Handbook of Computer Science(cs) and IT

1. Theory of Computation

Basics of Theory of Computation

            Computation is defined as any type of calculation. It is also defined as use of computer technology information processing. The theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm.

Basic Definitions

Symbol          A symbol is an abstract entity i.e., letters and digits

String            A String is a finite sequence of symbols.

Alphabet      An alphabet is a finite set of symbols, usually denoted by Σ.

 

Language    A formal language is a set of strings of symbols from some alphabet.

 

Key Points

• Generally a, b, c, ……used to denote symbols. Alphabets will represent the observable events of an automata.
• Generally w, x, y, z used to denote words. A word will represent the behavior of an automatation.

Kleene Closure

It Σ is the set of alphabets, then there is a language is which any string of letters from Σ is a word, even the null string. We call this Language closure of the alphabet. It is denoted by * (asterisk) after the name of the alphabet is Σ^*. This notation is also known as the Kleene Star.

e.g.,                            if Σ = {a}, then

Σ^* = {Λ, a, aa, aaa, …}

 

Where, Λ represents null String.

if                                  if Σ = {a, b}, then

Σ^* = {Λ, a, b, aa, ab, aaa, …}

 

Key Points

• By using Kleene Star operation, we can make an infinite language of strings of letters of of alphabet.
• The words in increasing order of length called lexicographic order.

 

Positive Closure

The ‘+’ (Plus Operation) is sometimes called positive closure. A+ (Plus) closure never contain null value.

If                     Σ = {a}, then Σ^* = {a, aa, aaa, …}

 

Note If S is the set of strings, then S^* is the language S^* without the word Λ.

 

Operations Over Words In Σ^*

• Concatenation if x, y ϵ Σ^*, then x concatenated with y is the word formed by the symbols of x followed by the symbols of y. This is denoted by xy.
• Substring A string v is a substring of a string ω if and only if there are strings x and y such that ω = xvy.
• Suffix if ω = xv for some string x, then v is suffix of ω.
• Prefix if ω = vy for some string y, then v is suffix of ω.
• Reversal Give a string ω, its reversal denoted by ω^R is the string speeled backwards.

e.g.,    (abcd)^R = dcba

Alphabet Ʃ^*

Ʃ^* It is the set of all words for a given alphabet Ʃ. This can be described inductively in atleast two different ways

• Basic case The empty word ^ is the Ʃ^* (notation : ^ ϵ Ʃ^* )

Inductive Step If a ϵ Ʃ and x ϵ Ʃ^* ,then ax ϵ Ʃ^*

                                    And also xa ϵ Ʃ^*

• Null set The language that has no words and can be represented by ɸ.