# Chaitin's construction

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 Revision as of 13:53, 3 August 2006 (edit)m (Spell-check with ispell)← Previous diff Current revision (21:34, 14 March 2009) (edit) (undo) (Workpage template'', warning about missing substantial details) (43 intermediate revisions not shown.) Line 1: Line 1: + :'''''Correction in process. There is a substantial point that is lacking yet, the formulae and the concepts are not correct without it.''''' + __TOC__ __TOC__ Line 4: Line 6: Are there any real numbers which are defined exactly, but cannot be computed? Are there any real numbers which are defined exactly, but cannot be computed? - This question leads us to [[exact real arithmetic]], foundations of [[mathematics]] and [[computer science]]. + This question leads us to [[exact real arithmetic]], and [[algorithmic information theory]], and foundations of [[mathematics]] and [[computer science]]. - Wikipedia article on [http://en.wikipedia.org/wiki/Chaitin%27s_constant Chaitin's construction], referring to e.g. + See Wikipedia article on [http://en.wikipedia.org/wiki/Chaitin%27s_constant Chaitin's construction], referring to e.g. * [http://www.expmath.org/expmath/volumes/11/11.3/Calude361_370.pdf Computing a Glimpse of Randomness] (written by Cristian S. Calude, Michael J. Dinneen, and Chi-Kou Shu) * [http://www.expmath.org/expmath/volumes/11/11.3/Calude361_370.pdf Computing a Glimpse of Randomness] (written by Cristian S. Calude, Michael J. Dinneen, and Chi-Kou Shu) * [http://www.plus.maths.org.uk/issue37/features/omega/index.html Omega and why math has no TOEs] (Gregory Chaitin). * [http://www.plus.maths.org.uk/issue37/features/omega/index.html Omega and why math has no TOEs] (Gregory Chaitin). Line 12: Line 14: == Basing it on combinatory logic == == Basing it on combinatory logic == - Some more direct relatedness to functional programming: we can base $\Omega$ on [[combinatory logic]] (instead of a [[Turing machine]]), see the prefix coding system described in [http://homepages.cwi.nl/~tromp/cl/cl.html Binary Lambda Calculus and Combinatory Logic] (page 20) written by John Tromp: + Some more direct relatedness to functional programming: we can base $\Omega$ on [[combinatory logic]] (instead of a [[Turing machine]]). + + === Coding === + + See the prefix coding system described in [http://homepages.cwi.nl/~tromp/cl/cl.html Binary Lambda Calculus and Combinatory Logic] (page 20) written by John Tromp: :$\widehat{\mathbf S} \equiv 00$ :$\widehat{\mathbf S} \equiv 00$ :$\widehat{\mathbf K} \equiv 01$ :$\widehat{\mathbf K} \equiv 01$ :$\widehat{\left(x y\right)} \equiv 1 \widehat x \widehat y$ :$\widehat{\left(x y\right)} \equiv 1 \widehat x \widehat y$ of course, $c$, $d$ are meta-variables, and also some other notations are changed slightly. of course, $c$, $d$ are meta-variables, and also some other notations are changed slightly. + + === Decoding === + + Having seen this, decoding is rather straightforward. + [[/Parser|Here is a parser]] for illustration, but it serves only didactical purposes: it will not be used in the final implementation, because a good term generator makes parsing superfluous at this task. + + === Chaitin's construction === Now, Chaitin's construction will be here Now, Chaitin's construction will be here Line 32: Line 45: :should mean the length of a bit sequence (not [[combinatory logic]] term evaluation!) :should mean the length of a bit sequence (not [[combinatory logic]] term evaluation!) - Here, $\mathrm{dc}$ is a partial function (from finite bit sequences). If this is confusing or annoying, then we can choose a more Haskell-like approach, making $\mathrm{dc}$ a total function: + === Table for small legths === + {| border="1" cellspacing="0" cellpadding="5" align="center" + ! Length ($n$) + ! All strings ($2^n$) + ! Decodable strings, ratio, their sum till now + ! Terminating, ratio, their sum till now + ! $\Omega$ approximated till now: mantissa -- binary, length-fitting binary, decimal + |- + | 0 + | 1 + | 0, 0, 0 + | 0, 0, 0 + | -, -, - + |- + | 1 + | 2 + | 0, 0, 0 + | 0, 0, 0 + | -, 0, 0 + |- + | 2 + | 4 + | 2, $\frac12$, $\frac12$ + | 2, $\frac12$, $\frac12$ + | 1, 10, 5 + |- + | 3 + | 8 + | 0, 0, $\frac12$ + | 0, 0, $\frac12$ + | 1, 100, 5 + |- + | 4 + | 16 + | 0, 0, $\frac12$ + | 0, 0, $\frac12$ + | 1, 1000, 5 + |- + | 5 + | 32 + | 4, $\frac18$, $\frac58$ + | 4, $\frac18$, $\frac58$ + | 101, 10100, 625 + |} + It illustrates nicely, that Chaitin's construction is a [http://en.wikipedia.org/wiki/Normal_number normal number], as if its digits (in binary representation) were generated by tossing a coin. + + == Eliminating any concept of code by handling [[combinatory logic]] terms directly == + + Chaitin's construction can be grasped also as + :$\sum_{p\in \mathrm{CL},\;\mathrm{hnf}\;p} 2^{-\left|\mathrm{dc}^{-1}\;p\right|}$ + + We can avoid referring to any code notion, if we modularize out function + :$\left|\cdot\right|\circ\mathrm{dc}^{-1}$ + and give it a separate name, e.g. + :$\left\Vert\cdot\right\Vert : \mathrm{CL}\to\mathbb N$ + and notice that it can be defined directly in terms of CL-terms (we need not use any decoding concept any longer): + + :$\left\Vert\mathbf K\right\Vert = 2$ + :$\left\Vert\mathbf S\right\Vert = 2$ + :$\left\Vert\left(x\;y\right)\right\Vert = 1 + \left\Vert x\right\Vert + \left\Vert y\right\Vert$ + + Thus, we transfer (lift) the notion of “length” from bit sequences to [[combinatory logic]] terms in an appropriate way. Let us call it, e.g. the “norm” of the term. + + Thus, Chaitin's construction is grasped also as + :$\sum_{p \in \mathrm{Dom}_{\mathrm{nf}}} 2^{-\left\Vert p\right\Vert}$ + where + :$\mathrm{nf} : \mathrm{CL} \supset\!\to \mathrm{CL}$ + is a partial function defined on CL terms, it attributes to each "terminating" term its normal form. + + Thus, we have no notions of “bit sequence”,“code”, “coding”, “decoding” at all. But their ghosts still haunt us: the definition of norm function looks rather strange without thinking on the fact that is was transferred from a concept of coding. + + === More natural norm functions (from CL terms) === + + Question: + If we already move away from the approaches referring to any code concept, then + could we define norm in other ways? E.g. + :$\left\Vert\cdot\right\Vert : \mathrm{CL}\to\mathbb N$ + :$\left\Vert\mathbf K\right\Vert = 1$ + :$\left\Vert\mathbf S\right\Vert = 1$ + :$\left\Vert\left(x\;y\right)\right\Vert = 1 + \left\Vert x\right\Vert + \left\Vert y\right\Vert$ + And is it worth doing it at all? The former one, at leat, had a good theoretical foundation (based on analysis, arithmetic and probability theory). This latter one is not so cleaner, that we should prefer it, so, lacking theoretical grounds. + + What I really want is to exclude conceptually the notion of coding, and with it the notion of “syntactically incorrect versus syntactically correct but diverging”. Thus, taking into account only syntactically correct things, seeing only the choice of terminating versus non-terminating. Thus taking only termination vs nontermination into account, when calculating Chaitin's construction. + + What I want to preserve: + * it can be interpreted as a probability + * it is a [http://en.wikipedia.org/wiki/Normal_number normal number], as if its digits (in binary representation) were generated by tossing a coin + thus I do not want to spoil these features. + + ==== Table for simpler CL-terms ==== + Let us not take into account coding and thus excluding the notion of “syntactically incorrect coding” even ''conceptually''. + Can we guess a good norm? + {| border="1" cellspacing="0" cellpadding="5" align="center" + ! Binary