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.

Language Outcomes of the Play and Language for Autistic Youngsters (PLAY) Project Home Consultation model—An Extended Analysis

The current study is a post-hoc analysis of data from the original randomized control trial of the Play and Language for Autistic Youngsters (PLAY) Home Consultation program, a parent-mediated, DIR/Floortime based early intervention program for children with ASD (Solomon, Van Egeren, Mahone, Huber, & Zimmerman, 2014). We examined 22 children from the original RCT who received the PLAY program. Children were split into two groups (high and lower functioning) based on the ADOS module administered prior to intervention. Fifteen-minute parent-child video sessions were coded through the use of CHILDES transcription software. Child and maternal language, communicative behaviors, and communicative functions were assessed in the natural language samples both pre- and post-intervention. Results demonstrated significant improvements in both child and maternal behaviors following intervention. There was a significant increase in child verbal and non-verbal initiations and verbal responses in whole group analysis. Total number of utterances, word production, and grammatical complexity all significantly improved when viewed across the whole group of participants; however, lexical growth did not reach significance. Changes in child communicative function were especially noteworthy, and demonstrated a significant increase in social interaction and a significant decrease in non-interactive behaviors. Further, mothers demonstrated an increase in responsiveness to the child’s conversational bids, increased ability to follow the child’s lead, and a decrease in directiveness. When separated for analyses within groups, trends emerged for child and maternal variables, suggesting greater gains in use of communicative function in both high and low groups over changes in linguistic structure. Additional analysis also revealed a significant inverse relationship between maternal responsiveness and child non-interactive behaviors; as mothers became more responsive, children’s non-engagement was decreased. Such changes further suggest that changes in learned skills following PLAY parent training may result in improvements in child social interaction and language abilities.