Lecture 16 - March 15, 2007

Recursion:

 

“A definition in which the item being defined appears as part of the definition is called in inductive definition  or a recursive definition…This works because there are two parts to a recursive definition: 

1.    A basis, where some simple cases of the item being defined are explicitly given

2.    An inductive or recursive step, where new cases of the item being defined are given in terms of previous cases.”   [1]

 

Examples:

A sequence S  is a list of objects that are enumerated in some order; there is a first such object, then a second, and so on.  S(k) denotes the kth object in the sequence.  A sequence is defined recursively by explicitly naming the first value (or the first few values) in the sequence and then defining later values in the sequence in terms of early values.

The sequence S is defined recursively by

1.    S(1) = 2

2.    S(n) = 2*S(n – 1) for n  >= 2

 A rule like that of statement 2 above, which defines a sequence value in terms of one or more earlier values, is called a  recurrence relation.

 

 

 

 

Here’s another:  The sequence T is defined recursively as follows:

1.    T(1) = 1

2.    T(n) = T(n – 1) + 3  for n >= 2

 

 

 

 

 

Here’s another:  The famous Fibonnaci sequence of numbers, introduced in the 13th century by an Italian merchant and mathematician, is defined recursively by

          F(1) = 1

          F(2) = 1

          F(n) = F(n – 2)  + F(n – 1) for n > 2

 

Here’s another:   Certain strings of statement letters, logical connectives, and parentheses such as  (A ^ B)’ V C, are considered legitimate, while other strings such as ^^A’’B, are not legitimate.  Here is a recursive definition of a well formed formula (wff).

1.    Any statement letter is a wff

2.    If P and Q are wffs, so are (P ^ Q), (P v Q), (P -> Q) , (P’), and (P <->  Q)

Here’s another:  A recursive definition for the set of people who are ancestors of James could have the following basis:

          James’s parents are ancestors of James

What is the inductive step?   

Every parent of an ancestor of James is an ancestor of James

 

 

Here’s another:  The set of all (finite-length ) strings of symbols over a finite alphabet A is denoted by A*.  The recursive definition of A* is

1.    The empty string l (string with no symbols) belongs to A*

2.    Any single member of A belongs to A*

3.    If x and y are strings in A*,  so is xy, the concatenation  of strings x and y.

Here’s another  In a given programming language, identifiers can be alphanumeric strings of arbitrary length but must begin with a letter.  A recursive definition for the set of such strings is

1.    A single letter is an identifier.

2.    If A is an identifier, so is the concatenation of A and any letter or digit.

A more symbolic notation for describing sets of strings that are recursively defined is called Backus-Naur form or BNF, which was originally developed by John Backus and Peter Naur to define the programming language ALGOL.   to be discussed next class

 

General Characteristics of Recursive methods

·       Recursive methods get their solutions by calling themselves.

·       They have what is known as a stopping case which keeps them from calling themselves forever.

·       They call themselves with a smaller problem each time.

 

Most common problem:

·       Infinite recursion

·       Caused by

o      Omitting stopping case

o      Not having successive cases getting smaller

 

TriangleArea

          PlayWithTriangle.java

          Triangle.java

 

.Dr Bernstein’s notes

https://users.cs.jmu.edu/bernstdh/web/common/lectures/slides_recursion.php

 

Link to BigJava materials

Website for BigJava text 2nd edition     http://www.wiley.com/college/horstmann   chapter 18

 

 


[1] Gersting, Judith L., Mathematical Structures for Computer Science, Fifth Edition, Freeman, 2003, pp. 120-1