### Revision(s):

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Revision #2 to TR05-051 | 25th August 2005 00:00
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#### ``On Complexity of Counting Fixed Point Configurations in Certain Classes of Graph Automata ''

**Abstract:**
We study computational complexity of counting the fixed point configurations (FPs) in certain discrete dynamical systems. We prove that counting FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) is computationally intractable, even when each node is required to update according to a symmetric Boolean function. We also show that the problems of counting the garden of Eden configurations (GEs), as well as all transient configurations, are just as hard in that setting. Moreover, the hardness of enumerating FPs holds even in some severely restricted cases, such as when the nodes of an SDS or SyDS use only two different symmetric Boolean update rules, and when each nodehas a neighborhood size bounded by a small constant.

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Revision #1 to TR05-051 | 10th May 2005 00:00
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#### On Complexity of Counting Fixed Point Configurations in Certain Classes of Graph Automata

**Abstract:**
We study computational complexity of counting the fixed point

configurations (FPs) in certain discrete dynamical systems. We

prove that counting FPs in Sequential and Synchronous Dynamical

Systems (SDSs and SyDSs, respectively) is computationally

intractable, even when each node is required to update according

to a symmetric Boolean function. We also show that counting the

garden of Eden configurations (GEs), as well as all transient

configurations, is just as hard in this setting. Moreover, the

hardness of enumerating FPs holds even in some severely restricted

cases, such as when the nodes of an SDS or SyDS use only two

different symmetric Boolean update rules, and when each node has a

neighborhood size bounded by a small constant.

### Paper:

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TR05-051 | 18th March 2005 00:00
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#### On Complexity of Counting Fixed Points in Certain Classes of Graph Automata

**Abstract:**
We study the computational complexity of counting the fixed point configurations in certain discrete dynamical systems. We prove that both exact and approximate counting in Sequential and Synchronous Dynamical Systems (SDSs and SyDS, respectrively) is computationally intractable, even when each node is required to update according to a symmetric Boolean function. We also show that the exact counting of Garden of Eden configurations, as well as of all transient configurations, is just as hard in this setting. Moreover, the hardness of counting holds even in the severely restricted cases, such as when the nodes of a symmetric SDS or SyDS use only two different update rules, and when each node has a neighborhood size bounded by a small constant.