# Motivation

UTXO growth is a serious concern for Bitcoin's long-term decentralization. To
run a competitive mining operation potentially the entire UTXO set must be in
RAM to achieve competitive latency; your larger, more centralized, competitors
will have the UTXO set in RAM. Mining is a zero-sum game, so the extra latency
of not doing so if they do directly impacts your profit margin. Secondly,
having possession of the UTXO set is one of the minimum requirements to run a
full node; the larger the set the harder it is to run a full node.

Currently the maximum size of the UTXO set is unbounded as there is no
consensus rule that limits growth, other than the block-size limit itself; as
of writing the UTXO set is 1.3GB in the on-disk, compressed serialization,
which expands to significantly more in memory. UTXO growth is driven by a
number of factors, including the fact that there is little incentive to merge
inputs, lost coins, dust outputs that can't be economically spent, and
non-btc-value-transfer "blockchain" use-cases such as anti-replay oracles and
timestamping.

We don't have good tools to combat UTXO growth. Segregated Witness proposes to
give witness space a 75% discount, in part of make reducing the UTXO set size
by spending txouts cheaper. While this may change wallets to more often spend
dust, it's hard to imagine an incentive sufficiently strong to discourage most,
let alone all, UTXO growing behavior.

For example, timestamping applications often create unspendable outputs due to
ease of implementation, and because doing so is an easy way to make sure that
the data required to reconstruct the timestamp proof won't get lost - all
Bitcoin full nodes are forced to keep a copy of it. Similarly anti-replay
use-cases like using the UTXO set for key rotation piggyback on the uniquely
strong security and decentralization guarantee that Bitcoin provides; it's very
difficult - perhaps impossible - to provide these applications with
alternatives that are equally secure. These non-btc-value-transfer use-cases
can often afford to pay far higher fees per UTXO created than competing
btc-value-transfer use-cases; many users could afford to spend $50 to register
a new PGP key, yet would rather not spend $50 in fees to create a standard two
output transaction. Effective techniques to resist miner censorship exist, so
without resorting to whitelists blocking non-btc-value-transfer use-cases as
"spam" is not a long-term, incentive compatible, solution.

A hard upper limit on UTXO set size could create a more level playing field in
the form of fixed minimum requirements to run a performant Bitcoin node, and
make the issue of UTXO "spam" less important. However, making any coins
unspendable, regardless of age or value, is a politically untenable economic
change.


# TXO Commitments

A merkle tree committing to the state of all transaction outputs, both spent
and unspent, we can provide a method of compactly proving the current state of
an output. This lets us "archive" less frequently accessed parts of the UTXO
set, allowing full nodes to discard the associated data, still providing a
mechanism to spend those archived outputs by proving to those nodes that the
outputs are in fact unspent.

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.

At an extreme, with TXO commitments we could even have no UTXO set at all,
entirely eliminating the UTXO growth problem. Transactions would simply be
accompanied by TXO commitment proofs showing that the outputs they wanted to
spend were still unspent; nodes could update the state of the TXO MMR purely
from TXO commitment proofs. However, the log2(n) bandwidth overhead per txin is
substantial, so a more realistic implementation is be to have a UTXO cache for
recent transactions, with TXO commitments acting as a alternate for the (rare)
event that an old txout needs to be spent.

Proofs can be generated and added to transactions without the involvement of
the signers, even after the fact; there's no need for the proof itself to
signed and the proof is not part of the transaction hash. Anyone with access to
TXO MMR data can (re)generate missing proofs, so minimal, if any, changes are
required to wallet software to make use of TXO commitments.


## Delayed Commitments

TXO commitments aren't a new idea - the author proposed them years ago in
response to UTXO commitments. However it's critical for small miners' orphan
rates that block validation be fast, and so far it has proven difficult to
create (U)TXO implementations with acceptable performance; updating and
recalculating cryptographicly hashed merkelized datasets is inherently more
work than not doing so. Fortunately if we maintain a UTXO set for recent
outputs, TXO commitments are only needed when spending old, archived, outputs.
We can take advantage of this by delaying the commitment, allowing it to be
calculated well in advance of it actually being used, thus changing a
latency-critical task into a much easier average throughput problem.

