Development of Toolkit for Formalizing the Programming of Canned Cycles on CNC Machine Tools

The analysis shows that the development of programming tools for CNC systems is focused on the development of new machine canned cycles, including those for processing complicated surfaces, their complex combinations or complex mutual arrangement, but does not concern such issues as the universalization of cycle parameters. This does not make it impossible to transfer control programs from one CNC system to another. This work proposes an approach to the development of tools for creating universal cycles of typical technological transitions of machining on machines with different CNC systems.

Classifying Rotationally-Closed Languages Having Greedy Universal Cycles

Let $\textbf{T}(n,k)$ be the set of strings of length $n$ over the alphabet $\Sigma=\{1,2,\ldots,k\}$. A universal cycle for $\textbf{T}(n,k)$ can be constructed using a greedy algorithm: start with the string $k^n$, and continually append the least symbol possible without repeating a substring of length $n$. This construction also creates universal cycles for some subsets $\textbf{S}\subseteq\textbf{T}(n,k)$; we will classify all such subsets that are closed under rotations.

On a Greedy Algorithm to Construct Universal Cycles for Permutations

A universal cycle for permutations of length [Formula: see text] is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length [Formula: see text], and containing all permutations of length [Formula: see text] as factors. It is well known that universal cycles for permutations of length [Formula: see text] exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, we offer a simple way to generate a universal cycle for permutations of length [Formula: see text], which is based on applying a greedy algorithm to a permutation of length [Formula: see text]. We prove that this approach gives a unique universal cycle [Formula: see text] for permutations, and we study properties of [Formula: see text].

Quasi-Eulerian Hypergraphs

We generalize the notion of an Euler tour in a graph in the following way. An Euler family in a hypergraph is a family of closed walks that jointly traverse each edge of the hypergraph exactly once. An Euler tourthus corresponds to an Euler family with a single component. We provide necessary and sufficient conditions for the existence of an Euler family in an arbitrary hypergraph, and in particular, we show that every 3-uniform hypergraph without cut edges admits an Euler family. Finally, we show that the problem of existence of an Euler family is polynomial on the class of all hypergraphs.This work complements existing results on rank-1 universal cycles and 1-overlap cycles in triple systems, as well as recent results by Lonc and Naroski, who showed that the problem of existence of an Euler tour in a hypergraph is NP-complete.

Generalizing the Classic Greedy and Necklace Constructions of de Bruijn Sequences and Universal Cycles

We present a class of languages that have an interesting property: For each language $\mathbf{L}$ in the class, both the classic greedy algorithm and the classic Lyndon word (or necklace) concatenation algorithm provide the lexicographically smallest universal cycle for $\mathbf{L}$. The languages consist of length $n$ strings over $\{1,2,\ldots ,k\}$ that are closed under rotation with their subset of necklaces also being closed under replacing any suffix of length $i$ by $i$ copies of $k$. Examples include all strings (in which case universal cycles are commonly known as de Bruijn sequences), strings that sum to at least $s$, strings with at most $d$ cyclic descents for a fixed $d>0$, strings with at most $d$ cyclic decrements for a fixed $d>0$, and strings avoiding a given period. Our class is also closed under both union and intersection, and our results generalize results of several previous papers.