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Chaitin's construction/Parser

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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

See combinatory logic term modules here.

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)

1.3.2 Parser

 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
where the use of
reveals that it is a partial function (of course, because not every bit sequence is a correct coding of a CL-term).

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

\sum_{p\in 2^*,\;\mathrm{maybe}\;\downarrow\;\mathrm{hnf}\;\left(\mathrm{dc}\;p\right)} 2^{-\left|p\right|}

where \downarrow 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.