For day 16 Advent of Code 2021, it was all about monadic parser combinators (whatever they are)! Just like Day 15, this one took me a lot longer to complete than I had wanted, this time it was all about learning Haskell’s approach to parsing text.

According to the story, we had to decode a message in a custom binary format, the Buoyancy Interchange Transmission System (BITS). BITS is a message format that encodes both literal values and operations.

Building the parser

The first part of the problem was to decode the hexadecimal input to a stream of bits. I felt I cheated a bit here, because I just went for simplicity:

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hex2bin :: String -> String
hex2bin s = s >>= hex2bin'

hex2bin' :: Char -> String
hex2bin' '0' = "0000"
hex2bin' '1' = "0001"
hex2bin' '2' = "0010"
hex2bin' '3' = "0011"
hex2bin' '4' = "0100"
hex2bin' '5' = "0101"
hex2bin' '6' = "0110"
hex2bin' '7' = "0111"
hex2bin' '8' = "1000"
hex2bin' '9' = "1001"
hex2bin' 'A' = "1010"
hex2bin' 'B' = "1011"
hex2bin' 'C' = "1100"
hex2bin' 'D' = "1101"
hex2bin' 'E' = "1110"
hex2bin' 'F' = "1111"

So this would just take a list of characters (0 - F) with the nibbles they represent. The bind operator (think flatMap in scala) is just the ticket there.

Now that I had to parse the input, I turned to monadic parser combinators. That sounds complicated, but is actually making parsing very easy (once I had understood the concept). Essentially, it allows you to build little parsing building blocks and because these parsing blocks are monads, they can be combined with monadic operators. I’m going to try to explain by building up the blocks.

First of all, we have the basic element of the parser, we’re looking at parsing 0s and 1s:

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p_bin :: Parsec String () Char
p_bin = oneOf "01"

This will parse either a 0 or a 1. To test it:

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*Main> parse p_bin "error msg" "0101"
Right '0'
*Main> parse p_bin "error msg" "23"
Left "error msg" (line 1, column 1):
unexpected "2"

The output of the parser is an Either - which is Left for an error condition or Right for a result.

This basic building block can then be used to build up more complex parsers:

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p_ver :: Parsec String () String
p_ver = count 3 p_bin

p_lit :: Parsec String () Packet
p_lit = do
  ver <- p_ver
  string "100"
  gs <- many $ char '1' >> count 4 p_bin
  lg <- char '0' >> count 4 p_bin
  return (Lit (bin2dec ver) (bin2dec ((concat gs) ++ lg))) 

To define the first type of packet - we’ve got a 3-bit field for the “version”, and another 3-bit field which is always set to 4 (“100”). Then a series of 5-bit fields. First there are many of those fields starting with '1' followed by a payload of a 4-bit nibble. Finally, there’s 5-bit field starting with '0' and another 4-bit nibble. All these payload fields are combined and then turn into a numeric literal.

As a way of an example, when parsing the string

0011001110001010
VVVTTTAAAAABBBBB

We’ve got the version (V), type (T) and two groups (A and B):

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*Main> parse p_lit "error msg" "0011001110001010"
Right (Lit 1 202)

Before I continue on the other packets, I’d just like to take a detour on how I’ve modelled the packets in Haskell:

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data Packet = Op Int Int [Packet] | Lit Int Int 
  deriving (Eq, Show)

Unpacking this:

• This defines a Packet data structure which can either be
• a Lit literal, which has a version and a value (both integers)
• an Op operator, which has a version, a type and a list of packets nested within it
• the deriving (Eq, Show) implements the ability to print or compare the data

Now we just need to parse the operator packets - but it is not as simple as that. Operator packets can have other subpackets inside them. To determine how many packets there are, they can be specified either by the number of subpackets, or the length (in bits) of all the subpackets. To check which is which, there is a bit that indicates this. If the first bit is '1', then the following 11 bits specify the number of packets:

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p_op_by_num :: Parsec String () [Packet]
p_op_by_num = do
  len <- try (char '1') >> count 11 p_bin
  ps <- count (bin2dec len) p_packet
  return ps

The Haskell do notation is really useful here (this is the equivalent of the scala for-comprehension), and essentially makes monadic binds easier to read. To explain the above a bit further:

• the parser tries to read the character 1 - if that is not present, then try ensure that the character is not consumed.
• following that (by using the >> combinator), it parses 11 binary characters and puts the output into len
• then it parses len packets using the p_packet parser (see further down)

For the other type of operator packet parsing - where the first bit is '0', then the following 15 bits specify the length of the payload.

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p_position :: Int -> Parsec String () [a]
p_position n = do
  pos <- getPosition
  guard $ (sourceColumn pos) >= n
  return []

p_op_by_len :: Parsec String () [Packet]
p_op_by_len = do
  len <- try (char '0') >> count 15 p_bin
  pos <- getPosition
  let nextN = (sourceColumn pos) + (bin2dec len)
  ps <- manyTill p_packet $ p_position nextN
  return ps

This does the following:

• the p_position parser takes a position and returns a parse failure if the position is less than n
• the p_op_by_len parser first reads in the character 0 - if it doesn’t match it fails - and then 15 binary characters. That makes up the len.
• then we get the current position and add the column value to len
• then we use the manyTill parser combinator to parse p_packet until the p_position parser gets a match.

