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Introduction of Theory Of Computation Notes

by admin in Uncategorized
Languages

Languages

Definition: Any set of strings over an alphabet ∑ is called a language. The set of all strings, including the empty string over an alphabet ∑ is denoted as ∑*. Infinite languages L are denoted as L ={wϵ ∑*: w has property P }
All Possible string is called a Language. A language is accepted by DFA is called a Regular Language.

Examples:

  • (a)   L1 = {w  ϵ {0,1}* : w has an equal number of 0′ s and 1′ s}
  • (b)  L2 ={wϵ S*: w =wR: where wR is the reverse string of w}

Descriptive definition of language

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

Example 1

The language  L of strings of odd length, defined over Σ={a}, can be written as

L={a, aaa, aaaaa,…..}

Example 2

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 3

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 4

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 5

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 6

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 7

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 8

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 9

The language {anbn  }, of strings defined over Σ={a,b}, as

{anbn   : n=1,2,3,…}, can be written as

{ab, aabb, aaabbb,aaaabbbb,…}

Example 10

The language {anbnan  }, of strings defined over Σ={a,b}, as {anbnan: n=1,2,3,…}, can be written as

{aba, aabbaa, aaabbbaaa,aaaabbbbaaaa,…}

Example 11

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 12

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 13

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 14

The language SQUARE, of strings defined over Σ={a}, as {an2 : n=1,2,3,…}, can be written as

{a, aaaa, aaaaaaaaa,…}

Example 15

The language DOUBLESQUARE, of strings defined over Σ={a,b}, as

{an2 bn2 : n=1,2,3,…}, can be written as {ab, aaaabbbb, aaaaaaaaabbbbbbbbb,…}

Example 16

The language PRIME, of strings defined over Σ={a}, as

{ap : p is prime}, can be written as {aa,aaa,aaaaa,aaaaaaa,aaaaaaaaaaa…}

Palindrome

Definition: A palindrome is a string which is the same whether written or read backward or forward. 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 = 22
  • The 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

Theory of Computation Notes

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Read More: InfoKhajana.com covered the following topics in these notes.

References –
cs.stanford.edu
Wikipedia