Problem Calculus: Index

• Introduction
• Motivation
• Some Terminology
• Semantically Homogeneous vs Semantically Heterogeneous
• Diagrams and Invalid Elements
• Complexity Projection
• Arrow Complexity
• Paths and Non-Overlapping Arrow Spaces

Problem Calculus: Paths and Non-Overlapping Arrow Spaces

In addition to everything we've covered so far, you can also encounter families of spaces that have properties that make it complex.

The first thing to consider is when you have more than one path that gets you to the same destination space when starting from a given source space.

In the diagram above, we're starting with a starting element called "A".  Two arrows can take A and go to one space or another.  Finally, they arrive at a final space.  The question is:  Are the final elements equal to each other.  Ideally, you want the final two elements to be equal.  This property allows you to worry less about the spaces you're going to and the spaces you're coming from.  If this holds for all elements in the space, then you can consider the multiple paths as if they were the same and it is easier to form an intuition about the system.

Additionally, you have to be concerned about arrow spaces (input and output) that are non-overlapping in any given main space.  The issue is that it becomes difficult to know when the output of one arrow can be used for the input of another arrow.  Additionally, when trying to decide what paths exist in your system, you have to be concerned about whether the non-overlapping arrow spaces allow a path to exist.  And some paths will only sometimes exist if the arrow spaces are only partially overlapping.

Here are two arrow spaces where the incoming arrows form partial overlapping spaces.

Here are two arrow spaces where the outgoing arrows form partial overlapping spaces.

Finally, here is an incoming arrow forming a partial overlapping space with an outgoing arrow space.

All of those spaces are arrow input or output spaces that live inside of a source/destination space.  The outer space is being left out in the previous examples.

Consider the following diagram.

This diagram shows two possible paths that go through the featured space.  In one instance the arrows form partial overlapping arrow spaces.  This path will only sometimes work because some elements in the output space will be outside of the input space for the next arrow.  Additionally, one of the output spaces is completely non-overlapping with the input space of the exiting arrow.  Even though the arrow looks like it should produce values that can be used for the exiting arrow when you make a further analysis you'll discover that the elements will never be usable.