General Recursive Definitions

1. General Recursive Definitions Lecture 37 Section 8.4 Fri, Mar 25, 2005

2. Recursively Defined Sets • The definition of a recursively defined set consists of three parts. • Base – Specific objects are in the set. • Recursion – Rules describing how to form new members from other members. • Restriction – No other elements are in the set.

3. Boolean Expressions • Define Boolean expressions as follows. • Base – Each lowercase letter of the alphabet represents a Boolean expression and T and F are Boolean expressions. • Recursion – If p and q are Boolean expressions, then so are pq, p q, p and (p). • Restriction – There are no other Boolean expressions.

4. Strings Not Containing the Substring 11 • Define a set of strings as follows. • Base – , 0, and 1 are in the set. • Recursion – If s is in the set, then so is 0s and 10s. • Restriction – No other strings are in the set. • The set consists of all strings that do not contain the substring 11.

5. The MIU System • Define the strings in the MIU system as follows. • Base – MI is in the MIU system.

6. The MIU System • Recursion • If xI is in the MIU system, then so is xIU. • If Mx is in the MIU system, then so is Mxx. • If xIIIy is in the MIU system, then so is xUy. • If xUUy is in the MIU system, then so is xUy. • Restriction – There are no other strings in the MIU system.

7. The MIU System • Derive a few strings in the MIU system. • MI  MIU • MI  MII  MIIII  MUI • MI  MII  MIIII  MIIIIIIII  MUIIIII  MUUII  MUII

8. The MU Puzzle • Is MU in the MIU system?

9. The MU Puzzle • Let n be the number of I’s in a string in the MIU system. • If we apply Rule 1, then the resulting string still contains n I’s. • If we apply Rule 2, then the resulting string contains 2n I’s. • If we apply Rule 3, then the resulting string contains n – 3 I’s. • If we apply Rule 4, then the resulting string still contains n I’s.

10. The MU Puzzle • Initially, n is 1. • Only Rules 2 and 3 change the number of I’s. • Thus, the following changes are possible. • n 2n. • nn – 3.

11. The MU Puzzle • The first change will map • A multiple of 3 to a multiple of 3, and • A non-multiple of 3 to a non-multiple of 3. • The second change will also map • A multiple of 3 to a multiple of 3, and • A non-multiple of 3 to a non-multiple of 3. • Given that n = 1 in the initial string and 1 is not a multiple of 3, it is impossible to eliminate I from the strings.

12. Arithmetic Expressions • Define a set of arithmetic expressions as follows. • Base – Any number is an arithmetic expression.

13. Arithmetic Expressions • Recursion – If x and y are arithmetic expressions, then so are • x + y • x – y • x * y • x / y • (x) • Restriction – There are no other arithmetic expressions.

14. Arithmetic Expressions • Derive the arithmetic expression (2 + 3) * (10 / 2 – 4)

15. Pointer Arithmetic • Pointer- and integer-valued expressions may be defined recursively. • Base • Any letter may represent a pointer-valued expression. • Any letter may represent an integer-valued expression.

16. Pointer Arithmetic • Recursion – Let p and q be pointer-valued expressions and i and j be integer-valued expressions. • i + j is an integer-valued expression. • i – j is an integer-valued expression • p – q is an integer-valued expression. • (i) is an integer-valued expression.

17. Pointer Arithmetic • p + i is a pointer-valued expression. • p – i is a pointer-valued expression. • (p) is a pointer-valued expression. • Restriction – No other expressions are in the set.

18. Pointer Arithmetic • Which expressions are legal expressions? • Which are integer-valued and which are pointer-valued? • p + (q + i) • (p + q) + i • (p – q) + i • (p – q) + (r – s) • (p + i) – (q – j) • (p – (q – j)) + i • (p – q) + (j + i)

19. Recursively Defined Functions • Mathematical functions may be defined recursively. • The classic example is the factorial function: • 0! = 1, • n! = n (n – 1)! for all n  1.

20. Recursively Defined Functions • Given the “successor” function succ(n) = n + 1, addition of nonnegative integers may be defined recursively. • sum(0, 0) = 0, • sum(m, 0) = succ(sum(m – 1, 0)) for all m 1, • sum(m, n) = succ(sum(m, n – 1)) for all n 1.

21. Recursively Defined Functions • Given the sum() function just defined, how could we define multiplication of nonnegative integers? • prod(m, n) = ?

22. The Ackermann Function • Define the Ackermann function A : ZnonnegZnonneg Z+ by • A(0, n) = n + 1, n 0. • A(m, 0) = A(m – 1, 1), m > 0. • A(m, n) = A(m – 1, A(m, n – 1)), m > 0, n > 0.

23. The Ackermann Function • A(0, 0) = 1. • A(0, 1) = 2. • A(0, 2) = 3. • A(0, 3) = 4. • In general, A(0, n) = n + 1.

24. The Ackermann Function • A(1, 0) = A(0, 1) = 2. • A(1, 1) = A(0, A(1, 0)) = A(0, 2) = 3. • A(1, 2) = A(0, A(1, 1)) = A(0, 3) = 4. • A(1, 3) = A(0, A(1, 2)) = A(0, 4) = 5. • In general, A(1, n) = n + 2.

25. The Ackermann Function • A(2, 0) = A(1, 1) = 3. • A(2, 1) = A(1, A(2, 0)) = A(1, 3) = 5. • A(2, 2) = A(1, A(2, 1)) = A(1, 5) = 7. • A(2, 3) = A(1, A(2, 2)) = A(1, 7) = 9. • In general, A(2, n) = 2n + 3.

26. The Ackermann Function • A(3, 0) = A(2, 1) = 5. • A(3, 1) = A(2, A(3, 0)) = A(2, 5) = 13. • A(3, 2) = A(2, A(3, 1)) = A(2, 13) = 29. • A(3, 3) = A(2, A(3, 2)) = A(2, 29) = 61. • In general, A(3, n) = 2n + 3 – 3.

27. The Ackermann Function • A(4, 0) = A(3, 1) = 13. • A(4, 1) = A(3, A(4, 0)) = A(3, 13) = 216 – 3 = 65533. • A(4, 2) = A(3, 65533) = 265533 – 3. • A(4, 3) = ? • A(4, 4) = ? • A(5, 5) = ?