Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on Theoretical Computer Science (ICTCS'2007

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24 versio

Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}. For a class F of boolean functions and a class G of graphs, an (F, G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #Pcomplete to compute the number of fixed points in an (F, G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas

Proceedings of AUTOMATA 2010: 16th International workshop on cellular automata and discrete complex systems

International audienceThese local proceedings hold the papers of two catgeories: (a) Short, non-reviewed papers (b) Full paper

Perspectives on the relationship between local interactions and global outcomes in spatially explicit models of systems of interacting individuals

Understanding the behaviour of systems of interacting individuals is a key aim of much research in the social sciences and beyond, and a wide variety of modelling paradigms have been employed in pursuit of this goal. Often, systems of interest are intrinsically spatial, involving interactions that occur on a local scale or according to some specific spatial structure. However, while it is recognised that spatial factors can have a significant impact on the global behaviours exhibited by such systems, in practice, models often neglect spatial structure or consider it only in a limited way, in order to simplify interpretation and analysis. In the particular case of individual-based models used in the social sciences, a lack of consistent mathematical foundations inevitably casts doubt on the validity of research conclusions. Similarly, in game theory, the lack of a unifying framework to encompass the full variety of spatial games presented in the literature restricts the development of general results and can prevent researchers from identifying important similarities between models. In this thesis, we address these issues by examining the relationship between local interactions and global outcomes in spatially explicit models of interacting individuals from two different conceptual perspectives. First, we define and analyse a family of spatially explicit, individual-based models, identifying and explaining fundamental connections between their local and global behaviours. Our approach represents a proof of concept, suggesting that similar methods could be effective in identifying such connections in a wider range of models. Secondly, we define a general model for spatial games of search and concealment, which unites many existing games into a single framework, and we present theoretical results on its optimal strategies. Our model represents an opportunity for the development of a more broadly applicable theory of spatial games, which could facilitate progress and highlight connections within the field

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 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

Queensland University of Technology: Handbook 1998

The Queensland University of Technology handbook gives an outline of the faculties and subject offerings available that were offered by QUT

Queensland University of Technology: Handbook 1997

The Queensland University of Technology handbook gives an outline of the faculties and subject offerings available that were offered by QUT