Motivated by the study of cellular automata algorithmic and dynamics, we investigate an extension of ultimately periodic words to twodimensional infinite words: collisions. A natural composition operation on tilings leads to a catenation operation on collisions. By existence of aperiodic tile sets, ultimately periodic tilings of the plane cannot generate all possible tilings but exhibit some us...

In this note, colorings of the plane by finite sequential machines are compared to previously introduced notions of ultimately periodic tilings of the plane. Finite automata with no counter characterize exactly biperiodic tilings. Finite automata with one counter characterize exactly particles — periodic colorings that are ultimately periodic in every direction. Finite automata with two counter...

Cameron has introduced a natural bijection between the set of one way infinite binary sequences and the set of sum-free sets (of positive integers), and observed that a sum-free set is ultimately periodic only if the corresponding binary sequence is ultimately periodic. He asked if the converse also holds. In this paper we present necessary and sufficient conditions for a sum-free set to be ult...

This paper introduces the concept of ultimately periodic functions for nite semigroups. Ultimately periodic functions are shown to be the same as regularity-preserving functions in automata theory. This characterization reveals the true algebraic nature of regularity-preserving functions. It makes it possible to prove properties of ultimately periodic functions through automata-theoretic method...

Given a morphism h prolongable on a and an integer p, we present an algorithm that calculates which letters occur infinitely often in congruent positions modulo p in the infinite word h(a). As a corollary, we show that it is decidable whether a morphic word is ultimately p-periodic. Moreover, using our algorithm we can find the smallest similarity relation such that the morphic word is ultimate...

It is decidable if a set of numbers, whose representation in a base b is a regular language, is ultimately periodic. This was established by Honkala in 1986. We give here a structural description of minimal automata that accept an ultimately periodic set of numbers. We then show that it can be verified in linear time if a given minimal automaton meets this description. This yields a O(n log(n))...

In this paper, we study the complexity of model-checking formulas of three important real-time logics (MTL, MITL, and TCTL) over restricted sets of timed paths. The classes of restricted sets of timed paths that we consider are (i) a single finite (or ultimately periodic) timed path, (ii) a infinite set of finite (or infinite) timed paths defined by a finite (or ultimately periodic) path in a r...

Abstract Network Calculus (NC) is an algebraic theory that represents traffic and service guarantees as curves in a Cartesian plane, order to compute performance for flows traversing network. NC uses transformation operations, e.g., min-plus convolution of two curves, model how the profile changes with traversal network nodes. Such while mathematically well-defined, can quickly become unmanagea...