PascGalois Zn Help: 1-D Cellular Automata Counts/Period Screen


The Counts/Period Screen

If you wish to count the number of elements in a sequence of rows, a sub-triangle, a sub-trapezoid, or find the period or death of a finite automaton this screen 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 rows that extend beyond the number of rows you input in the viewer. Furthermore, since the data is regenerated each time a count is done the system does not need to store more rows than what is needed and hence one can do counts on far more rows than would be possible to graph. This screen also has more advanced options for finding the period or death of a finite automaton.

At the top of the screen is a tab system that allows the user to select the mode for counting. One mode is to simply count the elements on a sequence of rows and the other is for counting the number of elements in a sub-triangle or a sub-trapezoid. Directly below that is the counts table and element list, similar to the setup of the counts dialog box that is displayed when the display counts option is invoked from the image screen. The selected column is, of course, the number of the elements counted in the range of rows.

The Counting Mode

There are two counting modes, Row Counts and Sub-Triangle Counts. With the Row Counts mode, pictured above, the user inputs the beginning row and ending row to count, this range is inclusive. When the user selects the # button the program will display the element counts for those rows.

The Sub-Triangle Counts tab is pictured below.

In this mode the user selects an anchor (or starting point) and then tells the program how to increase (or decrease) the number of columns that are in the count on each subsequent row. This mode can be a little tricky and may take some time to get used to. We would suggest that the user try smaller examples and refer to the graph to get a feel for the positioning before doing large counts without the aid of the graph.

We will discuss the options in general and then look at a few examples. The position numbering for this mode is exactly the same as the numbering of the graphed image. Also, the Sub-Triangle title of this mode is really a misnomer, this mode will also count sub-trapezoids.

First, the user must select the range of rows. The first row is the anchor row, this is where the triangle or trapezoid starts. Note that the first row is not necessarily the smallest numbered row. If the user inputs rows 5 to 9 then the anchor row is row 5 but if the user inputs the rows 9 to 5 then the anchor row is row 9. Furthermore, in the 5 to 9 case the triangle or trapezoid will grow from row 5 down to row 9 and in the 9 to 5 case the triangle or trapezoid will grow from row 9 up to row 5, that is a triangle pointing down.

Second, the user selects a beginning column and an ending column for the anchor. This makes all of the cells on the beginning row from the beginning column to the ending column the anchor. These cells are also in the counts.

Finally, the user selects how the counts will be done on subsequent lines using the left and right updates. The way this works is that the beginning column of the next row will be the old beginning column moved to the left the number of cells in the left box and the ending column of the next row will be the old ending column moved to the right the number of cells in the right box. So if the current row is counting from column 2 to 5 and both the left and right are set to 1 then the next row will count cells from 1 to 6 and then the row after that will count columns 0 to 7 and so on. Note that if a cell is to be counted and it is outside the automaton it will simply be ignored.

For example, in the image below the cell marked with a green "*" is at row 4 and column 5.


If we set the row range from 4 to 6, the beginning and ending columns to 5 and 5, the left increment to 0 and the right increment to 1 we would count the region in blue below.


If we set the row range from 4 to 6, the beginning and ending columns to 5 and 5, the left increment to 1 and the right increment to 1 we would count the region in blue below.


If we set the row range from 4 to 6, the beginning and ending columns to 5 and 7, the left increment to 1 and the right increment to 1 we would count the region in blue below.


If we set the row range from 4 to 6, the beginning and ending columns to 5 and 5, the left increment to 1 and the right increment to 2 we would count the region in blue below.


Finally, if we set the row range from 4 to 7, the beginning and ending columns to 5 and 7, the left increment to 1 and the right increment to -1 we would count the region in blue below.


Note that you can also have the anchor at the bottom of the region and count up to the top by having the larger row value first. For example, if we set the row range from 4 to 2, the beginning and ending columns to 5 and 7, the left increment to 1 and the right increment to -1 we would count the region in blue below.


The Counts/Period Toolbar

The Counts/Period Toolbar consists of seven options (five for infinite automata).

The Element List Toolbar

The element list toolbar has four simple options.


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