day13 - have a for-loop implementation. • The inverse of...

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Click to edit Master subtitle style 10/4/11 μ Recursive 4th Model A Simple Extension to Primitive Recursive
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10/4/11 Dec 2, © UCF (Charles E. 22 μ Recursive Concepts All primitive recursive functions are algorithms since the only iterator is bounded. That’s a clear limitation. There are algorithms like Ackerman’s function that cannot be represented by the class of primitive recursive functions. The class of recursive functions adds one more iterator, the minimization operator ( μ ), read “the least value such that.”
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10/4/11 Dec 2, © UCF (Charles E. 33 Ackermann’s Function A(1, j)=2j for j ≥ 1 A(i, 1)=A(i-1, 2) for i ≥ 2 A(i, j)=A(i-1, A(i, j-1)) for i, j ≥ 2 Wilhelm Ackermann observed in 1928 that this is not a primitive recursive function. Ackermann’s function grows too fast to
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Unformatted text preview: have a for-loop implementation. • The inverse of Ackermann’s function is important to analyze Union/Find algorithm. 10/4/11 Dec 2, © UCF (Charles E. 44 Union/Find • Start with a collection S of unrelated elements – singleton equivalence classes • Union(x,y), x and y are in S, merges the class containing x ([x]) with that containing y ([y]) • Find(x) returns the canonical element of [x] • Can see if x& y, by seeing if Find(x)==Find(y) 10/4/11 Dec 2, © UCF (Charles E. 55 The “ Operator • Minimization: If G is already known to be recursive, then so is F, where F(x1,…,xn) = μ y (G(y,x1,…,xn) == 1) • We also allow other predicates besides testing for one. In fact any predicate that is recursive can be...
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This note was uploaded on 10/03/2011 for the course COT 5310 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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day13 - have a for-loop implementation. • The inverse of...

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