What are the fundamental concepts of TOC?

Data Structure AlgorithmsComputer ScienceComputers

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 − yR = 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}

raja
Published on 11-Jun-2021 13:14:57
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