Mathematics and Computer Science: Coping with Finiteness

@article{Knuth1976MathematicsAC,
  title={Mathematics and Computer Science: Coping with Finiteness},
  author={Donald Ervin Knuth},
  journal={Science},
  year={1976},
  volume={194},
  pages={1235 - 1242}
}
  • D. Knuth
  • Published 17 December 1976
  • Philosophy
  • Science
By presenting these examples, I have tried to illustrate four main points. 1) Finite numbers can be really enormous, and the known universe is very small. Therefore the distinction between finite and infinite is not as relevant as the distinction between realistic and unrealistic. 2) In many cases there are subtle ways to solve very large problems quickly, in spite of the fact that they appear at first to require examination of too many possibilities. 3) There are also cases where we can prove… 
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References

SHOWING 1-10 OF 16 REFERENCES
A note on two problems in connexion with graphs
  • E. Dijkstra
  • Mathematics, Computer Science
    Numerische Mathematik
  • 1959
TLDR
A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
A method of factoring and the factorization of
The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is
The complexity of decision problems in automata theory and logic.
TLDR
This paper is intended to demonstrate the efforts of the Massachusetts Institute of Technology’s graduate students to improve the quality of their teaching and research.
The 11 possibilities for bar 8 are all identical, and Mozart gave only two distinct possibilities for bar 16, so the total number of waltzes is 2 x 1114 rather than 1116
    A determinant formula that specifies the number of spanning trees in a particular graph was discovered by
      See also A
      • The Design and Analysis ofComputer Algorithms
      • 1972
      Musikalisches Wdirfelspiel (Schott, Mainz, 1956), K 516 f Anh
      • C 30.01; this was first published in
      • 1793
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