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Algebra Logika, 2002, Volume 41, Number 6, Pages 639–681 (Mi al201)  

This article is cited in 55 scientific papers (total in 55 papers)

Computable Structure and Non-Structure Theorems

S. S. Goncharova, J. F. Knightb

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b University of Notre Dame

Abstract: In a lecture in Kazan (1977), Goncharov dubbed a number of problems regarding the classification of computable members of various classes of structures. Some of the problems seemed likely to have nice answers, while others did not. At the end of the lecture, Shore asked what would be a convincing negative result. The goal of the present article is to consider some possible answers to Shore's question. We consider structures $\mathcal A$ of some computable language, whose universes are computable sets of constants. In measuring complexity, we identify $\mathcal A$ with its atomic diagram $D(\mathcal A)$, which, via the Gödel numbering, may be treated as a subset of $\omega$. In particular, $\mathcal A$ is computable if $D(\mathcal A)$ is computable. If $K$ is some class, then $K^c$ denotes the set of computable members of $K$. A computable characterization for $K$ should separate the computable members of $K$ from other structures, that is, those that either are not in $K$ or are not computable. A computable classification (structure theorem) should describe each member of $K^c$ up to isomorphism, or other equivalence, in terms of relatively simple invariants. A computable non-structure theorem would assert that there is no computable structure theorem. We use three approaches. They all give the “correct” answer for vector spaces over $Q$, and for linear orderings. Under all of the approaches, both classes have a computable characterization, and there is a computable classification for vector spaces, but not for linear orderings. Finally, we formulate some open problems.

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English version:
Algebra and Logic, 2002, 41:6, 351–373

Bibliographic databases:

UDC: 510.53
Received: 08.03.2001

Citation: S. S. Goncharov, J. F. Knight, “Computable Structure and Non-Structure Theorems”, Algebra Logika, 41:6 (2002), 639–681; Algebra and Logic, 41:6 (2002), 351–373

Citation in format AMSBIB
\by S.~S.~Goncharov, J.~F.~Knight
\paper Computable Structure and Non-Structure Theorems
\jour Algebra Logika
\yr 2002
\vol 41
\issue 6
\pages 639--681
\jour Algebra and Logic
\yr 2002
\vol 41
\issue 6
\pages 351--373

