Difference between revisions of "Gödel Numbers and Lojban"

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Revision as of 11:14, 21 March 2014

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Now we can construct Gödel numbers for any Lojban Description text.

Therefore, if we can prove that the lojban grammar & morphology is

expressible

using lojban mekso EX pertaining to the Gödel numbers, then,

according to Gödel's

proof (1931), it would be proven that there exist lojban sentences that

cannot be proven grammatically/morphologically correct using a proof

expressed in lojban.

(not like na nei, mind you, I really mean correctness on the

syntactical view.) OK, but na nei was the kind of sentence whose

equivalent Gödel use to blow up sign systems semantically...

And given that Lojban's grammar & morphology seems to be parsable using

DFAs, I am pretty sure that it is expressible with arithmetical

statements and therefore MEX.

Of course, this is an annoying conclusion, because we feel that lojban

is very able to fully express its own syntax. But on the other hand, if

this conclusion proves to be false, then it would mean that Lojban's

grammar & morphology is *not* fully parsable using DFAs. And because we

already know that the grammar is (it is YACC-defined), it would mean

that the morphology is problematic. (which I was already pretty sure

of, dunno why ;-)

Although I may have not been precise enough for a mathematical proof,

I am sure that lojban's claim about being a "logical language" should

entail discussions involving Gödel's statements about logic.

Very interesting; the limits of Lojban! I lack the mathematical

power to join the discussion, but I am extremely interested in the

conclusions! --xod

I believe Richard Curnow has mentioned before that the DFA

(Deterministic Finite Automaton) for handling Lojban morphology has

somewhere between 900 and 1000 different states, and can't reliably be

human-generated. Instead, he describes it with a NFA (did I just forget

the acronym? argh) (Non-Deterministic Finite Automaton) and then

converts the NFA to a DFA. Lemme tell 'ya, the NFA is hideous, too. (the

whole point of this is to point out that while the grammar can be

proven, nobody is seems sure about the morphology yet) --[jbocre: Jay

Kominek|jay]

Correct me if I misunderstand, but the conclusion would be that

Lojban's grammar and morphology cannot be expressed by mekso or any

other formal system, though the grammar and morphology could still be

fully expressed using informal statements in Lojban.

Of course, as well as with informal statements in any other language.

But the point is, Lojban Description is supposed to be parseable using computers,

which means that its syntax ought to be fully expressible in computer

terms, and thus mekso.

It would be a major lose (correct me if I'm wrong) in regards to

lojban's goals...

So this seems to be saying that either Lojban's morphology is vague in

some area, or there are grammatically correct sentences which cannot be

proven to be grammatically correct. Why do you assume it is the first?

Number theory, a very powerful language indeed, contains true statements

which cannot be proven. Why should we expect Lojban Description to be

different?
Consider that these sentences which cannot be proven to be

grammatically correct are going to be huge, monstruously long sentences.

I don't think this defeats any of Lojban's goals. --rab.spir


An aside into provability:

Hey, how can you say "true statements which cannot be proven"! That's

completely meaningless. --xod

Gödel settled this in 1931.

[1] --

nitcion.

Yes, but this is rather tough. A useful analogy can be found at

Goedel's Record Player.

More information and clear explanation about this stuff can be found in

jbocre: Gödel, Escher, Bach by Hofstadter (see previous paragraph)


The class of languages to which Lojban's grammar belongs is well

understood. Assuming you can properly decompose a string of sounds or

symbols into their separate words, there is no string of words for which

membership in the language is undecidable. Whether or not that holds for

the morphology seems to be unproven at this point. Maybe the LLG should

investigate this and maybe stick their seal of approval on something as

the official machine description of the morphology? Maybe I should snarf

the lexer out of the official parser and see what it looks like...

--Jay

The lexer in the official parser is weak: about all it can do is divide

compound cmavo. It is not a full implementation of the morphology

algorithm.

