Classification from a computable viewpoint

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory. Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings. Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work

Trees of Scot rank ω\u3csub\u3e1\u3c/sub\u3e\u3csup\u3eCK\u3c/sub\u3e, and computable approximability

Makkai [10] produced an arithmetical structure of Scott rank ω 1 CK . In [9], Makkai\u27s example is made computable. Here we show that there are computable trees of Scott rank ω 1 CK . We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated “group trees” of [10] and [9], Using the same kind of trees, we obtain one of rank ω 1 CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind

Does ownership matter? Claimant characteristics and case outcomes in investor-state arbitration

The power of foreign investors has become a key component in debates on sovereignty and globalisation. Here the mechanism of investor-state dispute settlement (ISDS), which allows firms legal recourse against host governments, has come to the forefront of debates over corporate rights in the contemporary era. While proponents laud ISDS as a neutral and efficient means of dispute resolution, critics claim that it shields transnational corporations from the oversight of national legal systems while enhancing their ability to interfere in host state policy matters. Despite the rich literature on ISDS, we know relatively little about the identity of corporate actors who use these mechanisms. Who are these foreign firms that take governments to court? Do different companies use these mechanisms to different effect? This article examines the impact of corporate ownership on ISDS outcomes in 241 cases from 1996 to 2014. Our results suggest that ownership matters. While public and private companies demonstrate similar propensities to settle, respondent states are less likely to win cases that are brought by a publicly traded company. For host-governments, this finding is particularly significant given that public companies demand more – and are likely to win more – in damages than private companies.Introduction Explaining ISDS outcomes The costs of arbitration Public vs. private companies and investor state disputes Measuring ISDS outcomes Control variables The determinants of ISDS outcomes - ownership matters Conclusion Acknowledgements Footnotes Reference