advent-of-code/2020/day19/README.org

** --- Day 19: Monster Messages ---
You land in an airport surrounded by dense forest. As you walk to your
high-speed train, the Elves at the Mythical Information Bureau contact
you again. They think their satellite has collected an image of a /sea
monster/! Unfortunately, the connection to the satellite is having
problems, and many of the messages sent back from the satellite have
been corrupted.

They sent you a list of /the rules valid messages should obey/ and a
list of /received messages/ they've collected so far (your puzzle
input).

The /rules for valid messages/ (the top part of your puzzle input) are
numbered and build upon each other. For example:

#+BEGIN_EXAMPLE
  0: 1 2
  1: "a"
  2: 1 3 | 3 1
  3: "b"
#+END_EXAMPLE

Some rules, like =3: "b"=, simply match a single character (in this
case, =b=).

The remaining rules list the sub-rules that must be followed; for
example, the rule =0: 1 2= means that to match rule =0=, the text being
checked must match rule =1=, and the text after the part that matched
rule =1= must then match rule =2=.

Some of the rules have multiple lists of sub-rules separated by a pipe
(=|=). This means that /at least one/ list of sub-rules must match. (The
ones that match might be different each time the rule is encountered.)
For example, the rule =2: 1 3 | 3 1= means that to match rule =2=, the
text being checked must match rule =1= followed by rule =3= /or/ it must
match rule =3= followed by rule =1=.

Fortunately, there are no loops in the rules, so the list of possible
matches will be finite. Since rule =1= matches =a= and rule =3= matches
=b=, rule =2= matches either =ab= or =ba=. Therefore, rule =0= matches
=aab= or =aba=.

Here's a more interesting example:

#+BEGIN_EXAMPLE
  0: 4 1 5
  1: 2 3 | 3 2
  2: 4 4 | 5 5
  3: 4 5 | 5 4
  4: "a"
  5: "b"
#+END_EXAMPLE

Here, because rule =4= matches =a= and rule =5= matches =b=, rule =2=
matches two letters that are the same (=aa= or =bb=), and rule =3=
matches two letters that are different (=ab= or =ba=).

Since rule =1= matches rules =2= and =3= once each in either order, it
must match two pairs of letters, one pair with matching letters and one
pair with different letters. This leaves eight possibilities: =aaab=,
=aaba=, =bbab=, =bbba=, =abaa=, =abbb=, =baaa=, or =babb=.

Rule =0=, therefore, matches =a= (rule =4=), then any of the eight
options from rule =1=, then =b= (rule =5=): =aaaabb=, =aaabab=,
=abbabb=, =abbbab=, =aabaab=, =aabbbb=, =abaaab=, or =ababbb=.

The /received messages/ (the bottom part of your puzzle input) need to
be checked against the rules so you can determine which are valid and
which are corrupted. Including the rules and the messages together, this
might look like:

#+BEGIN_EXAMPLE
  0: 4 1 5
  1: 2 3 | 3 2
  2: 4 4 | 5 5
  3: 4 5 | 5 4
  4: "a"
  5: "b"

  ababbb
  bababa
  abbbab
  aaabbb
  aaaabbb
#+END_EXAMPLE

Your goal is to determine /the number of messages that completely match
rule =0=/. In the above example, =ababbb= and =abbbab= match, but
=bababa=, =aaabbb=, and =aaaabbb= do not, producing the answer /=2=/.
The whole message must match all of rule =0=; there can't be extra
unmatched characters in the message. (For example, =aaaabbb= might
appear to match rule =0= above, but it has an extra unmatched =b= on the
end.)

/How many messages completely match rule =0=?/

Your puzzle answer was =285=.

** --- Part Two ---
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:

#+BEGIN_EXAMPLE
  8: 42 | 42 8
  11: 42 31 | 42 11 31
#+END_EXAMPLE

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 [[https://en.wikipedia.org/wiki/Formal_grammar][significantly more
difficult]].)

For example:

#+BEGIN_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
#+END_EXAMPLE

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=

/After updating rules =8= and =11=, how many messages completely match
rule =0=?/

Your puzzle answer was =412=.

Both parts of this puzzle are complete! They provide two gold stars: **