Binding

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Funcons-beta : Binding.cbs | PLAIN | PDF

OUTLINE

Binding

Binding

\[\begin{align*} [ \ \KEY{Type} \quad & \NAMEREF{environments} \\ \KEY{Alias} \quad & \NAMEREF{envs} \\ \KEY{Datatype} \quad & \NAMEREF{identifiers} \\ \KEY{Alias} \quad & \NAMEREF{ids} \\ \KEY{Funcon} \quad & \NAMEREF{identifier-tagged} \\ \KEY{Alias} \quad & \NAMEREF{id-tagged} \\ \KEY{Funcon} \quad & \NAMEREF{fresh-identifier} \\ \KEY{Entity} \quad & \NAMEREF{environment} \\ \KEY{Alias} \quad & \NAMEREF{env} \\ \KEY{Funcon} \quad & \NAMEREF{initialise-binding} \\ \KEY{Funcon} \quad & \NAMEREF{bind-value} \\ \KEY{Alias} \quad & \NAMEREF{bind} \\ \KEY{Funcon} \quad & \NAMEREF{unbind} \\ \KEY{Funcon} \quad & \NAMEREF{bound-directly} \\ \KEY{Funcon} \quad & \NAMEREF{bound-value} \\ \KEY{Alias} \quad & \NAMEREF{bound} \\ \KEY{Funcon} \quad & \NAMEREF{closed} \\ \KEY{Funcon} \quad & \NAMEREF{scope} \\ \KEY{Funcon} \quad & \NAMEREF{accumulate} \\ \KEY{Funcon} \quad & \NAMEREF{collateral} \\ \KEY{Funcon} \quad & \NAMEREF{bind-recursively} \\ \KEY{Funcon} \quad & \NAMEREF{recursive} \ ] \end{align*}\] \[\begin{align*} \KEY{Meta-variables} \quad & \VAR{T} <: \NAMEHYPER{../../../Values}{Value-Types}{values} \end{align*}\]

Environments

\[\begin{align*} \KEY{Type} \quad & \NAMEDECL{environments} \leadsto \NAMEHYPER{../../../Values/Composite}{Maps}{maps} ( \NAMEREF{identifiers}, \NAMEHYPER{../../../Values}{Value-Types}{values}\QUERY ) \\ \KEY{Alias} \quad & \NAMEDECL{envs} = \NAMEREF{environments} \end{align*}\]

An environment represents bindings of identifiers to values. Mapping an identifier to $\SHADE{( \ )}$ represents that its binding is hidden.

Circularity in environments (due to recursive bindings) is represented using bindings to cut-points called $\SHADE{\NAMEHYPER{../.}{Linking}{links}}$. Funcons are provided for making declarations recursive and for referring to bound values without explicit mention of links, so their existence can generally be ignored.

\[\begin{align*} \KEY{Datatype} \quad \NAMEDECL{identifiers} \ ::= \ & \{ \_ : \NAMEHYPER{../../../Values/Composite}{Strings}{strings} \} \mid \NAMEDECL{identifier-tagged}( \_ : \NAMEREF{identifiers}, \_ : \NAMEHYPER{../../../Values}{Value-Types}{values}) \end{align*}\] \[\begin{align*} \KEY{Alias} \quad & \NAMEDECL{ids} = \NAMEREF{identifiers} \\ \KEY{Alias} \quad & \NAMEDECL{id-tagged} = \NAMEREF{identifier-tagged} \end{align*}\]

An identifier is either a string of characters, or an identifier tagged with some value (e.g., with the identifier of a namespace).

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{fresh-identifier} : \TO \NAMEREF{identifiers} \end{align*}\]

$\SHADE{\NAMEREF{fresh-identifier}}$ computes an identifier distinct from all previously computed identifiers.