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    1. White WM, “On the complexity of categoricity in computable structures”, Mathematical Logic Quarterly, 49:6 (2003), 603–614  crossref  mathscinet  zmath  isi  scopus
    2. Goncharov SS, Harizanov VS, Knight JF, et al, “Pi(1)(1) relations and paths through O”, Journal of Symbolic Logic, 69:2 (2004), 585–611  crossref  mathscinet  zmath  isi  scopus
    3. Calvert W, “The isomorphism problem for classes of computable fields”, Archive For Mathematical Logic, 43:3 (2004), 327–336  crossref  mathscinet  zmath  isi  scopus
    4. Khoussainov B., Nies A., Rubin S., Stephan F., “Automatic structures: Richness and limitations”, 19th Annual IEEE Symposium on Logic in Computer Science, Proceedings, Proceedings/Symposium on Logic in Computer Science, 2004, 44–53  crossref  mathscinet  isi
    5. Calvert W., “The isomorphism problem for computable Abelian p-groups of bounded length”, Journal of Symbolic Logic, 70:1 (2005), 331–345  crossref  mathscinet  zmath  isi  scopus
    6. W. Calvert, V. S. Harizanova, J. F. Knight, S. Miller, “Index sets of computable structures”, Algebra and Logic, 45:5 (2006), 306–325  mathnet  crossref  mathscinet  zmath
    7. N. T. Kogabaev, “Complexity of some natural problems on the class of computable $I$-algebras”, Siberian Math. J., 47:2 (2006), 291–297  mathnet  crossref  mathscinet  zmath  isi
    8. N. S. Vinokurov, “Index sets of classes of automatic structures”, Siberian Math. J., 47:5 (2006), 835–843  mathnet  crossref  mathscinet  zmath
    9. Calvert W., Knight J.F., “Classification from a computable viewpoint”, Bulletin of Symbolic Logic, 12:2 (2006), 191–218  crossref  mathscinet  zmath  isi  scopus
    10. Calvert W., Knight J.F., Millar J., “Computable trees of Scott rank omega(CK)(1), and computable approximation”, Journal of Symbolic Logic, 71:1 (2006), 283–298  crossref  mathscinet  zmath  isi  scopus
    11. E. B. Fokina, “Index sets of decidable models”, Siberian Math. J., 48:5 (2007), 939–948  mathnet  crossref  mathscinet  zmath  isi
    12. Khoussainov B., Nies A., Rubin S., Stephan F., “Automatic Structures: Richness and Limitations”, Logical Methods in Computer Science, 3:2 (2007), 2  crossref  mathscinet  zmath  isi  scopus
    13. Calvert W., Fokina E., Goncharov S.S., Knight J.F., Kudinov O., Morozov A.S., Puzarenko V., “Index sets for classes of high rank structures”, Journal of Symbolic Logic, 72:4 (2007), 1418–1432  crossref  mathscinet  zmath  isi  scopus
    14. Calvert W., Goncharov S.S., Knight J.F., “Computable structures of Scott rank omega(CK)(1) in familiar classes”, Advances in Logic, Contemporary Mathematics Series, 425, 2007, 49–66  crossref  mathscinet  zmath  adsnasa  isi
    15. Fokina E.B., “Index sets of computable structures with decidable theories”, Computation and Logic in the Real World, Proceedings, Lecture Notes in Computer Science, 4497, 2007, 290–296  crossref  mathscinet  zmath  isi  scopus
    16. E. N. Pavlovskii, “Estimation of the algorithmic complexity of classes of computable models”, Siberian Math. J., 49:3 (2008), 512–523  mathnet  crossref  mathscinet  zmath  isi
    17. Downey R., Montalban A., “The isomorphism problem for torsion-free Abelian groups is analytic complete”, Journal of Algebra, 320:6 (2008), 2291–2300  crossref  mathscinet  zmath  isi  scopus
    18. Rubin S., “Automata presenting structures: A survey of the finite string case”, Bulletin of Symbolic Logic, 14:2 (2008), 169–209  crossref  mathscinet  zmath  isi  scopus
    19. Greenberg N., Montalban A., “Ranked structures and arithmetic transfinite recursion”, Transactions of the American Mathematical Society, 360:3 (2008), 1265–1307  crossref  mathscinet  zmath  isi  scopus
    20. E. N. Pavlovskii, “Indeksnye mnozhestva prostykh modelei”, Sib. elektron. matem. izv., 5 (2008), 200–210  mathnet  mathscinet
    21. Khoussainov B., Minnes M., “Model theoretic complexity of automatic structures - (Extended abstract)”, Theory and Applications of Models of Computation, Proceedings, Lecture Notes in Computer Science, 4978, 2008, 514–525  crossref  mathscinet  zmath  isi  scopus