tree pattern + ! Maximal depth, vertices, edges + ! Leafs, branches + ! So many CL-terms = how to count it + ! Terminating, ratio + ! So many till now, ratio till now + |- + | $\cdot$ + | 0, 1, 0 + | 1, 0 + | $2 = 2$ + | 2, 1 + | 2, 1 + |- + | $\left(\right)$ + | 1, 3, 2 + | 2, 1 + | $4 = 2\cdot2$ + | 4, 1 + | 6, 1 + |- + | $\cdot\left(\right)$ + | 2, 5, 4 + | 3, 2 + | $8 = 2\cdot2^2$ + | 8, 1 + | 14, 1 + |- + | $\left(\right)\cdot$ + | 2, 5, 4 + | 3, 2 + | $8 = 2^2\cdot2$ + | 8, 1 + | 22, 1 + |- + | $\left(\right)\left(\right)$ + | 2, 7, 6 + | 4, 3 + | $16 = 2^2\cdot2^2$ + | 16, 1 + | 38, 1 + |} + + == Implementation == + + To do: + Writing a program in Haskell -- or in [[combinatory logic]]:-) -- which could help in making conjectures on [[combinatory logic]]-based Chaitin's constructions. It would make only approximations, in a similar way that most Mandelbrot plotting softwares work. The analogy: + * they ask for a maximum limit of iterations, so that they can make a conjecture on convergence of a series; + * this program will ask for the maximum limit of reducton steps, so that it can make a conjecture on termination (having-normal-form) of a CL term. + Explanation for this: non-termination of each actually examined CL-term cannot be proven by the program, but a good conjecture can be made: if termination does not take place in the given limit of reduction steps, then the actually examined CL-term is regarded as non-terminating. + + === Architecture === + + A CL term generator generates CL terms in “ascending order” (in terms of a theoretically appropriate “norm”), and by computing the norm of each CL-term, it approximates Chaitin's construction (at a given number of digits, and according to the given maximal limit of reduction steps). + + === User interface === + + chaitin --model-of-computation=cl --encoding=tromp --limit-of-reduction-steps=500 --digits=9 --decimal + chaitin --model-of-computation=cl --encoding=direct --limit-of-reduction-steps=500 --digits=9 --decimal + + === Term generator === + - dc :: [Bit] -> Maybe CL + module CLGen where + + import Generator (gen0) + import CL (k, s, apply) + + direct :: [CL] + direct = gen0 apply [s, k] + + + See [[/Combinatory logic|combinatory logic term modules here]]. + + + module Generator (gen0) where + + import PreludeExt (cross) + + gen0 :: (a -> a -> a) -> [a] -> [a] + gen0 f c = gen f c 0 + + gen :: (a -> a -> a) -> [a] -> Integer -> [a] + gen f c n = sizedGen f c n ++ gen f c (succ n) + + sizedGen :: (a -> a -> a) -> [a] -> Integer -> [a] + sizedGen f c 0 = c + sizedGen f c (n + 1) = map (uncurry f) + $+ concat [sizedGen f c i cross sizedGen f c (n - i) | i <- [0..n]] + + + + module PreludeExt (cross) where + + cross :: [a] -> [a] -> [(a, a)] + cross xs ys = [(x, y) | x <- xs, y <- ys] - then, Chaitin's construction will be - :$\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. == Related concepts == == Related concepts == Line 51: Line 248: == To do == == To do == - Writing a program in Haskell -- or in [[combinatory logic]]:-) -- which could help in making conjectures on [[combinatory logic]]-based Chaitin's constructions. It would make only approximations, in a similar way that most Mandelbrot plotting softwares work: it would ask for a maximum limit of iterations. + * Making tasks described in [[#Implementation]] - chaitin --computation=cl --coding=tromp --limit-of-iterations=5000 --digits=10 --decimal + * Making more natural norm functions (from CL-terms), see [[#More natural norm functions (from CL terms)]] ## Current revision Correction in process. There is a substantial point that is lacking yet, the formulae and the concepts are not correct without it. ## Contents ## 1 Introduction Are there any real numbers which are defined exactly, but cannot be computed? This