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

First Order Theories of Some Lattices of Open Sets

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic

The Rice-Shapiro theorem in Computable Topology

We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. We show that the relaxation of these requirements leads to the classes of effectively enumerable topological spaces where the Rice-Shapiro theorem does not hold. We propose two constructions that generate effectively enumerable topological spaces with particular properties from wn--families and computable trees without computable infinite paths. Using them we propose examples that give a flavor of this class

Aperiodic Subshifts of Finite Type on Groups

In this note we prove the following results: $\bullet$ If a finitely presented group $G$ admits a strongly aperiodic SFT, then $G$ has decidable word problem. More generally, for f.g. groups that are not recursively presented, there exists a computable obstruction for them to admit strongly aperiodic SFTs. $\bullet$ On the positive side, we build strongly aperiodic SFTs on some new classes of groups. We show in particular that some particular monster groups admits strongly aperiodic SFTs for trivial reasons. Then, for a large class of group $G$, we show how to build strongly aperiodic SFTs over $\mathbb{Z}\times G$. In particular, this is true for the free group with 2 generators, Thompson's groups $T$ and $V$, $PSL_2(\mathbb{Z})$ and any f.g. group of rational matrices which is bounded.Comment: New version. Adding results about monster group

The descriptive theory of represented spaces

This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

The Computable Universe Hypothesis

When can a model of a physical system be regarded as computable? We provide the definition of a computable physical model to answer this question. The connection between our definition and Kreisel's notion of a mechanistic theory is discussed, and several examples of computable physical models are given, including models which feature discrete motion, a model which features non-discrete continuous motion, and probabilistic models such as radioactive decay. We show how computable physical models on effective topological spaces can be formulated using the theory of type-two effectivity (TTE). Various common operations on computable physical models are described, such as the operation of coarse-graining and the formation of statistical ensembles. The definition of a computable physical model also allows for a precise formalization of the computable universe hypothesis--the claim that all the laws of physics are computable.Comment: 33 pages, 0 figures; minor change

Computational Processes and Incompleteness

We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process