On Wed, May 18, 2016 at 01:14:59PM +0200, Jorge Timón wrote:
> On May 17, 2016 15:23, "Peter Todd via bitcoin-dev" <
> bitcoin-dev@lists.linuxfoundation.org> wrote:
> > # TXO Commitments
> >
> 
> > Specifically TXO commitments proposes a Merkle Mountain Range¹ (MMR), a
> > type of deterministic, indexable, insertion ordered merkle tree, which
> allows
> > new items to be cheaply appended to the tree with minimal storage
> requirements,
> > just log2(n) "mountain tips". Once an output is added to the TXO MMR it is
> > never removed; if an output is spent its status is updated in place. Both
> the
> > state of a specific item in the MMR, as well the validity of changes to
> items
> > in the MMR, can be proven with log2(n) sized proofs consisting of a
> merkle path
> > to the tip of the tree.
> 
> How expensive it is to update a leaf from this tree from unspent to spent?

log2(n) operations.

I wrote a full MMR implementation with pruning support as part of my
proofchains work:

https://github.com/proofchains/python-proofmarshal/blob/master/proofmarshal/mmr.py

Documentation is a bit lacking, but I'd suggest reading the above source code
and the unit tests(1) to understand what's going on. As of writing item
retrieval by index is implemented(2), and if you follow how that works you'll
see it's log2(n) operations; changing elements in-place isn't yet
implemented(3) but would be a fun homework problem. I'll bet anyone a beer that
you'll find it can be done in k*log2(n) operations, with a reasonably small k. :)

Additionally, I also have a merkelized key:value prefix tree implementation
called a "merbinner tree" in the same library, again with pruning support. It
does implement changing elements in place(4) with log2(n) operations.

Incidentally, something I probably should have made more clear in my TXO
commitments post is that the original MMR scheme I developed for OpenTimestamps
(and independently reinvented for Certificate Transparency) is insufficient:
while you can easily extract a proof that an element is present in the MMR,
that inclusion proof doesn't do a good job of proving the position in the tree
very well. OpenTimestamps didn't need that kind of proof, and I don't think
Certificate Transparency needs it either. However many other MMR applications
do, including many types of TXO commitments.

My proofchains MMR scheme fixes this problem by making each inner node in the
MMR commit to the total number of elements under it(5) - basically it's a
merkle-sum-tree with the size of the tree being what's summed. There may be
more efficient ways to do this, but a committed sum-length is easy to
implement, and the space overhead is only 25% even in the least optimised
implementation possible.

1) https://github.com/proofchains/python-proofmarshal/blob/3f0ba0a9d46f36377ad6c1901de19273604e6fbc/proofmarshal/test/test_mmr.py
2) https://github.com/proofchains/python-proofmarshal/blob/3f0ba0a9d46f36377ad6c1901de19273604e6fbc/proofmarshal/mmr.py#L294
3) https://github.com/proofchains/python-proofmarshal/blob/3f0ba0a9d46f36377ad6c1901de19273604e6fbc/proofmarshal/mmr.py#L230
4) https://github.com/proofchains/python-proofmarshal/blob/3f0ba0a9d46f36377ad6c1901de19273604e6fbc/proofmarshal/merbinnertree.py#L140
5) https://github.com/proofchains/python-proofmarshal/blob/3f0ba0a9d46f36377ad6c1901de19273604e6fbc/proofmarshal/mmr.py#L139

> Wouldn't it be better to have both an append-only TXO and an append-only
> STXO (with all spent outputs, not only the latest ones like in your "STXO")?

Nope. The reason why this doesn't work is apparent when you ask how will the
STXO be indexed?

If it's indexed by outpoint - that is H(txid:n) - to update the STXO you need
he entire thing, as the position of any new STXO that you need to add to the
STXO tree is random.

OTOH, if you index the STXO by txout creation order, with the first txout ever
created having position #0, the second #1, etc. the data you may need to update
the STXO later has predictable locality... but now you have something that's
basically identical to my proposed insertion-ordered TXO commitment anyway.

Incidentally, it's interesting how if a merbinner tree is insertion-order
indexed you end up with a datastructure that's almost identical to a MMR.

-- 
https://petertodd.org 'peter'[:-1]@petertodd.org