question leads us to exact real arithmetic, and algorithmic information theory, and foundations of mathematics and computer science. See Wikipedia article on Chaitin's construction, referring to e.g. ## 2 Basing it on combinatory logic Some more direct relatedness to functional programming: we can base Ω on combinatory logic (instead of a Turing machine). ### 2.1 Coding See the prefix coding system described in Binary Lambda Calculus and Combinatory Logic (page 20) written by John Tromp: $\widehat{\mathbf S} \equiv 00$ $\widehat{\mathbf K} \equiv 01$ $\widehat{\left(x y\right)} \equiv 1 \widehat x \widehat y$ of course, c, d are meta-variables, and also some other notations are changed slightly. ### 2.2 Decoding Having seen this, decoding is rather straightforward. Here is a parser for illustration, but it serves only didactical purposes: it will not be used in the final implementation, because a good term generator makes parsing superfluous at this task. ### 2.3 Chaitin's construction Now, Chaitin's construction will be here $\sum_{p\in \mathrm{Dom}_\mathrm{dc},\;\mathrm{hnf}\left(\mathrm{dc}\;p\right)} 2^{-\left|p\right|}$ where hnf should denote an unary predicate “has normal form” (“terminates”) dc should mean an operator “decode” (a function from finite bit sequences to combinatory logic terms) $2\!\;^{*}$ should denote the set of all finite bit sequences Domdc should denote the set of syntactically correct bit sequences (semantically, they may either terminate or diverge), i.e. the domain of the decoding function, i.e. the range of the coding function. Thus, $\left\{00, 01, 1\;00\;00, 1\;00\;01, 1\;01\;00, 1\;01\;01, \dots\right\} = \mathrm{Dom}_{\mathrm{dc}} = \mathrm{Rng}_{\widehat\ }$ “Absolute value” should mean the length of a bit sequence (not combinatory logic term evaluation!) ### 2.4 Table for small legths Length (n) All strings (2n) Decodable strings, ratio, their sum till now Terminating, ratio, their sum till now Ω approximated till now: mantissa -- binary, length-fitting binary, decimal 0 1 0, 0, 0 0, 0, 0 -, -, - 1 2 0, 0, 0 0, 0, 0 -, 0, 0 2 4 2, $\frac12$, $\frac12$ 2, $\frac12$, $\frac12$ 1, 10, 5 3 8 0, 0, $\frac12$ 0, 0, $\frac12$ 1, 100, 5 4 16 0, 0, $\frac12$ 0, 0, $\frac12$ 1, 1000, 5 5 32 4, $\frac18$, $\frac58$ 4, $\frac18$, $\frac58$ 101, 10100, 625 It illustrates nicely, that Chaitin's construction is a normal number, as if its digits (in binary representation) were generated by tossing a coin. ## 3 Eliminating any concept of code by handling combinatory logic terms directly Chaitin's construction can be grasped also as $\sum_{p\in \mathrm{CL},\;\mathrm{hnf}\;p} 2^{-\left|\mathrm{dc}^{-1}\;p\right|}$ We can avoid referring to any code notion, if we modularize out function $\left|\cdot\right|\circ\mathrm{dc}^{-1}$ and give it a separate name, e.g. $\left\Vert\cdot\right\Vert : \mathrm{CL}\to\mathbb N$ and notice that it can be defined directly in terms of CL-terms (we need not use any decoding concept any longer): $\left\Vert\mathbf K\right\Vert = 2$ $\left\Vert\mathbf S\right\Vert = 2$ $\left\Vert\left(x\;y\right)\right\Vert = 1 + \left\Vert x\right\Vert + \left\Vert y\right\Vert$ Thus, we transfer (lift) the notion of “length” from bit sequences to combinatory logic terms in an appropriate way. Let us call it, e.g. the “norm” of the term. Thus, Chaitin's construction is grasped also as $\sum_{p \in \mathrm{Dom}_{\mathrm{nf}}} 2^{-\left\Vert p\right\Vert}$ where $\mathrm{nf} : \mathrm{CL} \supset\!