### Book

## Advances in Modal Logic

*Advances in Modal Logic* is a bi-annual international conference and book series in Modal Logic. The aim of the conference series is to report on important new developments in pure and applied modal logic, and to do so at varying locations throughout the world. The book series is based on the conferences. Please consult thebackground pages for further details.

With a set *S* of words in an alphabet *A *we associate the frame (*S*; *H*), where sHt iff *s* and *t* are words of the same length and *h*(*s*; *t*) = 1 for the Hamming distance *h*. We investigate some unimodal logics of these frames. We show that if the length of words *n* is fixed and finite, the logics are closely related to many-dimensional products of logic **S5**, so in many cases they are undecidable and not finitely axiomatizable. The relation *H* can be extended to infinite sequences. In this case we prove some completeness theorems characterizing the well-known modal logics **DB** and **TB **in terms of the Hamming distance.

We consider modal logics of products of neighborhood frames and prove that for any pair L and L' of logics from set {S4, D4, D, T} modal logic of products of L-neighborhood frames and L'-neighborhood frames is the fusion of L and L'.

Several interesting applications of provability logic in proof theory made use of a polymodal logic GLP due to Giorgi Japaridze. This system, although de- cidable, is not very easy to handle. In particular, it is not Kripke complete. It is complete w.r.t. neighborhood semantics, however this could only be established recently by rather complicated techniques [1]. In this talk we will advocate the use of a weaker system, called Re ection Calculus, which is much simpler than GLP, yet expressive enough to regain its main proof-theoretic applications, and more. From the point of view of modal logic, RC can be seen as a fragment of polymodal logic consisting of implications of the form A ! B, where A and B are formulas built-up from > and the variables using just ^ and the diamond modalities. In this paper we formulate it in a somewhat more succinct self-contained format. Further, we state its arithmetical interpretation, and provide some evidence that RC is much simpler than GLP. We then outline a consistency proof for Peano arithmetic based on RC and state a simple combinatorial statement, the so-called Worm principle, that was suggested by the use of GLP but is even more directly related to the Re ection Calculus.

A grammar logic refers to an extension of the multi-modal logic K in which the modal axioms are generated from a formal grammar. We consider a proof theory, in nested sequent calculus, of grammar logics with converse, i.e., every modal operator [a] comes with a converse [¯a]. Extending previous works on nested sequent systems for tense logics, we show all grammar logics (with or without converse) can be formalised in nested sequent calculi, where the axioms are internalised in the calculi as structural rules. Syntactic cut-elimination for these calculi is proved using a procedure similar to that for display logics. If the grammar is context-free, then one can get rid of all structural rules, in favor of deep inference and additional propagation rules. We give a novel semi-decision procedure for context-free grammar logics, using nested sequent calculus with deep inference, and show that, in the case where the given context-free grammar is regular, this procedure terminates. Unlike all other existing decision procedures for regular grammar logics in the literature, our procedure does not assume that a finite state automaton encoding the axioms is given

We discuss an example of recursively-enumerable Kripke-complete first-order modal logics that are not Kripke complete with respect to a first-order definable class of frames.

This volume is dedicated to Leo Esakia’s contributions to the theory of modal and intuitionistic systems. Leo Esakia was one of the pioneers in developing duality theory for modal and intuitionistic logics, and masterfully utilizing it to obtain some major results in the area. The volume consists of 10 chapters, written by leading experts, that discuss Leo’s original contributions and consequent developments that have shaped the current state of the field.

We consider modal logics of products of neighborhood frames and prove that for any pair L and L' of logics from set {S4, D4, D, T} modal logic of products of L-neighborhood frames and L'-neighborhood frames is the fusion of L and L'.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.