Predrag T. Tosic

Characterizing configuration spaces of simple threshold cellular automata

We study herewith the simple threshold cellular automata (CA), as perhaps the simplest broad class of CA with non-additive (that is, non-linear and non-affine) local update rules. We characterize all possible computations of the most interesting rule for such CA, namely, the Majority (MAJ) rule, both in the classical, parallel CA case, and in case of the corresponding sequential CA where the nodes update sequentially, one at a time. We compare and contrast the configuration spaces of arbitrary simple threshold automata in those two cases, and point out that some parallel threshold CA cannot be simulated by any of their sequential equivalents. We show that the temporal cycles exist only in case of (some) parallel simple threshold CA, but can never take place in sequential threshold CA. We also show that most threshold CA have very few fixed point configurations and few (if any) cycle configurations, and that, while the MAJ sequential and parallel CA may have many fixed points, nonetheless “almost all” configurations, in both parallel and sequential cases, are transient states. Finally, motivated by the contrasts between parallel and sequential simple threshold CA, we try to motivate the study of genuinely asynchronous CA.

On computational complexity of counting fixed points in symmetric boolean graph automata

We study computational complexity of counting the fixed point configurations (FPs) in certain classes of graph automata viewed as discrete dynamical systems. We prove that both exact and approximate counting of FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) are computationally intractable, even when each node is required to update according to a symmetric Boolean function. We also show that the problems of counting exactly the garden of Eden configurations (GEs), as well as all transient configurations, are in general intractable, as well. Moreover, exactly enumerating FPs or GEs remains hard 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 every node has a neighborhood size bounded by a small constant.

Concurrency vs. sequential interleavings in 1-D threshold cellular automata

Summary form only given. Cellular automata (CA) are an abstract model of fine-grain parallelism: the individual node update operations are rather simple, and therefore comparable to the basic operations of the computer hardware, yet the power of the model stems from the interaction and synergy of these simple local node computations that can often generate highly complex global behavior. In classical CA, all the nodes execute their operations in parallel, that is, (logically) simultaneously. We consider herein the sequential version of CA, or SCA, and compare and contrast SCA with the classical, parallel CA. We show that there are 1D CA with simple nonlinear node state update rules that cannot be simulated by any comparable SCA, irrespective of the node update ordering. While the result is trivial if one considers a single automatons computations, we find this property quite interesting and having important implications when applied to all possible computations of entire nontrivial classes of CA (SCA). We also share some thoughts on how to extend the results herein, and, in particular, we try to motivate the study of genuinely asynchronous cellular automata.

ON THE COMPLEXITY OF COUNTING FIXED POINTS AND GARDENS OF EDEN IN SEQUENTIAL DYNAMICAL SYSTEMS ON PLANAR BIPARTITE GRAPHS

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 the 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 show that counting these configurations is, in general, computationally intractable. We also show 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 discrete dynamical systems to the list of those problem domains where counting combinatorial structures of interest is intractable even when the related decision problems are known to be efficiently solvable.

Towards a hierarchical taxonomy of autonomous agents

Autonomous agents have become an influential and powerful paradigm in a great variety of disciplines, from sociology and economics to distributed artificial intelligence and software engineering to philosophy. Given that the paradigm has been around for awhile, one would expect a broadly agreed-upon, solid understanding of what autonomous agents are and what they are not. This, however, is not the case. We therefore join the ongoing debate on what are the appropriate notions of autonomous agency. We approach agents and agent ontology from a cybernetics and general systems perspective, in contrast to the much more common in the agent literature sociology, anthropology and/or cognitive psychology based approaches. We attempt to identify the most fundamental attributes of autonomous agents, and propose a tentative hierarchy of autonomous agents based on those attributes.

On modeling and analyzing sparsely networked large-scale multi-agent systems with cellular and graph automata

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.