Concretely each block B_i commits to the TXO set state as of block B_{i-n}, in
other words what the TXO commitment would have been n blocks ago, if not for
the n block delay. Since that commitment only depends on the contents of the
blockchain up until block B_{i-n}, the contents of any block after are
irrelevant to the calculation.


## Implementation

Our proposed high-performance/low-latency delayed commitment full-node
implementation needs to store the following data:

1) UTXO set

    Low-latency K:V map of txouts definitely known to be unspent. Similar to
    existing UTXO implementation, but with the key difference that old,
    unspent, outputs may be pruned from the UTXO set.


2) STXO set

    Low-latency set of transaction outputs known to have been spent by
    transactions after the most recent TXO commitment, but created prior to the
    TXO commitment.


3) TXO journal

    FIFO of outputs that need to be marked as spent in the TXO MMR. Appends
    must be low-latency; removals can be high-latency.


4) TXO MMR list

    Prunable, ordered list of TXO MMR's, mainly the highest pending commitment,
    backed by a reference counted, cryptographically hashed object store
    indexed by digest (similar to how git repos work). High-latency ok. We'll
    cover this in more in detail later.


### Fast-Path: Verifying a Txout Spend In a Block

When a transaction output is spent by a transaction in a block we have two
cases:

1) Recently created output

    Output created after the most recent TXO commitment, so it should be in the
    UTXO set; the transaction spending it does not need a TXO commitment proof.
    Remove the output from the UTXO set and append it to the TXO journal.

2) Archived output

    Output created prior to the most recent TXO commitment, so there's no
    guarantee it's in the UTXO set; transaction will have a TXO commitment
    proof for the most recent TXO commitment showing that it was unspent.
    Check that the output isn't already in the STXO set (double-spent), and if
    not add it. Append the output and TXO commitment proof to the TXO journal.

In both cases recording an output as spent requires no more than two key:value
updates, and one journal append. The existing UTXO set requires one key:value
update per spend, so we can expect new block validation latency to be within 2x
of the status quo even in the worst case of 100% archived output spends.


### Slow-Path: Calculating Pending TXO Commitments

In a low-priority background task we flush the TXO journal, recording the
outputs spent by each block in the TXO MMR, and hashing MMR data to obtain the
TXO commitment digest. Additionally this background task removes STXO's that
have been recorded in TXO commitments, and prunes TXO commitment data no longer
needed.

Throughput for the TXO commitment calculation will be worse than the existing
UTXO only scheme. This impacts bulk verification, e.g. initial block download.
That said, TXO commitments provides other possible tradeoffs that can mitigate
impact of slower validation throughput, such as skipping validation of old
history, as well as fraud proof approaches.


### TXO MMR Implementation Details

Each TXO MMR state is a modification of the previous one with most information
shared, so we an space-efficiently store a large number of TXO commitments
states, where each state is a small delta of the previous state, by sharing
unchanged data between each state; cycles are impossible in merkelized data
structures, so simple reference counting is sufficient for garbage collection.
Data no longer needed can be pruned by dropping it from the database, and
unpruned by adding it again. Since everything is committed to via cryptographic
hash, we're guaranteed that regardless of where we get the data, after
unpruning we'll have the right data.

Let's look at how the TXO MMR works in detail. Consider the following TXO MMR
with two txouts, which we'll call state #0:

      0
     / \
    a   b

If we add another entry we get state #1:

        1
       / \
      0   \
     / \   \
    a   b   c

Note how it 100% of the state #0 data was reused in commitment #1. Let's
add two more entries to get state #2:

            2
           / \
          2   \
         / \   \
        /   \   \
       /     \   \
      0       2   \
     / \     / \   \
    a   b   c   d   e

This time part of state #1 wasn't reused - it's wasn't a perfect binary
tree - but we've still got a lot of re-use.