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{fresh-identifier} \leadsto \NAMEREF{identifier-tagged} ( \STRING{generated}, \NAMEHYPER{../.}{Generating}{fresh-atom} ) \end{align*}\]

Current bindings

\[\begin{align*} \KEY{Entity} \quad & \NAMEDECL{environment}(\_ : \NAMEREF{environments}) \vdash \_ \TRANS \_ \end{align*}\] \[\begin{align*} \KEY{Alias} \quad & \NAMEDECL{env} = \NAMEREF{environment} \end{align*}\]

The environment entity allows a computation to refer to the current bindings of identifiers to values.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{initialise-binding}( \VAR{X} : \TO \VAR{T}) : \TO \VAR{T} \\&\quad \leadsto \NAMEHYPER{../.}{Linking}{initialise-linking} ( \NAMEHYPER{../.}{Generating}{initialise-generating} ( \NAMEREF{closed} ( \VAR{X} ) ) ) \end{align*}\]

$\SHADE{\NAMEREF{initialise-binding} ( \VAR{X} )}$ ensures that $\SHADE{\VAR{X}}$ does not depend on non-local bindings. It also ensures that the linking entity (used to represent potentially cyclic bindings) and the generating entity (for creating fresh identifiers) are initialised.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{bind-value}( \VAR{I} : \NAMEREF{identifiers}, \VAR{V} : \NAMEHYPER{../../../Values}{Value-Types}{values}) : \TO \NAMEREF{environments} \\&\quad \leadsto \{ \VAR{I} \mapsto \VAR{V} \} \\ \KEY{Alias} \quad & \NAMEDECL{bind} = \NAMEREF{bind-value} \end{align*}\]

$\SHADE{\NAMEREF{bind-value} ( \VAR{I}, \VAR{X} )}$ computes the environment that binds only $\SHADE{\VAR{I}}$ to the value computed by $\SHADE{\VAR{X}}$.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{unbind}( \VAR{I} : \NAMEREF{identifiers}) : \TO \NAMEREF{environments} \\&\quad \leadsto \{ \VAR{I} \mapsto ( \ ) \} \end{align*}\]

$\SHADE{\NAMEREF{unbind} ( \VAR{I} )}$ computes the environment that hides the binding of $\SHADE{\VAR{I}}$.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{bound-directly}( \_ : \NAMEREF{identifiers}) : \TO \NAMEHYPER{../../../Values}{Value-Types}{values} \end{align*}\]

$\SHADE{\NAMEREF{bound-directly} ( \VAR{I} )}$ returns the value to which $\SHADE{\VAR{I}}$ is currently bound, if any, and otherwise fails.

$\SHADE{\NAMEREF{bound-directly} ( \VAR{I} )}$ does not follow links. It is used only in connection with recursively-bound values when references are not encapsulated in abstractions.

\[\begin{align*} \KEY{Rule} \quad & \RULE{ & \NAMEHYPER{../../../Values/Composite}{Maps}{lookup} ( \VAR{$\rho$}, \VAR{I} ) \leadsto ( \VAR{V} : \NAMEHYPER{../../../Values}{Value-Types}{values} ) }{ & \NAMEREF{environment} ( \VAR{$\rho$} ) \vdash \NAMEREF{bound-directly} ( \VAR{I} : \NAMEREF{identifiers} ) \TRANS \VAR{V} } \\ \KEY{Rule} \quad & \RULE{ & \NAMEHYPER{../../../Values/Composite}{Maps}{lookup} ( \VAR{$\rho$}, \VAR{I} ) \leadsto ( \ ) }{ & \NAMEREF{environment} ( \VAR{$\rho$} ) \vdash \NAMEREF{bound-directly} ( \VAR{I} : \NAMEREF{identifiers} ) \TRANS \NAMEHYPER{../../Abnormal}{Failing}{fail} } \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{bound-value}( \VAR{I} : \NAMEREF{identifiers}) : \TO \NAMEHYPER{../../../Values}{Value-Types}{values} \\&\quad \leadsto \NAMEHYPER{../.}{Linking}{follow-if-link} ( \NAMEREF{bound-directly} ( \VAR{I} ) ) \\ \KEY{Alias} \quad & \NAMEDECL{bound} = \NAMEREF{bound-value} \end{align*}\]

$\SHADE{\NAMEREF{bound-value} ( \VAR{I} )}$ inspects the value to which $\SHADE{\VAR{I}}$ is currently bound, if any, and otherwise fails. If the value is a link, $\SHADE{\NAMEREF{bound-value} ( \VAR{I} )}$ returns the value obtained by following the link, if any, and otherwise fails. If the inspected value is not a link, $\SHADE{\NAMEREF{bound-value} ( \VAR{I} )}$ returns it.

