Skip to main content
Springer Nature Link
Log in

Definability, decidability, complexity

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

We survey how the definability problem in first-order logic was born and the relations between this problem and the question of decidability of logical theories. We also show present connections between definability and the important theoretical problems of computational complexity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Annalen 99(1928)118–133; English translation in [93] pp. 493–507.

    Google Scholar 

  2. Aristotle,Posterior Analytics, II, 3, 90 b 9 and 91 a 12; English translation inThe Works of Aristotle, 11 Vols., ed. W.D. Ross (Clarendon Press, Oxford, 1931); French translation by Jean Tricot,Les Seconds Analytiques (Vrin, 1947).

  3. E.W. Beth, On Padoa's method in the theory of definition, Koningklijke Nederlandse Akademie van Wetenschappen,Proc. of the Section of Sciences, Vol. 56 (1953), series A, Mathematical Sciences, pp. 330–339; also in Indagationes Mathematicae, Vol. 15, pp. 330–339.

    Google Scholar 

  4. Bourbaki,Théorie des Ensembles (Hermann, 1954) EI 14–15; English translationTheory of Sets (Hermann and Addison-Wesley, 1968) p. 16.

  5. D.P. Bovet and P. Crescenzi,Introduction to the Theory of Complexity (Prentice-Hall, 1994) p. XI + 282.

  6. S. Buss, Studies in proof theory,Bounded Arithmetic (Bibliopolis, Napoli, 1986) p. 221.

    Google Scholar 

  7. G. Cantor, Über eine elementare Frage der Mannigfaltigkeitslehere,Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 1 (1891) pp. 75–78; also in [8], pp. 278–280.

    Google Scholar 

  8. G. Cantor,Gesammelte Abhandlungen, Mathematischen und Philosophischen Inhalts (Springer, 1932, p. VII + 486 (reprinted: Hidesheim, Olms, 1966); (2nd ed., 1980) p. VII + 489.

  9. P. Cegielski, L'article fondateur de Julia Robinson sur la définissabilité, quarante ans après, LITP No. 90.41, Université Paris VII (1990) p. 15; also chap. VIII of [10].

  10. P. Cegielski, Thèse de doctorat d'état, LITP No. 90.77, Institut Blaise Pascal, Paris (1990) p. 310.

    Google Scholar 

  11. P. Cegielski, Le théorème de Dirchlet est finitiste, LITP 92.40 (1992) p. 131.

  12. P. Cegielski and D. Richard, Indécidabilité de la théorie des entiers naturels munis d'une énumération des premiers et de la divisibilité, Compte Rend. Acad. Sci. Paris 315, Série I (1992)1431–1434.

  13. P. Cegielski, Y. Matiyasevich and D. Richard, Definability and decidability issues in extensions of the integers with the divisibility predicate, to appear in J. Symb. Logic.

  14. A. Church, An unsolvable problem of elementary number theory, Amer. J. Math. 58(1936)345–363; reprinted in [16], pp. 88–107.

    Google Scholar 

  15. W. Craig, Review of two papers by E.W. Beth and of three papers by K.J.J. Hintikka, J. Symb. Logic 22 (1957).

  16. M. Davis (ed.),The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions (Raven Press, New York, 1965).

    Google Scholar 

  17. M. Davis, Hilbert's tenth problem is unsolvable, Amer. Math. Monthly 80(1973)233–269; also inComputability and Unsolvability (2nd ed., Dover, 1982) pp. 199–235.

    Google Scholar 

  18. R. Dedekind,Was sind und was sollen die Zahlen?, 1888 (Braunschweig, 6th ed., 1930); also inDedekind Gesammelte Mathematische Werke, Vol. III (Braunschweig, 1932) pp. 335–391. English translation by W.W. Beman, The nature and meaning of numbers, in:Essays on the Theory of Numbers (Open Court, Chicago, 1901, reed: Dover, 1963) p. 115; French translation by J. Milner and H. Sinaceur,Les Nombres, que sont-ils et à quoi servent-ils?, La bibliothèque d'Ornicar? (Diffusion Seuil, s.d., 1979) p. 142.

  19. Fibonacci, in fact Leonardo Pisa,Liber abaci (1202), Bullettino di Bibliografia e di Storia delle Scienze Mathematiche e Fisiche (Baldassare Boncompagni, Rome, 1868–1887) 20 Vols; no English or French translation. Partial English translation by D.J. Struik,A Source Book in Mathematics 1200–1800, (Princeton University Press, 1969) pp. 2–3.