Now suppose state #2 is committed into the blockchain by the most recent block.
Future transactions attempting to spend outputs created as of state #2 are
obliged to prove that they are unspent; essentially they're forced to provide
part of the state #2 MMR data. This lets us prune that data, discarding it,
leaving us with only the bare minimum data we need to append new txouts to the
TXO MMR, the tips of the perfect binary trees ("mountains") within the MMR:

            2
           / \
          2   \
               \
                \
                 \
                  \
                   \
                    e

Note that we're glossing over some nuance here about exactly what data needs to
be kept; depending on the details of the implementation the only data we need
for nodes "2" and "e" may be their hash digest.

Adding another three more txouts results in state #3:

                  3
                 / \
                /   \
               /     \
              /       \
             /         \
            /           \
           /             \
          2               3
                         / \
                        /   \
                       /     \
                      3       3
                     / \     / \
                    e   f   g   h

Suppose recently created txout f is spent. We have all the data required to
update the MMR, giving us state #4. It modifies two inner nodes and one leaf
node:

                  4
                 / \
                /   \
               /     \
              /       \
             /         \
            /           \
           /             \
          2               4
                         / \
                        /   \
                       /     \
                      4       3
                     / \     / \
                    e  (f)  g   h

If an archived txout is spent requires the transaction to provide the merkle
path to the most recently committed TXO, in our case state #2. If txout b is
spent that means the transaction must provide the following data from state #2:

            2
           /
          2
         /
        /
       /
      0
       \
        b

We can add that data to our local knowledge of the TXO MMR, unpruning part of
it:

                  4
                 / \
                /   \
               /     \
              /       \
             /         \
            /           \
           /             \
          2               4
         /               / \
        /               /   \
       /               /     \
      0               4       3
       \             / \     / \
        b           e  (f)  g   h

Remember, we haven't _modified_ state #4 yet; we just have more data about it.
When we mark txout b as spent we get state #5:

                  5
                 / \
                /   \
               /     \
              /       \
             /         \
            /           \
           /             \
          5               4
         /               / \
        /               /   \
       /               /     \
      5               4       3
       \             / \     / \
       (b)          e  (f)  g   h

Secondly by now state #3 has been committed into the chain, and transactions
that want to spend txouts created as of state #3 must provide a TXO proof
consisting of state #3 data. The leaf nodes for outputs g and h, and the inner
node above them, are part of state #3, so we prune them:

                  5
                 / \
                /   \
               /     \
              /       \
             /         \
            /           \
           /             \
          5               4
         /               /
        /               /
       /               /
      5               4
       \             / \
       (b)          e  (f)

Finally, lets put this all together, by spending txouts a, c, and g, and
creating three new txouts i, j, and k. State #3 was the most recently committed
state, so the transactions spending a and g are providing merkle paths up to
it. This includes part of the state #2 data:

                  3
                 / \
                /   \
               /     \
              /       \
             /         \
            /           \
           /             \
          2               3
         / \               \
        /   \               \
       /     \               \
      0       2               3
     /       /               /
    a       c               g

After unpruning we have the following data for state #5:

                  5
                 / \
                /   \
               /     \
              /       \
             /         \
            /           \
           /             \
          5               4
         / \             / \
        /   \           /   \
       /     \         /     \
      5       2       4       3
     / \     /       / \     /
    a  (b)  c       e  (f)  g

That's sufficient to mark the three outputs as spent and add the three new
txouts, resulting in state #6:

                        6
                       / \
                      /   \
                     /     \
                    /       \
                   /         \
                  6           \
                 / \           \
                /   \           \
               /     \           \
              /       \           \
             /         \           \
            /           \           \
           /             \           \
          6               6           \
         / \             / \           \
        /   \           /   \           6
       /     \         /     \         / \
      6       6       4       6       6   \
     / \     /       / \     /       / \   \
   (a) (b) (c)      e  (f) (g)      i   j   k

Again, state #4 related data can be pruned. In addition, depending on how the
STXO set is implemented may also be able to prune data related to spent txouts
after that state, including inner nodes where all txouts under them have been
spent (more on pruning spent inner nodes later).


