Personal tools

Chaitin's construction/Parser

From HaskellWiki

< Chaitin's construction(Difference between revisions)
Jump to: navigation, search
(Receiving contents from the mentioned page, and adjusting its section hierarchy)
Current revision (14:31, 4 August 2006) (edit) (undo)
(Moving combinatory logic term modules to a separate pege)
 
(One intermediate revision not shown.)
Line 5: Line 5:
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.
-
== Decoding module ==
+
== Decoding function illustrated as a parser ==
 +
 
 +
=== Decoding module ===
<haskell>
<haskell>
Line 29: Line 31:
</haskell>
</haskell>
-
== Combinatory logic term modules ==
+
=== Combinatory logic term modules ===
-
=== CL ===
+
See [[../Combinatory logic | combinatory logic term modules here]].
-
<haskell>
+
=== Utility 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
+
-
</haskell>
+
-
=== CL extension ===
+
==== Binary tree ====
-
 
+
-
<haskell>
+
-
module CLExt ((>>@)) where
+
-
 
+
-
import CL (CL, apply)
+
-
import Control.Monad (Monad, liftM2)
+
-
 
+
-
(>>@) :: Monad m => m CL -> m CL -> m CL
+
-
(>>@) = liftM2 apply
+
-
</haskell>
+
-
 
+
-
=== Base symbol ===
+
-
 
+
-
<haskell>
+
-
module BaseSymbol (BaseSymbol, kay, ess) where
+
-
 
+
-
data BaseSymbol = K | S
+
-
 
+
-
kay, ess :: BaseSymbol
+
-
kay = K
+
-
ess = S
+
-
</haskell>
+
-
 
+
-
== Utility modules ==
+
-
 
+
-
=== Binary tree ===
+
<haskell>
<haskell>
Line 83: Line 45:
</haskell>
</haskell>
-
=== Parser ===
+
==== Parser ====
<haskell>
<haskell>
Line 102: Line 64:
</haskell>
</haskell>
-
=== Prelude extension ===
+
==== Prelude extension ====
<haskell>
<haskell>
Line 110: Line 72:
bool thenC elseC t = if t then thenC else elseC
bool thenC elseC t = if t then thenC else elseC
</haskell>
</haskell>
 +
 +
== Using this parser for decoding ==
 +
 +
=== Approach based on decoding with partial function ===
 +
 +
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
 +
<haskell>
 +
dc :: [Bit] -> CL
 +
dc = fst . head . runParser clP
 +
</haskell>
 +
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).
 +
 +
=== Approach based on decoding with total function ===
 +
 +
If this is confusing or annoying, then we can choose another approach, making <math>\mathrm{dc}</math> a total function:
 +
<haskell>
 +
dc :: [Bit] -> Maybe CL
 +
dc = fst . head . runParser (neverfailing clP)
 +
</haskell>
 +
where
 +
<haskell>
 +
neverfailing :: MonadPlus m => m a -> m (Maybe a)
 +
neverfailing p = liftM Just p `mplus` return Nothing
 +
</haskell>
 +
then, Chaitin's construction will be
 +
:<math>\sum_{p\in 2^*,\;\mathrm{maybe}\;\downarrow\;\mathrm{hnf}\;\left(\mathrm{dc}\;p\right)} 2^{-\left|p\right|}</math>
 +
where <math>\downarrow</math> should denote false truth value.
 +
 +
== 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.
 +
 +
 +
 +
[[Category:Theoretical foundations]]

Current revision

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

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
head
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.

Retrieved from "http://www.haskell.org/haskellwiki/Chaitin%27s_construction/Parser"

Category: Theoretical foundations