$\SHADE{\NAMEREF{bound-value} ( \VAR{I} )}$ is used for references to non-recursive bindings and to recursively-bound values when references are encapsulated in abstractions.

Scope

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{closed}( \VAR{X} : \TO \VAR{T}) : \TO \VAR{T} \end{align*}\]

$\SHADE{\NAMEREF{closed} ( \VAR{X} )}$ ensures that $\SHADE{\VAR{X}}$ does not depend on non-local bindings.

\[\begin{align*} \KEY{Rule} \quad & \RULE{ & \NAMEREF{environment} ( \NAMEHYPER{../../../Values/Composite}{Maps}{map} ( \ ) ) \vdash \VAR{X} \TRANS \VAR{X}' }{ & \NAMEREF{environment} ( \_ ) \vdash \NAMEREF{closed} ( \VAR{X} ) \TRANS \NAMEREF{closed} ( \VAR{X}' ) } \\ \KEY{Rule} \quad & \NAMEREF{closed} ( \VAR{V} : \VAR{T} ) \leadsto \VAR{V} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{scope}( \_ : \NAMEREF{environments}, \_ : \TO \VAR{T}) : \TO \VAR{T} \end{align*}\]

$\SHADE{\NAMEREF{scope} ( \VAR{D}, \VAR{X} )}$ executes $\SHADE{\VAR{D}}$ with the current bindings, to compute an environment $\SHADE{\VAR{$\rho$}}$ representing local bindings. It then executes $\SHADE{\VAR{X}}$ to compute the result, with the current bindings extended by $\SHADE{\VAR{$\rho$}}$, which may shadow or hide previous bindings.

$\SHADE{\NAMEREF{closed} ( \NAMEREF{scope} ( \VAR{$\rho$}, \VAR{X} ) )}$ ensures that $\SHADE{\VAR{X}}$ can reference only the bindings provided by $\SHADE{\VAR{$\rho$}}$.

\[\begin{align*} \KEY{Rule} \quad & \RULE{ & \NAMEREF{environment} ( \NAMEHYPER{../../../Values/Composite}{Maps}{map-override} ( \VAR{$\rho$}\SUB{1}, \VAR{$\rho$}\SUB{0} ) ) \vdash \VAR{X} \TRANS \VAR{X}' }{ & \NAMEREF{environment} ( \VAR{$\rho$}\SUB{0} ) \vdash \NAMEREF{scope} ( \VAR{$\rho$}\SUB{1} : \NAMEREF{environments}, \VAR{X} ) \TRANS \NAMEREF{scope} ( \VAR{$\rho$}\SUB{1}, \VAR{X}' ) } \\ \KEY{Rule} \quad & \NAMEREF{scope} ( \_ : \NAMEREF{environments}, \VAR{V} : \VAR{T} ) \leadsto \VAR{V} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{accumulate}( \_ : ( \TO \NAMEREF{environments} )\STAR) : \TO \NAMEREF{environments} \end{align*}\]

$\SHADE{\NAMEREF{accumulate} ( \VAR{D}\SUB{1}, \VAR{D}\SUB{2} )}$ executes $\SHADE{\VAR{D}\SUB{1}}$ with the current bindings, to compute an environment $\SHADE{\VAR{$\rho$}\SUB{1}}$ representing some local bindings. It then executes $\SHADE{\VAR{D}\SUB{2}}$ to compute an environment $\SHADE{\VAR{$\rho$}\SUB{2}}$ representing further local bindings, with the current bindings extended by $\SHADE{\VAR{$\rho$}\SUB{1}}$, which may shadow or hide previous current bindings. The result is $\SHADE{\VAR{$\rho$}\SUB{1}}$ extended by $\SHADE{\VAR{$\rho$}\SUB{2}}$, which may shadow or hide the bindings of $\SHADE{\VAR{$\rho$}\SUB{1}}$.

