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Funcons-beta : Functions.cbs | PLAIN | PDF

Functions

\[\begin{align*} [ \ \KEY{Datatype} \quad & \NAMEREF{functions} \\ \KEY{Funcon} \quad & \NAMEREF{function} \\ \KEY{Funcon} \quad & \NAMEREF{apply} \\ \KEY{Funcon} \quad & \NAMEREF{supply} \\ \KEY{Funcon} \quad & \NAMEREF{compose} \\ \KEY{Funcon} \quad & \NAMEREF{uncurry} \\ \KEY{Funcon} \quad & \NAMEREF{curry} \\ \KEY{Funcon} \quad & \NAMEREF{partial-apply} \ ] \end{align*}\] \[\begin{align*} \KEY{Meta-variables} \quad & \VAR{T}, \VAR{T}', \VAR{T}\SUB{1}, \VAR{T}\SUB{2} <: \NAMEHYPER{../..}{Value-Types}{values} \end{align*}\] \[\begin{align*} \KEY{Datatype} \quad \NAMEDECL{functions}( \VAR{T} , \VAR{T}' ) \ ::= \ & \NAMEDECL{function}( \VAR{A} : \NAMEHYPER{../.}{Generic}{abstractions} ( \VAR{T} \TO \VAR{T}' )) \end{align*}\]

\(\SHADE{\NAMEREF{functions} ( \VAR{T}, \VAR{T}' )}\) consists of abstractions whose bodies may depend on a given value of type \(\SHADE{\VAR{T}}\), and whose executions normally compute values of type \(\SHADE{\VAR{T}'}\). \(\SHADE{\NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \VAR{X} ) )}\) evaluates to a function with dynamic bindings, \(\SHADE{\NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{closure} ( \VAR{X} ) )}\) computes a function with static bindings.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{apply}( \_ : \NAMEREF{functions} ( \VAR{T}, \VAR{T}' ), \_ : \VAR{T}) : \TO \VAR{T}' \end{align*}\]

\(\SHADE{\NAMEREF{apply} ( \VAR{F}, \VAR{V} )}\) applies the function \(\SHADE{\VAR{F}}\) to the argument value \(\SHADE{\VAR{V}}\). This corresponds to call by value; using thunks as argument values corresponds to call by name. Moreover, using tuples as argument values corresponds to application to multiple arguments.

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{apply} ( \NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \VAR{X} ) ), \VAR{V} : \VAR{T} ) \leadsto \NAMEHYPER{../../../Computations/Normal}{Giving}{give} ( \VAR{V}, \VAR{X} ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{supply}( \_ : \NAMEREF{functions} ( \VAR{T}, \VAR{T}' ), \_ : \VAR{T}) : \TO \NAMEHYPER{../.}{Thunks}{thunks} ( \VAR{T}' ) \end{align*}\]

\(\SHADE{\NAMEREF{supply} ( \VAR{F}, \VAR{V} )}\) determines the argument value of a function application, but returns a thunk that defers executing the body of the function.

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{supply} ( \NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \VAR{X} ) ), \VAR{V} : \VAR{T} ) \leadsto \NAMEHYPER{../.}{Thunks}{thunk} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{give} ( \VAR{V}, \VAR{X} ) ) ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{compose}( \_ : \NAMEREF{functions} ( \VAR{T}\SUB{2}, \VAR{T}' ), \_ : \NAMEREF{functions} ( \VAR{T}\SUB{1}, \VAR{T}\SUB{2} )) : \TO \NAMEREF{functions} ( \VAR{T}\SUB{1}, \VAR{T}' ) \end{align*}\]

\(\SHADE{\NAMEREF{compose} ( \VAR{F}\SUB{2}, \VAR{F}\SUB{1} )}\) returns the function that applies \(\SHADE{\VAR{F}\SUB{1}}\) to its argument, then applies \(\SHADE{\VAR{F}\SUB{2}}\) to the result of \(\SHADE{\VAR{F}\SUB{1}}\).

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{compose} ( \NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \VAR{Y} ) ), \NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \VAR{X} ) ) ) \leadsto \\&\quad \NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{give} ( \VAR{X}, \VAR{Y} ) ) ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{uncurry}( \VAR{F} : \NAMEREF{functions} ( \VAR{T}\SUB{1}, \NAMEREF{functions} ( \VAR{T}\SUB{2}, \VAR{T}' ) )) \\&\quad : \TO \NAMEREF{functions} ( \NAMEHYPER{../../Composite}{Tuples}{tuples} ( \VAR{T}\SUB{1}, \VAR{T}\SUB{2} ), \VAR{T}' ) \\&\quad \leadsto \NAMEREF{function} ( \\&\quad\quad\quad\quad \NAMEHYPER{../.}{Generic}{abstraction} ( \\&\quad\quad\quad\quad\quad \NAMEREF{apply} ( \\&\quad\quad\quad\quad\quad\quad \NAMEREF{apply} ( \VAR{F}, \NAMEHYPER{../../../Computations/Abnormal}{Failing}{checked} \ \NAMEHYPER{../../Composite}{Sequences}{index} ( 1, \NAMEHYPER{../../Composite}{Tuples}{tuple-elements} \ \NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ), \\&\quad\quad\quad\quad\quad\quad \NAMEHYPER{../../../Computations/Abnormal}{Failing}{checked} \ \NAMEHYPER{../../Composite}{Sequences}{index} ( 2, \NAMEHYPER{../../Composite}{Tuples}{tuple-elements} \ \NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{uncurry} ( \VAR{F} )}\) takes a curried function \(\SHADE{\VAR{F}}\) and returns a function that takes a pair of arguments..

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{curry}( \VAR{F} : \NAMEREF{functions} ( \NAMEHYPER{../../Composite}{Tuples}{tuples} ( \VAR{T}\SUB{1}, \VAR{T}\SUB{2} ), \VAR{T}' )) : \TO \NAMEREF{functions} ( \VAR{T}\SUB{1}, \NAMEREF{functions} ( \VAR{T}\SUB{2}, \VAR{T}' ) ) \\&\quad \leadsto \NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEREF{partial-apply} ( \VAR{F}, \NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{curry} ( \VAR{F} )}\) takes a function \(\SHADE{\VAR{F}}\) that takes a pair of arguments, and returns the corresponding ‘curried’ function.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{partial-apply}( \VAR{F} : \NAMEREF{functions} ( \NAMEHYPER{../../Composite}{Tuples}{tuples} ( \VAR{T}\SUB{1}, \VAR{T}\SUB{2} ), \VAR{T}' ), \VAR{V} : \VAR{T}\SUB{1}) : \TO \NAMEREF{functions} ( \VAR{T}\SUB{2}, \VAR{T}' ) \\&\quad \leadsto \NAMEREF{function} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEREF{apply} ( \VAR{F}, \NAMEHYPER{../../Composite}{Tuples}{tuple} ( \VAR{V}, \NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{partial-apply} ( \VAR{F}, \VAR{V} )}\) takes a function \(\SHADE{\VAR{F}}\) that takes a pair of arguments, and determines the first argument, returning a function of the second argument.

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