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Theory of Computation (Fall 2013): Minimalization
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Theory of Computation (Fall 2013): Minimalization

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  • 1. Theory of Computation Minimalization Vladimir Kulyukin www.vkedco.blogspot.com
  • 2. Outline ● ● ● ● ● Minimalization Bounded Minimalization Minimalization & PRC Classes Unbounded Minimalization Minimalization in Proofs of Primitive Recursiveness
  • 3. Minimalization
  • 4. Minimalization: Definition • Suppose P is a predicate, P(t, x1, … , xn) for n ≥ 0 • Minimalization is a technique for finding the minimal value of t for which P(t, x1, … , xn) = 1 • If there is such a t, then minimalization returns it • If there is no such t, then minimalization is undefined
  • 5. Example 0 1 P(0) = 0 P(1) = 0 2 P(2) = 0 3 P(3) = 1 4 P(4) = ? α ( P( 0) ) = 1 α ( P( 0 ) ) ⋅ α ( P(1) ) = 1 ⋅ 1 = 1 α ( P( 0 ) ) ⋅ α ( P(1) ) ⋅ α ( P( 2 ) ) = 1 ⋅ 1 ⋅ 1 = 1 α ( P( 0 ) ) ⋅ ... ⋅ α ( P( n ) ) = 0, n ≥ 3
  • 6. Bounded Minimalization
  • 7. Bounded Minimalization Suppose : 1. ( ∀t ) <t0 P (t , x1 ,..., x n ) = 0 2. P (t 0 , x1 ,..., x n ) = 1 Then u ∏α ( P( t , x ,..., x ) ) 1 t =0 n 1 if u < t 0 = 0 if u ≥ t 0
  • 8. Bounded Minimalization Let P(t , x1 ,..., xn ) be a predicate in some PRC class C. Let t0 be the smallest value for which P(t , x1 ,..., xn ) = 1. Consider y u g ( y, x1 ,..., xn ) = ∑∏ α ( P(t , x1 ,..., xn )). u =0 t =0
  • 9. Example Suppose t 0 = 3, y = 4. u 4 g ( y = 4, x1 ,..., x n ) = ∑∏α ( P (t , x1 ,..., x n )) = u =0 t =0 0 ∏α ( P(t , x ,..., x t =0 1 1 n 2 ∏α ( P(t , x ,..., x 1 n t =0 )) + ∏α ( P (t , x1 ,..., x n )) + t =0 4 1 t =0 3 t =0 ∏α ( P(t , x ,..., x )) +∏α ( P (t , x1 ,..., x n )) + n )) = 1 + 1 + 1 + 0 + 0 = 3
  • 10. Example Suppose t 0 = 3, y = 1. 1 u g (1, x1 ,..., x n ) = ∑∏α ( P (t , x1 ,..., x n )) = u =0 t =0 0 ∏α ( P(t , x ,..., x 1 t =0 1+1 = 2 1 n )) +∏α ( P (t , x1 ,..., x n )) = t =0
  • 11. Lemma ( ∀y ) <t g ( y, x1 ,..., x n ) = y + 1 0
  • 12. Proof Let y < t 0 . Then for every u in the range u [ 0, y ], ∏α ( P(t , x1 ,..., xn )) = 1. Since the t =0 range [ 0, y ] has y + 1 elements, y u g ( y, x1 ,..., x n ) = ∑∏α ( P (t , x1 ,..., x n )) = y + 1. u =0 t =0
  • 13. Corollary If t 0 ≤ y, then g ( y, x1 ,..., x n ) = ∑ 1 = t 0 . u <t0 g ( y, x1 ,..., x n ) is the least value of t for which P (t , x1 ,..., x n ) is true.
  • 14. Bounded Minimalization: Definition min P( t , x1 ,..., x n ) t≤ y  g ( t , x1 ,..., x n ) if ( ∃) ≤ y P ( t , x1 ,..., x n ) = otherwise 0 min P( t , x1 ,..., x n ) is the least value of t for which P( t , x1 ,..., x n ) = 1 t≤ y if t ∈ [ 0, y ] and 0 otherwise.
  • 15. Minimalization & PRC Classes
  • 16. Theorem 7.1 (Ch. 03) If P (t , x1 ,..., xn ) belongs to some PRC class C and f ( y, x1 ,..., xn ) = min P(t , x1 ,..., xn ), t≤ y then f ( y, x1 ,..., xn ) also belongs to C.
  • 17. Proof 7.1 (Ch. 03) g ( y, x1 ,..., x n ) ∈ C , because ∑ ,∏ , and α are in C. By Theorem 6.3 (bounded quantification), ( ∃t ) ≤ y P(t , x1 ,..., x n ) is also in C. By Theorem 5.4 (definition by cases), min P (t , x1 ,..., x n ) is t≤ y in C.
  • 18. Unbounded Minimalization
  • 19. Theorem 7.2 (Ch. 3) If P( t , x1 ,..., xn ) is a computable predicate, then g ( x1 ,..., xn ) = min P( y, x1 ,..., xn ) is partially computable y
  • 20. Proof 7.2 (Ch. 03) [A1] IF P(Y, X1, …, Xn) GOTO E Y←Y+1 GOTO A1
  • 21. Minimalization in Proofs of Primitive Recursiveness
  • 22. Example 01 x Show that   (integer part of the quotient x/y ) is  y primitive recursive.
  • 23. Example 01 x ⇔ min[ (t + 1) ⋅ y > x ]  y t≤x  
  • 24. Example 01 7  ⇔ min[ (t + 1) ⋅ 2 > 7] = 3, 2 t ≤7   because ( 3 + 1) ⋅ 2 > 7.
  • 25. Example 01 8  =4 2    9  =4 2    4  5  = 0   x  0  = 0 Why?  
  • 26. Example 02 Show that R(x,y), i.e. the remainder of the division of x by y, is primitive recursive.
  • 27. Example 02: Discovering Patterns 4 1 4 1 =1+ =   + 3 3 3 3 7 1 7  1 = 2+ =  + 3 3 3 3 7 3 7  3 =1+ =   + 4 4 4 4
  • 28. Example 02 x  x  R ( x, y ) 1. =   + y y y x  x  R ( x, y ) 2. −   = y y y x 3. x −   y = R ( x , y ) y x 4. R ( x, y ) ⇔ x −  y y Note that R ( x,0) = 0. •
  • 29. Reading Suggestions ● Ch. 3, Computability, Complexity, and Languages, 2nd Edition, by Davis, Weyuker, Sigal

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