Regex Analysis for String Generation
The regular expression is $(0|ε)1+2* (3|ε)$. We need to find the number of strings of length exactly 4.
Breakdown of components:
- $(0|ε)$: Matches '0' (length 1) or null string $ε$ (length 0).
- $1+$: Matches one or more '1's. Length $L_{1^+} \ge 1$.
- $2*$: Matches zero or more '2's. Length $L_{2^*} \ge 0$.
- $(3|ε)$: Matches '3' (length 1) or null string $ε$ (length 0).
The total length must be 4: $L_{(0|\epsilon)} + L_{1^+} + L_{2^*} + L_{(3|\epsilon)} = 4$.
Enumerating String Possibilities
We analyze cases based on the choices for the first $(0|ε)$ and last $(3|ε)$ parts.
| First Part Choice | Length ($L_{p1}$) | Last Part Choice | Length ($L_{p4}$) | Required Length ($L_{p2} + L_{p3}$) | Valid ($L_{p2}, L_{p3}$) Pairs | Example Strings (Length 4) | Count |
| $0$ | 1 | $3$ | 1 | $1 + L_{p2} + L_{p3} + 1 = 4 \implies L_{p2} + L_{p3} = 2$ | (1, 1), (2, 0) | $0123$, $0113$ | 2 |
| $0$ | 1 | $ε$ | 0 | $1 + L_{p2} + L_{p3} + 0 = 4 \implies L_{p2} + L_{p3} = 3$ | (1, 2), (2, 1), (3, 0) | $0122$, $0112$, $0111$ | 3 |
| $ε$ | 0 | $3$ | 1 | $0 + L_{p2} + L_{p3} + 1 = 4 \implies L_{p2} + L_{p3} = 3$ | (1, 2), (2, 1), (3, 0) | $1223$, $1123$, $1113$ | 3 |
| $ε$ | 0 | $ε$ | 0 | $0 + L_{p2} + L_{p3} + 0 = 4 \implies L_{p2} + L_{p3} = 4$ | (1, 3), (2, 2), (3, 1), (4, 0) | $1222$, $1122$, $1112$, $1111$ | 4 |
Total String Count
Summing the counts from all cases:
Total Strings = 2 + 3 + 3 + 4 = 12.
Thus, there are 12 unique strings of length 4 generated by the regex.
Important Questions from Regular Expression - Teaching
If $r_1$ and $r_2$ are regular expressions, then which of the following are correct.
A. $L(r_1 + r_2) = L(r_1) \cup L(r_2)$
B. $L(r_1 \cdot r_2) = L(r_1) L(r_2)$
C. $L((r_1)) = L(r_1)$
D. $L(r_1^*) = (L(r_1))^*$
