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Capitolo

From decidability to feasibility

  • Duccio Pianigiani

We focus on some classical results concerning formal arithmetic,dating back to the classical era of the 1930s and 1940s, then in subsequent chapters inviting a comparison with more recent results. In this regard, we strongly emphasise the acceleration imparted also to logical study by the development of computer science: in addition to decidability in principle, indeed, computer science has forced attention to be paid to the classification of mathematical problems according to their degree of difficulty and the computational means that can be used in practice.

  • Keywords:
  • Entscheidungsproblem,
  • Turing’s Halting Problem,
  • decidability,
  • polynomial-time computability,
  • Cook-Levin Theorem,
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Duccio Pianigiani

University of Siena, Italy - ORCID: 0000-0001-9441-7226

  1. Ausiello, Giorgio and Gambosi, Giorgio and d’Amore, Fabrizio. 2002. Linguaggi, Modelli, Complessità. Milano: Franco Angeli.
  2. Bernays, Paul. Letter to Gödel, 7 January 1970. Bernays Papers, ETH Zürich Library/WHS, Hs. 975:1745.
  3. Buss, Samuel R. 1995. “On Gödel’s Theorems on Lengths of Proofs II: Lower Bounds for Recognizing k Symbol Provability Bounded Arithmetic and Propositional Proof Complexity.” In Feasible Mathematics II, edited by P. Clote and J. Remmel 57-90. Basel: Birkhauser. DOI: 10.1007/978-1-4612-2566-9_4
  4. Church, Alonso. 1936. “An unsolvable Problem of Elementary Number Theory”. American Journal of Mathematics 58 (2): 345-363.
  5. Cook, Stephen. 1971. “The complexity of theorem-proving procedures.” In STOC ’71 Pro- ceedings of the third annual ACM symposium on Theory of computing, 151-158. New York: ACM Association for Computing Machinery. DOI: 10.7551/mitpress/12274.003.0036
  6. Davis, Martin and Sigal, Ron and Elaine, Jean. 1994. Computability, complexity and language. Burlington: Morgan Kaufmann.
  7. Dawson, John. 1997. Logical Dilemmas. The Life and Work of Kurt Gödel. Wellesley: A K Peters.
  8. Dedekind, Richard. 1888. Was sind und was sollen die Zahlen? 1. Auflage. Braunschweig: Vieweg.
  9. Du, Dingzhu and Ko, Ker-I. 2014. Theory of Computational Complexity. 2nd Edition. Hoboken: Wiley.
  10. Enderton, Herbert B. A mathematical introduction to logic (Second edition). Harcourt/Academic Press: San Diego. DOI: 10.1016/C2009-0-22107-6
  11. Feferman, Solomon. 2006. “Are There Absolutely Unsolvable Problems? Gödel’s Dichotomy”. Philosophia Mathematica (III) 14: 134-152. DOI: 10.1093/philmat/nkj003
  12. Feferman, Solomon. 2006. “Turing’s Thesis.” Notices of the AMS, Vol. 53 (10): 2-8. DOI: 10.2307/j.ctv1vbd2dr.5
  13. Gandy, Robin. 1980. “Church’s Thesis and Principles for Mechanisms.” In The Kleene Symposium, edited by J. Barwise, H. J. Keisler, K. Kunen, K. , 123-148. Amsterdam: North- Holland. DOI: 10.1016/S0049-237X(08)71257-6
  14. Gödel, Kurt. 1930. “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”. Translated by Stefan Bauer-Mengelberg. In Gödel’s Collected Works (1986-2005) 102–23
  15. Gödel, Kurt. 1931. “Über Formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme I.” Monatshefte für Mathematik und Physik 38: 173-198 (reprinted in Gödel’s Collected Works (1986-2005), Vol. I: 144-195 DOI: 10.1007/BF01700692
  16. Gödel, Kurt. 1986-2005. Collected Works I-V, edited by S. Feferman, J. W. Dawson, Stephen C. Kleene, W. Goldfarb, G. Moore, C. Parsons, R. Solovay and Jean Van Heijenoort. Oxford: Oxford University Press. DOI: 10.1093/oso/9780195072556.001.0001
  17. Hájek, Peter and Pudlák, Pavel. 1993. Metamathematics of first order arithmetic. Berlin: Springer. DOI: 10.1017/9781316717271
  18. Hartmanis, Juris and Stearns, Richard E. 1965. “On the computational complexity of algo- rithms.” Transactions of American Mathematical Society 117 (5): 285-306. DOI: 10.1090/S0002-9947-1965-0170805-7
  19. Henkin, Leon. (1950). “Completeness in the Theory of Types”. The Journal of Symbolic Logic”, 15 (2): 81-91. DOI: 10.2307/2268698
  20. Hilbert, David and Ackermann, Wilhelm. 1928. Grundzüge der theoretischen Logik. Berlin: Springer.
  21. Levin, Leonid. 1973. “Universal search problems.” Problems of Information Transmission 9 (3): 115-116. DOI: 10.1007/978-0-387-30164-8_603