We can decide whether the string is grammatically correct using

YACC. However, in the situation which started this discussion, we

don't have the option of using YACC. You have to use the numerical

rules, expressed in mekso, which determine whether a sentence is

grammatically correct. The implication of this is that there exist

monstrous sentences, probably involving mekso which include the

Godel numbers of other Lojban Description sentences, which cannot be proven to be

grammatically correct using those mekso rules. If you don't believe me,

read jbocre: Gödel, Escher, Bach and all will become clear. --rab.spir

...And is it possible to change the title of this discussion? Done


Now that I finally have my own copy of jbocre: Gödel, Escher, Bach, I've

looked into how Gödel-incompleteness works - and I don't believe

Lojban Description is affected by it. Mathematics (and particularly systems such

as Typographical Number Theory) attempt to determine which statements

are true and which are not in a formal way; these systems are incomplete

because it is possible to create statements which imply their own

falsehood.

Lojban Description, on the other hand, does not attempt to distinguish the true from

the false mathematically. It only distinguishes the grammatically

correct from the grammatically incorrect. For Gödel-incompleteness to

apply to Lojban Description, there would need to be a sentence for which

demonstrating it to be grammatically correct somehow required that it is

grammatically incorrect - without any regard to the meaning of the

sentence. I don't think that grammar alone can imply anything of the

sort.

Additionally, the link {img

src=http://kilby.stanford.edu/~rvg/154/handouts/incompleteness.html}

points to a theorem saying that the correctness of a theorem in a

sufficiently powerful system cannot be recognized by a Turing Machine.

We know that a YACC grammar can be recognized by a Turing

machine. I believe this proves that Lojban's grammar alone is not

"sufficiently powerful" to make statements about itself - which is a

good thing, because that is the job of semantics - and so is not

Gödel-incomplete.

rab.spir

Erm. Ok. Again, forget YACC, think LALR(1). Turing machines can

determine if any sentence does or does not belong to a LALR(1) language,

in polynomial time, given memory proportional to sentence length. The

question, then, is whether or not Lojban Description can express a Turing machine

for recognizing itself. C is capable of constructing a Turing machine

for recognizing itself, and thousands of other languages, and it has

significantly fewer primitives than Lojban. (Further, C can express

Turing machine rules for manipulating the semantics of itself) I'm

confident that if you were bored enough, you could describe, using

Lojban Description (and maybe even just MEX), a Turing machine which recognized

Lojban. Maybe, though, I'm misinterpreting what the conundrum is.

--Jay

Yes, and that is why neither C nor Lojban Description is Gödel-incomplete at this

level. Number theory is incomplete because it can mathematically

determine whether statements are true or false, which necessitates that

there are statements whose truth cannot be determined by number theory.

Lojban Description cannot mathematically determine whether statements are true or

false. Its mathematics only go as far as recognizing whether a statement

is grammatically correct or not. Hence, the mathematical structure of a

statement cannot make a statement about itself, and so no Gödel

sentence exists in Lojban. Similarly, it can be determined in C whether

a given C program (as long as it doesn't do weird stuff with the

preprocessor) will compile, because the act of being compiled does not

allow the program to make statements about itself.

You could go to a higher level and find incompleteness. For example, you

could set up a sub-grammar in MEX - not just the number subgrammar

we already have, but one which would sort out true mathematical

statements from false ones. This grammar would not be LALR(1). Then

if you tried to describe this system using itself, you would have a

Gödel-incomplete system. The system would be number theory, except

with Lojban Description cmavo instead of mathematical symbols.

On the same level, C (or any computer language) is

Gödel-incomplete when it comes to determining whether a program will

halt, because there the mechanism used to determine whether the program

halts is the same mechanism that the program uses.

It comes down to the grammatical/semantic distinction which is mentioned

at the very top of this discussion. If you take the semantics out of

Lojban Description (and assume that semantics includes the meaning of statements

made with mekso), you cannot express anything. So, in order for Lojban Description

to make statements about itself, it has to do so at the semantic level.

If you create a mathematical system which can process Lojban Description at the

semantic level, you have an AI, and no longer something which is defined

by the language itself; and to confuse this AI you might only need to

say na nei.

(If I were Hofstadter, I would be able to phrase this much more

clearly.)

rab.spir

Well, I've acquired GEB, will read that, and from there investigate the

original papers as needed to grok this topic, and then return. :)

--Jay