What are the fundamental concepts of TOC?

The basic definitions of the fundamental concepts in the Theory of Computation (TOC) along with the relevant examples are explained below −

Symbol

Symbols simply call it as a character.

It is an atomic unit, such as a digit, character, lowercase letter, etc. Sometimes it is also a word. The formal language does not deal with the “meaning” of the symbols.

For example,

• a,b,c,……………z
• 0,1,2,…………..9
• +,-,*,%,…………special characters.

Alphabet

The set of characters is called as the alphabet.

An alphabet is a finite, non-empty set of symbols. It is denoted by Σ or E.

For example,

• Σ ={0,1} set of binary alphabets.
• Σ ={a,b,c,……..,z} set of all lower case letters.
• Σ ={A,B,C,………Z} set of all upper case letters.
• Σ ={+,&,%,……….} set of all special characters.

String or Word

A string is a finite set sequence of symbols choose from some alphabets

For example,

• 00011001 is a string from the binary alphabet Σ={0,1} and aabbcabcd is a string from the alphabet Σ={a,b,c,d}.
• If, w = 0110 y = 0aa x = aabcaa z = 111. Then,
  • Special string − s (also denoted by X)
  • Concatenation − wz = 0110111
  • Length − |w| = 4 |s| = 0 |x| = 6
  • Reversal − y^R = aa0

Some special sets of strings are as follows −

• E* All strings of symbols from E
• E+ E* - {s}

For example,

• E = {0, 1}
• E* = {s, 0, 1, 00, 01, 10, 11, 000, 001,...}
• E+ = {0, 1, 00, 01, 10, 11, 000, 001,.}

Length of string

It is the number of symbols in the string or word. It is denoted by |w|.

For example,

• w=01011001 from binary alphabet Σ={0,1}

  |w| = 8

• X= abbaddabba from binary alphabet Σ={a,b}

  |X| = 10

Language

A language is a set of strings from some alphabet (finite or infinite). In other words, any subset L of E* is a language in TOC.

Some special languages are as follows −

• {} The empty set/language, containing no string.
• {s} A language containing one string, the empty string.

Examples

• E = {0, 1}

  L = {x | x is in E* and x contains an even number of 0’s}

• E = {0, 1, 2,., 9, .}

  L = {x | x is in E* and x forms a finite length real number}

  = {0, 1.5, 9.326,.}

• E = {a, b, c,., z, A, B,., Z}

  L = {x | x is in E* and x is a Pascal reserved word}

  = {BEGIN, END, IF,...}

• E = {Pascal reserved words} U { (, ), ., :, ;,...} U {Legal Pascal identifiers}

  L = {x | x is in E* and x is a syntactically correct Pascal program}

• E = {English words}

  L = {x | x is in E* and x is a syntactically correct English sentence}