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Computability-Theoretic Complexity of Countable Structures

Published online by Cambridge University Press:  15 January 2014

Valentina S. Harizanov
Valentina S. Harizanov*
Affiliation:
Department of Mathematics, The George Washington University, Washington, D.C. 20052, U.S.A. E-mail: harizanv@gwu.edu
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Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computable function can make van der Waerden's intuitive notion of an explicit field precise. This led to the notion of a computable structure. In 1960, Rabin proved that every computable field has a computable algebraic closure. However, not every classical result “effectivizes”. Unlike Vaught's theorem that no complete theory has exactly two nonisomorphic countable models, Millar's and Kudaibergenov's result establishes that there is a complete decidable theory that has exactly two nonisomorphic countable models with computable elementary diagrams. In the 1970's, Metakides and Nerode [58], [59] and Remmel [71], [72], [73] used more advanced methods of computability theory to investigate algorithmic properties of fields, vector spaces, and other mathematical structures.

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Research Article
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Bulletin of Symbolic Logic , Volume 8 , Issue 4 , December 2002, pp. 457 - 477
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Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1] Arana, A., Possible m-diagrams of models of arithmetic, to appear in a volume on reverse mathematics (S. G. Simpson, editor).Google Scholar
[2] Arana, A., Solovay's theorem cannot be simplified, Annals of Pure and Applied Logic, vol. 112 (2001), pp. 27–41.CrossRefGoogle Scholar
[3] Ash, C., Knight, J., Manasse, M., and Siaman, T., Generic copies of countable structures, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 195–205.CrossRefGoogle Scholar
[4] Ash, C. J., Jockusch, C. G. Jr., and Knight, J. F., Jumps of orderings, Transactions of the American Mathematical Society, vol. 319 (1990), pp. 573–599.CrossRefGoogle Scholar
[5] Ash, C. J. and Knight, J. F., Possible degrees in recursive copies II, Annals of Pure and Applied Logic, vol. 87 (1997), pp. 151–165.CrossRefGoogle Scholar
[6] Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.Google Scholar
[7] Ash, C. J. and Nerode, A., Intrinsically recursive relations, Aspects of effective algebra (Crossley, J. N., editor), Upside Down A Book Co., Yarra Glen, Australia, 1981, pp. 26–41.Google Scholar
[8] Barwise, J., Back and forth through infinitary logic, Studies in model theory (Morley, M. D., editor), MAA Studies in Mathematics 8, Buffalo, New York, 1973, pp. 5–34.Google Scholar
[9] Buechler, S., Essential stability theory, Springer-Verlag, Berlin, 1996.CrossRefGoogle Scholar
[10] Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1990.Google Scholar
[11] Chisholm, J., Effective model theory vs. recursive model theory, The Journal of Symbolic Logic, vol. 55 (1990), pp. 1168–1191.CrossRefGoogle Scholar
[12] Chisholm, J. and Moses, M., An undecidable linear order that is n-decidable for all n, Notre Dame Journal of Formal Logic, vol. 39 (1998), pp. 519–526.CrossRefGoogle Scholar
[13] Csima, B. F., Computability of prime and saturated models, Ph.D. thesis , University of Chicago, in preparation.Google Scholar
[14] Csima, B. F., Degree spectra of prime models, Abstracts of papers presented to the American Mathematical Society, vol. 23 (1), 2002, (Joint Mathematics Meetings, San Diego, 01 6–9, 2002), p. 14.Google Scholar