    22. E. N. Pavlovskii, “Slozhnost indeksnykh mnozhestv nekotorykh klassov modelei”, Vestn. NGU. Ser. matem., mekh., inform., 8:1 (2008), 71–76  mathnet
    23. E. B. Fokina, “Algoritmicheskie svoistva modelei signatury s dvumya odnomestnymi funktsionalnymi simvolami”, Vestn. NGU. Ser. matem., mekh., inform., 8:1 (2008), 90–101  mathnet
    24. S. S. Goncharov, N. T. Kogabaev, “O $\Sigma^0_1$-klassifikatsii otnoshenii na vychislimykh strukturakh”, Vestn. NGU. Ser. matem., mekh., inform., 8:4 (2008), 23–32  mathnet
    25. Khoussainov B., Minnes M., “Model-theoretic complexity of automatic structures”, Annals of Pure and Applied Logic, 161:3 (2009), 416–426  crossref  mathscinet  zmath  isi  scopus
    26. Fokina E.B., “Index sets for some classes of structures”, Annals of Pure and Applied Logic, 157:2–3 (2009), 139–147  crossref  mathscinet  zmath  isi  scopus
    27. Fokina E.B., Friedman S.-D., “Equivalence Relations on Classes of Computable Structures”, Mathematical Theory and Computational Practice, Lecture Notes in Computer Science, 5635, 2009, 198–207  crossref  mathscinet  zmath  isi  scopus
    28. Kuske D., Liu J., Lohrey M., “The Isomorphism Problem On Classes of Automatic Structures”, 25th Annual IEEE Symposium on Logic in Computer Science (LICS 2010), IEEE Symposium on Logic in Computer Science, 2010, 160–169  mathscinet  isi
    29. Kuske D., Liu J., Lohrey M., “The Isomorphism Problem for omega-Automatic Trees”, Computer Science Logic, Lecture Notes in Computer Science, 6247, 2010, 396–410  crossref  mathscinet  zmath  isi  scopus
    30. Melnikov A.G., “Computable Ordered Abelian Groups and Fields”, Programs, Proofs, Processes, Lecture Notes in Computer Science, 6158, 2010, 321–330  crossref  mathscinet  zmath  adsnasa  isi  scopus
    31. S. S. Goncharov, “Degrees of autostability relative to strong constructivizations”, Proc. Steklov Inst. Math., 274 (2011), 105–115  mathnet  crossref  mathscinet  isi
    32. J. Carson, E. Fokina, V. S. Harizanov, J. F. Knight, S. Quinn, C. Safranski, J. Wallbaum, “The computable embedding problem”, Algebra and Logic, 50:6 (2012), 478–493  mathnet  crossref  mathscinet  zmath  isi
    33. Fokina E.B., Friedman S.-D., “On S11 equivalence relations over the natural numbers”, MLQ Math Log Q, 58:1–2 (2012), 113–124  crossref  mathscinet  zmath  isi  scopus
    34. Fokina E.B. Friedman S.-D. Harizanov V. Knight J.F. McCoy Ch. Montalban A., “Isomorphism Relations on Computable Structures”, J. Symb. Log., 77:1 (2012), 122–132  crossref  mathscinet  zmath  isi  elib  scopus
    35. N. T. Kogabaev, “The complexity of isomorphism problem for computable projective planes”, J. Math. Sci., 203:4 (2014), 509–515  mathnet  crossref
    36. Kuske D., Liu J., Lohrey M., “The Isomorphism Problem for Omega-Automatic Trees”, Ann. Pure Appl. Log., 164:1 (2013), 30–48  crossref  mathscinet  zmath  isi  scopus
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    40. Montalban A., “Copyable Structures”, J. Symb. Log., 78:4 (2013), 1199–1217  crossref  mathscinet  zmath  isi  scopus
    41. Becker H., “Isomorphism of Computable Structures and Vaught's Conjecture”, J. Symb. Log., 78:4 (2013), 1328–1344  crossref  mathscinet  zmath  isi  scopus
    42. M. V. Dorzhieva, “Eliminatsiya metarekursii iz teoremy Ouinsa”, Vestn. NGU. Ser. matem., mekh., inform., 14:1 (2014), 35–43  mathnet
    43. Melnikov A.G., “Computable Abelian Groups”, Bull. Symb. Log., 20:3 (2014), 315–356  crossref  mathscinet  zmath  isi  elib
    44. Fokina E.B. Harizanov V. Melnikov A., “Computable Model Theory”, Turing'S Legacy: Developments From Turing'S Ideas in Logic, Lecture Notes in Logic, 42, ed. Downey R., Cambridge Univ Press, 2014, 124–194  mathscinet  isi
    45. S. S. Goncharov, N. A. Bazhenov, M. I. Marchuk, “The index set of Boolean algebras autostable relative to strong constructivizations”, Siberian Math. J., 56:3 (2015), 393–404  mathnet  crossref  crossref  mathscinet  isi  elib  elib
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    49. Calvert W., “Pac Learning, Vc Dimension, and the Arithmetic Hierarchy”, Arch. Math. Log., 54:7-8 (2015), 871–883  crossref  mathscinet  zmath  isi  elib  scopus
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