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 Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate.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

On Shehtman's two problems

. The Priestley space X = ( X, π, ≤ ) of a bounded distributive lattice L carries three natural topologies, the Stone topology π , the topology of open upsets, and the topology of open downsets. For each of these topologies, there are three hyperspace topologies on its closed sets, the hit, miss, and hit-or-miss topologies. We give a systematic account of these nine hyperspaces, along with concrete descriptions of the corresponding free constructions on bounded distributive lattices. There are two constructions of the Vietoris space of X , as a hyperspace on its convex closed sets, and as a quotient space of the closed sets using the Egli-Milner quasi-order. We show that these two constructions are order-homeomorphic. These results are used to give coalgebraic proofs of dualities for positive modal algebras. Existing accounts of Vietoris spaces in modal logic make essential use of results of Johnstone from pointfree topology. The results here are self-contained, rely only on direct order-theoretic and topological arguments, and are accessible to those familiar with Priestley duality.

/pdf/remarks-on-hyperspaces-for-priestley-spaces-28m5k6fk.pdf

Remarks on hyperspaces for Priestley spaces

We introduce modal compact Hausdorff spaces as generalizations of modal spaces, and show these are coalgebras for the Vietoris functor on compact Hausdorff spaces. Modal compact regular frames and modal de Vries algebras are introduced as algebraic counterparts of modal compact Hausdorff spaces, and dualities are given for the categories involved. These extend the familiar Isbell and de Vries dualities for compact Hausdorff spaces, as well as the duality between modal spaces and modal algebras. As the first step in the logical treatment of modal compact Hausdorff spaces, a version of Sahlqvist correspondence is given for the positive modal language.

/pdf/modal-compact-hausdorff-spaces-1o1q4je01p.pdf

Modal compact Hausdorff spaces

It is a classic result in modal logic that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality. Our goal is to further generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space. 
Our starting point is the well-known Gelfand duality between the category $KHaus$ of compact Hausdorff spaces and the category $ubal$ of uniformly complete bounded archimedean $\ell$-algebras. We endow a bounded archimedean $\ell$-algebra with a modal operator, which results in the category $mbal$ of modal bounded archimedean $\ell$-algebras. Our main result establishes a dual adjunction between $mbal$ and the category $KHK$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between $KHK$ and the reflective subcategory $mubal$ of $mbal$ consisting of uniformly complete objects of $mbal$. This generalizes both Gelfand duality and the duality for modal algebras.

Modal operators on rings of continuous functions

Integer partitions started to gain interest in 1674 when Leibniz investigated the number of ways one can write (partition) a positive integer n as a sum of positive integers. Brylawski showed that the set of all partitions of a positive integer n endowed with the dominance ordering is a complete lattice. These lattices are called lattices of integer partitions, or just partition lattices, for short. The lattices of integer partitions can be viewed as concept lat-tices; in this contribution we continue the investigations of Behrisch and et al. and Ganter of those lattices, and concentrate on the problem of determining their factor lattices as suggested by Ganter. This question is equivalent to characterising the compatible subcontexts, which in turn have a description in terms of so-called arrow relations. Moreover, we focus on results on these arrow relations and thus shed some light on the structure of lattices of integer partitions.

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