Computable categoricity of graphs with finite components

A computable graph is computably categorical if any two computable presentations of the graph are computably isomorphic. In this paper we investigate the class of computably categorical graphs. We restrict ourselves to strongly locally finite graphs; these are the graphs all of whose components are finite. We present a necessary and sufficient condition for certain classes of strongly locally finite graphs to be computably categorical. We prove that if there exists an infinite \Delta^0_2-set of components that can be properly embedded into infinitely many components of the graph then the graph is not computably categorical. We outline the construction of a strongly locally finite computably categorical graph with an infinite chain of properly embedded components

Degree spectra and computable dimensions in algebraic structures

AbstractWhenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings (with zero-divisors); integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories

Degree spectra of relations on structures of finite computable dimension

AbstractWe show that for every computably enumerable (c.e.) degree a>0 there is an intrinsically c.e. relation on the domain of a computable structure of computable dimension 2 whose degree spectrum is {0,a}, thus answering a question of Goncharov and Khoussainov (Dokl. Math. 55 (1997) 55–57). We also show that this theorem remains true with α-c.e. in place of c.e. for any α∈ω∪{ω}. A modification of the proof of this result similar to what was done in Hirschfeldt (J. Symbolic Logic, to appear) shows that for any α∈ω∪{ω} and any α-c.e. degrees a0,…,an there is an intrinsically α-c.e. relation on the domain of a computable structure of computable dimension n+1 whose degree spectrum is {a0,…,an}. These results also hold for m-degree spectra of relations

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity

Model Theory and Groups

The aim of the workshop was to discuss the connections between model theory and group theory. Main topics have been the interaction between geometric group theory and model theory, the study of the asymptotic behaviour of geometric properties on groups, and the model theoretic investigations of groups of finite Morley rank around the Cherlin-Zilber Conjecture