Skip to main content Accessibility help
×
Home

AUTOMATIC AND POLYNOMIAL-TIME ALGEBRAIC STRUCTURES

  • NIKOLAY BAZHENOV (a1), MATTHEW HARRISON-TRAINOR (a2), ISKANDER KALIMULLIN (a3), ALEXANDER MELNIKOV (a4) and KENG MENG NG (a5)...

Abstract

A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1 $ -complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

Copyright

References

Hide All
[1]Abu Zaid, F., Grädel, E., and Reinhardt, F., Advice automatic structures and uniformly automatic classes, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017) (Goranko, V. and Dam, M., editors), Leibniz International Proceedings in Informatics (LIPIcs), vol. 82, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2017, pp. 35:1–35:20.
[2]Alaev, P. E., Existence and uniqueness of structures computable in polynomial-time. Algebra and Logic, vol. 55 (2016), no. 1, pp. 7276.
[3]Ash, C. and Knight, J., Computable Structures and the Hyperarithmetical Hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland, Amsterdam, 2000.
[4]Bennett, C. H., Logical reversibility of computation. IBM Journal of Research and Development, vol. 17 (1973), pp. 525532.
[5]Blumensath, A. and Grädel, E., Automatic structures, 15th Annual IEEE Symposium on Logic in Computer Science (Santa Barbara, CA, 2000) (Williams, A. D., editor), IEEE Computer Society Press, Los Alamitos, CA, 2000, pp. 5162.
[6]Braun, G. and Strüngmann, L., Breaking up finite automata presentable torsion-free abelian groups. International Journal of Algebra and Computation, vol. 21 (2011), no. 8, pp. 14631472.
[7]Brumleve, D., Hamkins, J. D., and Schlicht, P., The mate-in-n problem of infinite chess is decidable, How the World Computes (Cooper, S. B., Dawar, A., and Löwe, B., editors), Lecture Notes in Computer Science, vol. 7318, Springer, Heidelberg, 2012, pp. 7888.
[8]Calvert, W., Fokina, E., Goncharov, S. S., Knight, J. F., Kudinov, O., Morozov, A. S., and Puzarenko, V., Index sets for classes of high rank structures, this Journal, vol. 72 (2007), no. 4, pp. 14181432.
[9]Carson, J., Harizanov, V., Knight, J., Lange, K., McCoy, C., Morozov, A., Quinn, S., Safranski, C., and Wallbaum, J., Describing free groups. Transactions of the American Mathematical Society, vol. 364 (2012), no. 11, pp. 57155728.
[10]Cenzer, D., Downey, R. G., Remmel, J. B., and Uddin, Z., Space complexity of abelian groups. Archive for Mathematical Logic, vol. 48 (2009), no. 1, pp. 115140.
[11]Cenzer, D. and Remmel, J., Polynomial-time versus recursive models. Annals of Pure and Applied Logic, vol. 54 (1991), no. 1, pp. 1758.
[12]Cenzer, D. and Remmel, J., Feasibly categorical abelian groups, Feasible mathematics, II (Ithaca, NY, 1992) (Clote, P. and Remmel, J. B., editors), Progress in Computer Science and Applied Logic, vol. 13, Birkhäuser Boston, Boston, MA, 1995, pp. 91153.
[13]Cenzer, D. and Remmel, J. B., Polynomial time versus computable boolean algebras, Recursion Theory and Complexity, Proceedings 1997 Kazan Workshop (Arslanov, M. and Lempp, S., editors), de Gruyter, Berlin, 1999, pp. 1553.
[14]Cenzer, D. A. and Remmel, J. B., Polynomial-time abelian groups. Annals of Pure and Applied Logic, vol. 56 (1992), no. 1–3, pp. 313363.
[15]Delhommé, C., Automaticité des ordinaux et des graphes homogènes. Comptes rendus de l’Académie des Sciences Paris, vol. 339 (2004), no. 1, pp. 510.
[16]Downey, R. and Melnikov, A. G., Computable completely decomposable groups. Transactions of the American Mathematical Society, vol. 366 (2014), no. 8, pp. 42434266.
[17]Downey, R. G., Kach, A. M., Lempp, S., Lewis-Pye, A. E. M., Montalbán, A., and Turetsky, D. D., The complexity of computable categoricity. Advances in Mathematics, vol. 268 (2015), pp. 423466.
[18]Downey, R. G. and Montalbán, A., The isomorphism problem for torsion-free abelian groups is analytic complete. Journal of Algebra, vol. 320 (2008), no. 6, pp. 22912300.
