# Chaitin's construction/Parser

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Some notes about the used parser library: I shall use the didactical approach read in paper [http://www.cs.nott.ac.uk/~gmh/bib.html#monparsing Monadic Parser Combinators] (written by [http://www.cs.nott.ac.uk/Department/Staff/gmh/ 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#Parser|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. |
Some notes about the used parser library: I shall use the didactical approach read in paper [http://www.cs.nott.ac.uk/~gmh/bib.html#monparsing Monadic Parser Combinators] (written by [http://www.cs.nott.ac.uk/Department/Staff/gmh/ 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#Parser|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. |
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− | == Decoding module == |
+ | == Decoding function illustrated as a parser == |

+ | |||

+ | === Decoding module === |
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<haskell> |
<haskell> |
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</haskell> |
</haskell> |
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− | == Combinatory logic term modules == |
+ | === Combinatory logic term modules === |

− | === CL === |
+ | ==== CL ==== |

<haskell> |
<haskell> |
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</haskell> |
</haskell> |
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− | === CL extension === |
+ | ==== CL extension ==== |

<haskell> |
<haskell> |
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</haskell> |
</haskell> |
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− | === Base symbol === |
+ | ==== Base symbol ==== |

<haskell> |
<haskell> |
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</haskell> |
</haskell> |
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− | == Utility modules == |
+ | === Utility modules === |

− | === Binary tree === |
+ | ==== Binary tree ==== |

<haskell> |
<haskell> |
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</haskell> |
</haskell> |
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− | === Parser === |
+ | ==== Parser ==== |

<haskell> |
<haskell> |
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</haskell> |
</haskell> |
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− | === Prelude extension === |
+ | ==== Prelude extension ==== |

<haskell> |
<haskell> |
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bool thenC elseC t = if t then thenC else elseC |
bool thenC elseC t = if t then thenC else elseC |
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</haskell> |
</haskell> |
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+ | |||

+ | == Using this parser for decoding == |
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+ | |||

+ | === Approach based on decoding with partial function === |
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+ | |||

+ | Seen above, <math>\mathrm{dc}</math> was a partial function (from finite bit sequences to [[combinatory logic]] terms). We can implement it e.g. as |
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+ | <haskell> |
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+ | dc :: [Bit] -> CL |
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+ | dc = fst . head . runParser clP |
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+ | </haskell> |
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+ | where the use of <hask>head</hask> reveals that it is a partial function (of course, because not every bit sequence is a correct coding of a CL-term). |
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+ | |||

+ | === Approach based on decoding with total function === |
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+ | |||

+ | If this is confusing or annoying, then we can choose another approach, making <math>\mathrm{dc}</math> a total function: |
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+ | <haskell> |
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+ | dc :: [Bit] -> Maybe CL |
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+ | dc = fst . head . runParser (neverfailing clP) |
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+ | </haskell> |
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+ | where |
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+ | <haskell> |
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+ | neverfailing :: MonadPlus m => m a -> m (Maybe a) |
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+ | neverfailing p = liftM Just p `mplus` return Nothing |
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+ | </haskell> |
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+ | then, Chaitin's construction will be |
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+ | :<math>\sum_{p\in 2^*,\;\mathrm{maybe}\;\downarrow\;\mathrm{hnf}\;\left(\mathrm{dc}\;p\right)} 2^{-\left|p\right|}</math> |
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+ | where <math>\downarrow</math> should denote false truth value. |
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+ | |||

+ | == Term generators instead of parsers == |
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+ | |||

+ | 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. |
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+ | |||

+ | |||

+ | |||

+ | [[Category:Theoretical foundations]] |

## Revision as of 13:11, 4 August 2006

## 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) 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)

#### 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

### 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

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.