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## A realization theorem for the Gödel-Löb provability logic

We present a new justification logic corresponding to the Gödel-Löb provability logic GL and prove the realization theorem connecting these two systems in such a way that all the realizations provided in the theorem are normal.

We consider a many-sorted variant of Japaridze’s polymodal provability logic (GLP). In this variant, which is denoted GLP∗, propositional variables are assigned sorts n≤ω, where variables of finite sort n<ω are interpreted as Π_{n+1}-sentences of the arithmetical hierarchy, while those of sort ω range over arbitrary ones. We prove that GLP∗is arithmetically complete with respect to this interpretation. Moreover, we relate GLP∗ to its one-sorted counterpart GLP and prove that the former inherits some well-known properties of the latter, like Craig interpolation and polynomial space (PSPACE) decidability. We also study a positive variant of GLP∗ that allows for an even richer arithmetical interpretation—variables are permitted to range over theories rather than single sentences. This interpretation in turn allows the introduction of a modality that corresponds to the full uniform reflection principle. We show that our positive variant of GLP∗ is arithmetically complete.

The provability logic GLP introduced by G. Japaridze is a propositional polymodal logic with important applications in proof theory, specificially, in ordinal analysis of arithmetic. Though being incomplete with respect to any class of Kripke frames, the logic GLP is complete for its neighbourhood interpretation. This completeness result, established by L. Beklemishev and D. Gabelaia, implies strong neighbourhood completeness of this system for the case of the so-called local semantic consequence relation. In the given article, we consider Hilbert-style non-well-founded derivations in the provability logic GLP and establish that GLP with the obtained derivability relation is strongly neighbourhood complete in the case of the global semantic consequence relation.

We study interpolation properties of provability logics. We prove the Lyndon interpolation for GL and the uniform interpolation for GLP.

The Gödel-Löb provability logic GL and establish that

We present a proof system for the provability logic GLP in the formalism of nested sequents and prove the cut elimination theorem for it. As an application, we obtain the reduction of GLP to its important fragment called J syntactically.

We present a sequent calculus for the weak Grzegorczyk logic 𝖦𝗈 allowing non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.

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.