945 research outputs found

    Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

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    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

    Reach Control on Simplices by Piecewise Affine Feedback

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    We study the reach control problem for affine systems on simplices, and the focus is on cases when it is known that the problem is not solvable by continuous state feedback. We examine from a geometric viewpoint the structural properties of the system which make continuous state feedbacks fail. This structure is encoded by so-called reach control indices, which are defined and developed in the paper. Based on these indices, we propose a subdivision algorithm and associated piecewise affine feedback. The method is shown to solve the reach control problem in all remaining cases, assuming it is solvable by open-loop controls

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

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    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

    Minimal Reachability is Hard To Approximate

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    In this note, we consider the problem of choosing which nodes of a linear dynamical system should be actuated so that the state transfer from the system's initial condition to a given final state is possible. Assuming a standard complexity hypothesis, we show that this problem cannot be efficiently solved or approximated in polynomial, or even quasi-polynomial, time

    Singular Switched Systems in Discrete Time: Solvability, Observability, and Reachability Notions

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    Discrete-time singular (switched) systems, also known as(switched) difference-algebraic equations and discrete-time (switched)descriptor systems, have in general three solvability issues:inconsistent initial values, nonexistence ornonuniqueness of solutions, and noncausalities, which are generallynot desired in applications. To deal with those issues, newsolvability notions are proposed in the study, and the correspondingnecessary and sufficient conditions have been derived with the help of(strictly) index-1 notions. Furthermore, surrogate (switched)systems--ordinary (switched) systems that have equivalentbehavior--have also been established for solvable systems. Byutilizing those surrogate systems, fundamental analysis includingobservability, determinability, reachability, and controllability has also beencharacterized for singular linear (switched) systems. The solvabilitystudy has been extended to singular nonlinear (switched) systems, andmoreover, Lyapunov and incremental stability analyses have beenderived via single and switched Lyapunov function approaches

    Reachability in Dynamical Systems with Rounding

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    We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix M∈Qd×dM \in \mathbb{Q}^{d \times d}, an initial vector x∈Qdx\in\mathbb{Q}^{d}, a granularity g∈Q+g\in \mathbb{Q}_+ and a rounding operation [⋅][\cdot] projecting a vector of Qd\mathbb{Q}^{d} onto another vector whose every entry is a multiple of gg, we are interested in the behaviour of the orbit O=\mathcal{O}={}, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability---whether a given target y∈Qdy \in\mathbb{Q}^{d} belongs to O\mathcal{O}---is PSPACE-complete for hyperbolic systems (when no eigenvalue of MM has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.Comment: To appear at FSTTCS'2
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