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Compositional closure

Compositional closure of a quadruple (A,B,C,d) means that there are 2 constants and 2 binary operators as follows. The binary operators $\star$ (binding, or tree-forest composition) and $\frown$(concatenation, or path-extension composition) are written infix.

\begin{displaymath}\providecommand{\foo}[2]{
\begin{array}[t]{@{}l@{}}
\underl...
...a,b,c),\phi(c)) : C(a, b\star\phi)
\end{array} \\
\end{array}\end{displaymath}

The following laws should hold:

\begin{displaymath}\begin{array}{l}
d(a,\ensuremath{1_B} ,\ensuremath{1_C} ) = ...
...
d(a,b\star\phi,c \frown c') = d(d(a,b,c),c') \\
\end{array}\end{displaymath}

The remaining laws are directly analogous to the laws of a category:

\begin{displaymath}\begin{array}{ll}
\ensuremath{1_B}\star \phi = \phi(\ensurem...
...(c \frown c') \frown c'' = c \frown (c' \frown c'')
\end{array}\end{displaymath}

The two `star' constructions yield quadruples which are compositionally closed. The definitions of the concatenation and binding operators are as follows, by recursion of the first argument:

\begin{displaymath}\begin{array}{l}
\ensuremath{\ensuremath{\mathit{Nil}} \frown...
...hop{\mathit{Cons}(c,\ensuremath{\mathit{cs}})}})}))}\end{array}\end{displaymath}



Peter Hancock
1998-04-08