### Consensus and Pruning

It's important to note that pruning behavior is consensus critical: a full node
that is missing data due to pruning it too soon will fall out of consensus, and
a miner that fails to include a merkle proof that is required by the consensus
is creating an invalid block. At the same time many full nodes will have
significantly more data on hand than the bare minimum so they can help wallets
make transactions spending old coins; implementations should strongly consider
separating the data that is, and isn't, strictly required for consensus.

A reasonable approach for the low-level cryptography may be to actually treat
the two cases differently, with the TXO commitments committing too what data
does and does not need to be kept on hand by the UTXO expiration rules. On the
other hand, leaving that uncommitted allows for certain types of soft-forks
where the protocol is changed to require more data than it previously did.


### Consensus Critical Storage Overheads

Only the UTXO and STXO sets need to be kept on fast random access storage.
Since STXO set entries can only be created by spending a UTXO - and are smaller
than a UTXO entry - we can guarantee that the peak size of the UTXO and STXO
sets combined will always be less than the peak size of the UTXO set alone in
the existing UTXO-only scheme (though the combined size can be temporarily
higher than what the UTXO set size alone would be when large numbers of
archived txouts are spent).

TXO journal entries and unpruned entries in the TXO MMR have log2(n) maximum
overhead per entry: a unique merkle path to a TXO commitment (by "unique" we
mean that no other entry shares data with it). On a reasonably fast system the
TXO journal will be flushed quickly, converting it into TXO MMR data; the TXO
journal will never be more than a few blocks in size.

Transactions spending non-archived txouts are not required to provide any TXO
commitment data; we must have that data on hand in the form of one TXO MMR
entry per UTXO. Once spent however the TXO MMR leaf node associated with that
non-archived txout can be immediately pruned - it's no longer in the UTXO set
so any attempt to spend it will fail; the data is now immutable and we'll never
need it again. Inner nodes in the TXO MMR can also be pruned if all leafs under
them are fully spent; detecting this is easy the TXO MMR is a merkle-sum tree,
with each inner node committing to the sum of the unspent txouts under it.

When a archived txout is spent the transaction is required to provide a merkle
path to the most recent TXO commitment. As shown above that path is sufficient
information to unprune the necessary nodes in the TXO MMR and apply the spend
immediately, reducing this case to the TXO journal size question (non-consensus
critical overhead is a different question, which we'll address in the next
section).

Taking all this into account the only significant storage overhead of our TXO
commitments scheme when compared to the status quo is the log2(n) merkle path
overhead; as long as less than 1/log2(n) of the UTXO set is active,
non-archived, UTXO's we've come out ahead, even in the unrealistic case where
all storage available is equally fast. In the real world that isn't yet the
case - even SSD's significantly slower than RAM.


### Non-Consensus Critical Storage Overheads

Transactions spending archived txouts pose two challenges:

1) Obtaining up-to-date TXO commitment proofs

2) Updating those proofs as blocks are mined

The first challenge can be handled by specialized archival nodes, not unlike
how some nodes make transaction data available to wallets via bloom filters or
the Electrum protocol. There's a whole variety of options available, and the
the data can be easily sharded to scale horizontally; the data is
self-validating allowing horizontal scaling without trust.

While miners and relay nodes don't need to be concerned about the initial
commitment proof, updating that proof is another matter. If a node aggressively
prunes old versions of the TXO MMR as it calculates pending TXO commitments, it
won't have the data available to update the TXO commitment proof to be against
the next block, when that block is found; the child nodes of the TXO MMR tip
are guaranteed to have changed, yet aggressive pruning would have discarded that
data.

Relay nodes could ignore this problem if they simply accept the fact that
they'll only be able to fully relay the transaction once, when it is initially
broadcast, and won't be able to provide mempool functionality after the initial
relay. Modulo high-latency mixnets, this is probably acceptable; the author has
previously argued that relay nodes don't need a mempool² at all.

