CS402 Assignment No.2 Solution


     CS402 Assignment No.2 Solution

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Introduction to Computer Theory                                 
Summary
Introduction to the course title, Formal and In-formal languages, Alphabets, Strings, Null string, Words, Valid and In-valid alphabets, length of a string, Reverse of a string, Defining languages, Descriptive definition of languages, EQUAL, EVEN-EVEN, INTEGER, EVEN, { an bn},  { an bn an }, factorial, FACTORIAL, DOUBLEFACTORIAL, SQUARE, DOUBLESQUARE, PRIME, PALINDROME.
             
What does automata mean?
It is the plural of automaton, and it means  “something that works automatically”

Introduction to languages
There are two types of languages

                Formal Languages (Syntactic languages)
                Informal Languages (Semantic languages)

Alphabets Definition
A finite non-empty set of symbols (called letters), is called an alphabet. It is denoted by Σ ( Greek letter sigma).

Example
                Σ = {a,b}
     Σ = {0,1}  (important as this is the language which the computer understands.)
                Σ = {i,j,k}

Note Certain version of language ALGOL has 113 letters.
Σ (alphabet) includes letters, digits and a variety of operators including sequential operators such as GOTO and IF

Strings
Definition
                Concatenation of finite number of letters from the alphabet is called a string. 

Example
                If Σ = {a,b} then
                a, abab, aaabb, ababababababababab

Note
Empty string or null string
Sometimes a string with no symbol at all is used, denoted by (Small Greek letter Lambda) λ or (Capital Greek letter Lambda) Λ, is called an empty string or null string.
The capital lambda will mostly be used to denote the empty string, in further discussion.

Words
Definition
Words are strings belonging to some language.

Example
If Σ= {x} then a language L can be defined as  L={xn : n=1,2,3,…..} or L={x,xx,xxx,….}
Here x,xx,… are the words of L

Note
All words are strings, but not all strings are words.
Valid/In-valid alphabets
While defining an alphabet, an alphabet may contain letters consisting of group of symbols for example Σ1= {B, aB, bab, d}.
Now  consider an alphabet 
Σ2= {B, Ba, bab, d} and a string BababB.

This string can be tokenized in two different ways 
(Ba), (bab), (B)
(B), (abab), (B)
Which shows that the second group cannot be identified as a string, defined over  Σ = {a, b}.
As when this string is scanned by the compiler (Lexical Analyzer), first symbol B is identified as a letter belonging to Σ, while for the second letter the lexical analyzer would not be able to identify, so while defining an alphabet it should be kept in mind that ambiguity should not be created.

Remarks
While defining an alphabet of letters consisting of more than one symbols, no letter should be started with the letter of the same alphabet i.e. one letter should not be the prefix of another. However, a letter may be ended in a letter of same alphabet.

Conclusion
Σ1= {B, aB, bab, d}
Σ2= {B, Ba, bab, d}
Σ1 is a valid alphabet while Σ2 is an in-valid alphabet.

Length of Strings
Definition
The length of string s, denoted by |s|, is the number of letters in the string.

Example Σ={a,b} s=ababa
|s|=5

Example
Σ= {B, aB, bab, d}
s=BaBbabBd
Tokenizing=(B), (aB), (bab), (B), (d)
|s|=5

Reverse of a String Definition
The reverse of a string s denoted by Rev(s) or  sr,  is obtained by writing the letters of s in reverse order.

Example
If s=abc is a string defined over Σ={a,b,c} then Rev(s) or sr = cba

Example
Σ= {B, aB, bab, d} s=BaBbabBd
Rev(s)=dBbabaBB

Defining Languages
The languages can be defined in different ways , such as Descriptive definition, Recursive definition, using Regular Expressions(RE) and using Finite Automaton(FA) etc.

Descriptive definition of language
The language is defined, describing the conditions imposed on its words. 

Example
The language  L of strings of odd length, defined over Σ={a}, can be written as
L={a, aaa, aaaaa,…..}

Example
The language L of strings that does not start with a, defined over Σ ={a,b,c}, can be written as          L ={L, b, c, ba, bb, bc, ca, cb,  cc, …}

Example
The language L of strings of length 2, defined over Σ ={0,1,2}, can be written as
L={00, 01, 02,10, 11,12,20,21,22}

Example
The language L of strings ending in 0, defined over  Σ ={0,1}, can be written as
L={0,00,10,000,010,100,110,…}

Example 
The language EQUAL, of strings with number of a’s equal to number of b’s, defined over Σ={a,b}, can be written as
{Λ ,ab,aabb,abab,baba,abba,…}

Example 
The language EVEN-EVEN, of strings with even number of a’s and even number of b’s, defined over Σ={a,b}, can be written as
{Λ, aa, bb, aaaa,aabb,abab, abba, baab, baba, bbaa, bbbb,…}

Example
The language INTEGER, of strings defined over Σ={-,0,1,2,3,4,5,6,7,8,9}, can be written as  INTEGER = {…,-2,-1,0,1,2,…} 

Example
The language EVEN, of stings defined over Σ={-,0,1,2,3,4,5,6,7,8,9}, can be written as  EVEN = { …,-4,-2,0,2,4,…}

Example
The language {anbn }, of strings defined over Σ={a,b}, as 
{ab: n=1,2,3,…}, can be written as
{ab, aabb, aaabbb,aaaabbbb,…}

Example
The language {anbnan }, of strings defined over Σ={a,b}, as 
{an bn an: n=1,2,3,…}, can be written as
{aba, aabbaa, aaabbbaaa,aaaabbbbaaaa,…}

Example
The language factorial, of strings defined over Σ={0,1,2,3,4,5,6,7,8,9} i.e.
{1,2,6,24,120,…}

Example 
The language FACTORIAL, of strings defined over Σ={a}, as 
{an! : n=1,2,3,…}, can be written as
{a,aa,aaaaaa,…}. It is to be noted that the language FACTORIAL can be defined over any single letter alphabet.

Example 
The language DOUBLEFACTORIAL, of strings defined over Σ={a, b}, as 
{an!bn! : n=1,2,3,…}, can be written as 
{ab, aabb, aaaaaabbbbbb,…}

Example 
The language SQUARE, of strings defined over Σ={a}, as 
{an 2 : n=1,2,3,…}, can be written as
{a, aaaa, aaaaaaaaa,…}

Example 
The language DOUBLESQUARE, of strings defined over Σ={a,b}, as 
{an 2 bn 2 : n=1,2,3,…}, can be written as
{ab, aaaabbbb, aaaaaaaaabbbbbbbbb,…}

Example 
The language PRIME, of strings defined over Σ={a}, as 
{ap : p is prime}, can be written as
{aa,aaa,aaaaa,aaaaaaa,aaaaaaaaaaa…}

An Important language PALINDROME
The language consisting of Λ and the strings s defined over Σ  such that Rev(s)=s. It is to be denoted that the words of PALINDROME are called palindromes.

Example
For Σ={a,b}, 
PALINDROME={Λ , a, b, aa, bb, aaa, aba, bab, bbb, ...}

Remark
There are as many palindromes of length 2n as there are of length 2n-1. To prove the above remark, the following is to be noted:

Note
Number of strings of length ‘m’ defined over   alphabet of ‘n’ letters is nm.

Examples
The language of strings of length 2, defined over Σ={a,b} is L={aa, ab, ba, bb} i.e. number of strings = 2The language of strings of length 3, defined over Σ={a,b} is L={aaa, aab, aba, baa, abb, bab, bba, bbb} i.e. number of strings = 23


To calculate the number of palindromes of  length(2n), consider the following diagram,  












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