Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-14T12:21:56.864Z Has data issue: false hasContentIssue false
  • English
  • Français

Enumerating Types of Boolean Functions

Published online by Cambridge University Press:  15 January 2014

Alasdair Urquhart
Alasdair Urquhart*
Affiliation:
Department of Computer Science, University of Toronto, Toronto, Ontario M5S 3G4, CanadaE-mail: urquhart@cs.toronto.edu
Get access Rights & Permissions [Opens in a new window]

Extract

Abstract.

The problem of enumerating the types of Boolean functions under the group of variable permutations and complementations was first stated by Jevons in the 1870s, but not solved in a satisfactory way until the work of Pólya in 1940. This paper explains the details of Pólya's solution, and also the history of the problem from the 1870s to the 1970s.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 15 , Issue 3 , September 2009 , pp. 273 - 299
Copyright
Copyright © Association for Symbolic Logic 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Acerbi, F., On the shoulders of Hipparchus: A reappraisal of ancient Greek combinatorics, Archive for History of Exact Sciences, vol. 57 (2003), pp. 465–503.Google Scholar
[2] Aigner, Martin, A course in enumeration, Graduate Texts in Mathematics, no. 238, Springer-Verlag, 2007.Google Scholar
[3] Bergeron, F., Labelle, G., and Leroux, P., Combinatorial species and tree-like structures, Encyclopedia of mathematics and its applications, no. 67, Cambridge University Press, 1998.Google Scholar
[4] Biermann, Kurt-R. and Mau, Jürgen, Überprüfung einer frühen Anwendung der Kombinatorik in der Logik, The Journal of Symbolic Logic, vol. 23 (1958), pp. 129–132.CrossRefGoogle Scholar
[5] Biggs, Norman L., Lloyd, E. Keith, and Wilson, Robin J., Graph theory 1736–1936, Oxford University Press, 1976.Google Scholar
[6] Cameron, Peter J., Combinatorics: Topics, techniques, algorithms, Cambridge University Press, 1994.Google Scholar
[7] Cameron, Peter J., Permutation groups, London Mathematical Society Student Texts, no. 45, Cambridge University Press, 1999.Google Scholar
[8] Clifford, William Kingdon, On the types of compound statement involving four classes, Memoirs of the Literary and Philosophical Society of Manchester, Session 1876–1877, vol. 16 (1877), pp. 88–101, reprinted as Paper 1 in [9].Google Scholar
[9] Clifford, William Kingdon, Mathematical papers, Macmillan and Co., 1882, edited by Tucker, Robert, with an introduction by Smith, H. J. Stephen. Reprinted, Chelsea Publishing Company, New York 1968.Google Scholar
[10] de Bruijn, N. G., Pólya's theory of counting, Applied combinatorial mathematics (Beckenbach, Edwin F., editor), John Wiley and Sons, 1964, pp. 144–184.Google Scholar
[II] De Morgan, Augustus, Formal logic: or, the calculus of inference, necessary and probable, Taylor and Walton, London, 1847.Google Scholar
[12] Golomb, Solomon W., On factoring Jevons' number, Cryptologia, vol. 20 (1996), pp. 243–246.CrossRefGoogle Scholar
[13] Grea, R. and Higonnet, R., Logical design of electrical circuits, Macmillan Co., New York, 1958.Google Scholar
[14] Habsieger, Laurent, Kazarian, Maxim, and Lando, Sergei, On the second number of Plutarch, American Mathematical Monthly, vol. 105 (1998), p. 446.Google Scholar
[15] Harary, Frank, On the number of bi-colored graphs, Pacific Journal of Mathematics, vol. 8 (1958), pp. 743–755.Google Scholar
[16] Harary, Frank, Exponentiation of permutation groups, American Mathematical Monthly, vol. 66 (1959), pp. 572–575.Google Scholar
[17] Harary, Frank and Palmer, Edgar M., Graphical enumeration, Academic Press, New York and London, 1973.Google Scholar
[18] Harrison, Michael A., The number of transitivity sets of Boolean functions, Journal of the Society for Industrial and Applied Mathematics, vol. 11 (1963), pp. 806–828.CrossRefGoogle Scholar
[19] Harrison, Michael A., On the number of classes of (n, k) switching networks, Journal of the Franklin Institute, vol. 276 (1963), pp. 313–327.Google Scholar
[20] Harrison, Michael A., Introduction to switching and automata theory, McGraw-Hill Book Company, 1965.Google Scholar
[21] Harrison, Michael A. and High, Robert G., On the cycle index of a product of permutation groups, Journal of Combinatorial Theory, vol. 4 (1968), pp. 277–299.CrossRefGoogle Scholar
[22] Heath, Thomas, A history of Greek mathematics, Clarendon Press, Oxford, 1921, reprinted by Dover Publications, 1981.Google Scholar
[23] Jevons, W. Stanley, The principles of science: A treatise on logic and scientific method, vol. 1, Macmillan and Co., London, 1874.Google Scholar
[24] Jevons, W. Stanley, The principles of science: A treatise on logic and scientific method, second revised ed., Macmillan and Co., London, 1877.Google Scholar
