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Funcons-beta : Trees.cbs | PLAIN | PDF


Trees

\[\begin{align*} [ \ \KEY{Datatype} \quad & \NAMEREF{trees} \\ \KEY{Funcon} \quad & \NAMEREF{tree} \\ \KEY{Funcon} \quad & \NAMEREF{tree-root-value} \\ \KEY{Funcon} \quad & \NAMEREF{tree-branch-sequence} \\ \KEY{Funcon} \quad & \NAMEREF{single-branching-sequence} \\ \KEY{Funcon} \quad & \NAMEREF{forest-root-value-sequence} \\ \KEY{Funcon} \quad & \NAMEREF{forest-branch-sequence} \\ \KEY{Funcon} \quad & \NAMEREF{forest-value-sequence} \ ] \end{align*}\] \[\begin{align*} \KEY{Meta-variables} \quad & \VAR{T} <: \NAMEHYPER{../..}{Value-Types}{values} \end{align*}\] \[\begin{align*} \KEY{Datatype} \quad \NAMEDECL{trees}( \VAR{T} ) \ ::= \ & \NAMEDECL{tree}( \_ : \VAR{T}, \_ : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR) \end{align*}\]

\(\SHADE{\NAMEREF{trees} ( \VAR{T} )}\) consists of finitely-branching trees with elements of type \(\SHADE{\VAR{T}}\). When \(\SHADE{\VAR{V} : \VAR{T}}\), \(\SHADE{\NAMEREF{tree} ( \VAR{V} )}\) is a leaf, and \(\SHADE{\NAMEREF{tree} ( \VAR{V}, \VAR{B}\SUB{1}, \cdots, \VAR{B}\SUB{n} )}\) is a tree with branches \(\SHADE{\VAR{B}\SUB{1}}\), …, \(\SHADE{\VAR{B}\SUB{n}}\).

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{tree-root-value}( \_ : \NAMEREF{trees} ( \VAR{T} )) : \TO ( \VAR{T} )\QUERY \\ \KEY{Rule} \quad & \NAMEREF{tree-root-value} \ \NAMEREF{tree} ( \VAR{V} : \VAR{T}, \_\STAR : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR ) \leadsto \VAR{V} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{tree-branch-sequence}( \_ : \NAMEREF{trees} ( \VAR{T} )) : \TO ( \NAMEREF{trees} ( \VAR{T} ) )\STAR \\ \KEY{Rule} \quad & \NAMEREF{tree-branch-sequence} \ \NAMEREF{tree} ( \_ : \VAR{T}, \VAR{B}\STAR : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR ) \leadsto \VAR{B}\STAR \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{single-branching-sequence}( \_ : \NAMEREF{trees} ( \VAR{T} )) : \TO \VAR{T}\PLUS \end{align*}\]

\(\SHADE{\NAMEREF{single-branching-sequence} \ \VAR{B}}\) extracts the values in \(\SHADE{\VAR{B}}\) starting from the root, provided that \(\SHADE{\VAR{B}}\) is at most single-branching; otherwise it fails.

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{single-branching-sequence} \ \NAMEREF{tree} ( \VAR{V} : \VAR{T} ) \leadsto \VAR{V} \\ \KEY{Rule} \quad & \NAMEREF{single-branching-sequence} \ \NAMEREF{tree} ( \VAR{V} : \VAR{T}, \VAR{B} : \NAMEREF{trees} ( \VAR{T} ) ) \leadsto \\&\quad \NAMEHYPER{../../../Computations/Normal}{Flowing}{left-to-right} ( \VAR{V}, \NAMEREF{single-branching-sequence} \ \VAR{B} ) \\ \KEY{Rule} \quad & \NAMEREF{single-branching-sequence} \ \NAMEREF{tree} ( \_ : \VAR{T}, \_ : \NAMEREF{trees} ( \VAR{T} ), \_\PLUS : ( \NAMEREF{trees} ( \VAR{T} ) )\PLUS ) \leadsto \NAMEHYPER{../../../Computations/Abnormal}{Failing}{fail} \end{align*}\]

A sequence of trees corresponds to a forest, and the selector funcons on trees \(\SHADE{\VAR{B}}\) extend to forests \(\SHADE{\VAR{B}\STAR}\):

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{forest-root-value-sequence}( \_ : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR) : \TO \VAR{T}\STAR \\ \KEY{Rule} \quad & \NAMEREF{forest-root-value-sequence} ( \VAR{B} : \NAMEREF{trees} ( \VAR{T} ), \VAR{B}\STAR : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR ) \leadsto \\&\quad ( \NAMEREF{tree-root-value} \ \VAR{B}, \NAMEREF{forest-root-value-sequence} \ \VAR{B}\STAR ) \\ \KEY{Rule} \quad & \NAMEREF{forest-root-value-sequence} ( \ ) \leadsto ( \ ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{forest-branch-sequence}( \_ : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR) : \TO \VAR{T}\STAR \\ \KEY{Rule} \quad & \NAMEREF{forest-branch-sequence} ( \VAR{B} : \NAMEREF{trees} ( \VAR{T} ), \VAR{B}\STAR : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR ) \leadsto \\&\quad ( \NAMEREF{tree-branch-sequence} \ \VAR{B}, \NAMEREF{forest-branch-sequence} \ \VAR{B}\STAR ) \\ \KEY{Rule} \quad & \NAMEREF{forest-branch-sequence} ( \ ) \leadsto ( \ ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{forest-value-sequence}( \_ : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR) : \TO \VAR{T}\STAR \end{align*}\]

\(\SHADE{\NAMEREF{forest-value-sequence} \ \VAR{B}\STAR}\) provides the values from a left-to-right pre-order depth-first traversal.

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{forest-value-sequence} ( \NAMEREF{tree} ( \VAR{V} : \VAR{T}, \VAR{B}\SUB{1}\STAR : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR ), \VAR{B}\SUB{2}\STAR : ( \NAMEREF{trees} ( \VAR{T} ) )\STAR ) \leadsto \\&\quad ( \VAR{V}, \NAMEREF{forest-value-sequence} \ \VAR{B}\SUB{1}\STAR, \NAMEREF{forest-value-sequence} \ \VAR{B}\SUB{2}\STAR ) \\ \KEY{Rule} \quad & \NAMEREF{forest-value-sequence} ( \ ) \leadsto ( \ ) \end{align*}\]

Other linearizations of trees can be added: breadth-first, right-to-left, C3, etc.