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**11 - 18**of**18**### On complexity of counting fixed point configurations in certain classes of graph automata

- ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 2005

"... 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 updat ..."

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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 node has a neighborhood size bounded by a small constant.

### On Convergence Properties of One-Dimensional Cellular Automata with Majority Cell Update Rule

"... Abstract — We are interested in simple cellular automata (CA) and their computational and dynamical properties. In our past and ongoing work, we have been investigating (i) asymptotic dynamics of various types of CA and (ii) different communication models for CA. In this paper, we specifically focus ..."

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Abstract — We are interested in simple cellular automata (CA) and their computational and dynamical properties. In our past and ongoing work, we have been investigating (i) asymptotic dynamics of various types of CA and (ii) different communication models for CA. In this paper, we specifically focus on the convergence properties of a very simple kind of totalistic CA, namely, those defined on one-dimensional arrays where each cell or node updates according to the Boolean Majority function: the new state of a cell becomes 1 if and only if a simple majority of its inputs are currently in state 1, and it becomes 0 otherwise. We have observed in our prior work that such CA tend to have relatively simple asymptotic dynamics: a short transient chain followed by convergence to a “fixed point”. We now provide solid statistical evidence for these conjectures, based on our recent extensive computer simulations of Majority 1-D CA. In particular, we study the convergence properties of such CA for two communication models: one is the classical, parallel CA model with perfectly synchronous cell updates, and the other are CA whose cells update sequentially, one at a time; we consider two variants of such sequential update regimes. We simulate CA whose sizes range up to 1,000 cells, and demonstrate very fast (in particular, sublinear), and very slowly decreasing with an increase in the total number of cells, speeds of convergence. Finally, we draw conclusions based on our extensive simulations and outline some interesting questions to be considered in the future work.

### Modeling and Analysis of the Collective Dynamics of Large-Scale Multi-Agent Systems: A Cellular and Network Automata based Approach

, 2006

"... This technical report addresses a particular approach to modeling and analysis of the behavior of large-scale multi-agent systems. A broad variety of multi-agent systems are modeled as appropriate variants of cellular and network automata. Several fundamental properties of the collective dynamics of ..."

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This technical report addresses a particular approach to modeling and analysis of the behavior of large-scale multi-agent systems. A broad variety of multi-agent systems are modeled as appropriate variants of cellular and network automata. Several fundamental properties of the collective dynamics of those cellular and network automata are then formally analyzed. Various loosely coupled large-scale distributed information systems are of an increasing interest in a variety of areas of computer science and its applications – areas as diverse as team robotics, intelligent transportation systems, open distributed software environments, disaster response management, distributed databases and information retrieval, and computational theories of language evolution. A popular paradigm for abstracting such distributed infrastructures is that of multi-agent systems (MAS) made of typically a large number of autonomous agents that locally interact with each other. This report is an attempt at a cellular and network automata based mathematical and computational theory of such MAS. The

### Counting Fixed Points and Gardens of Eden of Sequential Dynamical Systems on Planar Bipartite Graphs

- ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 91 (2005)
, 2005

"... We study counting various types of configurations in certain classes of graph automata viewed as discrete dynamical systems. The graph automata models of our interest are Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively). These models have been proposed as a mathematical fo ..."

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We study counting various types of configurations in certain classes of graph automata viewed as discrete dynamical systems. The graph automata models of our interest are Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively). These models have been proposed as a mathematical foundation for a theory of large-scale simulations of complex multi-agent systems. Our emphasis in this paper is on the computational complexity of counting the fixed point and the garden of Eden configurations in Boolean SDSs and SyDSs. We have shown in [47] that counting fixed points is, in general, computationally intractable. We show in the present report that this intractability still holds when both the underlying graphs and the node update rules of these SDSs and SyDSs are severely restricted. In particular, we prove that the problems of exactly counting fixed points, gardens of Eden and two other types of S(y)DS configurations are all #P-complete, even if the SDSs and SyDSs are defined over planar bipartite graphs, and each of their nodes updates its state according to a monotone update rule given as a Boolean formula. We thus add these formal discrete dynamical systems to the list of those problem domains for which counting the combinatorial structures of interest is intractable even when the related decision problems are known to be efficiently solvable.

### On the Computational Complexity of Predicting Dynamical Evolution of Large Agent Ensembles

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### PARALLEL vs. SEQUENTIAL THRESHOLD CELLULAR AUTOMATA: Comparison and Contrast

"... Cellular automata (CA) are an abstract model of a distributed dynamical system, as well as of fine-grain parallelism in computing. In a classical cellular automaton, all the nodes execute their operations in parallel and in perfect synchrony. We consider herewith the sequential version of CA, called ..."

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Cellular automata (CA) are an abstract model of a distributed dynamical system, as well as of fine-grain parallelism in computing. In a classical cellular automaton, all the nodes execute their operations in parallel and in perfect synchrony. We consider herewith the sequential version of CA, called SCA, and compare those SCA with the classical, parallel CA. In particular, we show that there are 1D CA with very simple node update rules that cannot be simulated by any comparable SCA, irrespective of the node update ordering. Consequently, the granularity of the basic CA operations and, therefore, the fine-grain parallelism of the classical, synchronous CA, insofar as the “interleaving semantics” is concerned, turns out to be not fine enough. We also study in some detail the properties of the cellular automata whose nodes update their states according to the Majority update rule. Finally, we share some thoughts on how to extend the presented results, and, in particular, we try to motivate the study of genuinely asynchronous cellular automata.

### Computational Complexity of Some Enumeration Problems About Uniformly Sparse Boolean Network Automata

"... We study the computational complexity of counting the fixed point configurations (FPs), the predecessor configurations and the ancestor configurations in certain classes of network automata viewed as discrete dynamical systems. Some early results of this investigation are presented in [38, 39]. In p ..."

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We study the computational complexity of counting the fixed point configurations (FPs), the predecessor configurations and the ancestor configurations in certain classes of network automata viewed as discrete dynamical systems. Some early results of this investigation are presented in [38, 39]. In particular, it is proven in [39] that both by�Ç exact and approximate counting of FPs in the two closely related classes of Boolean network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems when each node is required to update according to a monotone Boolean function. In the present paper, we further strengthen those results by showing that the intractability of exact enumeration of FPs of a monotone Boolean SDS or SyDS still holds even when (i) the monotone update rules are restricted to linear threshold functions, and (ii) the underlying graph is uniformly sparse. By uniform sparseness we mean that every node in the graph has its degree bounded for a small value of constant. In particular, we prove that exactly enumerating FPs in such SDSs and SyDSs remains #P-complete even when no node degree exceeds�. Among other consequences, we show that this result also implies intractability of determining the exact memory capacity of discrete Hopfield networks with uniformly sparse and nonnegative integer weight matrices.