    Google Scholar 

  20. G. Frege,Die Grundlagen der Arithmetik (Breslau, 1884); English translation:The Foundations of Arithmetic (Blackwell, Oxford, 1950); French trannslation by C. Imbert,Les Fondements de l'Arithmétique (Seuil, 1969, deuxième édition s.d., 1979) p. 235.

  21. J. Gergonne, Essai sur la théorie des définitions, Ann. de Math. Pures et Appliq. (1818) 1–35.

  22. A. Girard,L'Arithmétique de Simon Stevin de Bruges (Leyde, 1634) p. 677.

  23. K. Gödel, Über formal unentscheidbare Sätze derPrincipia mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38(1931)173–198; English translation in [93] and inCollected Works, Vol. 1 (Oxford University Press, 1986); French translation inLe Théorème de Gödel (Seuil, 1989) p. 184.

    Google Scholar 

  24. K. Gödel, Letter to Zermelo dated 12 October 1932, edited by Grattan-Guinness in Historia Mathematica 6(1979)294–304.

  25. R.L. Goodstein, Hilbert's tenth problem and the independence of recursive difference, J. London Math. Soc., 2nd Series, 10(1975)175–176.

    Google Scholar 

  26. S. Grigorieff and D. Richard, Contribution à l'étude d'une conjecture de théorie des nombres par le codage ZBV, L'Enseignement Mathématique 35(1989)125–189.

    Google Scholar 

  27. R. Guy,Unsolved Problems in Number Theory (Springer, 1981) pp. 25–28.

  28. P. Hájek and P. Pudlak,Metamathematics of First-Order Arithmetic (Springer, 1993) p. 460.

  29. J. Herbrand, Sur le problème fondamental de la logique mathématique, Sprawozdania z posiedzen Towarzystwa Naukowego Warszawskiego wydzial, III, 24(1931)12–56; reprinted inÉcrits Logiques, ed. Jean van Heijenoort (P.U.F., 1968); English translation inLogical Writings (Reidel, 1971).

    Google Scholar 

  30. D. Hilbert, Mathematische Probleme,Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen (1990) pp. 253–297; French translation with emendations and additions inCompte Rendu du Deuxième Congrès Int. des Mathématiciens, Paris, 12 August, 1900 (Gauthier-Villars, 1902) pp. 58–114, réédition GABBAY (1992); English translation in Bull. Amer. Math. Soc. 8(1902)437–479, reprinted inMathematical Developments Arising from Hilbert Problems, ed. Browder,Proc. Symposia in Pure Mathematics, Vol. 28 (American Mathematical Society, 1976) pp. 1–34.

  31. J.P. Jones, Universal Diophantine equation, J. Symb. Logic 47(1982)549–571.

    Google Scholar 

  32. J.P. Jones, D. Sato, H. Wada and D. Wiens, Diophantine representation of the set of prime numbers, Amer. Math. Monthly 83(1976)449–464.

    Google Scholar 

  33. S.C. Kleene, Recursive predicates and quantifiers, Trans. Amer. Math. Soc. 53(1943)41–73; reprinted in [16] pp. 254–287.

    Google Scholar 

  34. J. Krajíček, P. Pudlák and G. Takeuti, Bounded arithmetic and the polynomial hierarchy, Ann. Pure and Appl. Logic 52(1991). [Result communicated by Professor Pudlák atSecondes Journées sur les Arithmétiques Faibles, held at LITP, Université Paris VII, December 1990; no proceedings published to date.]

  35. C.H. Langford, Some theorems on deducibility, Ann. Math 28, 2nd Series (1926–1927) 16–40; 459–471.

    Google Scholar 

  36. S. Leśniewski, Über die Definitionen der sagenannten Theorie der Deduktion, C.R. Soc. Sci. Lett. Varsovie Cl: 3(1932)289–309;The Collected Workds (University of Notre Dame Press, 1967).

    Google Scholar 

  37. Y. Matiyasevich, Diophantine representation of recursively enumerable predicates,Actes du Congrès Int. des Mathématiciens, Nice, 1970, Vol. 1 (Gauthier-Villars, 1971) pp. 235–238.

    Google Scholar 

  38. Y. Matiyasevich, Diofantovo predstavlenie mnozhestva prostykh chisel, Dokl. Akad. Nauk SSSR 196(1971)770–773; English translation: Diophantine representation of the set of prime numbers, Sov. Math. Dokl. 12(1971)249–254.