$\SHADE{\NAMEREF{accumulate} ( \_, \_ )}$ is associative, with $\SHADE{\NAMEHYPER{../../../Values/Composite}{Maps}{map} ( \ )}$ as unit, and extends to any number of arguments.

\[\begin{align*} \KEY{Rule} \quad & \RULE{ & \VAR{D}\SUB{1} \TRANS \VAR{D}\SUB{1}' }{ & \NAMEREF{accumulate} ( \VAR{D}\SUB{1}, \VAR{D}\SUB{2} ) \TRANS \NAMEREF{accumulate} ( \VAR{D}\SUB{1}', \VAR{D}\SUB{2} ) } \\ \KEY{Rule} \quad & \NAMEREF{accumulate} ( \VAR{$\rho$}\SUB{1} : \NAMEREF{environments}, \VAR{D}\SUB{2} ) \leadsto \NAMEREF{scope} ( \VAR{$\rho$}\SUB{1}, \NAMEHYPER{../../../Values/Composite}{Maps}{map-override} ( \VAR{D}\SUB{2}, \VAR{$\rho$}\SUB{1} ) ) \\ \KEY{Rule} \quad & \NAMEREF{accumulate} ( \ ) \leadsto \NAMEHYPER{../../../Values/Composite}{Maps}{map} ( \ ) \\ \KEY{Rule} \quad & \NAMEREF{accumulate} ( \VAR{D}\SUB{1} ) \leadsto \VAR{D}\SUB{1} \\ \KEY{Rule} \quad & \NAMEREF{accumulate} ( \VAR{D}\SUB{1}, \VAR{D}\SUB{2}, \VAR{D}\PLUS ) \leadsto \NAMEREF{accumulate} ( \VAR{D}\SUB{1}, \NAMEREF{accumulate} ( \VAR{D}\SUB{2}, \VAR{D}\PLUS ) ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{collateral}( \VAR{$\rho$}\STAR : \NAMEREF{environments}\STAR) : \TO \NAMEREF{environments} \\&\quad \leadsto \NAMEHYPER{../../Abnormal}{Failing}{checked} \ \NAMEHYPER{../../../Values/Composite}{Maps}{map-unite} ( \VAR{$\rho$}\STAR ) \end{align*}\]

$\SHADE{\NAMEREF{collateral} ( \VAR{D}\SUB{1}, \cdots )}$ pre-evaluates its arguments with the current bindings, and unites the resulting maps, which fails if the domains are not pairwise disjoint.

$\SHADE{\NAMEREF{collateral} ( \VAR{D}\SUB{1}, \VAR{D}\SUB{2} )}$ is associative and commutative with $\SHADE{\NAMEHYPER{../../../Values/Composite}{Maps}{map} ( \ )}$ as unit, and extends to any number of arguments.

Recurse

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{bind-recursively}( \VAR{I} : \NAMEREF{identifiers}, \VAR{E} : \TO \NAMEHYPER{../../../Values}{Value-Types}{values}) : \TO \NAMEREF{environments} \\&\quad \leadsto \NAMEREF{recursive} ( \{ \VAR{I} \}, \NAMEREF{bind-value} ( \VAR{I}, \VAR{E} ) ) \end{align*}\]

$\SHADE{\NAMEREF{bind-recursively} ( \VAR{I}, \VAR{E} )}$ binds $\SHADE{\VAR{I}}$ to a link that refers to the value of $\SHADE{\VAR{E}}$, representing a recursive binding of $\SHADE{\VAR{I}}$ to the value of $\SHADE{\VAR{E}}$. Since $\SHADE{\NAMEREF{bound-value} ( \VAR{I} )}$ follows links, it should not be executed during the evaluation of $\SHADE{\VAR{E}}$.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{recursive}( \VAR{SI} : \NAMEHYPER{../../../Values/Composite}{Sets}{sets} ( \NAMEREF{identifiers} ), \VAR{D} : \TO \NAMEREF{environments}) : \TO \NAMEREF{environments} \\&\quad \leadsto \NAMEREF{re-close} ( \NAMEREF{bind-to-forward-links} ( \VAR{SI} ), \VAR{D} ) \end{align*}\]

$\SHADE{\NAMEREF{recursive} ( \VAR{SI}, \VAR{D} )}$ executes $\SHADE{\VAR{D}}$ with potential recursion on the bindings of the identifiers in the set $\SHADE{\VAR{SI}}$ (which need not be the same as the set of identifiers bound by $\SHADE{\VAR{D}}$).