[15] Csima, B. F., Hirschfeldt, D. R., Knight, J. F., and Soare, R. I., Bounding prime models, preprint.Google Scholar
[16] Downey, R. and Jockusch, C. G. Jr., Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871–880.CrossRefGoogle Scholar
[17] Downey, R. and Knight, J. F., Orderings with αth jump degree 0 (α) , Proceedings of the American Mathematical Society, vol. 114 (1992), pp. 545–552.Google Scholar
[18] Downey, R. G., On presentations of algebraic structures, Complexity, logic, and recursion theory (Sorbi, A., editor), Lecture Notes in Pure and Applied Mathematics 187, Marcel Dekker, New York, 1997, pp. 157–205.Google Scholar
[19] Ershov, Yu. L., Decidability problems and constructive models, Nauka, Moscow, 1980, Russian.Google Scholar
[20] Ershov, Yu. L. and Goncharov, S. S., Constructive models, Siberian School of Algebra and Logic, Kluwer Academic/Plenum Publishers, 2000, English translation.CrossRefGoogle Scholar
[21] Feferman, S., Arithmetically definable models of formalized arithmetic, Notices of the American Mathematical Society, vol. 5, 1958, pp. 679–680.Google Scholar
[22] Goncharov, S. S., Restricted theories of constructive Boolean algebras, Sibirskii Matematicheskii Zhurnal, vol. 17 (1976), pp. 601–611, English translation.Google Scholar
[23] Goncharov, S. S., Constructive models of ℵ1-categorical theories, Mathematical Notes, vol. 23 (1978), pp. 885–888, Russian.CrossRefGoogle Scholar
[24] Goncharov, S. S., Strong constructivizability of homogeneous models, Algebra and Logic, vol. 17 (1978), pp. 247–263, English translation.CrossRefGoogle Scholar
[25] Goncharov, S. S., A totally transcendental decidable theory without constructivizable homogeneous models, Algebra and Logic, vol. 19 (1980), pp. 85–93, English translation.CrossRefGoogle Scholar
[26] Goncharov, S. S., Countable Boolean algebras and decidability, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1997.Google Scholar
[27] Goncharov, S. S., Harizanov, V. S., Laskowski, M. C., Lempp, S., and McCoy, C. F. D., Trivial, strongly minimal theories are model complete after naming constants, Proceedings of the American Mathematical Society, (to appear).Google Scholar
[28] Goncharov, S. S. and Khoussainov, B. M., Complexity of theories of computable models, Algebra and Logic, (to appear).Google Scholar
[29] Goncharov, S. S. and Nurtazin, A. T., Constructive models of complete solvable theories, Algebra and Logic, vol. 12 (1973), pp. 67–77, English translation.CrossRefGoogle Scholar
[30] Harizanov, V. S., Some effects of Ash-Nerode and other decidability conditions on degree spectra, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 51–65.CrossRefGoogle Scholar
[31] Harizanov, V. S., Pure computable model theory, Handbook of recursive mathematics (Ershov, Yu. L., Goncharov, S. S., Nerode, A., Remmel, J. B., and Marek, V. W. (assoc.), editors), vol. 1, Elsevier, Amsterdam, 1998, pp. 3–114.Google Scholar
[32] Harizanov, V. S., Knight, J. F., and Morozov, A. S., Sequences of n-diagrams, The Journal of Symbolic Logic, (to appear).Google Scholar
[33] Harrington, L., Recursively presentable prime models, The Journal of Symbolic Logic, vol. 39 (1974), pp. 305–309.CrossRefGoogle Scholar
[34] Herwig, B., Lempp, S., and Ziegler, M., Constructive models of uncountably categorical theories, Proceedings of the American Mathematical Society, vol. 127 (1999), pp. 3711–3719.CrossRefGoogle Scholar
[35] Hirschfeldt, D. R., Prime models of theories of computable linear orderings, Proceedings of the American Mathematical Society, vol. 129 (2001), pp. 3079–3083.CrossRefGoogle Scholar