[19]Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S., and Thurston, W. P., Word Processing in Groups, Jones and Bartlett Publishers, Boston, MA, 1992.
[20]Ershov, Y. and Goncharov, S., Constructive Models, Siberian School of Algebra and Logic, Consultants Bureau, New York, 2000.
[21]Fokina, E. B., Index sets of decidable models. Sibirskii Matematicheskii Zhurnal, vol. 48 (2007), no. 5, pp. 11671179.
[22]Fokina, E. B., Goncharov, S. S., Kharizanova, V., Kudinov, O. V., and Turetski, D., Index sets of n-decidable structures that are categorical with respect to m-decidable representations. Algebra Logika, vol. 54 (2015), no. 4, pp. 520528, 544–545, 547–548.
[23]Fröhlich, A. and Shepherdson, J. C., Effective procedures in field theory. Philosophical Transactions of the Royal Society of London. Series A, vol. 248 (1956), pp. 407432.
[24]Fuchs, L., Infinite Abelian Groups. vol. I, Pure and Applied Mathematics, vol. 36, Academic Press, New York, 1970.
[25]Fuchs, L., Infinite Abelian Groups. vol. II, Pure and Applied Mathematics. vol. 36-II, Academic Press, New York, 1973.
[26]Gončarov, S. S., The number of nonautoequivalent constructivizations. Algebra i Logika, vol. 16 (1977), no. 3, pp. 257282, 377.
[27]Goncharov, S. S., Bazhenov, N. A., and Marchuk, M. I., The index set of Boolean algebras that are autostable relative to strong constructivizations. Sibirskii Matematicheskii Zhurnal, vol. 56 (2015), no. 3, pp. 498512.
[28]Goncharov, S. S., Bazhenov, N. A., and Marchuk, M. I., Index sets of constructive models of natural classes that are autostable with respect to strong constructivizations. Doklady Akademii Nauk, vol. 464 (2015), no. 1, pp. 1214.
[29]Goncharov, S. S. and Naĭt, D., Computable structure and antistructure theorems. Algebra Logika, vol. 41 (2002), no. 6, pp. 639681, 757.
[30]Grigorieff, S., Every recursive linear ordering has a copy in dtime-space(n, log(n)), this Journal, vol. 55 (1990), no. 1, pp. 260276.
[31]Harrison, J., Recursive pseudo-well-orderings. Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526543.
[32]Harrison-Trainor, M., There is no classification of the decidably presentable structures. Journal of Mathematical Logic, vol. 18 (2018), no. 2, 1850010.
[33]Hjorth, G., The isomorphism relation on countable torsion free abelian groups. Fundamenta Mathematicae, vol. 175 (2002), no. 3, pp. 241257.
[34]Hodgson, B. R., On direct products of automaton decidable theories. Theoretical Computer Science, vol. 19 (1982), no. 3, pp. 331335.
[35]Jain, S., Khoussainov, B., and Stephan, F., Finitely generated semiautomatic groups. Computability, vol. 7 (2018), no. 2–3, pp. 273287.
[36]Jain, S., Khoussainov, B., Stephan, F., Teng, D., and Zou, S., Semiautomatic structures. Theory of Computing Systems, vol. 61 (2017), no. 4, pp. 12541287.
[37]Kalimullin, I., Melnikov, A., and Ng, K. M., Algebraic structures computable without delay. Theoretical Computer Science, vol. 674 (2017), pp. 7398.
[38]Kalimullin, I. S., Melnikov, A. G., and Ng, K. M., The diversity of categoricity without delay. Algebra and Logic, vol. 56 (2017), no. 2, pp. 171177.
[39]Kharlampovich, O., Khoussainov, B., and Myasnikov, A., From automatic structures to automatic groups. Groups, Geometry, and Dynamics, vol. 8 (2014), no. 1, pp. 157198.
[40]Khoussainov, B., Liu, J., and Minnes, M., Unary automatic graphs: An algorithmic perspective. Mathematical Structures in Computer Science, vol. 19 (2009), no. 1, pp. 133152.
[41]Khoussainov, B. and Minnes, M., Model-theoretic complexity of automatic structures. Annals of Pure and Applied Logic, vol. 161 (2009), no. 3, pp. 416426.
[42]Khoussainov, B. and Minnes, M., Three lectures on automatic structures, Logic Colloquium 2007 (Delon, F., Kohlenbach, U., Maddy, P., and Stephan, F., editors), Lecture Notes in Logic, vol. 35, Association for Symbolic Logic, La Jolla, CA, 2010, pp. 132176.
[43]Khoussainov, B. and Nerode, A., Automatic presentations of structures, Logic and Computational Complexity (Indianapolis, IN, 1994) (Leivant, D., editor), Lecture Notes in Computer Science, vol. 960, Springer, Berlin, 1995, pp. 367392.
