Deconstructing the VMS : A positionless ledger (part 1)

[Dunsel]

If the ledger can generate infinitely many strings, it must have a cycle.  That's because a set of infinitely many words must have arbitrarily long words.   If a finite state automaton with N states generates a word with N or more letters, it must go twice or more through some state.  That means the automaton has a cycle.  Then traversing that cycle M times generates a word that contains a substring that is repeated M times, for any M.

On the other hand, if the alphabet has N letters, it is possible to create a word with ~N^K letters that does not repeat any substring of K letters.  These words are called "De Bruijn sequences".   For example, with 10 letters one can create a text of a million letters that does not repeat any substring of 6 or more letters. 

Ok, I guess I didn't quite understand your definition of cycles, and I've let my replies drift into territory I'm not actually claiming.  I need to clarify.

Does the ledger have cycles? By itself, no.

Is it an FSM? By itself, no.

Could you use it to create an FSM with cycles? Of course. It's the inventory you could use to create an FSM. The inventory of letters and consonant/vowel structure in English is not an FSM. It tells you how words can be assembled. You could use that inventory and those constraints to construct an FSM.

The ledger is only one component. It isn't the production mechanism. It provides the constraint space for constructing words. The scribe is the other component and provides the rules for operating inside that space: what transition to choose, when to reset, when to stop, what to copy or what to mutate.

So could a procedure that simply walked the ledger's allowed transitions produce arbitrarily long strings? Sure. I could take English word rules and create one. But that's a property of a traversal procedure, not of the ledger or English rules. Like English, the scribes clearly weren't just blindly walking every available transition.

A De Bruijn sequence, as I understand it, would just be one possible mathematically defined way of traversing the set of transitions. The scribes had their own method.
That's the part I've been trying to get at:

The ledger is only half of the rule system. The scribal policy is the other half.