Eliminating Unbounded Search in Computable Algebra

Klaimullin, Melnikov and Ng [KMNa] have recently suggested a new systematic approach to algorithms in algebra which is intermediate between computationally feasible algebra [CR91, KNRS07] and abstract computable structure theory [AK00, EG00]. In this short survey we discuss some of the key results and ideas of this new topic [KMNa, KMNc, KMNb]. We also suggest several open problems.

[1]  André Nies,et al.  Automatic Structures: Richness and Limitations , 2004, LICS.

[2]  A. Montalbán,et al.  A computability theoretic equivalent to Vaught’s conjecture , 2012, 1206.5682.

[3]  Richard Lawrence Smith,et al.  Two theorems on autostability in p-Groups , 1981 .

[4]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

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

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

[7]  Anil Nerode,et al.  Automatic Presentations of Structures , 1994, LCC.

[8]  André Nies,et al.  Finite automata presentable abelian groups , 2009, Ann. Pure Appl. Log..

[9]  S. Goncharov.  Countable Boolean Algebras and Decidability , 1997 .

[10]  Antonio Montalbán,et al.  COMPUTABLE POLISH GROUP ACTIONS , 2018, The Journal of Symbolic Logic.

[11]  A. Nies,et al.  FA-presentable groups and rings , 2008 .

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

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

[14]  R. Dedekind,et al.  Was sind und was sollen die Zahlen , 1961 .

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

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

[17]  P. E. Alaev.  Existence and Uniqueness of Structures Computable in Polynomial Time , 2016 .

[18]  Ana Sokolova,et al.  Generic Trace Semantics via Coinduction , 2007, Log. Methods Comput. Sci..

[19]  G. Higman.  Subgroups of finitely presented groups , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[20]  Universityof Aukland,et al.  Effective Model Theory : The Number of Models and Their Complexity 1 , 1999 .

[21]  Dugald Macpherson,et al.  A survey of homogeneous structures , 2011, Discret. Math..

[22]  C. Ash,et al.  Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees , 1986 .

[23]  W. W. Boone,et al.  THE WORD PROBLEM. , 1958, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Todor Tsankov,et al.  The additive group of the rationals does not have an automatic presentation , 2009, The Journal of Symbolic Logic.

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

[26]  Gábor Braun,et al.  Breaking up Finite Automata Presentable torsion-Free Abelian Groups , 2011, Int. J. Algebra Comput..

[27]  Douglas A. Cenzer,et al.  Polynomial-Time versus Recursive Models , 1991, Ann. Pure Appl. Log..

[28]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[29]  B. L. Waerden.  Eine Bemerkung über die Unzerlegbarkeit von Polynomen , 1930 .

[30]  V. P. Dobritsa.  Some constructivizations of Abelian groups , 1983 .

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

[32]  Jeffrey B. Remmel,et al.  Polynomial-time versus computable Boolean algebras , 1999 .

[33]  J. Shepherdson,et al.  Effective procedures in field theory , 1956, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

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

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

[36]  Jeffrey B. Remmel,et al.  Recursive Boolean algebras with recursive atoms , 1981, Journal of Symbolic Logic.

[37]  Denis R. Hirschfeldt,et al.  A computably categorical structure whose expansion by a constant has infinite computable dimension , 2003, J. Symb. Log..