[36] Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimension in algebraic structures, Annals ofPure and Applied Logic, vol. 115 (2002), pp. 71–113.CrossRefGoogle Scholar
[37] Jockusch, C. G. and Soare, R. I., Boolean algebras, Stone spaces, and the iterated Turing jump, The Journal of Symbolic Logic, vol. 59 (1994), pp. 1121–1138.CrossRefGoogle Scholar
[38] Jockusch, C. G. Jr. and Soare, R. I., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[39] Jockusch, C. G. Jr. and Soare, R. I., Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39–64.CrossRefGoogle Scholar
[40] Kaye, R., Models of Peano Arithmetic, Clarendon Press, Oxford, 1991.CrossRefGoogle Scholar
[41] Khisamiev, N. G., On strongly constructive models of a decidable theory, Izvestii Akademii Nauk Kazakhstan SSR, vol. 35, Ser. Fiz.-Mat., no. 1, 1974, Russian, pp. 83–84.Google Scholar
[42] Khoussainov, B., Nies, A., and Shore, R. A., Computable models of theories with few models, Notre Dame Journal of Formal Logic, vol. 38 (1997), pp. 165–178.CrossRefGoogle Scholar
[43] Khoussainov, B. and Shore, R. A., Effective model theory: the number of models and their complexity, Models and computability, invited papers from Logic Colloquium '97 (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Notes Series 259, Cambridge University Press, Cambridge, 1999, pp. 193–239.CrossRefGoogle Scholar
[44] Knight, J., Lachlan, A. H., and Soare, R. I., Two theorems on degrees of models of true arithmetic, The Journal of Symbolic Logic, vol. 49 (1984), pp. 425–436.CrossRefGoogle Scholar
[45] Knight, J. F., Models of arithmetic: quantifiers and complexity, (S. G. Simpson, editor), to appear in a volume on reverse mathematics.Google Scholar
[46] Knight, J. F., Sequences of degrees associated with models of arithmetic, to appear in a volume on Logic Colloquium 2001.Google Scholar
[47] Knight, J. F., Degrees coded in jumps of orderings, The Journal of Symbolic Logic, vol. 51 (1986), pp. 1034–1042.CrossRefGoogle Scholar
[48] Knight, J. F., True approximations and models of arithmetic, Models and computability, invited papers from Logic Colloquium '97 (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Note Series 259, Cambridge University Press, 1999, pp. 255–278.CrossRefGoogle Scholar
[49] Knight, J. F., Minimality and completions of PA, The Journal of Symbolic Logic, vol. 66 (2001), pp. 1447–1457.CrossRefGoogle Scholar
[50] Knight, J. F. and Stob, M., Computable Boolean algebras, The Journal of Symbolic Logic, vol. 65 (2000), pp. 1605–1623.CrossRefGoogle Scholar
[51] Kudaibergenov, K. Zh., Constructivizable models of undecidable theories, Siberian Mathematical Journal, vol. 21 (1980), no. 5, pp. 155–158, Russian.Google Scholar
[52] Lachlan, A. H. and Soare, R. I., Models of arithmetic and upper bounds for arithmetic sets, The Journal of Symbolic Logic, vol. 59 (1994), pp. 977–983.CrossRefGoogle Scholar
[53] Lachlan, A. H. and Soare, R. I., Models of arithmetic and subuniform bounds for the arithmetic sets, The JournalofSymbolic Logic, vol. 63 (1998), pp. 59–72.CrossRefGoogle Scholar
[54] Lerman, M. and Schmerl, J. H., Theories with recursive models, The Journal of Symbolic Logic, vol. 44 (1979), pp. 59–76.CrossRefGoogle Scholar
[55] Macintyre, A. and Marker, D., Degrees of recursively saturated models, Transactions of the American Mathematical Society, vol. 282 (1984), pp. 539–554.CrossRefGoogle Scholar
[56] Marker, D., Degrees of models of true arithmetic, Proceedings of the Herbrand symposium, Logic Colloquium '81 (Stern, J., editor), North-Holland, Amsterdam, 1982, pp. 233–242.CrossRefGoogle Scholar
