Question

given an alphabet σ = {0, 1} construct a regular expression for strings with at most three 1’s e.g 0, 01, 1001, 11001, etc

Answer 1

The construction of a regular expression for strings containing at most three '1's over the alphabet {0, 1} involves creating parts that represent strings with varying numbers of '1's. Combining these parts with asterisks allows for all valid string combinations.

Step-by-step explanation:

The question asks how to construct a regular expression that matches strings over the alphabet σ = {0, 1}, which contain at most three '1's. The regular expression that satisfies this condition can be constructed by considering parts: strings with no '1's, with one '1', two '1's, and three '1's. The following is a regular expression that fits the criteria:

(0*1?0*)*(0*10*10*)*(0*10*1?0*)*

This regular expression breaks down as follows:

• '0*1?0*' accounts for sections of the string that contain at most one '1', with any number of '0's before and after.
• '0*10*10*' deals with a part of the string that contains exactly two '1's.
• '0*10*1?0*' covers the part that contains at most three '1's.

Putting all these parts together with '*' allows for any combination of these parts to occur, including none, which covers strings from no '1's to a maximum of three '1's.