Question

Let \(L_1\) be the language represented by the regular expression \(b^*ab^*(ab^*ab^*)^*\) and \(L_2 = \{ w \in (a + b)^* \mid |w| \leq 4 \\), where \(|w|\) denotes the length of string \(w\). The number of strings in \(L_2\) which are also in \(L_1\) is ............}

Hint:

When intersecting a regular expression and a finite language, enumerate all strings in the finite language and check which satisfy the regular expression.

Answer 1

Step 1: Define the set of language symbols.
Let the set of language symbols be: \[ \Sigma = \{a, b\} \implies |\Sigma| = 2. \] Step 2: Define the regular language \(L_1\).
The regular language \(L_1\) is defined as the set of strings where the number of \(a\)'s is odd. Step 3: Analyze the strings with an odd number of \(a\)'s.
For all strings with the alphabet set \(\{a, b\}\) of length \(n\), half of them will have an odd number of \(a\)'s. This can be expressed as: \[ \frac{2^n}{2}, \] where the numerator \(2^n\) represents the total number of strings of length \(n\) (based on the number of input symbols). Step 4: Calculate the total number of strings in \(L_1\) for all lengths up to 4.
We sum up half of the total strings of all lengths (up to 4) in \(L_1\): \[ \frac{2^0}{2} + \frac{2^1}{2} + \frac{2^2}{2} + \frac{2^3}{2} + \frac{2^4}{2}. \] Simplify the terms: \[ 0 + 1 + 2 + 4 + 8 = 15. \] Final Answer: The total number of strings in \(L_1\) (up to length 4) is: \[ \boxed{15}. \]