# Chaitin's construction/Parser

(Difference between revisions)
 Revision as of 13:11, 4 August 2006 (edit) (Moving text and modifying section hierarchy. Mentioning term generators)← Previous diff Current revision (14:31, 4 August 2006) (edit) (undo) (Moving combinatory logic term modules to a separate pege) Line 33: Line 33: === Combinatory logic term modules === === Combinatory logic term modules === - ==== CL ==== + See [[../Combinatory logic | combinatory logic term modules here]]. - + - + - 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 + - + - + - ==== CL extension ==== + - + - + - module CLExt ((>>@)) where + - + - import CL (CL, apply) + - import Control.Monad (Monad, liftM2) + - + - (>>@) :: Monad m => m CL -> m CL -> m CL + - (>>@) = liftM2 apply + - + - + - ==== Base symbol ==== + - + - + - module BaseSymbol (BaseSymbol, kay, ess) where + - + - data BaseSymbol = K | S + - + - kay, ess :: BaseSymbol + - kay = K + - ess = S + - + === Utility modules === === Utility modules ===

## 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

### 1.3 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.