\[\begin{align*} \KEY{Auxiliary Funcon} \quad & \NAMEDECL{re-close}( \VAR{M} : \NAMEHYPER{../../../Values/Composite}{Maps}{maps} ( \NAMEREF{identifiers}, \NAMEHYPER{../.}{Linking}{links} ), \VAR{D} : \TO \NAMEREF{environments}) : \TO \NAMEREF{environments} \\&\quad \leadsto \NAMEREF{accumulate} ( \NAMEREF{scope} ( \VAR{M}, \VAR{D} ), \NAMEHYPER{../.}{Flowing}{sequential} ( \NAMEREF{set-forward-links} ( \VAR{M} ), \NAMEHYPER{../../../Values/Composite}{Maps}{map} ( \ ) ) ) \end{align*}\]

$\SHADE{\NAMEREF{re-close} ( \VAR{M}, \VAR{D} )}$ first executes $\SHADE{\VAR{D}}$ in the scope $\SHADE{\VAR{M}}$, which maps identifiers to freshly allocated links. This computes an environment $\SHADE{\VAR{$\rho$}}$ where the bound values may contain links, or implicit references to links in abstraction values. It then sets the link for each identifier in the domain of $\SHADE{\VAR{M}}$ to refer to its bound value in $\SHADE{\VAR{$\rho$}}$, and returns $\SHADE{\VAR{$\rho$}}$ as the result.

\[\begin{align*} \KEY{Auxiliary Funcon} \quad & \NAMEDECL{bind-to-forward-links}( \VAR{SI} : \NAMEHYPER{../../../Values/Composite}{Sets}{sets} ( \NAMEREF{identifiers} )) : \TO \NAMEHYPER{../../../Values/Composite}{Maps}{maps} ( \NAMEREF{identifiers}, \NAMEHYPER{../.}{Linking}{links} ) \\&\quad \leadsto \NAMEHYPER{../../../Values/Composite}{Maps}{map-unite} ( \\&\quad\quad\quad\quad \NAMEHYPER{../.}{Giving}{interleave-map} ( \\&\quad\quad\quad\quad\quad \NAMEREF{bind-value} ( \NAMEHYPER{../.}{Giving}{given}, \NAMEHYPER{../.}{Linking}{fresh-link} ( \NAMEHYPER{../../../Values}{Value-Types}{values} ) ), \\&\quad\quad\quad\quad\quad \NAMEHYPER{../../../Values/Composite}{Sets}{set-elements} ( \VAR{SI} ) ) ) \end{align*}\]

$\SHADE{\NAMEREF{bind-to-forward-links} ( \VAR{SI} )}$ binds each identifier in the set $\SHADE{\VAR{SI}}$ to a freshly allocated link.

\[\begin{align*} \KEY{Auxiliary Funcon} \quad & \NAMEDECL{set-forward-links}( \VAR{M} : \NAMEHYPER{../../../Values/Composite}{Maps}{maps} ( \NAMEREF{identifiers}, \NAMEHYPER{../.}{Linking}{links} )) : \TO \NAMEHYPER{../../../Values/Primitive}{Null}{null-type} \\&\quad \leadsto \NAMEHYPER{../.}{Flowing}{effect} ( \\&\quad\quad\quad\quad \NAMEHYPER{../.}{Giving}{interleave-map} ( \\&\quad\quad\quad\quad\quad \NAMEHYPER{../.}{Linking}{set-link} ( \NAMEHYPER{../../../Values/Composite}{Maps}{map-lookup} ( \VAR{M}, \NAMEHYPER{../.}{Giving}{given} ), \NAMEREF{bound-value} ( \NAMEHYPER{../.}{Giving}{given} ) ), \\&\quad\quad\quad\quad\quad \NAMEHYPER{../../../Values/Composite}{Sets}{set-elements} ( \NAMEHYPER{../../../Values/Composite}{Maps}{map-domain} ( \VAR{M} ) ) ) ) \end{align*}\]

For each identifier $\SHADE{\VAR{I}}$ in the domain of $\SHADE{\VAR{M}}$, $\SHADE{\NAMEREF{set-forward-links} ( \VAR{M} )}$ sets the link to which $\SHADE{\VAR{I}}$ is mapped by $\SHADE{\VAR{M}}$ to the current bound value of $\SHADE{\VAR{I}}$.

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