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I am studying a distance CS course, but there is no tutor available, so I would appreciate your help...

Consider the language $S = \{a, aa, ab\}$ How many distinct words of length $n$ will appear in $S^*$?

Is there a way to do this without having to "enumerate" $S^*$ until the answer is found?

xskxzr
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mikey3433
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2 Answers2

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The language $S^*$ is regular. Therefore, it can be expressed by a DFA. Now use Why isn't it simple to count the number of words in a regular language? or https://cstheory.stackexchange.com/q/8200/5038.

D.W.
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D.W. has shown how to solve the problem in a general setting. Let me show the simplest way for this particular problem.

What is $S^*$?

$S^*=\{w\in\{a,b\}^*: w\text{ does not contain }bb\text{ and }w \text{ does not start with } b\}$.

The description above can be shown, for example, by demonstrating that the right hand side contains $\epsilon, a, aa, ab$ and its concatenation with itself.

Recurrence relation

Let $f_n$ be the number of words of length $n$ in $S^*$. Then $f_0=1$ and $f_1=1$.

Let $w$ be a word of length $n+1$, $n\ge1$.

  • $w$ ends with $a$. Then $w=xa$ for $x\in S^*$ of length $n$.
  • Otherwise, $w$ ends with $b$. Then $w=yab$ for $y\in S^*$ of length $n-1$.

The above shows that $f_{n+1}=f_n+f_{n-1}$. So $f_n$ is the famous Fibonacci sequence.

Final formula

$$f_n=\frac{\left(\dfrac{1+\sqrt5}2\right)^n-\left(\dfrac{1-\sqrt5}2\right)^n}{\sqrt5}$$

Exercise. (One minute or less if you find the shortcut.) How many distinct words of length $n$ appear in $\{a,bb\}^*$?

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John L.
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