Question

Write a regular expression for a language over the alphabet {a, b} that accepts all strings starting with 'a' and ending with 'b'.

Answer 1

Let's dive deep into the regular expression a(a|b)*b and understand exactly how it works, component by component.

The Goal

First, let's restate the rule we want to enforce:
1. The string must begin with the character a.
2. The string must end with the character b.
3. Any characters in between the first a and the last b can be either a or b.

The regular expression a(a|b)*b is a concise way to write these rules. We can think of it as having three distinct parts that correspond perfectly to our requirements.

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The Three-Part Breakdown

Think of the expression as: [The Beginning] [The Middle] [The End]

| Part | Regex Component | What it Means |
| :------------ | :-------------- | :--------------------------------------------- |
| Beginning | a | The string must start with a literal 'a'. |
| Middle | (a|b)* | Between the start and end, there can be anything. |
| End | b | The string must end with a literal 'b'. |

Now, let's analyze each component in detail.

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Part 1: The Beginning (a)

a(a|b)*b ^

This is the simplest part. In regular expressions, when a character doesn't have a special meaning (like *, +, |, (, )), it is a literal.

• a: This is a literal match. It asserts that the pattern must start with the character 'a'. The regular expression engine will first look at the very beginning of a string. If it's not an 'a', the string is immediately rejected without checking the rest.

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Part 2: The Middle ((a|b)*)

This is the most complex and flexible part of the expression. Let's break it down from the inside out.

a(a|b)*b ^^^^^

The Characters: a|b (The Alternation)

• The pipe character | is the OR operator in regular expressions. It's used to specify alternatives.
• a|b means "match a single character that is either an 'a' or a 'b'".
• This defines the "alphabet" or the set of allowed characters for this part of the string.

The Group: (...) (The Parentheses)

• Parentheses are used for grouping. They bundle the expression inside them into a single logical unit.
• In our case, (a|b) treats the "a or b" choice as one whole piece. This is crucial because we want to apply the next operator (the *) to the entire choice, not just to the b.
  • Without parentheses, a|b* would mean "match an 'a' OR match a 'b' zero or more times", which is a completely different rule.

*The Quantifier: `` (The Kleene Star)**

• The asterisk * is a quantifier. It applies to the element immediately preceding it, which is our group (a|b).
• The * means "match the preceding element zero or more times".
• This is the key to the middle section's flexibility:
  • Zero times: It can match an empty string. This is how the shortest possible string, ab, is accepted. The middle part simply doesn't exist.
  • One time: It can match a single a or a single b (e.g., in aab or abb).
  • More times: It can match any sequence of as and bs of any length (e.g., aaabbab in the string aaabbabb).

Putting the middle part together: (a|b)* means "match a sequence containing any number of 'a's and 'b's, including no characters at all".

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Part 3: The End (b)

`
a(a|b)*b

   ^

`

• b: Just like the a at the beginning, this is a literal match. It asserts that the character at the very end of the string must be a 'b'. If a string starts with 'a' and has a valid middle but ends in 'a', it will be rejected.

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A Walkthrough Analogy: The Sandwich

A great way to visualize this is to think of it as building a sandwich.
a: This is the top slice of bread. Every valid sandwich must have it.
b: This is the bottom slice of bread. Every valid sandwich must have it.
`(a|b)`: These are the fillings.

*   The `(a|b)` part says you are only allowed to use 'a' and 'b' as ingredients. No 'c', no 'd'.
*   The `*` part says you can have as many layers of filling as you want (`ababab...`), or you can have **no filling at all** (just the two slices of bread, `ab`).

Summary of How the Regex Engine Evaluates a String

When a regex engine sees the string aaabb against the pattern a(a|b)*b:

1. Check Start: Does the string start with a? Yes, aaabb does. Proceed.
2. Check End: Does the string end with b? Yes, aaabb does. Proceed.
3. Check Middle: Now, look at what's between the first a and the last b. The substring is aab.
4. Can (a|b)* match aab?
   • Can (a|b) match the first a? Yes.
   • Can (a|b) match the second a? Yes.
   • Can (a|b) match the b? Yes.
5. Since all parts of the rule are satisfied, the string aaabb is a match.

When it sees the string aba:

1. Check Start: Does it start with a? Yes.
2. Check End: Does it end with b? No, it ends with a.
3. The string is immediately rejected.