\to \mathrm{CL}$ is a partial function defined on CL terms, it attributes to each "terminating" term its normal form. Thus, we have no notions of “bit sequence”,“code”, “coding”, “decoding” at all. But their ghosts still haunt us: the definition of norm function looks rather strange without thinking on the fact that is was transferred from a concept of coding. ### 3.1 More natural norm functions (from CL terms) Question: If we already move away from the approaches referring to any code concept, then could we define norm in other ways? E.g. $\left\Vert\cdot\right\Vert : \mathrm{CL}\to\mathbb N$ $\left\Vert\mathbf K\right\Vert = 1$ $\left\Vert\mathbf S\right\Vert = 1$ $\left\Vert\left(x\;y\right)\right\Vert = 1 + \left\Vert x\right\Vert + \left\Vert y\right\Vert$ And is it worth doing it at all? The former one, at leat, had a good theoretical foundation (based on analysis, arithmetic and probability theory). This latter one is not so cleaner, that we should prefer it, so, lacking theoretical grounds. What I really want is to exclude conceptually the notion of coding, and with it the notion of “syntactically incorrect versus syntactically correct but diverging”. Thus, taking into account only syntactically correct things, seeing only the choice of terminating versus non-terminating. Thus taking only termination vs nontermination into account, when calculating Chaitin's construction. What I want to preserve: • it can be interpreted as a probability • it is a normal number, as if its digits (in binary representation) were generated by tossing a coin thus I do not want to spoil these features. #### 3.1.1 Table for simpler CL-terms Let us not take into account coding and thus excluding the notion of “syntactically incorrect coding” even conceptually. Can we guess a good norm? Binary tree pattern Maximal depth, vertices, edges Leafs, branches So many CL-terms = how to count it Terminating, ratio So many till now, ratio till now $\cdot$ 0, 1, 0 1, 0 2 = 2 2, 1 2, 1 $\left(\right)$ 1, 3, 2 2, 1 $4 = 2\cdot2$ 4, 1 6, 1 $\cdot\left(\right)$ 2, 5, 4 3, 2 $8 = 2\cdot2^2$ 8, 1 14, 1 $\left(\right)\cdot$ 2, 5, 4 3, 2 $8 = 2^2\cdot2$ 8, 1 22, 1 $\left(\right)\left(\right)$ 2, 7, 6 4, 3 $16 = 2^2\cdot2^2$ 16, 1 38, 1 ## 4 Implementation To do: Writing a program in Haskell -- or in combinatory logic:-) -- which could help in making conjectures on combinatory logic-based Chaitin's constructions. It would make only approximations, in a similar way that most Mandelbrot plotting softwares work. The analogy: • they ask for a maximum limit of iterations, so that they can make a conjecture on convergence of a series; • this program will ask for the maximum limit of reducton steps, so that it can make a conjecture on termination (having-normal-form) of a CL term. Explanation for this: non-termination of each actually examined CL-term cannot be proven by the program, but a good conjecture can be made: if termination does not take place in the given limit of reduction steps, then the actually examined CL-term is regarded as non-terminating. ### 4.1 Architecture A CL term generator generates CL terms in “ascending order” (in terms of a theoretically appropriate “norm”), and by computing the norm of each CL-term, it approximates Chaitin's construction (at a given number of digits, and according to the given maximal limit of reduction steps). ### 4.2 User interface chaitin --model-of-computation=cl --encoding=tromp --limit-of-reduction-steps=500 --digits=9 --decimal chaitin --model-of-computation=cl --encoding=direct --limit-of-reduction-steps=500 --digits=9 --decimal  ### 4.3 Term generator module CLGen where import Generator (gen0) import CL (k, s, apply) direct :: [CL] direct = gen0 apply [s, k] module Generator (gen0) where import PreludeExt (cross) gen0 :: (a -> a -> a) -> [a] -> [a] gen0 f c = gen f c 0 gen :: (a -> a -> a) -> [a] -> Integer -> [a] gen f c n = sizedGen f c n ++ gen f c (succ n) sizedGen :: (a -> a -> a) -> [a] -> Integer -> [a] sizedGen f c 0 = c sizedGen f c (n + 1) = map (uncurry f)$
concat [sizedGen f c i cross sizedGen f c (n - i) | i <- [0..n]]
module PreludeExt (cross) where
cross xs ys = [(x, y) | x <- xs, y <- ys]