Variable neighborhood search (VNS) is a recent metaheuristic for solving combinatorial and global optimization problems whose basic idea is systematic change of neighborhood within a local search. In this survey paper we present basic rules of VNS and some of its extensions. Moreover, applications are briefly summarized. They comprise heuristic solution of a variety of optimization problems, ways to accelerate exact algorithms and to analyze heuristic solution processes, as well as computer-assisted discovery of conjectures in graph theory.

Recently, Araujo and De la Pe~na [1] gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which are trees and the cyclomatic number of the graphs. The ramification index of a tree is first shown to be equal to the number of pending vertices minus 2. Then, in view of extremal graphs obtained with the system AutoGraphiX, all bounds of Araujo and De la Pe\~na [1] are improved, yielding tight bounds, and in one case corrected. Moreover, chemical trees of given order and number of pending vertices with minimum and with maximum connectivity index are characterized.

Automated theory formation involves the production of examples, concepts and hypotheses about the concepts. The HR program performs automated theory formation and has used to form theories in mathematical domains. In addition to providing a plausible model for automated theory formation, HR has been applied to some applications in machine learning. We discuss HR's application to inducing de nitions from examples, scienti c discovery, problem solving and puzzle generation. For each problem, we look at how theory formation was applied, and mention some initial results from using HR.

Abstract. The application of Inductive Logic Programming to scientific datasets has been highly successful. Such applications have led to breakthroughs in the domain of interest and have driven the development of ILP systems. The application of AI techniques to mathematical discovery tasks, however, has largely involved computer algebra systems and theorem provers rather than machine learning systems. We discuss here the application of the HR and Progol machine learning programs to discovery tasks in mathematics. While Progol is an established ILP system, HR has historically not been described as an ILP system. However, many applications of HR have required the production of first order hypotheses given data expressed in a Prolog-style manner, and the core functionality of HR can be expressed in ILP terminology. In (Colton, 2003), we presented the first partial description of HR as an ILP system, and we build on this work to provide a full description here. HR performs a novel ILP routine called Automated Theory Formation, which combines inductive and deductive reasoning to form clausal theories consisting of classification rules and association rules. HR generates definitions using a set of production rules,

Abstract. The first attempt at automating mathematical conjecture-making appeared in the late-1950s. It was not until the mid-1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlying this program is the Principle of the Strongest Conjecture: make the strongest conjecture for which no counterexample is known. These two programs as well as other attempts to automate mathematical conjecture-making are surveyed—the success of a conjecture-making program, it is found, correlates strongly whether the program is designed to produce statements that are relevant to answering or advancing our mathematical questions. 1.

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

The connectivity index wα(G) of a graph G is the sum of the weights (d(u)d(v)) α of all edges uv of G, where α is a real number (α � = 0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w1(T). Also, for −1 ≤ α < 0, we give a sharp lower bound and a upper bound for wα(T).

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Abstract. We discuss what constitutes knowledge in pure mathematics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body of knowledge in pure mathematics. We discuss to what extent the output from certain programs can be considered a discovery in pure mathematics. This enables us to assess the state of the art with respect to Newell and Simon’s prediction that a computer would discover and prove an important mathematical theorem. 1