Related papers: Topological complexity of the work map
This chapter serves as an introduction to systems engineering focused on the broad issues surrounding realizing complex integrated systems. What is a system? We pose a number of possible definitions and perspectives, but leave open the…
Robotic manipulation in complex, constrained spaces is vital for widespread applications but challenging, particularly when navigating narrow passages with elongated objects. Existing planning methods often fail in these low-clearance…
Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering.…
The topological organization of several world cities are studied according to respective representations by complex networks. As a first step, the city maps are processed by a recently developed methodology that allows the most significant…
The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of…
We survey results on the topological complexity of classical configuration spaces of distinct ordered points in orientable surfaces and related spaces, including certain orbit configuration spaces and Eilenberg-Mac Lane spaces associated to…
Transitivity, the existence of periodic points and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that for graphs that are not trees, for every $\varepsilon>0,$ there exist (complicate)…
We perform a refined complexity-theoretic analysis of three classical problems in the context of Hierarchical Task Network Planning: the verification of a provided plan, whether an executable plan exists, and whether a given state can be…
Topological effects produce chaos in multiagent simulation and distributed computation. We explain this result by developing three themes concerning complex systems in the natural and social sciences: (i) Pragmatically, a system is complex…
Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…
The ability to autonomously navigate in unknown environments is important for mobile robots. The map is the core component to achieve this. Most map representations rely on drift-free state estimation and provide a global metric map to…
We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove…
Genetic regulatory networks are defined by their topology and by a multitude of continuously adjustable parameters. Here we present a class of simple models within which the relative importance of topology vs. interaction strengths becomes…
We prove that the topological complexity of (a motion planning algorithm on) the complement of generic complex essential hyperplane arrangement of $n$ hyperplanes in an $r$-dimensional linear space is min$\{n+1,2r\}$.
We develop the properties of the $n$-th sequential topological complexity $TC_n$, a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in…
We present a method for computing the topological entropy of one-dimensional maps. As an approximation scheme, the algorithm converges rapidly and provides both upper and lower bounds.
Identifying and categorizing specific robot tasks, behaviors, and resources is an essential precursor to reproducing and evaluating robotics experiments across laboratories and platforms. Without some means of capturing how one environment,…
We use an alternative definition of topological complexity to show that the topological complexity of the mapping telescope of a sequence $X_1\rightarrow X_2\rightarrow X_3\rightarrow...$ is bounded above by $2max{TC(X_i); i=1,2,...}$.
Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display…
Network or graph structures are ubiquitous in the study of complex systems. Often, we are interested in complexity trends of these system as it evolves under some dynamic. An example might be looking at the complexity of a food web as…