Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of theorems in many finitely axiomatisable theories is nonrecursive, but the set of theorems for any finitely axiomatisable complete theory is recursive. Finitely presented groups might have an nonrecursive word problem, but finitely presented simple groups have a recursive word problem. In this article we introduce a topological framework based on closure spaces to show that many of these proofs can be obtained in a similar setting. We will show in particular that these statements can be generalized to cover arbitrary structures, with no finite or recursive presentation/axiomatization. This generalizes in particular work by Kuznetsov and others. Examples from first order logic and symbolic dynamics will be discussed at length

The Typical Constructible Object

International audienceBaire Category is an important concept in mathematical analysis. It provides a way of identifying the properties of typical objects and proving the existence of objects with specified properties avoiding explicit constructions. For instance it has been extensively used to better understand and separate classes of real functions such as analytic and smooth functions. Baire Category proves very useful in computability theory and computable analysis, again to understand the properties of typical objects and to prove existence results. However it cannot be used directly when studying classes of computable or computably enumerable objects: those objects are atypical. Here we show how Baire Category can be adapted to such small classes, and how one can define typical computably enumerable sets or lower semicomputable real numbers for instance

Ultrafilter spaces on the semilattice of partitions

The Stone-Cech compactification of the natural numbers bN, or equivalently, the space of ultrafilters on the subsets of omega, is a well-studied space with interesting properties. If one replaces the subsets of omega by partitions of omega, one can define corresponding, non-homeomorphic spaces of partition ultrafilters. It will be shown that these spaces still have some of the nice properties of bN, even though none is homeomorphic to bN. Further, in a particular space, the minimal height of a tree pi-base and P-points are investigated

Subshifts as Models for MSO Logic

We study the Monadic Second Order (MSO) Hierarchy over colourings of the discrete plane, and draw links between classes of formula and classes of subshifts. We give a characterization of existential MSO in terms of projections of tilings, and of universal sentences in terms of combinations of "pattern counting" subshifts. Conversely, we characterise logic fragments corresponding to various classes of subshifts (subshifts of finite type, sofic subshifts, all subshifts). Finally, we show by a separation result how the situation here is different from the case of tiling pictures studied earlier by Giammarresi et al.Comment: arXiv admin note: substantial text overlap with arXiv:0904.245

Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of theorems in many finitely axiomatisable theories is nonrecursive, but the set of theorems for any finitely axiomatisable complete theory is recursive. Finitely presented groups might have an nonrecursive word problem, but finitely presented simple groups have a recursive word problem. In this article we introduce a topological framework based on closure spaces to show that many of these proofs can be obtained in a similar setting. We will show in particular that these statements can be generalized to cover arbitrary structures, with no finite or recursive presentation/axiomatization. This generalizes in particular work by Kuznetsov and others. Examples from first order logic and symbolic dynamics will be discussed at length

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page