  22. Lindström, Per. 1969. “On Extensions of Elementary Logic.” Theoria 35: 1-11 DOI: 10.1111/j.1755-2567.1969.tb00356.x
  23. Lucas, John R. 1961. “Minds, machines and Godel”. Philosophy 36 (137): 112-127. DOI: 10.1017/s0031819100057983
  24. Kleene, Stephen C. 1967. Mathematical Logic. New York: Wiley
  25. Kreisel, Georg. 1971. “Some Reasons for Generalizing Recursion Theory.” In Studies in Logic and the Foundations of Mathematics 61, edited by R.O. Gandy and C.M.E. Yates 139-198. Amsterdam: Elsevier. DOI: 10.2307/2271903
  26. Mal’cev, Anatoli Ivanovic. 1971. The Metamathematics of Algebraic Systems. Collected Papers: 1936–1967. Studies in Logic and the Foundations of Mathematics, Volume 66. Amsterdam: Elsevier.
  27. McCulloch, Warren S. and Pitts, Walter. 1943. “A Logical Calculus of the Ideas Immanent in Nervous Activity”. Bulletin of Mathematical Biophysics 5: 115-133. DOI: 10.2307/2268029
  28. Nelson, Edward. 1986. Predicative Arithmetic. Princeton: Princeton Univ. Press. DOI: 10.2307/2274590
  29. Nielsen, Michael A. and Chuang, Isaac L. 2010. Quantum Computation and Quantum Infor- mation: 10th Anniversary Edition. Cambridge: Cambridge University Press.
  30. Post, E.L. 1936. “Finite Combinatory Processes - Formulation 1”. The Journal of Symbolic Logic 1: 103-105. DOI: 10.2307/2269031
  31. Ryll-Nardzewski, Czesław. 1952. “The role of the axiom of induction in elementary arithmetic”. Fundamenta Mathematicae 39: 239-63.
  32. Odifreddi, Piergiorgio. 1989. Classical Recursion Theory I. Amsterdam: North Holland.
  33. Sipser, Michael. 2006. Introduction to the Theory of Computation, second edition. Boston: Course Technology.
  34. Soare, Robert I. 1996. “Computability and Recursion.” The Bulletin of Symbolic Logic 2 (3) DOI: 10.2307/420992
  35. Soare, Robert I. 2009. “Turing oracle machines, online computing, and three displacements in computability theory”. Annals of Pure and Applied Logic 160 (3): 368-399. DOI: 10.1016/j.apal.2009.01.008
  36. Soare, Robert I. 2014. “Turing and the discovery of computability”. In Turing’s Legacy: Developments from Turing’s Ideas in Logic. Lecture Notes in Logic, edited by R. Downey 467-492. Cambridge: Cambridge University Press. DOI: 10.1017/CBO9781107338579.001
  37. Tait, William. 1981. “Finitism.” Journal of Philosophy 78 (9): 524-546. DOI: 10.2307/2026089
  38. Tarski, Alfred and Mostowski, Andrzej and Robinson, Raphael M. 1953. Undecidable Theories. Amsterdam: Elsevier.
  39. Turing, Alan M. 1936. “On computable numbers, with application to the Entscheidungsproblem”, in Proceedings of London Math. Soc. 42: 230-265, 544-546. DOI: 10.1112/plms/s2-42.1.230
  40. Van Emde Boas, Peter. 1990. “Machine Models and Simulations.” In Handbook of Theoretical Computer Science, Algorithms and Complexity, edited by L. Van Leeuwen, 1-66. Amsterdam: Elsevier.
  41. Visser, Albert. 2009. “Why the theory R is special”. Logic Group preprint series, 279. Utrecht Univ. Now in In Friedman, H. and Tennant, N. (eds.) 2014. Foundational Adventures. Essay in honour of Harvey Friedman: 7-23. London: College Publications.
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  • Anno di pubblicazione: 2025
  • Pagine: 7-32

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  • Anno di pubblicazione: 2025

Informazioni sul capitolo

Titolo del capitolo

From decidability to feasibility

Autori

Duccio Pianigiani

Lingua

Inglese

DOI

10.36253/979-12-215-0778-2.04

Opera sottoposta a peer review

Anno di pubblicazione

2025

Copyright

© 2025 Author(s)

Licenza d'uso

CC BY-SA 4.0

Licenza dei metadati

CC0 1.0

Informazioni bibliografiche

Titolo del libro

Lectures in Proof Theory and Complexity

Autori

Duccio Pianigiani

Opera sottoposta a peer review

Anno di pubblicazione

2025

Copyright

© 2025 Author(s)

Licenza d'uso

CC BY-SA 4.0

Licenza dei metadati

CC0 1.0

Editore

Firenze University Press, USiena Press

DOI

10.36253/979-12-215-0778-2

eISBN (pdf)

979-12-215-0778-2

eISBN (xml)

979-12-215-0779-9

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UNIverSI. Ricerca e Didattica all’Università di Siena

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3035-5915

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3035-5931

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