As you look over the list of messages, you realize your matching rules aren’t quite right. To fix them, completely replace rules 8: 42 and 11: 42 31 with the following:

8: 42 | 42 8
11: 42 31 | 42 11 31

This small change has a big impact: now, the rules do contain loops, and the list of messages they could hypothetically match is infinite. You’ll need to determine how these changes affect which messages are valid.

Fortunately, many of the rules are unaffected by this change; it might help to start by looking at which rules always match the same set of values and how those rules (especially rules 42 and 31) are used by the new versions of rules 8 and 11.

(Remember, you only need to handle the rules you have; building a solution that could handle any hypothetical combination of rules would be significantly more difficult¹.)

For example:

42: 9 14 | 10 1
9: 14 27 | 1 26
10: 23 14 | 28 1
1: "a"
11: 42 31
5: 1 14 | 15 1
19: 14 1 | 14 14
12: 24 14 | 19 1
16: 15 1 | 14 14
31: 14 17 | 1 13
6: 14 14 | 1 14
2: 1 24 | 14 4
0: 8 11
13: 14 3 | 1 12
15: 1 | 14
17: 14 2 | 1 7
23: 25 1 | 22 14
28: 16 1
4: 1 1
20: 14 14 | 1 15
3: 5 14 | 16 1
27: 1 6 | 14 18
14: "b"
21: 14 1 | 1 14
25: 1 1 | 1 14
22: 14 14
8: 42
26: 14 22 | 1 20
18: 15 15
7: 14 5 | 1 21
24: 14 1

abbbbbabbbaaaababbaabbbbabababbbabbbbbbabaaaa
bbabbbbaabaabba
babbbbaabbbbbabbbbbbaabaaabaaa
aaabbbbbbaaaabaababaabababbabaaabbababababaaa
bbbbbbbaaaabbbbaaabbabaaa
bbbababbbbaaaaaaaabbababaaababaabab
ababaaaaaabaaab
ababaaaaabbbaba
baabbaaaabbaaaababbaababb
abbbbabbbbaaaababbbbbbaaaababb
aaaaabbaabaaaaababaa
aaaabbaaaabbaaa
aaaabbaabbaaaaaaabbbabbbaaabbaabaaa
babaaabbbaaabaababbaabababaaab
aabbbbbaabbbaaaaaabbbbbababaaaaabbaaabba

Without updating rules 8 and 11, these rules only match three messages: bbabbbbaabaabba, ababaaaaaabaaab, and ababaaaaabbbaba.

However, after updating rules 8 and 11, a total of 12 messages match:

• bbabbbbaabaabba
• babbbbaabbbbbabbbbbbaabaaabaaa
• aaabbbbbbaaaabaababaabababbabaaabbababababaaa
• bbbbbbbaaaabbbbaaabbabaaa
• bbbababbbbaaaaaaaabbababaaababaabab
• ababaaaaaabaaab
• ababaaaaabbbaba
• baabbaaaabbaaaababbaababb
• abbbbabbbbaaaababbbbbbaaaababb
• aaaaabbaabaaaaababaa
• aaaabbaabbaaaaaaabbbabbbaaabbaabaaa
• aabbbbbaabbbaaaaaabbbbbababaaaaabbaaabba

Assignment

After updating rules 8 and 11, how many messages completely match rule 0?? This is done in the following way:

• Write a function match that takes the two arguments: i) a message (String) and ii) the pathname (String) of a text file containing the rules valid messages should obey. The function must return a Boolean value (Bool) that indicates whether the given message matches all of rule 0 after updating rules 8 and 11.
• Write a function matches that takes the two arguments: i) the pathname (String) of a text file containing a list of received messages that have been collected so far and ii) the pathname (String) of a text file containing the rules valid messages should obey. The function must return how many (Int) messages completely match rule 0 after updating rules 8 and 11.

Example

In this interactive session we assume the text files messages.txt² and rules.txt³ to be located in the current directory.

> match "bbabbbbaabaabba" "rules.txt"
True
> match "babbbbaabbbbbabbbbbbaabaaabaaa" "rules.txt"
True
> match "aaabbbbbbaaaabaababaabababbabaaabbababababaaa
" "rules.txt")
True
> match "abbbbbabbbaaaababbaabbbbabababbbabbbbbbabaaaa" "rules.txt"
False
> match "aaaabbaaaabbaaa" "rules.txt"
False
> match "babaaabbbaaabaababbaabababaaab" "rules.txt"
False

> matches "messages.txt" "rules.txt"
12