Finally, we parse a whole Op packet:

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p_op :: Parsec String () Packet
p_op = do 
  ver <- p_ver
  typeId <- p_type
  ps <- p_op_by_len <|> p_op_by_num
  return (Op (bin2dec ver) (bin2dec typeId) ps)

We get the version and the type id - followed by the packet payload. Now this can be either p_op_by_len or p_op_by_num - and the <|> operator allows to choose between them. If the first one succeeds, the second one is not used.

Lastly, we define how to parse a packet:

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p_packet :: Parsec String () Packet
p_packet = try p_lit <|> p_op

A packet can either be a literal or an operand.

Part One

For part one, all that was asked was to add up the version ids in each of the packets:

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sumVersions' :: Packet -> Int
sumVersions' (Lit v _) = v
sumVersions' (Op v _ ps) = v + (sum $ map sumVersions' ps)

Again, the Haskell pattern matching shines through here. If we have a Lit packet, then we just return the version integer, if we have an Op packet, then we return the version and add up the results of recursively calling the function itself.

Part Two

For part two, we had to evaluate the operators. Depending on the operator, the result would be a different arithmetic operation. As it turned out, with the parsing completed, the hard part was already done. To evaluate, I just needed something slightly more complicated than the sumVersions from part one:

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evaluate :: Packet -> Int
evaluate (Lit _ n) = n
evaluate (Op _ 0 ps) = evaluateAll sum ps
evaluate (Op _ 1 ps) = evaluateAll product ps
evaluate (Op _ 2 ps) = evaluateAll minimum ps
evaluate (Op _ 3 ps) = evaluateAll maximum ps
evaluate (Op _ 5 ps) = evaluateAll (boolOp (>)) ps
evaluate (Op _ 6 ps) = evaluateAll (boolOp (<)) ps
evaluate (Op _ 7 ps) = evaluateAll (boolOp (==)) ps

evaluateAll :: ([Int] -> Int) -> [Packet] -> Int
evaluateAll f ps = f $ map evaluate ps

So to evaluate an operation, we need to evaluate each of the types of packet.

• For a Lit literal, the value was just whatever was parsed in.
• For an Op operator, the result would depend on the type id. Using pattern matching and a helper function, I am able to express this quite succinctly.
• The evaluateAll takes a function that takes a list of integers as input and a single integer as output, it also takes a list of packets. It then evaluates each packet and feeds that to the function.
• So operation 1 sums up all the value using the built in function sum
• Same with product, minimum and maximum

For the last remaining operators - they are boolean operators that only work on lists of size two:

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boolOp :: (Int -> Int -> Bool) -> [Int] -> Int
boolOp f [a, b] = bool2num $ f a b

bool2num :: Bool -> Int
bool2num True = 1
bool2num False = 0

So boolOp is a function that takes two parameters. First, a function that compares two integers. Then a list of integers and it returns an integer. By using currying, we turn boolOp (<) into a function that is [Int] -> Int with the first parameter filled in by < (operators in Haskell are just functions - calling a < b is just syntactic sugar for the function call (<) a b). That way I can express any boolean operation easily. For this puzzle, I just needed <, > and ==.

To get the result of this puzzle, I just combined the parsing with the evaluating:

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binStr = hex2bin input
packet = parse p_packet "" binStr
evaluate' packet

Gotchas

When I initially implemented the first part, I was a little unfocused when reading about the subpackets in the operator. I skimmed over the fact that the two different types were talking about two different lengths. Number of packets or number of bits in the payload.

My first implementation of the Op parser was

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p_op :: Parsec String () Packet
p_op = do 
  ver <- p_ver
  typeId <- p_type
  len <- (p_len_t_15 >> p_len_15) <|> (p_len_t_11 >> p_len_11)
  ps <- many (try p_packet)
  return (Op (bin2dec ver) (bin2dec typeId) ps)

Basically, I would read in the version and the type, the would read in the length - which can be defined as either a 11 bit length field or a 15 bit length field - then completely ignore the length and read in as many sub packets as I could.

Now this worked for part 1. But it was a fluke. When it was reading in subpackets, it did not stop reading until it couldn’t read anymore packets. That meant that when operator packets were inside other operator packets, some that packets that belonged to other operators would be associated with the wrong “parent”. But because the answer for part 1 was just the sum of all the version values it did not matter whether a packet was associated with the right operator.

For part 2, that mattered a lot. It was only then that I came up with the idea of using the manyTill parser combinator.

Another fun gotcha was that I initially forgot to add the return to the parsers when I was using do notation. This meant that the parsers just kept giving me compile errors. return is a bit of a funny one when you initially encounter it coming from imperative programming. It is NOT a return statement. It is a function that wraps the value in the monad being used in the do notation.

The type signature of return is:

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return :: Monad a => b -> a b

But missing that off meant that instead of dealing with a Parser String, the function was returning a String - and that understandably didn’t make the compiler happy.

Rest of the solution on GitHub

Conclusion

I found this introduction to parsing in Haskell absolutely fascinating. Before doing this exercise I probably would have reached for regular expressions or writing functions to operate on the source string, but I found that the code is more succinct and clear than the specification of the issue. Furthermore, non of these constructs are extensions to the language or parsing specific, all is achieved by using monads. And this is what is making me appreciate the simplicity and beauty of Haskell even more.

And no, I’m not going to do a post trying to explain what monads are! Not yet anyway.

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