Abduction, unpredictability and Garden of Eden

Kolmogorov Complexity and the Garden of Eden Theorem

Suppose τ is a cellular automaton over an amenable group and a finite alphabet. Celebrated Garden of Eden theorem states, that pre-injectivity of τ is equivalent to non-existence of Garden of Eden configuration. In this paper we will prove, that imposing some mild restrictions , we could add another equivalent assertion: non-existence of Garden of Eden configuration is equivalent to preservatio...

The Garden of Eden theorem: old and new

We review topics in the theory of cellular automata and dynamical systems that are related to the Moore-Myhill Garden of Eden theorem.

Constrained Eden

Let CA be a one-dimensional cellular automaton. A computational complexity problem, the Constrained Eden problem, denoted C-EDEN(CA), is introduced. C-EDEN(CA) is a finitary variant of the Garden of Eden problem. Even for certain elementary cellular automata, C-EDEN(CA) is NP-complete, providing the first examples of NP-complete problems associated with cellular automata over a twoelement alpha...

Some Extensions of the Moore-Myhill Garden of Eden Theorem

Shift-Induced Dynamical Systems on Partitions and Compositions

The rules of “Bulgarian solitaire” are considered as an operation on the set of partitions to induce a finite dynamical system. We focus on partitions with no preimage under this operation, known as Garden of Eden points, and their relation to the partitions that are in cycles. These are the partitions of interest, as we show that starting from the Garden of Eden points leads through the entire...