Morphic words and equidistributed sequences

The problem we consider is the following: Given an infinite word $w$ on an ordered alphabet, construct the sequence $\nu_w=(\nu[n])_n$, equidistributed on $[0,1]$ and such that $\nu[m]<\nu[n]$ if and only if $\sigma^m(w)<\sigma^n(w)$, where $\sigma$ is the shift operation, erasing the first symbol of $w$. The sequence $\nu_w$ exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of $\nu_w$ for the case when the subshift of $w$ is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence $\nu_w$ in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism $\varphi$, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of $\varphi$

Ten Conferences WORDS: Open Problems and Conjectures

In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

Optimal path planning for surveillance with temporal-logic constraints

In this paper we present a method for automatically generating optimal robot paths satisfying high-level mission specifications. The motion of the robot in the environment is modeled as a weighted transition system. The mission is specified by an arbitrary linear temporal-logic (LTL) formula over propositions satisfied at the regions of a partitioned environment. The mission specification contains an optimizing proposition, which must be repeatedly satisfied. The cost function that we seek to minimize is the maximum time between satisfying instances of the optimizing proposition. For every environment model, and for every formula, our method computes a robot path that minimizes the cost function. The problem is motivated by applications in robotic monitoring and data-gathering. In this setting, the optimizing proposition is satisfied at all locations where data can be uploaded, and the LTL formula specifies a complex data-collection mission. Our method utilizes Büchi automata to produce an automaton (which can be thought of as a graph) whose runs satisfy the temporal-logic specification. We then present a graph algorithm that computes a run corresponding to the optimal robot path. We present an implementation for a robot performing data collection in a road-network platform.This material is based upon work supported in part by ONR-MURI (award N00014-09-1-1051), ARO (award W911NF-09-1-0088), and Masaryk University (grant numbers LH11065 and GD102/09/H042), and other funding sources (AFOSR YIP FA9550-09-1-0209, NSF CNS-1035588, NSF CNS-0834260). (N00014-09-1-1051 - ONR-MURI; W911NF-09-1-0088 - ARO; LH11065 - Masaryk University; GD102/09/H042 - Masaryk University; FA9550-09-1-0209 - AFOSR YIP; CNS-1035588 - NSF; CNS-0834260 - NSF