    Google Scholar 

  39. Y. Matiyasevich, Some purely mathematical results inspired by mathematical logic,Logic, Foundations of Mathematics and Computability, Vol. 1,Proc. 5th Int. Congres of Logic, Methodology and Philosophy of Sciences (Reidel, 1977) pp. 121–127.

    Google Scholar 

  40. Y. Matiyasevich, Prostye chisla perechislyayutya polinomon ot 10 peremennykh, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 68(1977)62–82; English translation: Primes are nonnegative values of a polynomial in 10 variables, J. Sov. Math. 15(1981)33–44.

    Google Scholar 

  41. Y. Matiyasevich,Desyataya problema Gil'berta [in Russian] (Nauka, 1993) p. 223; English translation:Hilbert's Tenth Problem (MIT Press, 1993) chap. XXII and p. 264; French translation:Le Dixième Problème de Hilbert (Masson, 1995).

  42. Y. Matiyasevich and J. Robinson, Reduction of an arbitrary Diophantine equation to one in 13 unknowns, Acta Arithm. 27(1975)521–553.

    Google Scholar 

  43. K. McAloon, On the complexity of models of arithmetic, J. Symb. Logic 47(1982)403–415.

    Google Scholar 

  44. J.C.C. McKinsey, On the independence of undefined ideas, Bull. Amer. Math. Soc. 41(1935)291–297.

    Google Scholar 

  45. E. Mendelson,Introduction to Mathematical Logic (Van Nostrand, 1964); (2nd ed., 1979) p. VIII + 328; (Wadsworth, CA, 1987, 3rd ed.) p. X + 342.

  46. A.R. Meyer and L.J. Stockmeyer, The equivalence problem for regular expression with squaring requires exponential time,Proc. 13th IEEE Symp. on Switching and Automata Theory (1973) pp. 125–129.

  47. J.D. Monk,Mathematical Logic (Springer, 1976) p. X + 531.

  48. J.-F. Pabion and D. Richard, Synonymy and re-interpretation for some sublanguages of Peano arithmetic, in:Open Days in Model Theory and Set Theory, eds. W. Guzicki, W. Marek, A. Pelc and C. Rauszer, Proc. of a conference held September 1981 at Jadwisin, near Warsaw, Poland (Leeds University Press).

  49. A. Padoa, Essai d'une théorie algébrique des nombres entiers, précédé d'une introduction logique à une théorie déductive quelconque,Bibliothèque du Congrès International de Philosophie, Paris 1900, Vol. 3 (Armand Colin, 1901) pp. 309–365; partial English translation in [93] pp. 118–123.

    Google Scholar 

  50. B. Pascal, De l'esprit géométrique et de l'art de persuader, numerous editions, for instance inOeuvres Complètes, ed. Lafuma (Seuil, Paris, 1963) pp. 348–359; English translation: Provincial Letters, Pensées, Scientific Treatise, Encylopædia Britannica “Great Books of the Western World” (1952).

    Google Scholar 

  51. G. Peano,I Principii di Geometria (Bocca, Turin, 1889) p. 40.

    Google Scholar 

  52. G. Peano, Le definizioni in matematica,Periodico di Matematiche (1921) pp. 175–189; English translation:Selected Works of Giuseppe Peano, translated and edited by H.C. Kennedy (University of Toronto Press, 1973) p. XI + 249.

  53. R. Péter,Rekursive Functionnen (Verlag der Ungarischen Akademie der Wissenschaften, Budapest, 1951); English translation:Recursive Functions (Academic Press, 1967).

    Google Scholar 

  54. M. Pieri, Delle geometria elementaire come sistema ipotetico-deduttivo: monografia del punto e del mote,Memorie della Reale Accademia delle Scienze di Torino (1899).

  55. B. Poizat,Cours de Théorie des Modèles (Nur al-mantiq wal-ma'rifah, Diffusion Offilib, 1985) p. 584.

  56. E.L. Post, Degrees of recursive unsolvability, Bull. Amer. Math. Soc. 54(1948)641–642.

    Google Scholar 

  57. M. Presburger, Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt,Sprawozdanie z I Kongresu matematyków krajów slowianskich, Warsawa, 1929, Warsaw (1930) pp. 92–101; English translation and study by J. Zygmunt in History and Philosophy of Logic. 12(1991)211–233.

