Abstract
Autostability spectra for Boolean algebras and their various enrichments are explored. The basic notions and definitions of computable structure theory used in the paper can be found in [1, 2]. As usual, we write “c.e.” as an abbreviation for “computably enumerable.” Recall that a computable structure M is said to be decidable if its full diagram FD(M) is computable. A structure M is strongly constructivizable if M has a decidable copy. Let d be a Turing degree. A constructivizable structure A is said to be d-autostable if, for any computable copies B and C of A, there exists a d-computable isomorphism f : B → C. A strongly constructivizable structure A is d-autostable relative to strong constructivizations if, for any decidable copies B and C for A, there exists a d-computable isomorphism f : B → C. If A is a constructivizable structure, then the autostability spectrum of A is the set of all Turing degrees d for which the structure A is d-autostable. The autostability spectrum relative to strong constructivizations for a strongly constructivizable structure A is the set of all degrees d such that A is d-autostable relative to strong constructivizations. Denote the autostability spectrum of a structure A by AutSpec (A), and the autostability spectrum relative to strong constructivizations for A by SCAutSpec (A). We say that a Turing degree d is the degree of autostability (degree of autostability relative to strong constructivizations) for A if d is least in AutSpec (A) (in SCAutSpec (A)). Note that in ∗Supported by RFBR (project No. 14-01-00376), by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-860.2014.1), and by the Russian Ministry of Education and Science (gov. contract No. 2014/139).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have