Link Search Menu Expand Document
CBS-beta
\( % cbs-katex.sty % \newcommand{\STYLE}[2]{\htmlClass{cbs-#1}{#2}} \newcommand{\DECL}[3]{\htmlId{#1:#2}{#3}} \newcommand{\REF}[3]{\href{###1:#2}{#3}} \newcommand{\HYPER}[5]{\href{#1/#2/index.html###3:#4}{#5}} % \SHADE{MATH} can be defined to produce a shaded background to highlight % inline MATH in running text: \newcommand{\SHADE}[1]{#1} % \KEY{TEXT}, \STRING{TEXT}, \ATOM{TEXT}, \LEX{TEXT} can be used in math mode: \newcommand{\KEY}[1]{\textsf{\textit{\STYLE{Key}{#1}}}} \newcommand{\STRING}[1]{\textsf{``\texttt{#1}''}} \newcommand{\ATOM}[1]{\textsf{`\texttt{#1}'}} \newcommand{\LEX}[1]{\textsf{\STYLE{Key}{`}\texttt{#1}\STYLE{Key}{'}}} % The following commands produce ASCII characters that are treated specially by LaTeX: \newcommand{\HASH}{\char`\#} \newcommand{\DOLLAR}{\char`\$} \newcommand{\PERCENT}{\char`\%} \newcommand{\AMPERSAND}{\char`\&} \newcommand{\APOSTROPHE}{\char`\'} \newcommand{\BACKSLASH}{\char`\\} \newcommand{\CARET}{\char`\^} \newcommand{\UNDERSCORE}{\char`\_} \newcommand{\GRAVE}{\char`\`} \newcommand{\LEFTBRACE}{\char`\{} \newcommand{\RIGHTBRACE}{\char`\}} \newcommand{\TILDE}{\textasciitilde} % {\char`\~} % \NAME{name} highlights the name; % \NAMEDECL{name} declares Name.name as the target of links to name; % \NAMEREF{name} links name to the target Name.name in the current file; % \NAMEHYPER{url}{file}{name} links name to Name.name at url/file/file.pdf. % Similarly for \VAR{partvariable}, \SYN{syntaxname}, \SEM{semanticsName}, % and \SECT{sectionnumber} % The kerns in \SUB and \VAR avoid overlaps with primes: \newcommand{\SUB}[1]{_{\kern-2mu\STYLE{PartVariable}{\textsf{#1}}}} % PLAIN \newcommand{\VAR}[1]{\STYLE{PartVariable}{\textsf{\textit{#1}\kern2mu}}} \newcommand{\NAME}[1]{\STYLE{Name}{\textsf{#1}}} \newcommand{\SYN}[1]{\STYLE{SyntaxName}{\textsf{#1}}} \newcommand{\SEM}[1]{\STYLE{SemanticsName}{\textsf{#1}}} \newcommand{\SECT}[1]{\STYLE{SectionNumber}{\textsf{#1}}} % DECL \newcommand{\VARDECL}[1]{\DECL{PartVariable}{#1}{\VAR{#1}}} \newcommand{\NAMEDECL}[1]{\DECL{Name}{#1}{\NAME{#1}}} \newcommand{\SYNDECL}[1]{\DECL{SyntaxName}{#1}{\SYN{#1}}} \newcommand{\SEMDECL}[1]{\DECL{SemanticsName}{#1}{\SEM{#1}}} \newcommand{\SECTDECL}[1]{\DECL{SectionNumber}{#1}{\textsf{#1}}} % REF \newcommand{\VARREF}[1]{\REF{PartVariable}{#1}{\VAR{#1}}} \newcommand{\NAMEREF}[1]{\REF{Name}{#1}{\NAME{#1}}} \newcommand{\SYNREF}[1]{\REF{SyntaxName}{#1}{\SYN{#1}}} \newcommand{\SEMREF}[1]{\REF{SemanticsName}{#1}{\SEM{#1}}} \newcommand{\SECTREF}[1]{\REF{SectionNumber}{#1}{\SECT{#1}}} % HYPER \newcommand{\VARHYPER}[3]{\HYPER{#1}{#2}{PartVariable}{#3}{\VAR{#3}}} \newcommand{\NAMEHYPER}[3]{\HYPER{#1}{#2}{Name}{#3}{\NAME{#3}}} \newcommand{\SYNHYPER}[3]{\HYPER{#1}{#2}{SyntaxName}{#3}{\SYN{#3}}} \newcommand{\SEMHYPER}[3]{\HYPER{#1}{#2}{SemanticsName}{#3}{\SEM{#3}}} \newcommand{\SECTHYPER}[3]{\HYPER{#1}{#2}{SectionNumber}{#3}{\SECT{#3}}} % \LEFTPHRASE MATH \RIGHTPHRASE produces [[ MATH ]] with proper brackets: \newcommand{\LEFTPHRASE}{\llbracket} \newcommand{\RIGHTPHRASE}{\rrbracket} % \LEFTGROUP MATH \RIGHTGROUP produces ( MATH ) where the parentheses are % highlighted the same as keywords: \newcommand{\LEFTGROUP}{\STYLE{Key}{(}} \newcommand{\RIGHTGROUP}{\STYLE{Key}{)}} % MATH\PLUS produces a superscript + % MATH\STAR produces a superscript * % MATH\QUERY produces a superscript ? \newcommand{\PLUS}{{}^{\texttt{+}}} \newcommand{\STAR}{{}^{\texttt{*}}} \newcommand{\QUERY}{{}^{\texttt{?}}} % \RULE{& PREMISE \\ & ...}{& FORMULA ... \\ & ...} produces an inference rule % with separately aligned premises and conclusion % PREMISE % ... % ----------- % FORMULA ... % ... \newcommand{\RULE}[2] {\frac{\begin{aligned}#1\end{aligned}}{\begin{aligned}#2\end{aligned}}} % \AXIOM{& FORMULA ... \\ & ...} produces an aligned formula % % FORMULA ... % ... \newcommand{\AXIOM}[1]{\begin{aligned}#1\end{aligned}} % \TO TYPE produces => TYPE \newcommand{\TO}{\mathop{\Rightarrow}} % TERM \TRANS TERM produces TERM ---> TERM \newcommand{\TRANS}{\longrightarrow} % TERM \xrightarrow{LABEL} TERM puts the label above the long arrow % \)

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.

From the PLanCompS Project CBS-beta issues… Suggest an improvement…

← ↑ →