1. CS 250, Discrete Structures, Fall 2014 Nitesh Saxena
Lecture 3.3: Recursion
CS 250, Discrete Structures, Fall 2014
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag

2. Course Admin Mid-Term 1 Graded To be distributed today
HW2 being graded
Solution provided
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Lecture Recursion

3. Lecture 3.2: Induction and Strong Induction (contd)
Course Admin
Overall grades
Recall: they will be relative, based on overall performance of the class
Further improvement possible in the upcoming HWs and two exams
Please continue to work hard. It will pay off.
Don’t hesitate to ask for extra help
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Lecture 3.2: Induction and Strong Induction (contd)

4. Course Admin HW3 posted Mid Term 2: Nov 4 (Tues)
Covers “Induction and Recursion” (Chapter 5)
Due in Thu, Nov 6
Mid Term 2: Nov 4 (Tues)
Review Oct 30 (Thu)
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Lecture Recursion

5. Outline Some practice: strong induction Recursion
Recursive Functions and Definitions
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Lecture Recursion

6. Strong Induction Example (Rosen)
Prove that every integer > 1 can be expressed as a product of prime numbers
[This is referred to as the fundamental theorem of arithmetic]
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Lecture Recursion

7. Recursively Defined Sequences
Often it is difficult to express the members of an object or numerical sequence explicitly.
EG: The Fibonacci sequence:
{fn } = 0,1,1,2,3,5,8,13,21,34,55,…
There may, however, be some “local” connections that can give rise to a recursive definition –a formula that expresses higher terms in the sequence, in terms of lower terms.
EG: Recursive definition for {fn }:
INITIALIZATION: f0 = 0, f1 = 1
RECURSION: fn = fn-1+fn-2 for n > 1.
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Lecture Recursion

8. Recursive Functions
It is possible to think of any function with domain N as a sequence of numbers, and vice-versa.
Simply set: fn =f (n)
For example, our Fibonacci sequence becomes the Fibonacci function as follows:
f (0) = 0, f (1) = 1, f (2) = 1, f (3) = 2,…
Such functions can then be defined recursively by using recursive sequence definition. EG:
INITIALIZATION: f (0) = 0, f (1) = 1
RECURSION: f (n) = f (n -1) +f (n -2), for n > 1.
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Lecture Recursion

9. Recursive Functions: Factorial
A simple example of a recursively defined function is the factorial function:
n! = 1· 2· 3· 4 ···(n –2)·(n –1)·n
i.e., the product of the first n positive numbers (by convention, the product of nothing is 1, so that 0! = 1).
Q: Find a recursive definition for n!
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Lecture Recursion

10. Recursive Functions: Factorial
A:INITIALIZATION: 0!= 1
RECURSION: n != n · (n -1)!
To compute the value of a recursive function, e.g. 5!, one plugs into the recursive definition obtaining expressions involving lower and lower values of the function, until arriving at the base case.
EG: 5! =
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Lecture Recursion

11. Recursive Functions: Factorial
A:INITIALIZATION: 0!= 1
RECURSION: n != n · (n -1)!
To compute the value of a recursive function, e.g. 5!, one plugs into the recursive definition obtaining expressions involving lower and lower values of the function, until arriving at the base case.
EG: 5! = 5 · 4! = 5 · 4 · 3! = 5 · 4 · 3 · 2!
= 5 · 4 · 3 · 2 · 1! = 5 · 4 · 3 · 2 · 1 · 0! = 120
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Lecture Recursion

12. Recursive Functions: gcd
Euclid’s algorithm makes use of the fact that gcd(x,y ) = gcd(y, x mod y)
(here we assume that x > 0)
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Lecture Recursion

13. Recursive Definitions: Mathematical Notation
Definition of summation notation:
There is also a general product notation :
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Lecture Recursion

14. Recursive Definitions: Mathematical Notation
Q: Find a recursive definition for the product notation
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Lecture Recursion

15. Recursive Definitions: Mathematical Notation
A: This is very similar to definition of summation notation.
Note: Initialization is argument for “product of nothing” being 1, not 0.
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Lecture Recursion

16. Recursively Defined Sets
Our examples so far have been inductively defined functions.
Sets can be defined inductively, too.
Give an inductive definition of
S = {x: x is a multiple of 3}
3  S
x,y  S  x + y  S
Base Case
Recursive Case
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Lecture Recursion

17. Strings: Recursive Definition
Let  be a finite set called an alphabet.
The set of strings on , denoted * is defined as:
  *, where  denotes the null or empty string.
If x  , and w  *, then wx  *, where wx is the concatenation of string w with symbol x.
Example: Let  = {a, b, c}. Then
* = {, a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, aab,…}
How big is *?
Countably infinite
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Lecture Recursion

18. Length of a String: Recursive Definition
Recursive definition of the length of strings (the length of string w is |w|.):
|| = 0
If x  , and w  *, then |wx| = |w| + 1
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Lecture Recursion

19. Well Formed Formulae (WFF) for Propositions
A set of wff is defined as follows:
T is a wff
F is a wff
p is a wff for any propositional variable p
If p is a wff, then (p) is a wff
If p and q are wffs, then (p  q), is a wff
If p and q are wffs, then (p  q) is a wff
For example, a statement like ((r)  (p  r)) can be proven to be a wff by arguing that (r) and (p  r) are wffs by recursion and then applying rule 5.
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Lecture Recursion

20. Today’s Reading
Rosen 5.3
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Lecture Recursion