Introduction to Automata

Power Of Sigma (Σ) in TOC

Sigma is a set of symbols that are always finite. It is denoted with “Σ”. An example of Sigma with four values is given below.

Σ = {a,0,b,k}    Where a, 0, b, and k are sigma values.

Powers of ‘ Σ ‘ Formula

The following is the formula of Power of Sigma

KN

Where K tells about the total number of Sigma values, and N is any positive Integer that tells about the maximum length of the string.

Power Of Sigma Examples

Following are the various examples of the power of Sigma. Suppose there are two alphabets of Sigma, i.e. (Σ= {a,b}). Hence, K=2. The power of Sigma with different values is given below.

Sigma with power Zero (K=2, N=0)

Σ0Σ= 20  = 1

Set of all strings over Σ of length 0 is only epsilon {ε}.

Sigma with Power 1 (K=2, N=1)

Σ(a,b)1 = Σ= 21 = 2

It means there are only two strings over Σ of a single length. i.e.{a, b} where ‘a’ is a single length

Sigma with Power 2 (K=2, N=2)

Σ(a,b)2 = Σ= 22 = 4

It means there are only four strings over Σ of length 2. i.e.{aa, ab, ba, bb}  where ‘aa’ is a double length.

Sigma with Power 3 (K=2, N=3)

Σ(a,b)3 = Σ= 23 = 8

It means there are only eight strings over Σ of length 3. i.e. {aaa, aab, baa………bbb} where ‘ aaa’ is a triple length.

In the same way, Sigma with power 4, 5, 6, and so on.

Power of Sigma when (K=4, N=2)

If K=4 and N=2, then it means there are four sigma values, i.e., Σ(a,b,c,d), and Power 2 tells every string of language may contain a maximum length of 2 as given below

Σ(a,b,c,d)2 = Σ= 42 = 16

It means there are 16  strings over Σ of length 2. i.e.{aa, bb, cc, dd, ab, ac, ad, ………} where ‘aa’ is a string of length 2.

Note:Similarly, Unlimited cases can be seen by changing the values of sigma and its power