Chaitin's construction/Parser

Contents

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

1.2.1 CL

 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)

(>>@) :: 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)

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)

where

 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.