  58. W.V.O. Quine, Vagaries of definition, Ann. New York Acad. Sci. (1973); also inThe Ways of Paradox and Other Essays, 2nd ed. (Harvard University Press, 1976) p. X + 335.

  59. J.-P. Ressayre, Polynomial time uniformization and nonstandard methods,Secondes Journées sur les Arithmétiques Faibles, LITP, Université Paris VII (1990). Also in this volume, Ann. of Math. and AI 16(1996).

  60. D. Richard, De la structure additive à la saturation des modèles de Peano et à la classification des sous-languages de l'Arithmétique,Model Theory and Arithmetic, eds. Berline, K. McAloon and J.-P. Ressayre, Lecture Notes in Mathematics No. 890 (Springer, 1981) pp. 270–296.

  61. D. Richard, La théorie sans égalité du successeur et de la coprimarité des entiers naturels est indécidable. Le prédicat de primarité est définissable dans le language de cette théorie, C.R. Acad. Sci. Paris 294, série I (1982)143–145.

    Google Scholar 

  62. D. Richard (1984), first publication in [9].

  63. D. Richard, The arithmetics as theories of two orders, Ann. Discr. Math. 23(1984)287–312.

    Google Scholar 

  64. D. Richard, All arithmetical sets of powers of primes are first-order definable in terms of the successor function and the coprimeness predicate, Discr. Math. 53(1985)221–247.

    Google Scholar 

  65. D. Richard, Answer to a problem raised by J. Robinson: The arithmetic of positive or negative integers is definable from successor and divisibility, J. Symb. Logic 50(1985)135–143.

    Google Scholar 

  66. D. Richard, Définissabilité en arithmétique et méthode de codage ZBV appliquée à langages avec successuer et coprimarité, Thèse d'État, Université Lyon-I, No. 85-16 (1985).

  67. D. Richard, Equivalence of some questions in mathematical logic with some conjectures in number theory,Number Theory and Applications, ed. R. Mollin, NATO ASI Series, Series C: Math. Phys. Sci., Vol. 265 (1988) pp. 529–545.

    Google Scholar 

  68. D. Richard, Definability in terms of the successor function and the coprimeness predicate in the set of arbitrary integers, J. Symb. Logic 54(1989)1253–1287.

    Google Scholar 

  69. J. Robinson, Definability and decision problems in arithmetic, J. Symb. Logic 14(1949)98–114.

    Google Scholar 

  70. R. Robinson, An essentially undecidable axiom system,Proc. Int. Congress of Mathematics, Vol. 1 (1950) pp. 729–730.

    Google Scholar 

  71. J.B. Rosser, Extensions of some theorems of Gödel and Church, J. Symb. Logic 1(1936)87–91; reprinted in [16] pp. 230–235.

    Google Scholar 

  72. B. Russel, Letter to Frege (1902), first published in [93] pp. 124–125; French translation inLogique et Fondements des Mathématiques: Anthologie (1850–1914), eds. F. Rivenc and P. de Rouilhan (Payot, 1992) p. 447.

  73. C.L. Siegel, Zur Theorie der quadratischen Formen,Nachrichten der Akademie der Wissenschaften in Göttingen II Mathematisch-Physikalisch Klass (1972) pp. 21–46.

  74. T. Skolem, Begründung der elementaren Arithmetik durch die rekurrierende Dekweise ohne Anwendung scheinbarer Veränderlichen mit unendlicem Ausdehnungsberiech, Videnskabselskabets Skrifter, I. Matematisk-naturivdenskabelig Klasse No. 6 (1923) pp. 1–38; reprinted inSelected Works in Logic (Universitetsforlaget, Oslo, 1970) pp. 153–188; English translation in [93] pp. 302–333.

  75. T. Skolem, Über einige satzfunktionen in der arithmetik, Videnskabselskabet i Kristiana Skriften, 1 Klasse, No. 7, Oslo (1930); reprinted inSelected Works in Logic, ed. J.E. Fenstad (Universitetsforlag, Oslo, 1970) pp. 281–306; French translation by C. Richard (1979).

  76. T. Skolem, Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar vieler Aussagen mit ausschliessich Zahlenvariablen, Fundamenta Mathematica 23(1934)150–161.

