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Partial constructions

Peter Hancock

Basically very old, with a few updates near the front

In Chapter 16 (``General Trees') in ``Programming in Martin-Löf's Type Theory'', Nordström. Petersson and Smith describe a inductive construction for certain indexed families of sets of well-founded trees. The whole family is defined by simultaneous induction, in such a way that the definition of one member of the family may depend on the definition of any other member. Indeed, part of the interest of the construction is that it is a paradigm for (i.e. sums up, distills, ..) a rather large class of inductive definitions in which there is mutual dependency. The construction, described in section 1 starts with a quadruple (A,B,C,d), where:

Such a structure is equivalent to a function which assigns to each element of the base set a family of families of elements of the base set, provided that ``family of A's'' is taken to mean ``function into A defined on some set''. Here is how these things are defined in (informal) constructive type theory:

\begin{displaymath}\begin{array}{lcl}
\mathit{Fam}\; (A : \mathsf{Type}) & = & ...
...mathit{Fam}\; (\mathit{Fam}\; A))^A : \mathsf{Type}
\end{array}\end{displaymath}

Quadruples form the objects of various categories.

These notions of structure and map derive their interest from their wide range of applications, notably to state machines 4, rule systems 5, and topology 6.

These pages collect together some notes about (A,B,C,d) structures, related notions, and their applications. They are rough, and should be read sceptically, if at all!


  
Figure 1: A detail from a picture of a quadruple. The boxes contain elements a of A. There is a exit line from a box for each element of B(a), and that line has a fan-out indexed by C(a,b). Taking the arrow indexed by c in that fanout leads to a box containing d(a,b,c). The detail omits all exit lines but b:B(a), and all entry arrows but c:C(a,b)
\begin{figure}\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}
(75,6...
...){\framebox (20,10)[c]{$d(a,b,c) : A$ }}
\end{picture}\end{center}
\end{figure}

What may be new in these notes, compared to the original Petersson-Synek concept is



 
next up previous
Next: Inductive construction
Peter Hancock
1998-04-08