Strong jump inversion Article Swipe
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· 2018
· Open Access
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· DOI: https://doi.org/10.1093/logcom/exy025
· OA: W2888323212
We say that a structure $\\mathcal{A}$ admits \\emph{strong jump inversion}\nprovided that for every oracle $X$, if $X'$ computes $D(\\mathcal{C})'$ for some\n$\\mathcal{C}\\cong\\mathcal{A}$, then $X$ computes $D(\\mathcal{B})$ for some\n$\\mathcal{B}\\cong\\mathcal{A}$. Jockusch and Soare \\cite{JS} showed that there\nare low linear orderings without computable copies, but Downey and Jockusch\n\\cite{DJ} showed that every Boolean algebra admits strong jump inversion. More\nrecently, D.\\ Marker and R.\\ Miller \\cite{MM} have shown that all countable\nmodels of $DCF_0$ (the theory of differentially closed fields of characteristic\n$0$) admit strong jump inversion. We establish a general result with sufficient\nconditions for a structure $\\mathcal{A}$ to admit strong jump inversion. Our\nconditions involve an enumeration of $B_1$-types, where these are made up of\nformulas that are Boolean combinations of existential formulas. Our general\nresult applies to some familiar kinds of structures, including some classes of\nlinear orderings and trees. We do not get the result of Downey and Jockusch for\narbitrary Boolean algebras, but we do get a result for Boolean algebras with no\n$1$-atom, with some extra information on the complexity of the isomorphism. Our\ngeneral result gives the result of Marker and Miller. In order to apply our\ngeneral result, we produce a computable enumeration of the types realized in\nmodels of $DCF_0$. This also yields the fact that the saturated model of\n$DCF_0$ has a decidable copy.\n
