Structural Highness Notions

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of backand-forth relations), jump hierarchies, and computing descending sequences in linear orders.

[1]  Julia A. Knight,et al.  Computable structures and the hyperarithmetical hierarchy , 2000 .

[2]  Richard A. Shore,et al.  1 1 relations and paths through O , 2004 .

[3]  Danna Zhou,et al.  d. , 1840, Microbial pathogenesis.

[4]  Robert I. Soare,et al.  The Friedberg-Muchnik Theorem Re-Examined , 1972, Canadian Journal of Mathematics.

[5]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[6]  Barbara Csima.  Degrees of Categoricity and Related Notions , 2013 .

[7]  Johanna N. Y. Franklin,et al.  Taking the path computably traveled , 2019, J. Log. Comput..

[8]  Theodore A. Slaman,et al.  When oracles do not help , 1991, COLT '91.

[9]  Haim Judah,et al.  Set Theory: On the Structure of the Real Line , 1995 .

[10]  Johanna N. Y. Franklin,et al.  Lowness for isomorphism, countable ideals, and computable traceability , 2019, Math. Log. Q..

[11]  Johanna N. Y. Franklin,et al.  Degrees of and lowness for isometric isomorphism , 2019, J. Log. Anal..

[12]  Julia F. Knight,et al.  Isomorphism relations on computable structures , 2012, J. Symb. Log..

[13]  Johanna N. Y. Franklin,et al.  Degrees that Are Low for Isomorphism , 2014, Comput..

[14]  G. Sacks.  Higher recursion theory , 1990 .

[15]  Johanna N. Y. Franklin,et al.  Relativizations of randomness and genericity notions , 2011 .

[16]  Johanna N. Y. Franklin,et al.  Lowness for isomorphism and degrees of genericity , 2018, Comput..

[17]  Johanna N. Y. Franklin.  LOWNESS AND HIGHNESS PROPERTIES FOR RANDOMNESS NOTIONS , 2009 .

[18]  Stuart A. Kurtz,et al.  Notions of Weak Genericity , 1983, J. Symb. Log..

[19]  Stephen G. Simpson,et al.  A degree-theoretic definition of the ramified analytical hierarchy , 1976 .

[20]  Julia F. Knight,et al.  Computable trees of Scott rank ω1CK, and computable approximation , 2006, Journal of Symbolic Logic.

[21]  Michael Nedzelsky,et al.  Recursion Theory I , 2008, Arch. Formal Proofs.