PascGalois JE Help: 2-D Cellular Automata Counts/Period Tab
The Counts/Period Tab
If you wish to count the number of elements in a sequence of levels,
the number of elements in a sub-pyramid or frustum, or find the
period or death of a finite automaton this tab
may be of use to you. The calculations here are independent of those in the
image panel so you can count elements in sequences of levels that extend
beyond the number of levels you input in the options tab. Furthermore,
since the data is regenerated each time a count is done the system
does not need to store more levels than what is needed and hence one
can do counts on far more levels than would be possible to graph.
This tab also has more advanced options for finding the period or death of a
finite automaton.
At the top you will see a tab system that allows the user to select a counting mode.
The modes are level counting, sub-pyramid counting or frustum counting.
The Counting Modes
In the level counting mode at the top of the tab is the range of levels to be used.
This range is inclusive. In this mode the program will count all of the elements
on each level in the range.
In the sub-pyramid counting mode, pictured below, the user selects the range
of levels, an anchor position and the way that the counting region is increased
as you move to the next level.
The first entry in the level range is the anchor level, this is where the sub-pyramid
will start. As with the counting of the one-dimensional automata if the beginning
level is larger than the ending level the sub-pyramid will be pointing downward.
The user then selects the position of the "top" of the sub-pyramid. This can be
a little tricky because if you change the level range the position numbering of a point
may also change. For example, use the default 4-rule with a single seed element
and graph 20 levels. If you place the mouse over the element on the first level
in the image tab you will see that the position is row 20 and column 20. Now if
you change the number of levels to 30 and regraph the automaton the position
of this "top" point is row 30 and column 30. So care must be taken when selecting this
position. The position numbering system that is used in the counting is the same as the
numbering system used in the graphing routines. Furthermore, the numbering
that is used is the same as with the image of the largest level in the range of
levels that are counted. That is, if you are counting a sub-pyramid that is
between levels 23 and 47 the numbering that is used is exactly the same as if you
were to graph the first 47 levels of the automaton. The user should use the
graphs to get a feel for the numbering system before trying to count a large
pyramid without the aid of a graph.
The region increments allow the user to specify how the counting region
is increased as you progress to the next level. The region can also be expanded
as either a rectangle or a diamond. For example, say that the green "*" is at
row 10 and column 10. If we set all increments to 1 and use the rectangular
region the counting regions will be in blue.
Under the same setup but using the diamond regions our counting regions would be.
As with the one-dimensional automaton counts the increments can be 0 and
negative as well. For example, if you set the down to -1, keep the rest
of the increments as 1 and use a rectangular region you will count the elements
on one face of the pyramid.
In the frustum counting mode, pictured below, the user selects the range
of levels, an anchor rectangle and the way that the counting region is increased
as you move to the next level.
This mode works the same way that the sub-pyramid counting mode works except that
the increase in region is always rectangular and the anchor is a rectangular
region instead of a single cell. The rectangular region is selected by specifing
the upper left position and lower right position of the rectangle.
The Counts/Period Toolbar
The Counts/Period Toolbar consists of seven options (five for infinite automata).
- The first generates the data and displays the new counts in the table.
- The second and third save the count data to either a text file or a LaTeX
file.
- The fourth and fifth copy the counts to the clipboard as either text or LaTeX code.
- When the automaton is finite there will be a sixth option to find the period. The
program will start at the beginning level, memorize a sequence of levels (with depth the same as
the depth of the update rule) and search until the ending level for that sequence of levels.
If found, the program will return the period and if not it will notify you that a period could
not be found.
In general, what you want to do is set the beginning row fairly large, and then set the ending row very large.
This should increase your chances of finding the period.
- When the automaton is finite there will also be a seventh option to find the death
of the automaton. The calculation of the automaton death searches for a level (or set of n levels
if the depth of the update rule is n) that match the group identity. If a set of identity
levels are found the program will tell you where the death starts. That is, the last level number
of the first block of identity levels that are the same depth as the update rule.
The Element List Toolbar
The element list toolbar has four simple options.
- The first option refreshes the element list. Theoretically you should never need
this option but as with the image tab the calculations are done in separate threads
and if the element list for some reason does not appear this option will display it.
- The second option selects all of the elements in the list.
- The third option unselects all of the elements in the list.
- The fourth option selects the subgroup generated by the currently selected elements.
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