The Borel Complexity of Isomorphism for Some First Order Theories

In this work we consider several instances of the following problem: "how complicated can the isomorphism relation for countable models be?"' Using the Borel reducibility framework, we investigate this question with regard to the space of countable models of particular complete first-order theories. We also investigate to what extent this complexity is mirrored in the number of back-and-forth inequivalent models of the theory. We consider this question for two large and related classes of theories. First, we consider o-minimal theories, showing that if T is o-minimal, then the isomorphism relation is either Borel complete or Borel. Further, if it is Borel, we characterize exactly which values can occur, and when they occur. In all cases Borel completeness implies lambda-Borel completeness for all lambda. Second, we consider colored linear orders, which are (complete theories of) a linear order expanded by countably many unary predicates. We discover the same characterization as with o-minimal theories, taking the same values, with the exception that all finite values are possible except two. We characterize exactly when each possibility occurs, which is similar to the o-minimal case. Additionally, we extend Schirrman's theorem, showing that if the language is finite, then T is countably categorical or Borel complete. As before, in all cases Borel completeness implies lambda-Borel completeness for all lambda.

THE RELATIVE SIGNIFICANCE OF SYNTACTIC KNOWLEDGE AND VOCABULARY KNOWLEDGE IN SECOND LANGUAGE LISTENING COMPREHENSION

The main purpose of the current study was to examine the role of vocabulary knowledge (VK) and syntactic knowledge (SK) in L2 listening comprehension, as well as their relative significance. Unlike previous studies, the current project employed assessment tasks to measure aural and proceduralized VK and SK. In terms of VK, to avoid under-representing the construct, measures of both breadth (VB) and depth (VD) were included. Additionally, the current study examined the role of VK and SK by accounting for individual differences in two important cognitive factors in L2 listening: metacognitive knowledge (MK) and working memory (WM). Also, to explore the role of VK and SK more fully, the current study accounted for the negative impact of anxiety on WM and L2 listening. The study was carried out in an English as a Foreign Language (EFL) context, and participants were 263 Iranian learners at a wide range of English proficiency from lower-intermediate to advanced. Participants took a battery of ten linguistic, cognitive and affective measures. Then, the collected data were subjected to several preliminary analyses, but structural equation modeling (SEM) was then used as the primary analysis method to answer the study research questions. Results of the preliminary analyses revealed that MK and WM were significant predictors of L2 listening ability; thus, they were kept in the main SEM analyses. The significant role of WM was only observed when the negative effect of anxiety on WM was accounted for. Preliminary analyses also showed that VB and VD were not distinct measures of VK. However, the results also showed that if VB and VD were considered separate, VD was a better predictor of L2 listening success. The main analyses of the current study revealed a significant role for both VK and SK in explaining success in L2 listening comprehension, which differs from findings from previous empirical studies. However, SEM analysis did not reveal a statistically significant difference in terms of the predictive power of the two linguistic factors. Descriptive results of the SEM analysis, along with results from regression analysis, indicated to a more significant role for VK.