Let us describe the seen language with a LL(1) grammar, and let us make use of the lack of backtracking, lack of look-ahead, when deciding which parser approach to use.
Some notes about the used parser library: I shall use the didactical approach read in paper Monadic Parser Combinators (written by Graham Hutton and Erik Meier). The optimalisations described in the paper are avoided here. Of course, we can make optimalisations, or choose sophisticated parser libraries (Parsec, arrow parsers). A pro for this simpler parser: it may be easier to augment it with other monad transformers. But, I think, the task does not require such ability. So the real pro for it is that it looks more didactical for me. Of couse, it may be inefficient at many other tasks, but I hope, the LL(1) grammar will not raise huge problems.
1 Decoding function illustrated as a parser
1.1 Decoding module
module Decode (clP) where import Parser (Parser, item) import CL (CL, k, s, apply) import CLExt ((>>@)) import PreludeExt (bool) clP :: Parser Bool CL clP = item >>= bool applicationP baseP applicationP :: Parser Bool CL applicationP = clP >>@ clP baseP :: Parser Bool CL baseP = item >>= bool k s kP, sP :: Parser Bool CL kP = return k sP = return s
1.2 Combinatory logic term modules
module CL (CL, k, s, apply) where import Tree (Tree (Leaf, Branch)) import BaseSymbol (BaseSymbol, kay, ess) type CL = Tree BaseSymbol k, s :: CL k = Leaf kay s = Leaf ess apply :: CL -> CL -> CL apply = Branch
1.2.2 CL extension
module CLExt ((>>@)) where import CL (CL, apply) import Control.Monad (Monad, liftM2) (>>@) :: Monad m => m CL -> m CL -> m CL (>>@) = liftM2 apply
1.2.3 Base symbol
module BaseSymbol (BaseSymbol, kay, ess) where data BaseSymbol = K | S kay, ess :: BaseSymbol kay = K ess = S
1.3 Utility modules
1.3.1 Binary tree
module Tree (Tree (Leaf, Branch)) where data Tree a = Leaf a | Branch (Tree a) (Tree a)
module Parser (Parser, runParser, item) where import Control.Monad.State (StateT, runStateT, get, put) type Parser token a = StateT [token]  a runParser :: Parser token a -> [token] -> [(a, [token])] runParser = runStateT item :: Parser token token item = do token : tokens <- get put tokens return token
1.3.3 Prelude extension
module PreludeExt (bool) where bool :: a -> a -> Bool -> a bool thenC elseC t = if t then thenC else elseC
2 Using this parser for decoding
2.1 Approach based on decoding with partial function
Seen above, dc was a partial function (from finite bit sequences to combinatory logic terms). We can implement it e.g. as
dc :: [Bit] -> CL dc = fst . head . runParser clP
2.2 Approach based on decoding with total function
If this is confusing or annoying, then we can choose another approach, making dc a total function:
dc :: [Bit] -> Maybe CL dc = fst . head . runParser (neverfailing clP)
neverfailing :: MonadPlus m => m a -> m (Maybe a) neverfailing p = liftM Just p `mplus` return Nothing
then, Chaitin's construction will be
where should denote false truth value.
3 Term generators instead of parsers
All these are illustrations -- they will not be present in the final application. The real software will use no parsers at all -- it will use term generators instead. It will generate directly “all” combinatory logic terms in an “ascending length” order, attribute “length” to them, and approximating Chaitin's construct this way. It will not use strings / bit sequences at all: it will handle combinatory logic-terms directly.