# 1D Totalistic Cellular Automaton

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## WHAT IS IT?

This program is a one-dimensional totalistic cellular automata. In a totalistic CA, the value of the next cell state is determined by the sum of the current cell and its neighbors, not by the values of each individual neighbor. The model allows you to explore the behavior of random totalistic CAs.

This model is intended for the more sophisticated users who are already familiar with basic 1D CA's. If you are exploring CA for the first time, we suggest you first look at one of the simpler CA models such as CA 1D Rule 30.

## HOW IT WORKS

Each cell may have one of several colors with the values 0 to STATES - 1. The next state of a cell is determined by taking the sum value of the center and the neighbors on each side (as determined by RADIUS). This sum is used as an index into a state-transition table, the "rule," which defines the new state of that cell.

## HOW TO USE IT

STATES: Defines the number of states of each cell.
RADIUS: Defines the radius on both sides of a cell used to define its new state.

SET RANDOM SEED: By setting the random seed you can repeat experiments.
RANDOM RULE: Generates a random transition rule, with all states being equally likely. The rule is displayed below (RULE CODE, which shows the new state for each neighborhood total), and its ENTROPY and LAMBDA parameters are computed.
ENTER RULE: This allows you to enter a rule as a list of state values.
QUIESCENCE: If this is turned on (the usual case), then the quiescent (0) state will be forced to map into the quiescent state. If it is not set, then the quiescent state is permitted to map into any state.
DECIMATE: Zeros one of the non-zero entries in the rule, thus causing that neighborhood sum to map into the quiescent state.

SETUP SINGLE: Sets up a single color-two cell centered in the top row.
SETUP RANDOM: Sets up cells of uniformly random colors across the top row.
INPUT INITIAL STATE: Sets up cells of specified state values/colors near center of top row.
AUTO-CONTINUE?: Automatically continue the CA from the top once it reaches the bottom row.
GO: Run the CA. If GO is clicked again after a run, the run continues from the top.

START TEST: Clears the output area, creates a random rule, and generates a random initial state for a decimation run.
RUN TEST: Equivalent to GO, i.e., runs the CA as above. It is possible to reset the initial state (e.g., randomly or to specified values) and RUN TEST again.
CLASSIFY & DECIMATE: Based on the test run, the user types in a string descibing the behavior (e.g., "IV" or "II (long transient)"). The classification, parameters (lambda etc.), and the rule are written in the output area. The rule is automatically decimated and the initial state randomized in preparation for another RUN TEST. At the end of a decimation run (when the rule is all zeros), the output area can be copied and pasted into a text file when running under NetLogo (but not as an applet). Note that you will be alternating between RUN TEST and CLASSIFY & DECIMATE.

OPEN RECORD-FILE: Open a file to receive the record of a decimation run (exactly the same information displayed in the output area, described above). You are requested to enter a filename or path. Note that you will have to have write access to directory from which this program is running or to the path. The alternative is to copy and past from the Output area, as described above.
CLOSE RECORD-FILE: Close the record-file and write it to disk. If you open a record-file then a previously opened record-file will be closed automatically. Note, however, that if you quit the program without closing the record file, you will lose the file's contents!

## THINGS TO NOTICE

How does the complexity of the multicolor totalistic CA differ from the two-color CA? (see the CA 1D Elementary model)

Do most rules lead to constantly repeating patterns, nesting, randomness, or more complex behavior (Wolfram Class IV)? What does this tell you about the nature of complexity?

## THINGS TO TRY

Observe the behavior of a rule under different initial conditions (single point or random initial state). Do different random initial states affect its qualitative behavior.

Start with a random rule and observe its behavior. Then decimate the rule, pick a new random initial state, and observe again. Continue progressively decimating the rule and look for changes in behavior (e.g., different Wolfram classes). Note if qualitative changes of behavior happen at particular values of the LAMBDA or ENTROPY parameters.

Do this decimation experiment with a number of random rules to see if you can determine which parameter best predicts the CAs qualitative behavior.

Explore the effects of different numbers of states and different neighborhood sizes on the CA's qualitative behavior. What conditions seem to be necessary for complex (Class IV) behavior to emerge?

## EXTENDING THE MODEL

Try making a two-dimensional cellular automaton. The neighborhood could be the eight cells around it, or just the cardinal cells (the cells to the right, left, above, and below).

## RELATED MODELS

Life - an example of a two-dimensional cellular automaton
CA 1D Rule 30 - the basic rule 30 model
CA 1D Rule 30 Turtle - the basic rule 30 model implemented using turtles
CA 1D Rule 90 - the basic rule 90 model
CA 1D Rule 250 - the basic rule 250 model
CA 1D Elementary - a simple one-dimensional 2-state cellular automata model
CA 1D Totalistic - a simple one-dimensional 3-state, unit-radius CA model
CA Continuous - a totalistic continuous-valued cellular automata with thousands of states

## CREDITS AND REFERENCES

Thanks to Ethan Bakshy for his help with this model.

The first cellular automaton was conceived by John Von Neumann in the late 1940's for his analysis of machine reproduction under the suggestion of Stanislaw M. Ulam. It was later completed and documented by Arthur W. Burks in the 1960's. Other two-dimensional cellular automata, and particularly the game of "Life," were explored by John Conway in the 1970's. Many others have since researched CA's. In the late 1970's and 1980's Chris Langton, Tom Toffoli and Stephen Wolfram did some notable research. Wolfram classified all 256 one-dimensional two-state single-neighbor cellular automata. In his recent book, "A New Kind of Science," Wolfram presents many examples of cellular automata and argues for their fundamental importance in doing science.

Von Neumann, J. and Burks, A. W., Eds, 1966. Theory of Self-Reproducing Automata. University of Illinois Press, Champaign, IL.

Toffoli, T. 1977. Computation and construction universality of reversible cellular automata. J. Comput. Syst. Sci. 15, 213-231.

Langton, C. 1984. Self-reproduction in cellular automata. Physica D 10, 134-144

Langton, C. 1990. Computation at the Edge of Chaos: Phase Transitions and Emergent Computation. In Emergent Computation, ed. Stephanie Forrest. North-Holland.

Langton, C. 1992. Life at the Edge of Chaos. In Artificial Life II, ed. Langton et al. Addison-Wesley.

Wolfram, S. 1986. Theory and Applications of Cellular Automata: Including Selected Papers 1983-1986. World Scientific Publishing Co., Inc., River Edge, NJ.

Bar-Yam, Y. 1997. Dynamics of Complex Systems. Perseus Press. Reading, Ma.

Wolfram, S. 2002. A New Kind of Science. Wolfram Media Inc. Champaign, IL.

This model was modified 2007-08-24 by Bruce MacLennan from Uri Wilensky's CA 1D Totalistic model to compute entropy and lambda values, and to allow antering rules and decimation, setting the random seed, specification of initial state, and control and recording of decimation runs.

To refer to the original model in academic publications, please use: Wilensky, U. (2002). NetLogo CA 1D Totalistic model. http://ccl.northwestern.edu/netlogo/models/CA1DTotalistic. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.