Lecture 02-13-17

Ackermann’s Function

• “Strange lecture”
• Not responsible for anything but more for “cultural knowledge as CS majors”
• Invented to find out what a computer could/could not do
• Computer can calculate non-primitive recursive function
• Related to running time of union-find
• Grows fast

What is it?

A(m, n) = 
    n + 1                       if m = 0
    A(m - 1, 1)                 if m > 0 and n = 0
    A(m - 1, A(m, n - 1))       if m > 0 and n > 0

• Shortcuts:
  • A(0, n) = n + 1
  • A(1, n) = n + 2
  • A(2, n) = 2n + 3
• Tarjan’s Inverse Function (simplified)
  • Alpha(n) = smallest i such that A(i, i) >= n
  • Running time of n union-find operations on n element set: n * alpha(n)
  • Alpha function also appears:
    • Davenport sevel sequences
    • “Envelope of sequence of lines”
      • Used in computation geometry for walking around some configuration

Computability of Ackermann’s Function

• Claim: A(i, j) is always computable
• Proof:
  • Double induction
  • BWOC
    • Suppose not, then there is some smallest A(x, y) which is not computable
    • Lexicographically smallest
    • A(x, y) = A(x - 1, A(x, y - 1))
      • A(x, y - 1) is smaller and thus computable & contradiction