[44]Khoussainov, B. and Nerode, A., Open questions in the theory of automatic structures. Bulletin of the European Association for Theoretical Computer Science (EATCS), (2008), no. 94, pp. 181204.
[45]Khoussainov, B., Nies, A., Rubin, S., and Stephan, F., Automatic structures: Richness and limitations. Logical Methods in Computer Science, vol. 3 (2007), no. 2, pp. 2:2, 18.
[46]Khoussainov, B. and Rubin, S., Automatic structures: Overview and future directions. Journal of Automata, Languages and Combinatorics, vol. 8 (2003), no. 2, pp. 287301.
[47]Lachlan, A. H., On some games which are relevant to the theory of recursively enumerable sets. Annals of Mathematics (2), vol. 91 (1970), pp. 291310.
[48]Lempp, S. and Slaman, T. A., The complexity of the index sets of $\aleph _0 $-categorical theories and of Ehrenfeucht theories, Advances in Logic (Gao, S., Jackson, S., and Zhang, Y., editors), Contemporary Mathematics, vol. 425, American Mathematical Society, Providence, RI, 2007, pp. 4347.
[49]Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprinted of the 1977 edition.
[50]Mal’cev, A., Constructive algebras. I. Uspekhi Matematicheskikh Nauk, vol. 16 (1961), no. 3 (99), pp. 360.
[51]McCoy, C. and Wallbaum, J., Describing free groups, Part II:${\rm{\Pi }}_4^0 $ hardness and no ${\rm{\Sigma }}_2^0 $ basis . Transactions of the American Mathematical Society , vol. 364 (2012), no. 11, pp. 57295734.
[52]Melnikov, A. G., Eliminating unbounded search in computable algebra, Unveiling Dynamics and Complexity (Kari, J., Manea, F., and Petre, I., editors), Lecture Notes in Computer Science, vol. 10307, Springer, Cham, 2017, pp. 7787.
[53]Nerode, A. and Remmel, J. B., Polynomial time equivalence types, Logic and Computation (Pittsburgh, PA, 1987) (Sieg, W., editor), Contemporary Mathematics, vol. 106, American Mathematical Society, Providence, RI, 1990, pp. 221249.
[54]Nies, A. and Semukhin, P., Finite automata presentable abelian groups. Annals of Pure and Applied Logic, vol. 161 (2009), no. 3, pp. 458467.
[55]Nurtazin, A. T., Strong and weak constructivization and computable families. Algebra and Logic, vol. 13 (1974), no. 3, pp. 177184.
[56]Oliver, G. P. and Thomas, R. M., Automatic presentations for finitely generated groups, STACS 2005 (Diekert, V. and Durand, B., editors), Lecture Notes in Computer Science, vol. 3404, Springer, Berlin, 2005, pp. 693704.
[57]Rabin, M., Computable algebra, general theory and theory of computable fields. Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341360.
[58]Riggs, K., The decomposability problem for torsion-free abelian groups is analytic complete. Proceedings of the American Mathematical Society, vol. 143 (2015), no. 8, pp. 36313640.
[59]Rogers, H., Theory of Recursive Functions and Effective Computability, second ed., MIT Press, Cambridge, MA, 1987.
[60]Selivanov, V. L., Enumerations of families of general recursive functions. Algebra and Logic, vol. 15 (1976), no. 2, pp. 128141.
[61]Soare, R., Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987, A study of computable functions and computably generated sets.
[62]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151163.
[63]Thomas, S., The classification problem for torsion-free abelian groups of finite rank. Journal of the American Mathematical Society, vol. 16 (2003), no. 1, pp. 233258.
[64]Tsankov, T., The additive group of the rationals does not have an automatic presentation, this Journal, vol. 76 (2011), no. 4, pp. 13411351.
[65]van der Waerden, B., Eine Bemerkung über die Unzerlegbarkeit von Polynomen. Mathematische Annalen, vol. 102 (1930), no. 1, pp. 738739.

Keywords

AUTOMATIC AND POLYNOMIAL-TIME ALGEBRAIC STRUCTURES

  • NIKOLAY BAZHENOV (a1), MATTHEW HARRISON-TRAINOR (a2), ISKANDER KALIMULLIN (a3), ALEXANDER MELNIKOV (a4) and KENG MENG NG (a5)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.