Multidimensional cellular automata and generalization of Fekete's lemma

Fekete's lemma is a well known combinatorial result on number sequences: we extend it to functions defined on $d$-tuples of integers. As an application of the new variant, we show that nonsurjective $d$-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support's boundary determined by the automaton's neighbourhood index.Comment: 6 pages, no figures, LaTeX. Improved some explanations; revised structure; added examples; renamed "hypercubes" into "right polytopes"; added references to Arratia's paper on EJC, Calude's book, Cook's proof of Rule 110 universality, and arXiv paper 0709.117

Fekete's lemma for componentwise subadditive functions of two or more real variables

We prove an analogue of Fekete's subadditivity lemma for functions of several real variables which are subadditive in each variable taken singularly. This extends both the classical case for subadditive functions of one real variable, and a result in a previous paper by the author. While doing so, we prove that the functions with the property mentioned above are bounded in every closed and bounded subset of their domain. The arguments follows those of Chapter 6 in E. Hille's 1948 textbook.Comment: 22 pages. Revised and expanded. Longer introduction, more detailed background, statement of main theorem extende

Path Integrals on Euclidean Space Forms

In this paper we develop a quantization method for flat compact manifolds based on path integrals. In this method the Hilbert space of holomorphic functions in the complexification of the manifold is used. This space is a reproducing kernel Hilbert space. A definition of the Feynman propagator, based on the reproducing property of this space, is proposed. In the $\mathbb{R}^n$ case the obtained results coincide with the known expressions

DOP: Deep Optimistic Planning with Approximate Value Function Evaluation

Research on reinforcement learning has demonstrated promising results in manifold applications and domains. Still, efficiently learning effective robot behaviors is very difficult, due to unstructured scenarios, high uncertainties, and large state dimensionality (e.g. multi-agent systems or hyper-redundant robots). To alleviate this problem, we present DOP, a deep model-based reinforcement learning algorithm, which exploits action values to both (1) guide the exploration of the state space and (2) plan effective policies. Specifically, we exploit deep neural networks to learn Q-functions that are used to attack the curse of dimensionality during a Monte-Carlo tree search. Our algorithm, in fact, constructs upper confidence bounds on the learned value function to select actions optimistically. We implement and evaluate DOP on different scenarios: (1) a cooperative navigation problem, (2) a fetching task for a 7-DOF KUKA robot, and (3) a human-robot handover with a humanoid robot (both in simulation and real). The obtained results show the effectiveness of DOP in the chosen applications, where action values drive the exploration and reduce the computational demand of the planning process while achieving good performance

Q-CP: Learning Action Values for Cooperative Planning

Research on multi-robot systems has demonstrated promising results in manifold applications and domains. Still, efficiently learning an effective robot behaviors is very difficult, due to unstructured scenarios, high uncertainties, and large state dimensionality (e.g. hyper-redundant and groups of robot). To alleviate this problem, we present Q-CP a cooperative model-based reinforcement learning algorithm, which exploits action values to both (1) guide the exploration of the state space and (2) generate effective policies. Specifically, we exploit Q-learning to attack the curse-of-dimensionality in the iterations of a Monte-Carlo Tree Search. We implement and evaluate Q-CP on different stochastic cooperative (general-sum) games: (1) a simple cooperative navigation problem among 3 robots, (2) a cooperation scenario between a pair of KUKA YouBots performing hand-overs, and (3) a coordination task between two mobile robots entering a door. The obtained results show the effectiveness of Q-CP in the chosen applications, where action values drive the exploration and reduce the computational demand of the planning process while achieving good performance