For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted GLP 0 ω . Later, Icard defined a topological model for GLP 0 ω which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each Θ, Λ we build a Kripke model I Θ Λ and a topological model T Θ Λ , and show that GLP 0 Λ is sound for both of these structures, as well as complete, provided Θ is large enough.

We introduce the logics GLP Λ , a generalization of Japaridze's polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength.We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variablefree fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLP Λ .

The intuitive notion of evidence has both semantic and syntactic features. In this paper, we develop an evidence logic for epistemic agents faced with possibly contradictory evidence from different sources. The logic is based on a neighborhood semantics, where a neighborhood N indicates that the agent has reason to believe that the true state of the world lies in N . Further notions of relative plausibility between worlds and beliefs based on the latter ordering are then defined in terms of this evidence structure, yielding our intended models for evidence-based beliefs. In addition, we also consider a second more general flavor, where belief and plausibility are modeled using additional primitive relations, and we prove a representation theorem showing that each such general model is a p-morphic image of an intended one. This semantics invites a number of natural special cases, depending on how uniform we make the evidence sets, and how coherent their total structure. We give a structural study of the resulting 'uniform' and 'flat' models. Our main result are sound and complete axiomatizations for the logics of all four major model classes with respect to the modal language of evidence, belief and safe belief. We conclude with an outlook toward logics for the dynamics of changing evidence, and the resulting language extensions and connections with logics of plausibility change.

In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions.Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β . These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions.Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α . Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations.Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.

Abashidze and Blass independently proved that the modal logic GL is complete for its topological interpretation over any ordinal greater than or equal to ω ω equipped with the interval topology. Icard later introduced a family of topologies I λ for λ < ω, with the purpose of providing semantics for Japaridze's polymodal logic GLPω. Icard's construction was later extended by Joosten and the second author to arbitrary ordinals λ ≥ ω.We further generalize Icard topologies in this article. Given a scattered space X = (X, τ ) and an ordinal λ, we define a topology τ +λ in such a way that τ+0 is the original topology τ and τ +λ coincides with I λ when X is an ordinal endowed with the left topology.We then prove that, given any scattered space X and any ordinal λ > 0 such that the rank of (X, τ ) is large enough, GL is strongly complete for τ +λ . One obtains the original Abashidze-Blass theorem as a consequence of the special case where X = ω ω and λ = 1.

This paper studies the transfinite propositional provability logics GLPΛ and their corresponding algebras. These logics have for each ordinal ξ < Λ a modality α . We will focus on the closed fragment of GLPΛ (i.e., where no propositional variables occur) and worms therein. Worms are iterated consistency expressions of the form ξn . . . ξ1 ⊤. Beklemishev has defined well-orderings < ξ on worms whose modalities are all at least ξ and presented a calculus to compute the respective order-types.In the current paper we present a generalization of the original < ξ orderings and provide a calculus for the corresponding generalized ordertypes o ξ . Our calculus is based on so-called hyperations which are transfinite iterations of normal functions.Finally, we give two different characterizations of those sequences of ordinals which are of the form o ξ (A) ξ∈On for some worm A. One of these characterizations is in terms of a second kind of transfinite iteration called cohyperation.Definition 2.1. For Λ an ordinal, the logic GLP Λ is the propositional normal modal logic that has for each α < Λ a modality [α] and is axiomatized by all propositional logical tautologies together witht the following schemata: The rules of inference are Modus Ponens and necessitation for each modality:ψ [α]ψ . By GLP we denote the class-size logic that has a modality [α] for each ordinal α and all the corresponding axioms and rules. The classic Gödel-Löb provability logic GL is denoted by GLP 1 . Japaridze algebrasThe relations < α do not give proper linear orders on W α , given that different worms may be equivalent in GLP and hence undistinguishable in the ordering. We remedy this by passing to the Lindenbaum algebra of GLP -that is, the quotient of the language of GLP modulo provable equivalence. This algebra is a Japaridze algebra, as described below:Definition 2.2 (Japaridze algebra). A Japaridze algebra is a structure J = D, {[α]} α<Λ , ∧, ¬, 0, 1 such that

Given a computable ordinal Λ, the transfinite provability logic GLP Λ has for each ξ < Λ a modality [ξ] intended to represent a provability predicate within a chain of increasing strength. One possibility is to read [ξ]φ as φ is provable in T using ω-rules of depth at most ξ, where T is a second-order theory extending ACA 0. In this paper we will formalize such iterations of ω-rules in second-order arithmetic and show how it is a special case of what we call uniform provability predicates. Uniform provability predicates are similar to Ignatiev's strong provability predicates except that they can be iterated transfinitely. Finally, we show that GLP Λ is sound and complete for any uniform provability predicate.

In the generalized Russian cards problem, Alice, Bob and Cath draw a, b and c cards, respectively, from a deck of size a + b + c. Alice and Bob must then communicate their entire hand to each other, without Cath learning the owner of a single card she does not hold. Unlike many traditional problems in cryptography, however, they are not allowed to encode or hide the messages they exchange from Cath. The problem is then to find methods through which they can achieve this. We propose a general four-step solution based on finite vector spaces, and call it the “colouring protocol”, as it involves colourings of lines. Our main results show that the colouring protocol may be used to solve the generalized Russian cards problem in cases where a is a power of a prime, c = O(a2) and b = O(c2). This improves substantially on the set of parameters for which solutions are known to exist; in particular, it had not been shown previously that the problem could be solved in cases where the eavesdropper has more cards than one of the communicating players.Ministerio de Economía y Competitividad FFI2011-15945-EEuropean Research Council ERC Starting Grant EPS 313360Junta de Andalucía P08-HUM-0415