The problem asks us to count the number of tokens produced by a lexical analyzer for a given string, based on specific token definitions. We need to exclude whitespace tokens ($ws$) from the final count.

Token Definitions Analysis

The lexical analyzer uses the following rules:

• $letter$ $\rightarrow$ $[A-Z a-z]$ (Any uppercase or lowercase letter)
• $digit$ $\rightarrow$ $[0-9]$ (Any digit from 0 to 9)
• $id$ $\rightarrow$ $letter(letter | digit)*$ (Starts with a letter, followed by zero or more letters or digits)
• $number$ $\rightarrow$ $digit+$ (One or more digits)
• $ws$ $\rightarrow$ $(blank | tab | newline)+$ (One or more whitespace characters)

The lexical analyzer typically operates by scanning the input string from left to right and identifying the longest possible match for a token at each position.

String Tokenization Process

Let's trace the tokenization for the input string: $x1 23mm 78 y 7z zz5 14A 8H AaYcD$

1. x1: Starts with 'x' (a $letter$). Followed by '1' (a $digit$). This matches the $id$ rule. Token 1 ($id$).
2. (space): Matches $ws$. Ignored as per the question.
3. 23mm:
   • Starts with '2' (a $digit$). The longest sequence of digits is '23'. Matches the $number$ rule. Token 2 ($number$).
   • Remaining string is 'mm'. Starts with 'm' (a $letter$). Followed by 'm' (a $letter$). Matches the $id$ rule. Token 3 ($id$).
4. (space): Matches $ws$. Ignored.
5. 78: Starts with '7' (a $digit$). The longest sequence of digits is '78'. Matches the $number$ rule. Token 4 ($number$).
6. (space): Matches $ws$. Ignored.
7. y: Starts with 'y' (a $letter$). Matches the $id$ rule (base case: letter followed by zero characters). Token 5 ($id$).
8. (space): Matches $ws$. Ignored.
9. 7z:
   • Starts with '7' (a $digit$). Matches the $number$ rule. Token 6 ($number$).
   • Remaining character is 'z'. Starts with 'z' (a $letter$). Matches the $id$ rule. Token 7 ($id$).
10. (space): Matches $ws$. Ignored.
11. zz5: Starts with 'z' (a $letter$). Followed by 'z' (a $letter$) and '5' (a $digit$). Matches the $id$ rule. Token 8 ($id$).
12. (space): Matches $ws$. Ignored.
13. 14A:
    • Starts with '1' (a $digit$). The longest sequence of digits is '14'. Matches the $number$ rule. Token 9 ($number$).
    • Remaining character is 'A'. Starts with 'A' (a $letter$). Matches the $id$ rule. Token 10 ($id$).
14. (space): Matches $ws$. Ignored.
15. 8H:
    • Starts with '8' (a $digit$). Matches the $number$ rule. Token 11 ($number$).
    • Remaining character is 'H'. Starts with 'H' (a $letter$). Matches the $id$ rule. Token 12 ($id$).
16. (space): Matches $ws$. Ignored.
17. AaYcD: Starts with 'A' (a $letter$). Followed by 'a', 'Y', 'c', 'D' (all $letters$). Matches the $id$ rule. Token 13 ($id$).

Final Token Count

Summing up the identified tokens (excluding $ws$): 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 13 tokens.