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# Theory of Computation: Lecture 13

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1) Bounded minimalization
2) Bounded minimalization through bounded products and summation
3) Primitive recursive functions

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### Theory of Computation: Lecture 13

1. 1. CS 5000: Theory of Computation Lecture 13 Vladimir Kulyukin Department of Computer Science Utah State Universitywww.youtube.com/vkedco
2. 2. Outline ● Review ● Minimalizationwww.youtube.com/vkedco
4. 4. Review: Corollary 5.5 (Ch. 3) Let C be a PRC class, let n - ary functions g 1 ,..., g m , h and predicates P1 ,..., Pm belong to C , and let Pi ( x1 ,..., x n ) ⋅ Pj ( x1 ,..., x n ) = 0, for all 1 ≤ i < j ≤ m and all x1 ,..., x n . Let  g 1 ( x1 ,..., x n ) if P1 ( x1 ,..., x n )  ...  f ( x1 ,..., x n ) =   g m ( x1 ,..., x n ) if Pm ( x1 ,..., x n )  h( x1 ,..., x n ) otherwise.  Then f ∈ C.www.youtube.com/vkedco
5. 5. Review: Theorem 6.1 (Ch. 3) Let C be a PRC class. If f ( x1 ,..., xn ) ∈ C , then so do the functions y g ( y, x1 ,..., xn ) = ∑ t= 0 f (t ,x1 ,..., xn ) and y h( y, x1 ,..., xn ) = ∏ t= 0 f (t , x1 ,..., xn ).www.youtube.com/vkedco
6. 6. Review: Corollary 6.2 (Ch. 3) If f (t , x1 ,..., x n ) ∈ C and C is PRC, then so do these functions : y g ( y, x1 ,..., x n ) = ∑ t= 1 f (t ,x1 ,..., x n ) y h( y, x1 ,..., x n ) = ∏ t= 1 f (t , x1 ,..., x n )www.youtube.com/vkedco
7. 7. Review: Theorem 6.3 (Ch. 3) If the predicate P (t , x1 ,..., x n ) belongs to some PRC class C , then so do the predicates (∀ t ) ≤ y P (t , x1 ,..., x n ) and (∃ t ) t ≤ y P (t , x1 ,..., x n )www.youtube.com/vkedco
8. 8. Review: Corollary of Theorem 6.3 (Ch. 3) Theorem 6.3 is valid for strict bounded quantification : (∀ t ) < y P (t , x1 ,..., x n ) ⇔ (∀ t ) ≤ y [ t = y ∨ P (t , x1 ,..., x n )] ( ∃ t ) < y P(t , x1 ,..., x n ) ⇔ (∃ t ) ≤ y [α (t = y ) & P (t , x1 ,..., x n )]www.youtube.com/vkedco
9. 9. Minimalization • 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 undefinedwww.youtube.com/vkedco
10. 10. Example 1 0 1 2 3 4 P(0) = 0 P(1) = 0 P(2) = 0 P(3) = 1 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 ≥ 3www.youtube.com/vkedco
11. 11. Bounded Minimalization Suppose : 1. ( ∀ t ) < t0 P (t , x1 ,..., x n ) = 0 2. P (t 0 , x1 ,..., x n ) = 1 Then u  1 if u < t 0 ∏ α ( P( t , x ,..., x ) ) 1 n =  t= 0  0 if u ≥ t 0www.youtube.com/vkedco
12. 12. 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 ) = ∑∏ u= 0 t= 0 α ( P (t , x1 ,..., xn )).www.youtube.com/vkedco
13. 13. Example 2 Suppose t 0 = 3, y = 4. 4 u g ( y = 4, x1 ,..., x n ) = ∑∏ u= 0 t= 0 α ( P (t , x1 ,..., x n )) = 0 1 ∏t= 0 α ( P (t , x1 ,..., x n )) + ∏ α ( P(t , x1 ,..., x n )) + t= 0 2 3 ∏t= 0 α ( P (t , x1 ,..., x n )) + ∏ t= 0 α ( P(t , x1 ,..., x n )) + 4 ∏t= 0 α ( P (t , x1 ,..., x n )) = 1 + 1 + 1 + 0 + 0 = 3www.youtube.com/vkedco
14. 14. Example 3 Suppose t 0 = 3, y = 1. 1 u g (1, x1 ,..., x n ) = ∑∏ u= 0 t= 0 α ( P (t , x1 ,..., x n )) = 0 1 ∏ t= 0 α ( P (t , x1 ,..., x n )) + ∏ α ( P (t , x1 ,..., x n )) = t= 0 1+ 1 = 2www.youtube.com/vkedco
15. 15. Lemma ( ∀ y ) < t g ( y, x1 ,..., x n ) = 0 y+1www.youtube.com/vkedco
16. 16. Proof Let y < t 0 . Then for every u in the range u [ 0, y ], ∏ α ( P (t , x1 ,..., x n )) = 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= 0www.youtube.com/vkedco
17. 17. Corollary If t 0 ≤ y, then g ( y, x1 ,..., x n ) = ∑ u < t0 1 = t0 . g ( y, x1 ,..., x n ) is the least value of t for which P (t , x1 ,..., x n ) is true.www.youtube.com/vkedco
18. 18. Bounded Minimalization: Definition  g ( y, x1 ,..., x n ) if ( ∃ t ) ≤ y P ( t , x1 ,..., x n ) min P ( t , x1 ,..., x n ) =  t≤ y 0 otherwise 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.www.youtube.com/vkedco
19. 19. Theorem 7.1 (Ch. 3) 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.www.youtube.com/vkedco
20. 20. Proof 7.1 (Ch. 3) 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.www.youtube.com/vkedco
21. 21. Example 4  x Show that   (integer part of the quotient x/y ) is  y primitive recursive.www.youtube.com/vkedco
22. 22. Example 4  x   ⇔ min[ (t + 1) ⋅ y > x ]  y t≤ xwww.youtube.com/vkedco
23. 23. Example 4  7  2 ⇔ min[ (t + 1) ⋅ 2 > 7] = 3,   t≤ 7 because ( 3 + 1) ⋅ 2 > 7.www.youtube.com/vkedco
24. 24. More Examples  8  2 = 4    9  2 = 4    4  5 = 0    x  0  = 0 Why?  www.youtube.com/vkedco
25. 25. Example 5 Show that R(x,y), i.e. the remainder of the division of x by y, is primitive recursive.www.youtube.com/vkedco
26. 26. Example 5: Patterns 4 1  4 1 = 1+ = +  3 3 3 3   7 1  7 1 = 2+ =   + 3 3  3 3 7 3  7 3 = 1+ =   + 4www.youtube.com/vkedco 4  4 4
27. 27. Example 5 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) = xwww.youtube.com/vkedco
28. 28. Reading Suggestions nd ● Ch. 3, Computability, Complexity, and Languages, 2 Edition, by Davis, Weyuker, Sigalwww.youtube.com/vkedco