[25] Kerber, Adalbert, Der Zykelindex der Exponentialgruppe, Mitteilungen aus dem Mathemematischen Seminar Giessen, vol. 98 (1973), pp. 5–20.Google Scholar
[26] Kerber, Adalbert, Representations of permutation groups II, Lecture Notes in Mathematics, no. 495, Springer-Verlag, Berlin and Heidelberg and New York, 1975.CrossRefGoogle Scholar
[27] Kerber, Adalbert, Applied finite group actions, second, revised and expanded ed., Algorithms and Combinatorics, no. 19, Springer-Verlag, 1999.Google Scholar
[28] Kneale, William and Kneale, Martha, The development of logic, Oxford University Press, 1962.Google Scholar
[29] Ladd, Christine, On the algebra of logic, (Charles Saunders Peirce, editor), Studies in Logic by Members of the Johns Hopkins University, Little, Brown, and Company, Boston, 1883, pp. 17–71.Google Scholar
[30] Lehmer, D. N., A theorem in the theory of numbers, Bulletin of the American Mathematical Society, vol. 13 (1907), pp. 501–502.Google Scholar
[31] Lothaire, M., Combinatorics on words, Cambridge University Press, 1997, corrected reissue of the 1983 edition: Encyclopedia of Mathematics and its Applications, vol. 17.CrossRefGoogle Scholar
[32] Mates, Benson, Stoic logic, University of California Press, 1953.Google Scholar
[33] Nagy, A., Über das Jevons-Clifford'sche Problem, Monatshefte für Mathematik und Physik, vol. 5 (1894), pp. 331–345.Google Scholar
[34] Olver, Peter J., Classical invariant theory, London Mathematical Society Student Texts, no. 44, Cambridge University Press, 1999.Google Scholar
[35] Ore, Oystein, Theory of monomial groups, Transactions of the American Mathematical Society, vol. 51 (1942), pp. 15–64.Google Scholar
[36] O'Toole, Robert R. and Jennings, Raymond E., The Megarians and the Stoics, Handbook of the history of logic, volume 1: Greek, Indian and Arabic Logic (Gabbay, Dov M. and Woods, John, editors), Elsevier North Holland, 2004, pp. 397–522.Google Scholar
[37] Palmer, E. M. and Robinson, R. W., Enumeration under two representations of the wreath product, Acta Mathematica, vol. 131 (1973), pp. 123–143.Google Scholar
[38] Plutarch, , Moralia, vol. XIII, Part II, Harvard University Press, 1976, with an English translation by Harold Cherniss.Google Scholar
[39] Pólya, George, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica, vol. 68 (1937), pp. 145–253, reprinted in [41].Google Scholar
[40] Pólya, George, Sur les types des propositions composées, The Journal of Symbolic Logic, vol. 5 (1940), pp. 98–103, reprinted in [41].Google Scholar
[41] Pólya, George, Collected papers, volume IV: Probability; combinatorics; teaching and learning in mathematics, (Rota, Gian-Carlo, editor), The MIT Press, Cambridge, Massachusetts, 1984.Google Scholar
[42] Pólya, George and Read, Ronald C., Combinatorial enumeration of groups, graphs and chemical compounds, Springer-Verlag, 1987.Google Scholar
[43] Redfield, J. H., The theory of group-reduced distributions, American Journal of Mathematics, vol. 49 (1927), pp. 433–455.Google Scholar
[44] Richards, Mark and Alderman, John, Core memory: A visual survey of vintage computers, Chronicle Books, San Francisco, 2007.Google Scholar
[45] Schröder, Ernst, Vorlesungen über die Algebra der Logik (exakte Logik), vol. I, B. G. Teubner, Leipzig, 1890.Google Scholar
[46] Shannon, Claude E., A symbolic analysis of relay and switching circuits, Transactions of the American Institute of Electrical Engineers, vol. 57 (1938), pp. 713–723, reprinted in [48].Google Scholar
[47] Shannon, Claude E., The synthesis of two-terminal switching circuits, Bell System Technical Journal, vol. 28 (1949), pp. 59–98, reprinted in [48].CrossRefGoogle Scholar
[48] Shannon, Claude E., Collected papers, (Sloane, N. J. A. and Wyner, Aaron D., editors), IEEE Press, 1993.Google Scholar
[49] Slepian, David, On the number of symmetry types of Boolean functions of n variabless, Canadian Journal of Mathematics, vol. 5 (1953), pp. 185–193.Google Scholar
[50] Sloane, N. J. A. and Plouffe, Simon, The encyclopedia of integer sequences, Academic Press, 1995, a greatly expanded version of the Encyclopedia is available on-line at http://www.research.att.com/~njas/sequences/.Google Scholar
[51] Stanley, Richard P., Enumerative combinatorics, vol. 1, Wadsworth and Brooks/Cole, Monterey, California, 1986.CrossRefGoogle Scholar
[52] Stanley, Richard P., Hipparchus, Plutarch, Schröder and Hough, American Mathematical Monthly, vol. 104 (1997), pp. 344–350.Google Scholar
[53] Thompson, Thomas M., From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, no. 21, Mathematical Association of America, 1983.Google Scholar
[54] Venn, John, Symbolic logic, second ed., Macmillan, London, 1894, reprinted Chelsea Publishing Company, Bronx, New York, 1971.Google Scholar
[55] Young, Alfred, On quantitative substitutional analysis (fifth paper), Proceedings of the London Mathematical Society, vol. 31 (1930), pp. 273–288.CrossRefGoogle Scholar