The study of multi-agent systems (MAS) focuses on systems in which many intelligent agents interact with each other. These agents are considered to be autonomous entities such as software programs or robots. Their interactions can either be cooperative (for example as in an ant colony) or selfish (as in a free market economy). This book assumes only basic knowledge of algorithms and discrete maths, both of which are taught as standard in the first or second year of computer science degree programmes. A basic knowledge of artificial intelligence would useful to help understand some of the issues, but is not essential. The books main aims are: To introduce the student to the concept of agents and multi-agent systems, and the main applications for which they are appropriate To introduce the main issues surrounding the design of intelligent agents To introduce the main issues surrounding the design of a multi-agent society To introduce a number of typical applications for agent technology

An Introduction to MultiAgent Systems.

What is modal logic

General Topology-I

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

Topology and Geometry

By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras. In our recent article, we have extended de Vries duality to completely regular spaces by generalizing de Vries algebras to de Vries extensions. To illustrate the utility of this point of view, we show how to use this new duality to obtain algebraic counterparts of normal and locally compact Hausdorff spaces in the form of de Vries extensions that are subject to additional axioms which encode the desired topological property. This, in particular, yields a different perspective on de Vries duality. As a further application, we show that a duality for locally compact Hausdorff spaces due to Dimov can be obtained from our approach.

De Vries duality for normal spaces and locally compact Hausdorff spaces

It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure (and hence modal box as interior), then $$\mathsf S4$$
 is the logic of any dense-in-itself metrizable space. The McKinsey–Tarski Theorem relies heavily on a metric that gives rise to the topology. We give a new and more topological proof of the theorem, utilizing Bing’s Metrization Theorem.

A New Proof of the McKinsey–Tarski Theorem

Let $${\beta(\mathbb{N})}$$
 denote the Stone–Cech compactification of the set $${\mathbb{N}}$$
 of natural numbers (with the discrete topology), and let $${\mathbb{N}^\ast}$$
 denote the remainder $${\beta(\mathbb{N})-\mathbb{N}}$$
 . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of $${\mathbb{N}^\ast}$$
 is S4 and that the modal logic of $${\beta(\mathbb{N})}$$
 is S4.1.2.

The modal logic of {beta(mathbb{N})}

A lattice P is transferable for a class of lattices $$\mathcal {K}$$
 if whenever P can be embedded into the ideal lattice $$\mathfrak {I}K$$
 of some $$K\in \mathcal {K}$$
 , then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice $$\mathfrak {I}K$$
 with the MacNeille completion $$\overline{K}$$
 . Basic properties of MacNeille transferability are developed. Particular attention is paid to MacNeille transferability in the class of Heyting algebras where it relates to stable classes of Heyting algebras, and hence to stable intermediate logics.

/pdf/macneille-transferability-and-stable-classes-of-heyting-1av5vr291c.pdf

MacNeille transferability and stable classes of Heyting algebras

On modal logics arising from scattered locally compact Hausdorff spaces

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