State Transition Diagram for Multiple Chained L.D.A.s

Three instances of the state transition diagram which applies to an individual l.d.a.  are presented side by side to suggest how a series of l.d.a.s may be chained together through their extensions to form Collatz itineraries consisting of a number of l.d.a.s connected up through extensions.

The graph for an individual l.d.a.  has been modified to bring its header nodes  (denoted A and B)  to the forefront and to give it a slimmer appearance.  This causes the graph to appear non-planar,  but it is the same old graph.  The colors of nodes and edges have the same meanings as that earlier graph.

Recall that every node in the graph has two potential outgoing steps not specifically pictured.  Any node may produce a leaf node as the next predecessor and every node has an extension.  We may regard this graph as a road map for tracing out the path to any integer through l.d.a.s and extensions.   If the integer is not a leaf node,  the path terminates somewhere inside an l.d.a.  If the integer is a leaf node,  the path will fall out of an l.d.a.  through an edge to the leaf node not pictured in the graphs.  Imagine a brown edge out of every node leading to a leaf node.  The brown brackets at the bottom serve to collect all such paths. These collectors can only operate when a leaf node is the terminating element of the predecessor path.

Similarly,  you may imagine there is a bright blue connection pointing rightwards from every node to its extension,  and that all these connections to extensions are collected by the bright blue bracket at the right of each instance of the l.d.a.  state transition graph.  The extensions will be from leaves  (collected by the brown bracket)  or from internal nodes to the A or the B header node of the next l.d.a.

An example has been prepared which follows the passage of a path segment through three l.d.a.s.  The points to note are that l.d.a.s are not used in their entirety in the path segments which use them , that progress to the nextd l.d.a. can occur from internal node's l.d.a.s or through leaf node's l.d.a.s,  and that the usage of available edges within l.d.a.s within l.d.a.s is relatively sparse  (even though the example uses two relatively long l.d.a.s).

This diagram serves to further clarify the relationship between l.d.a.s,  extensions,  and the larger Collatz itineraries.  It depicts a portion of the state transition diagram for the predecessor graph viewed from a vantage which includes all these components.


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