    Google Scholar 

  77. L.J. Stockmeyer, The polynomial-time hierarchy, Theor. Comp. Sci. 3(1977)1–22.

    Google Scholar 

  78. P. Suppes,Introduction to Logic (Van Nostrand, 1957) p. XVIII + 312; (reed: Dover, 1993).

  79. L. Svenonius, A theorem on permutations in models, Theoria 25(1959)173–179.

    Google Scholar 

  80. A. Tarski, Sur les ensembles définissables de nombres réels I, Fundamenta Mathematica 17(1931)210–239; reprinted in [88], Vol. 1, chap. VI; [89], Vol. 1, pp. 517–548.

    Google Scholar 

  81. A. Tarski, Z badán metodologicznych nad definiowalnosca terminów, Przegl ad filozoficzny 37(1934)438–460.

    Google Scholar 

  82. A. Tarski, Einige methodologische Untersuchungen über die Definierbarkeit der Begriffe, Erkenntnis 5(1935)80–100; reprinted in [89], Vol. 3, pp. 637–659; French translation in [88], Vol. 2, chap. X; English translation in [87], pp. 296–319.

    Google Scholar 

  83. A. Tarski, Der Wahrheisbegriff in den formalisierten Sprachen, Studia Philosophica 1(1936)261–405; English translation in [87] pp. 157–278; French translation in [88] chap. VIII, pp. 159–269. [Interesting result for us in section 5, theorem 1].

    Google Scholar 

  84. A. Tarski, A problem concerning the notion of definability, J. Symb. Logic. 13(1948)107–111; reprinted in [89], Vol. 3, pp. 163–170 (this is an abstract of [80], second part, unpublished).

    Google Scholar 

  85. A. Tarski,A Decision Method for Elementary Algebra and Geometry, 2nd Ed. (Rand Corporation, Santa Monica, 1948) p. III+60; (California University Press, Berkeley, 1951) p. III + 63; French translation of the 1st edition (unpublished) in [88], Vol. 2, pp. 203–242.

    Google Scholar 

  86. A. Tarski, On essential undecidability, J. Symb. Logic 14(1949)75–76.

    Google Scholar 

  87. A. Tarski,Logic, Semantics, Metamathematics (Oxford University Press, 1956).

  88. A. Tarski,Logique, Sémantique, Métamathématique 1923–1944 (Armand Colin, Paris) Vol. 1 (1972), Vol. 2 (1974).

    Google Scholar 

  89. A. Tarski,Collected Papers, eds. Givant and McKenzie (Birkhäuser, 1986) 4 Vols.

  90. A. Tarski, A. Mostowski and R. Robinson,Undecidable Theories (North-Holland, 1953).

  91. S. Tennenbaum, Non-archimedean models for arithmetic,Notices of the Amer. Math. Soc. (1959) p. 270.

  92. A.M. Turing, On computable numbers, with an application to the Entscheidungsproblem,Proc. London Math. Soc. Vol. 42 (1936) pp. 230–265; erratum, ibid., Vol. 43, pp. 544–546; reprinted in [16] pp. 116–154; French translation in: A. Turing and J.-Y. Girard,La Machine de Turing (Seuil, 1995) pp. 47–102.

    Google Scholar 

  93. J. Van Heijenoort (ed.),From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Harvard University Press, 1967; 4th printing: 1981, corrected).

  94. H. Wada, Polynomial representations of prime numbers (in Japanese), Sûgaku 27(1975)160–161.

    Google Scholar 

  95. A. Woods, Some problems in logic and number theory and their connections, Thesis, University of Manchester (1981).

  96. C. Wrathall, Complete sets and the polynomial time hierarchy, Theor. Comp. Sci. 3(1977)23–33.

    Google Scholar 

  97. E. Zermelo, Untersuchungen über die Grundlagen der Mengenlehre, Math. Annal. 65, pp. 261–281; English translation: Investigations in the foundations of set theory I, in:From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, ed. J. Van Heijenoort (Harvard University Press, 1967; 4th printing, 1981, corrected) pp. 199–215; French translation: Recherches sur les fondements de la théorie des ensembles, in:Logique et Fondements des Mathématiques (Payot, 1992) pp. 367–378.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LITP, Institut Blaise Pascal, Université Paris 12, IUT de Fontainebleau, Route Hurtault, 77300, Fontainebleu, France

    Patrick Cegielski

Authors
  1. Patrick Cegielski
    View author publications

    You can also search for this author inPubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cegielski, P. Definability, decidability, complexity. Ann Math Artif Intell 16, 311–341 (1996). https://doi.org/10.1007/BF02127802

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02127802

Keywords