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OUTLINE

• Binding
  • Environments
  • Current bindings
  • Scope
  • Recurse

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Binding

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

Environments

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

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

Circularity in environments (due to recursive bindings) is represented using bindings to cut-points called \(\FUNHYP{../.}{Linking}{links}\). 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.

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

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

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

\(\FUNREF{fresh-identifier}\) computes an identifier distinct from all previously computed identifiers.

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

Current bindings

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

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

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

\(\FUNREF{initialise-binding} ( \VAR{X} )\) ensures that \(\VAR{X}\) 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.

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

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

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

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

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

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

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

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

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

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

Scope

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

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

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

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

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

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

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

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

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

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

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

Recurse

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

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

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

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

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

\(\FUNREF{re-close} ( \VAR{M}, \VAR{D} )\) first executes \(\VAR{D}\) in the scope \(\VAR{M}\), which maps identifiers to freshly allocated links. This computes an environment \(\VAR{\rho}\) 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 \(\VAR{M}\) to refer to its bound value in \(\VAR{\rho}\), and returns \(\VAR{\rho}\) as the result.

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

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

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

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

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