[57] Marker, D., Non Σn axiomatizable almost strongly minimal theories, The Journal of Symbolic Logic, vol. 54 (1989), pp. 921–927.CrossRefGoogle Scholar
[58] Metakides, G. and Nerode, A., Recursively enumerable vector spaces, Annals of Mathematical Logic, vol. 11 (1977), pp. 147–171.CrossRefGoogle Scholar
[59] Metakides, G. and Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289–320.CrossRefGoogle Scholar
[60] Millar, T., Homogeneous models and decidability, Pacific Journal of Mathematics, vol. 91 (1980), pp. 407–418.CrossRefGoogle Scholar
[61] Metakides, G. and Nerode, A., Omitting types, type spectrums, and decidability, The Journal of Symbolic Logic, vol. 48 (1983), pp. 171–181.Google Scholar
[62] Millar, T. S., Foundations of recursive model theory, Annals of Mathematical Logic, vol. 13 (1978), pp. 45–72.CrossRefGoogle Scholar
[63] Metakides, G. and Nerode, A., Pure recursive model theory, Handbook of computability theory (Griffor, E. R., editor), Elsevier, Amsterdam, 1999, pp. 507–532.Google Scholar
[64] Miller, R., The -spectrum of a linear order, The Journal of Symbolic Logic, vol. 66 (2001), pp. 470–486.CrossRefGoogle Scholar
[65] Morley, M., Decidable models, Israel Journal of Mathematics, vol. 25 (1976), pp. 233–240.CrossRefGoogle Scholar
[66] Moses, M., Recursive linear orders with recursive successivities, Annals of Pure and Applied Logic, vol. 27 (1984), pp. 253–264.CrossRefGoogle Scholar
[67] Moses, M., n-recursive linear orders without (n + 1)-recursive copies, Logical methods (Crossley, J. N., Remmel, J. B., Shore, R. A., and Sweedler, M. E., editors), Birkhäuser, Boston, 1993, pp. 572–592.CrossRefGoogle Scholar
[68] Nies, A., A new spectrum of recursive models, Notre Dame Journal of Formal Logic, vol. 40 (1999), pp. 307–314.CrossRefGoogle Scholar
[69] Peretyat'kin, M., Turing machine computations in finitely axiomatizable theories, Algebra and Logic, vol. 21 (1982), pp. 272–295, English translation.CrossRefGoogle Scholar
[70] Peretyat'kin, M. G., Criterion for strong constructivizability of a homogeneous model, Algebra and Logic, vol. 17 (1978), pp. 290–301, English translation.CrossRefGoogle Scholar
[71] Remmel, J. B., Maximal and cohesive vector spaces, The Journal of Symbolic Logic, vol. 42 (1977), pp. 400–418.CrossRefGoogle Scholar
[72] Remmel, J. B., Recursively enumerable Boolean algebras, Annals of Mathematical Logic, vol. 15 (1978), pp. 75–107.CrossRefGoogle Scholar
[73] Remmel, J. B., Recursive isomorphism types of recursive Boolean algebras, The Journal of Symbolic Logic, vol. 46 (1981), pp. 572–594.CrossRefGoogle Scholar
[74] Richter, L. J., Degrees of structures, The Journal of Symbolic Logic, vol. 46 (1981), pp. 723–731.CrossRefGoogle Scholar
[75] Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Proceedings of the fifth symposium in pure mathematics (Dekker, J. C. E., editor), Recursive Function Theory, American Mathematical Society, Providence, Rhode Island, 1962, pp. 117–121.Google Scholar
[76] Slaman, T. A., Relative to any nonrecursive set, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2117–2122.CrossRefGoogle Scholar
[77] Slaman, T. A. and Soare, R. I., Extension of embeddings in the computably enumerable degrees, Annals of Mathematics, vol. 154 (2001), pp. 1–43.CrossRefGoogle Scholar
[78] Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[79] Thurber, J. J., Every low2 Boolean algebra has a recursive copy, Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 3859–3866.Google Scholar
[80] Wehner, S., Enumerations, countable structures, and Turing degrees, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2131–2139.CrossRefGoogle Scholar