TESTSHEET2

# TESTSHEET2 - →Ν#L.0#L(d ∅...

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Different sets —types— of lists (sequences or strings) are distinguished by their base types . For example, The set of String (Lists) of Integers: 1. λ Z* 2. if x Z and w Z* then w x Z* Examples— a few useful functions defined on lists: 1. length (#) :: Z* N . # λ = 0 # ( w x) = 1 + # w …i.e. suc(#x) 2. concatenation (^) :: (Z*, Z*) Z* w ^ λ = w w 1 ^ w 2 x = ( w 1 ^ w 2 )x 3. reverse :: Z* Z* reverse ( λ ) = λ reverse ( w x) = xreverse ( w) An inductive definition of trees: 1. (d, , ) is a binary tree. 2. If t1 and t2 are binary trees, then (d,t1,t2) is a binary tree. The tree’s height, h:: t →Ν h.0 h(d, , )=0. h.1 h (d,t1,t2)= Max(h(t1), h(t2)) +1 The number of nodes of a binary tree, #N:: t →Ν #N.0 #N(d, , )=1. #N.1 #N(d,t1,t2)= #N(t1)+ #N(t2) +1 The number of leaves of a binary tree, #L:: t
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Unformatted text preview: →Ν #L.0 #L(d, ∅ , ∅ )=l. #L.1 #L(d,t1,t2)= #L(t1)+ #L(t2)) The number of internal nodes of a binary tree, #I:: t →Ν #I.0 #I(d, ∅ , ∅ )=0. #I.1 #I(d,t1,t2)= #I(t1)+ #I(t2) +1 DEFINITIONS Let f and g be functions from N to R. Then f is asymptotically dominated by g (we say that f is O(g)) iff ( ∃ C,k|: ( ∀ x| x>k: |f(x)| ≤ C|g(x)|)) Let f and g be functions from N to R. Then f asymptotically dominates g (we say that f is Ω (g)) iff ( ∃ C,k|: ( ∀ x| x>k: |f(x)| ≥ C|g(x)|)) Let f and g be functions from N to R. If f is O(g)) and f is Ω (g)) then f is of order of g and we say f is Θ (g). ( Σ i| 1 ≤ i ≤ n:i) = n(n+1)/2 ( Σ i| 0 ≤ i ≤ n:a i ) = (a n+1-1)/(a-1)...
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## This note was uploaded on 11/30/2010 for the course CS 336 taught by Professor Myers during the Fall '08 term at University of Texas.

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