Recursive Function To define a function, f, on a recursively defined set R, define Recursive • f(b) explicitly for each base case b ∈ R Functions • f(c(x)) for each constructor, c, in terms of x and f(x) recursivef recursivefunctions.1 Albert R Meyer, March 1, 2013 Albert R Meyer, March 1, 2013 unctions 2 N k n recursive function on Recursive function on M expt(k, 0) ::= 1 Def. tree-depth(s) for s �� M expt(k, n+1) ::= k � expt(k,n) td( λ ) ::= 0 --uses recursive def of N : td( [ s ] t ) ::= • 0 ∈ � N 1 + max{td(s), td(t)} • if n ∈ � N , then n + 1 ∈ � N Albert R Meyer, March 1, 2013 recursivefunctions.3 Albert R Meyer, March 1, 2013 recursivefunctions.4 1
Recursive Functions Length versus Depth Lemma: |r| + 2 ≤ 2 td(r)+1 summary: for all r �� M f: Data � Values Proof by Structural Induction f(b) def’d directly for base b Base case: [ r = λ ] f(cnstr(x)) def’d using f(x), x | λ |+2 = 0+2 = 2 = 2 0+1 = 2 td( λ )+1 OK! recursivefunctions.6 Albert R Meyer, March 1, 2013 recursivefunctions.5 Albert R Meyer, March 1, 2013 Size versus Depth Size versus Depth |r|+2 = | [ s ] t| + 2 def. of r Constructor case: [r = [ s ] t] = (|s|+|t| +2) + 2 def. of length by ind. hypothesis: = (|s|+2)+(|t|+2) ≤ 2 td(s)+1 + 2 td(t)+1 induction hyp. |s| + 2 ≤ 2 td(s)+1 ≤ 2 max(td(s),td(t))+1 + 2 max(td(s),td(t))+1 = 2·2 max(td(s),td(t))+1 ≤ 2·2 td(r) def. of d(r) |t| + 2 ≤ 2 td(t)+1 = 2 td(r)+1 QED! Albert R Meyer, March 1, 2013 recursivefunctions.7 Albert R Meyer, March 1, 2013 recursivefunctions.8 2
positive powers of two log 2 of PP2 log 2 (2) ::= 1 2 ∈ PP2 log 2 (x � y) ::= log 2 (x) + log 2 (y) if x,y ∈ PP2, then x � y ∈ PP2 for x,y � � PP2 2, 2 � 2, 4 � 2, 4 � 4, 4 � 8, … log 2 (4) = log 2 (2 � 2) = 1 + 1 = 2 log 2 (8) = log 2 (2 � 4) = log 2 (2) + log 2 (4) 2 4 8 16 32 … ∈ � � PP2 = 1 + 2 = 3 Albert R Meyer, March 1, 2013 Albert R Meyer, March 1, 2013 recursivefunctions.25 recursivefunctions.24 loggy function on PP2 loggy function on PP2 loggy(2)::= 1 loggy(16) = loggy(8 � 2) = 9 loggy(x � y) ::= x + loggy(y) for x,y � � PP2 WAIT A SEC!: loggy(4) = loggy(2 � 2) = 2 + 1 = 3 loggy(16) = loggy(2 � 8) loggy(8) = loggy(2 � 4) = 2 + loggy(4) = 2 + loggy(8) = 2 + 5 = 2 + 3 = 5 loggy(16) = loggy(8 � 2) = 8 + loggy(2) = 7 = 8 + 1 = 9 Albert R Meyer, March 1, 2013 recursivefunctions.26 Albert R Meyer, March 1, 2013 recursivefunctions.27 3
ambiguous constructors ambiguous recursive defs The Problem: more than one way to problem to watch out for: construct elements of PP2 from recursive function on datum, e, cnstrct(x,y) = x � y is defined according to what 16 = cnstrct(8,2) but also constructor created e. 16 = cnstrct(2,8) If 2 or more ways to construct e, ambiguous then which definition to use? Albert R Meyer, March 1, 2013 Albert R Meyer, March 1, 2013 recursivefunctions.29 recursivefunctions.28 4
MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 20 15 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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