A family of grammars for Voynichese

[oshfdk]

Suppose we have the following set of words: AB, ABC, BC, AC, BA, BAB. Let's say, I'm trying to analyze it using a slot grammar. The degenerate grammars that you mentioned are [A|B|C|∅] [A|B|C|∅] [A|B|C|∅] (any letter in each slot), or [AB|ABC|BC|AC|BA|BAB] (all words in a single slot, 6 elements in 1 slot). These are not interesting, they won't tell us anything. There are not that many non-trivial slot grammars that we can have for this text.

[B|∅][AB|B|A][C|∅], this one can produce 12 strings: BABC, BAB, BBC, BB, BAC, BA, ABC, AB, BC, B, AC, A, covers the whole text and has 7 elements in 3 slots.

[BA|A|B][A|B|BC|C], this one can produce 12 strings as well: BAA, BAB, BABC, BAC, AA, AB, ABC, AC, BA, BB, BBC, BC, it covers the whole text and has the same stats as above, but with 7 elements in 2 slots

Is any one of these grammars better than the other in any way?

The example is right. In this case I would say those two grammars are equivalent in all respects (the number of slots I think is irrelevant).

Are they better than the trivial 6 elements in 1 slot grammar?

Sure they are, they are not trivial, they actually say something about the structure of words AB, ABC etc., using less information.

This is interesting. If the number of slots is irrelevant, then the trivial grammar comes at the top on each of three metrics: coverage (they all have 100%), efficiency and the number of elements. My intuition would be that the number of slots matters and my intuition would be that grammar [B|∅][AB|B|A][C|∅] is subjectively better than [BA|A|B][A|B|BC|C], because it includes two optional elements, effectively showing prefix-infix-suffix form (which may or may not correspond to some actual feature of the underlying language or could just be a statistical fluke), while grammar [BA|A|B][A|B|BC|C] appears to assign some chunks to prefixes and suffixes with no clear path for a language model. E.g., why BA is a prefix and BC is a suffix? Note that either grammar could be the true grammar of the original language, it's just that one of them attempts to extract functional elements from the text, and the other doesn't.

If we assume the number of slots is irrelevant, maybe the total combined length of all element strings is relevant? But in my opinion, all this is just guesswork, until the more fundamental problem below is addressed.

 [Edit]: I just realized, that my main point got diluted in the text, so I'll try to reiterate: as far as I understand, we can only have a satisfactory quantitative metric for the quality of a grammar if it's tied to some quantitative task wrt the text. If we identify this task, then we can try to work backwards from it to the properties of grammars and find a grammar that would solve this task better. However, I'm still in the woods when it comes to understanding the simple question of what this task is exactly ¯\_(ツ)_/¯

Ah that's the good question. I'll try to be as schematic as possible:

- The VMS has a word structure which is remarkable and unheard of.
- So it's very probable the word structure has something to do with the method used to write it (be it a meaningful or meaningless text, does not matter).
- So getting a better grab on this structure should be of help in studying VMS (and many people have tried to do this since, I don't know exactly, the 60s?). Will this be enough? I very much doubt it, but who knows
- It's very difficult to say if a grammar is 'good'. But having a reliable method to rank them according to some factor would help.

Once one has settled on a grammar the task could be "imagine what the word structure could encode and how, then try demonstrate it does". Very, very difficult. Btw I tried again to try to 'decode' Voynich as a Latin syllabary (by trying to fit the more common Voynich words to the most common Latin ones). Nada de nada, it does not work at all. But that was just one possibility, the problem is there are too many of them

I think this is a very good explanation of what is of interest qualitatively, the problem is if we are trying to optimize the quantitative (numeric) metrics of the grammars, we need a quantitative metric of the end task. We can try comparing the statistical or structural properties of the predicted and the actual sets. Something like (I'm just throwing in examples randomly): "median length of a grammar word divided by the median length of a text word", "median Levenshtein distance between a grammar word and the nearest text word", "total number of character bigrams predicted by the grammar but absent in the text", "same as the last one, but reformulated as probabilities: the probability of the actual count of a character bigram in the word list given the expected count of a character bigram in the grammar, assuming all the outputs of the grammar are equally likely", etc. There are many and many potential metrics of how well the predicted set matches the actual text, but without choosing a specific metric it's really hard to evaluate the grammars.

[Edit]: Maybe we even can reframe the metric as a linear algebra problem: given a slot grammar assign probabilities to each element (so that the probabilities sum to 100% for each slot) in such a way as to minimize the difference between the probabilities of the words of the predicted set (computed as the product of the probabilities of respective elements taken from each slot, I guess) vs the relative counts of the same words in the observed set (computed as the actual word count divided by the total number of tokens, 0 for missing words). This way we will try to mimic the actual text token distribution. Then compare the grammars using their respective optimal element probabilities with this metric (the difference between the probabilities of the words of the predicted set vs the relative counts of the same words in the observed set).