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Accessible part of a relation

The accessible part of a relation R on a set A is a certain subset of A. It consists of firstly of the objects with no immediate predecessors, i.e. initial objects; next the objects whose immediate predecessors are all initial; next the objects whose immediate predecessors are all the preceding kinds, and so on. More precisely, it is the least subset X of A which contains a : A whenever it includes the set $\{ \; b : A \; \vert \; b R a \; \}$ of all immediate predecessors of a. The relation is well-founded if the whole set A is accessible.

When the relation R is given by a triple, the immediate predecessors of an element a:A are given as an indexed set: $\{\;c(a,b) \; \vert \; b : B(a,b) \; \}$. One can define a family of sets of well-founded trees ( $\ensuremath{\mathit{Acc}} (a)$ for a:A) , the set $\ensuremath{\mathit{Acc}} (a)$ representing the proposition that a is accessible, as the least solution (with respect to pointwise inclusion) of

\begin{displaymath}\forall a : A . \ensuremath{\mathit{Acc}} (a)
= \{ \; \ensur...
...\phi : \Pi b : B(a) . \ensuremath{\mathit{Acc}} (c(a,b)) \; \}
\end{displaymath}

where $\ensuremath{\mathop{\mathit{mkAcc}(\makebox[1em]{\_})}} $ is a constructor. If a:A, then an element of $\ensuremath{\mathit{Acc}} (a)$ can be thought of as a proof that a belongs to the accessible, or well-founded part of the relation represented by the triple.

A slight generalisation of this construction leads to the notion of the least accessible closure of a subset of A. First we define a family of trees which may involve leaves.

\begin{displaymath}\forall a : A . \ensuremath{\mathit{AccT}} (a)
= \begin{arra...
...\\
\cup & \{ \; \ensuremath{\mathit{mkAcc0}}\; \}
\end{array}\end{displaymath}

Next for each such tree we define the set of chains in it which start at the root and finish at a leaf:

\begin{displaymath}\begin{array}{l}
\forall a : A, b : \ensuremath{\mathit{AccT}...
...t{AccC}} (c(a,b),\phi(b)) \}
\end{array}\end{array}\end{array}\end{displaymath}

Next for each such chain we define its terminal point:

\begin{displaymath}\begin{array}{lcl}
\ensuremath{\mathit{AccCt}} (a,\ensuremat...
... \ensuremath{\mathit{AccCt}} (c(a,b), \phi(b), bs)
\end{array}\end{displaymath}

Not sure where following concepts belong.

Now the accessible closure of a family of sets $P(a):\ensuremath{\mathit{Set}} $ indexed by a:A is the family:

\begin{displaymath}\overline{P}(a:A) = \Sigma_{ b : \ensuremath{\mathit{AccT}} (...
...thit{AccC}} (a,b) } .
P(\ensuremath{\mathit{AccCt}} (a,b,c))
\end{displaymath}

Property that a is barred by P - but note that a machine may be forced to stop without attaining a state with property P. This idea is close to eventuality.

The following dual notion (I don't know a common name for it) suggests itself. Again, given a family of sets $P(a):\ensuremath{\mathit{Set}} $ indexed by a:A, we define the family:

\begin{displaymath}\underline{P}(a:A) = \Pi_{b : \ensuremath{\mathit{AccT}} (a)}...
...athit{AccC}} (a,b)} .
P(\ensuremath{\mathit{AccCt}} (a,b,c))
\end{displaymath}

This idea is close to always (or potential infinity). Starting in a state a such that $\underline{P}(a)$, we can always find a path to an a' such that P(a'). Note the similarity with the maximality condition for a property of neighbourhoods to be a point: for all covering trees of any neighbourhood with the property P, there is a neighbourhood in that cover which includes one with the property P. (In the topological interpretation, the property is restricted to be monotone so, equivalently: for all covering trees of any neighbourhood with the property, there is a neighbourhood in that cover with the property.)


next up previous
Next: About this document ... Up: What about triples? Previous: Free category on a
Peter Hancock
1998-04-08