Non-density in punctual computability

Abstract In computable structure theory, one considers computable presentations of abstract structures such as graphs or groups, and one thinks of two different computable presentations as being essentially the same if there is a computable isomorphism between them. Because the inverse of a computable function is also computable, the relation of being computably isomorphic is an equivalence relation, and so the only structure on the set of computable presentations is the number of non-equivalent presentations. Recently there has been increased interest in primitive recursive presentations of structures, and in this setting, the inverse of a primitive recursive function is not necessarily primitive recursive, and so we get a relation of reducibility between structures which induces a partial pre-ordering on the primitive recursive presentations of a structure. Whenever we have a reducibility notion, one of the natural first questions is whether it is dense. We show that it is not dense: There are primitive recursive presentations A ≅ B of the same abstract structure, such that A is reducible to B (there is a primitive recursive isomorphism A → B ) but B is not reducible to A (there is no primitive recursive isomorphism B → A ), and for any third primitive recursive presentation M of the same structure, if A is reducible to M and M is reducible to B , then either M is reducible to A or B is reducible to M .

[1]  M. Rabin Computable algebra, general theory and theory of computable fields. , 1960 .

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

[3]  A. G. Melnikov,et al.  The Diversity of Categoricity Without Delay , 2017 .

[4]  Denis R. Hirschfeldt Degree spectra of intrinsically c.e. relations , 2001, Journal of Symbolic Logic.

[5]  Hal A. Kierstead,et al.  On-Line Coloring and Recursive Graph Theory , 1994, SIAM J. Discret. Math..

[6]  Serge Grigorieff,et al.  Every recursive linear ordering has a copy in DTIME-SPACE(n,log(n)) , 1990, Journal of Symbolic Logic.

[7]  Keng Meng Ng,et al.  The back-and-forth method and computability without delay , 2019, Israel Journal of Mathematics.

[8]  Matthew Harrison-Trainor,et al.  AUTOMATIC AND POLYNOMIAL-TIME ALGEBRAIC STRUCTURES , 2019, The Journal of Symbolic Logic.

[9]  Keng Meng Ng,et al.  Online presentations of finitely generated structures , 2020, Theor. Comput. Sci..

[10]  N. S. Romanovskii,et al.  Nilpotent groups of finite algorithmic dimension , 1989 .

[11]  Douglas A. Cenzer,et al.  Space complexity of Abelian groups , 2009, Arch. Math. Log..

[12]  PUNCTUAL CATEGORICITY AND UNIVERSALITY , 2020, The Journal of Symbolic Logic.

[13]  Charles F. D. McCoy Finite computable dimension does not relativize , 2002, Arch. Math. Log..

[14]  P. E. Alaev,et al.  Structures Computable in Polynomial Time. I , 2017 .

[15]  Iskander Sh. Kalimullin,et al.  FOUNDATIONS OF ONLINE STRUCTURE THEORY , 2019, The Bulletin of Symbolic Logic.

[16]  S. S. Goncharov,et al.  Problem of the number of non-self-equivalent constructivizations , 1980 .

[17]  Douglas A. Cenzer,et al.  Polynomial-Time Abelian Groups , 1992, Ann. Pure Appl. Log..

[18]  Anil Nerode,et al.  Open Questions in the Theory of Automatic Structures , 2008, Bull. EATCS.

[19]  Keng Meng Ng,et al.  Algebraic structures computable without delay , 2017, Theor. Comput. Sci..

[20]  Douglas Cenzer,et al.  Complexity Theoretic Model Theory and Algebra , 2013 .

[21]  Arkadii M. Slinko,et al.  Degree spectra and computable dimensions in algebraic structures , 2002, Ann. Pure Appl. Log..

[22]  K. M. Ng,et al.  A structure of punctual dimension two , 2020 .

[23]  Steffen Lempp,et al.  The computable dimension of ordered abelian groups , 2003 .

[24]  A. I. Mal'tsev CONSTRUCTIVE ALGEBRAS I , 1961 .