Wadge-like reducibilities on arbitrary quasi-Polish spaces

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called \Delta^0_\alpha-reductions, and try to find for various natural topological spaces X the least ordinal \alpha_X such that for every \alpha_X \leq \beta < \omega_1 the degree-structure induced on X by the \Delta^0_\beta-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that \alpha_X \leq {\omega} for every quasi-Polish space X, that \alpha_X \leq 3 for quasi-Polish spaces of dimension different from \infty, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.Comment: 50 pages, revised version, accepted for publication on Mathematical Structures in Computer Scienc

Categorical Comprehensions and Recursion

A new categorical setting is defined in order to characterize the subrecursive classes belonging to complexity hierarchies. This is achieved by means of coercion functors over a symmetric monoidal category endowed with certain recursion schemes that imitate the bounded recursion scheme. This gives a categorical counterpart of generalized safe composition and safe recursion.Comment: Comments are welcom

Borrowing in Apparent Time: With some comments on attitudes and universals

Borrowing is often seen as a threat by speakers of minority or endangered languages (King 2008, Dubois and Melançons 1997) but linguists may be more likely to see it as a natural, and potentially revealing, resource of bilingual speakers. This paper uses the sociolinguistic construct of apparent time to explore borrowing in an endangered language further. If borrowing is an index of communal language shift, we might expect to find differences in apparent time (cf. Labov 2008, Meakins 2011). Data comes from Hog Harbour, a community in Vanuatu, where the 1000 speakers are concerned about the continued vitality of their local language and point to the borrowing of Bislama words as a sign of its decline. We show that there is no clear sociolinguistic evidence that borrowing is increasing over time in the community: it is possible that younger speakers’ use of Bislama words may be a developmental phenomenon, not communal change in progress. We suggest that Matras’ (2012) analysis of interactional and cognitive pressure points in conversation accounts very well for the patterns observed

Introduction to the Baire space

This report reproduces the chapter 0 of my PhD dissertation, "Reducibility and determinateness on the Baire Space" . I have produced it as a Warwick Theory of Computation report because infinite games and the computational approach to topology presented here is, I feel, very relevant to computer science. The basic connection between topology and computability, explained in section E, is as follows: a function from the Baire space to itself is continuous if it can be computed by a continuously operating numeric 'filter' which has access to a countably infinite database. It used to be thought that infinite computations (not to mention infinite games) were of little relevance to practical computing, which (it was thought) was inherently finitary. The emergence of the dataflow model of computation (among other factors) has changed all this; computer scientists are now keenly interested in the behaviour and properties of continuously operating (nonterminatinq) devices. The entire history of the input to, or output from, such a device will normally be an infinite sequence of finite objects. Such a history can be 'coded up' as an infinite sequence of natural numbers; i.e., as an element of the Baire Space. The study of this space could therefore prove to be as important in computer science as it has already proved to be in (say) statistics. Of course I make no claims to have discovered the material presented here. It has been known for many years now that "computability" and "continuity" are closely related. However, I hope that this report will in a small way help to popularise the 'topological' approach to computation. I would like to thank John Addison for Introducing me to the 'operational' approach to topology. Readers interested in the dissertation itself are referred to Theory of Computation report no 44, which ls a collection of the more important parts