On Counting Fixed Point Configurations in Star Networks

Tošić, Predrag T.

doi:10.1109/ipdps.2005.303

Cited by 2 publications

(14 citation statements)

References 18 publications

(12 reference statements)

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“…Problems related to the existence of fixed point configurations are studied in [2]. Our own subsequent work further builds on the results in [2] by addressing the problems of exact and approximate enumeration of various structures such as the fixed points, the predecessor configurations, and the garden of Eden configurations [19,20,21,23,25]. Due to space constraints, we will only summarize the main findings related to the computational complexity of enumerating FP configurations for the two restricted classes of S(y)DSs -those whose nodes update according to (i) monotone and (ii) linear threshold Boolean-valued functions.…”

Section: On the Hardness Of Counting Fixed Points Of Boolean S(y)dssmentioning

confidence: 99%

“…Monotone Boolean functions, formulae and circuits [27] have been extensively studied in many areas of computer science, from machine learning to connectionist AI to VLSI design. Cellular and other graph automata with the local update rules restricted to monotone Boolean functions have also been of a considerable interest (e.g., [2,19,22]). The problem of counting the FPs in monotone Boolean SDSs and SyDSs is originally addressed in [19].…”

Section: On the Hardness Of Counting Fixed Points Of Boolean S(y)dssmentioning

confidence: 99%

“…Cellular and other graph automata with the local update rules restricted to monotone Boolean functions have also been of a considerable interest (e.g., [2,19,22]). The problem of counting the FPs in monotone Boolean SDSs and SyDSs is originally addressed in [19]. In general, counting FPs of monotone Boolean S(y)DSs either exactly or approximately is computationally intractable.…”

Section: On the Hardness Of Counting Fixed Points Of Boolean S(y)dssmentioning

confidence: 99%

“…In general, counting FPs of monotone Boolean S(y)DSs either exactly or approximately is computationally intractable. This intractability holds even for the graphs that are simultaneously bipartite, planar, and very sparse on average [19]. An example of such graphs are the star graphs, in which a single central node is connected to everyone else, and each non-central node is linked only to the central node.…”

Section: On the Hardness Of Counting Fixed Points Of Boolean S(y)dssmentioning

confidence: 99%

“…We analyze some computational complexity implications of extending the classical CA model by allowing a limited degree of heterogeneity in individual agent behaviors, as well as in the underlying communication network, in [19,20,21,23,25]. We study some implications of the nature of inter-agent communication in [22,24].…”

Section: Introductionmentioning

confidence: 99%

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On Modeling and Analyzing Sparsely Networked Large-Scale Multi-agent Systems with Cellular and Graph Automata

Tošić

2006

Computational Science – ICCS 2006

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Abstract. Modeling, designing and analyzing large scale multi-agent systems (MAS) with anywhere from tens of thousands to millions of autonomous agents will require mathematical and computational theories and models substantially different from those underlying the study of small-to medium-scale MAS made of only dozens, or perhaps hundreds, of agents. In this paper, we study certain aspects of the global behavior of large ensembles of simple reactive agents. We do so by analyzing the collective dynamics of several related models of discrete complex systems based on cellular automata. We survey our recent results on dynamical properties of the complex systems of interest, and discuss some useful ways forward in modeling and analysis of large-scale MAS via appropriately modified versions of the classical cellular automata.

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Section: On the Hardness Of Counting Fixed Points Of Boolean S(y)dssmentioning

confidence: 99%

Section: On the Hardness Of Counting Fixed Points Of Boolean S(y)dssmentioning

confidence: 99%

Section: On the Hardness Of Counting Fixed Points Of Boolean S(y)dssmentioning

confidence: 99%

Section: On the Hardness Of Counting Fixed Points Of Boolean S(y)dssmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Modeling and Analyzing Sparsely Networked Large-Scale Multi-agent Systems with Cellular and Graph Automata

Tošić

2006

Computational Science – ICCS 2006

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On the complexity of enumerating possible dynamics of sparsely connected Boolean network automata with simple update rules

Tošić¹

2010

Discrete Mathematics &Amp; Theoretical Computer Science

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International audience We study how hard is to determine some fundamental properties of dynamics of certain types of network automata. We address the computational complexity of determining how many different possible dynamic evolutions can arise from some structurally very simple, deterministic and sparsely connected network automata. In this as well as our prior, related work, we try to push the limits on the underlying simplicity of two structural aspects of such network automata: (i) the uniform sparseness of their topologies, and (ii) severely restricted local behaviors of the individual agents (that is, the local update rules of the network nodes). In this endeavor, we prove that counting the Fixed Point (FP) configurations and the predecessor and ancestor configurations in two classes of network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems. Moreover, this intractability is shown to hold when each node in such a network is required to update according to (i) a monotone Boolean function, (ii) a symmetric Boolean function, or even (iii) a simple threshold function that is both monotone and symmetric. Furthermore, the hardness of the exact enumeration of FPs and other types of configurations of interest remains to hold even in some severely restricted cases with respect to both the network topology and the diversity (or lack thereof) of individual node's local update rules. Namely, we show that the counting problems of interest remain hard even when the nodes of an SDS or SyDS use at most two different update rules from a given restricted class, and, additionally, when the network topologies are constrained so that each node has only $c = O(1)$ neighbors for small values of constant $c$. Our results also have considerable implications for other discrete dynamical system models studied in applied mathematics, physics, biology and computer science, such as Hopfield networks and spin glasses. In particular, one corollary of our results is that determining the memory capacity of sparse discrete Hopfield networks (viewed as associative memories) remains computationally intractable even when the interconnection and dependence structure among the nodes of a Hopfield network is severely restricted.

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On Counting Fixed Point Configurations in Star Networks

Cited by 2 publications

References 18 publications

On Modeling and Analyzing Sparsely Networked Large-Scale Multi-agent Systems with Cellular and Graph Automata

On Modeling and Analyzing Sparsely Networked Large-Scale Multi-agent Systems with Cellular and Graph Automata

On the complexity of enumerating possible dynamics of sparsely connected Boolean network automata with simple update rules

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