{~}c@{~}l} |
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\return~x &\mapsto& (x, st);\\ |
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\OpF\,(\Get~k) &\mapsto& \runState~st~(k~st);\\ |
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\OpF\,(\Put\,\Record{st';k}) &\mapsto& \runState~st'~k |
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\}\ea |
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\el |
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\] |
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% |
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\[ |
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\runState~4~(\incrEven~\Unit) \reducesto^+ \Record{\True;5} |
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\] |
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\subsection{Back to direct-style} |
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\subsection{Direct-style revolution} |
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\subsubsection{Effect tracking} |
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\section{Contributions} |
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