Primitive Recursive Ordered Fields and Some Applications

We establish primitive recursive versions of some known facts about computable ordered fields of reals and computable reals, and then apply them to proving primitive recursiveness of some natural problems in linear algebra and analysis. In particular, we find a partial primitive recursive analogue of Ershov-Madison's theorem about real closures of computable ordered fields, relate the corresponding fields to the primitive recursive reals, give sufficient conditions for primitive recursive root-finding, computing normal forms of matrices, and computing solution operators of some linear systems of PDE.

[1]  R. Loos.  Computing in Algebraic Extensions , 1983 .

[2]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[3]  Martin Ziegler,et al.  A Computable Spectral Theorem , 2000, CCA.

[4]  Akitoshi Kawamura,et al.  Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

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

[6]  V. L. Selivanov,et al.  On Constructive number fields and computability of solutions of PDEs , 2017 .

[7]  Victor L. Selivanov,et al.  Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs , 2013, Log. Methods Comput. Sci..

[8]  Martin Ziegler,et al.  Bit-Complexity of Solving Systems of Linear Evolutionary Partial Differential Equations , 2021, CSR.

[9]  I. Sh. Kalimullin,et al.  Primitive Recursive Fields and Categoricity , 2019, Algebra and Logic.

[10]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[11]  Akitoshi Kawamura,et al.  Parameterized Complexity for Uniform Operators on Multidimensional Analytic Functions and ODE Solving , 2018, WoLLIC.

[12]  I. M. Gelʹfand.  Lectures on linear algebra , 1963 .

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

[14]  A. I. Mal'cev.  Algorithms and Recursive Functions , 1970 .

[15]  Russell G. Miller.  Is it harder to factor a polynomial or to find a root , 2010 .

[16]  Mohd. Zubair Khan.  Lectures on linear algebra , 2008 .

[17]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

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

[19]  Viggo Stoltenberg-Hansen,et al.  Computable Rings and Fields , 1999, Handbook of Computability Theory.

[20]  Akitoshi Kawamura,et al.  On the computational complexity of the Dirichlet Problem for Poisson's Equation , 2016, Mathematical Structures in Computer Science.

[21]  Victor L. Selivanov,et al.  Polynomial-Time Presentations of Algebraic Number Fields , 2018, CiE.

[22]  P. E. Alaev,et al.  Fields of Algebraic Numbers Computable in Polynomial Time. I , 2020, Algebra i logika.

[23]  V. Selivanov,et al.  Fields of algebraic numbers computable in polynomial time. I , 2020, Algebra i logika.

[24]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[25]  Martin Ziegler,et al.  Computational Complexity of Real Powering and Improved Solving Linear Differential Equations , 2019, CSR.

[26]  Amaury Pouly,et al.  Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains , 2011, MFCS.

[27]  K. Mahler.  An inequality for the discriminant of a polynomial. , 1964 .

[28]  Russell Miller,et al.  Degree spectra of real closed fields , 2019, Arch. Math. Log..

[29]  Andreas Knobel,et al.  Constructive λ-models , 1992 .

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

[31]  Walid Gomaa.  Algebraic Characterizations of Computable Analysis Real Functions , 2011, Int. J. Unconv. Comput..

[32]  Eugene W. Madison,et al.  A note on computable real fields , 1970, Journal of Symbolic Logic.

[33]  Qingliang Chen,et al.  Primitive recursive real numbers , 2007, Math. Log. Q..

[34]  Victor L. Selivanov,et al.  Bit Complexity of Computing Solutions for Symmetric Hyperbolic Systems of PDEs (Extended Abstract) , 2018, CiE.

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

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

[37]  Victor L. Selivanov,et al.  Bit Complexity of Computing Solutions for Symmetric Hyperbolic Systems of PDEs with Guaranteed Precision , 2018, Comput..