I posted this previously about Andrew Wuensche's and Mike Lesser's gorgeous coffee table book, "The Global Dynamics of Cellular Automata" (I've updated the links here), which uses a "general method for running CA backwards in time to compute a state's predecessors with a direct reverse algorithm":

https://news.ycombinator.com/item?id=14468707

There's a thing called a "Garden of Eden" configuration that has no predecessors, which is impossible to get to from any other possible state.
For a rule like Life, there are many possible configurations that must have been created by God or somebody with a bitmap editor (or somebody who thinks he's God and uses Mathematica as a bitmap editor, like Stephen Wolfram ;), because it would have been impossible for the Life rule to evolve into those states. For example, with the "Life" rule, no possible configuration of cells could ever evolve into all cells with the value "1".

https://en.wikipedia.org/wiki/Garden_of_Eden_(cellular_autom...

For a rule that simply sets the cell value to zero, all configurations other than pure zeros are garden of eden states, and they all lead directly into a one step attractor of all zeros which always evolves back into itself, all zeros again and again (the shortest possible attractor loop that leads directly to itself).

There is a way of graphically visualizing that global rule state space, which gives insight into the behavior of the rule and the texture and complexity of its state space!

Andrew Wuensche and Mike Lesser published a gorgeous coffee table book entitled "The Global Dynamics of Cellular Automata" that plots out the possible "Garden of Eden" states and the "Basins of Attraction" they lead into of many different one-dimensional cellular automata like rule 30.

https://web.archive.org/web/20180228130331/https://uncomp.uw...

The beautiful color plates begin on page 79 in the free pdf file:

https://donhopkins.com/home/global_dynamics_of_CA.pdf

I've uploaded the money shots to imgur:

http://imgur.com/gallery/s3dhz

Those are not pictures of 1-d cellular automata rule cell states on a grid themselves, but they are actually graphs of the abstract global state space, showing merging and looping trajectories (but not branching since the rules are deterministic -- time flows from the garden of eden leaf tips around the perimeter into (then around) the basin of attractor loops in the center, merging like springs (GOE) into tributaries into rivers into the ocean (BOA)).

The rest of the book is an atlas of all possible 1-d rules of a particular rule numbering system (like rule 30, etc), and the last image is the legend.

He developed a technique of computing and plotting the topology network of all possible states a CA can get into -- tips are "garden of eden" states that no other states can lead to, and loops are "basins of attraction".

Here is the illustration of "rule 30" from page 144 (the legend explaining it is the last photo in the above album). [I am presuming it's using the same rule numbering system as Wolfram but I'm not sure -- EDIT: I visually checked the "space time pattern from a singleton seed" thumbnail against the illustration in the article, and yes it matches rule 30!]

http://imgur.com/a/lKAbP

"The Global Dynamics of Cellular Automata introduces a powerful new perspective for the study of discrete dynamical systems. After first looking at the unique trajectory of a system's future, an algoritm is presented that directly computes the multiple merging trajectories of the systems past. A given cellular automaton will "crystallize" state space into a set of basins of attraction that typically have a topology of trees rooted on attractor cycles. Portraits of these objects are made accessible through computer generated graphics. The "Atlas" presents a complete class of such objects, and is inteded , with the accompanying software, as an aid to navigation into the vast reaches of rule behaviour space. The book will appeal to students and researchers interested in cellular automata, complex systems, computational theory, artificial life, neural networks, and aspects of genetics."

https://en.wikipedia.org/wiki/Attractor

"Basins of attraction in cellular automata", by Andrew Wuensche:

http://onlinelibrary.wiley.com/doi/10.1002/1099-0526(200007/...

"To achieve the global perspective. I devised a general method for running CA backwards in time to compute a state's predecessors with a direct reverse algorithm. So the predecessors of predecessors, and so on, can be computed, revealing the complete subtree including the "leaves," states without predecessors, the so-called “garden-of-Eden" states.

Trajectories must lead to attractors in a finite CA, so a basin of attraction is composed of merging trajectories, trees, rooted on the states making up the attractor cycle with a period of one or more. State-space is organized by the "physics" underlying the dynamic behavior into a number of these basins of attraction, making up the basin of attraction field."

If you like the book, you'll love the code!

http://www.ddlab.com/

http://www.ddlab.com/screensave3.png

https://web.archive.org/web/20160304115404/https://uncomp.uw...

https://web.archive.org/web/20190823063636/http://uncomp.uwe...

https://web.archive.org/web/20190821013401/http://uncomp.uwe...

https://web.archive.org/web/20190717234849/http://uncomp.uwe...