For a miner though not having the data necessary to update the proofs as blocks
are found means potentially losing out on transactions fees. So how much extra
data is necessary to make this a non-issue?

Since the TXO MMR is insertion ordered, spending a non-archived txout can only
invalidate the upper nodes in of the archived txout's TXO MMR proof (if this
isn't clear, imagine a two-level scheme, with a per-block TXO MMRs, committed
by a master MMR for all blocks). The maximum number of relevant inner nodes
changed is log2(n) per block, so if there are n non-archival blocks between the
most recent TXO commitment and the pending TXO MMR tip, we have to store
log2(n)*n inner nodes - on the order of a few dozen MB even when n is a
(seemingly ridiculously high) year worth of blocks.

Archived txout spends on the other hand can invalidate TXO MMR proofs at any
level - consider the case of two adjacent txouts being spent. To guarantee
success requires storing full proofs. However, they're limited by the blocksize
limit, and additionally are expected to be relatively uncommon. For example, if
1% of 1MB blocks was archival spends, our hypothetical year long TXO commitment
delay is only a few hundred MB of data with low-IO-performance requirements.


## Security Model

Of course, a TXO commitment delay of a year sounds ridiculous. Even the slowest
imaginable computer isn't going to need more than a few blocks of TXO
commitment delay to keep up ~100% of the time, and there's no reason why we
can't have the UTXO archive delay be significantly longer than the TXO
commitment delay.

However, as with UTXO commitments, TXO commitments raise issues with Bitcoin's
security model by allowing relatively miners to profitably mine transactions
without bothering to validate prior history. At the extreme, if there was no
commitment delay at all at the cost of a bit of some extra network bandwidth
"full" nodes could operate and even mine blocks completely statelessly by
expecting all transactions to include "proof" that their inputs are unspent; a
TXO commitment proof for a commitment you haven't verified isn't a proof that a
transaction output is unspent, it's a proof that some miners claimed the txout
was unspent.

At one extreme, we could simply implement TXO commitments in a "virtual"
fashion, without miners actually including the TXO commitment digest in their
blocks at all. Full nodes would be forced to compute the commitment from
scratch, in the same way they are forced to compute the UTXO state, or total
work. Of course a full node operator who doesn't want to verify old history can
get a copy of the TXO state from a trusted source - no different from how you
could get a copy of the UTXO set from a trusted source.

A more pragmatic approach is to accept that people will do that anyway, and
instead assume that sufficiently old blocks are valid. But how old is
"sufficiently old"? First of all, if your full node implementation comes "from
the factory" with a reasonably up-to-date minimum accepted total-work
thresholdⁱ - in other words it won't accept a chain with less than that amount
of total work - it may be reasonable to assume any Sybil attacker with
sufficient hashing power to make a forked chain meeting that threshold with,
say, six months worth of blocks has enough hashing power to threaten the main
chain as well.

That leaves public attempts to falsify TXO commitments, done out in the open by
the majority of hashing power. In this circumstance the "assumed valid"
threshold determines how long the attack would have to go on before full nodes
start accepting the invalid chain, or at least, newly installed/recently reset
full nodes. The minimum age that we can "assume valid" is tradeoff between
political/social/technical concerns; we probably want at least a few weeks to
guarantee the defenders a chance to organise themselves.

With this in mind, a longer-than-technically-necessary TXO commitment delayʲ
may help ensure that full node software actually validates some minimum number
of blocks out-of-the-box, without taking shortcuts. However this can be
achieved in a wide variety of ways, such as the author's prev-block-proof
proposal³, fraud proofs, or even a PoW with an inner loop dependent on
blockchain data. Like UTXO commitments, TXO commitments are also potentially
very useful in reducing the need for SPV wallet software to trust third parties
providing them with transaction data.

i) Checkpoints that reject any chain without a specific block are a more
   common, if uglier, way of achieving this protection.

j) A good homework problem is to figure out how the TXO commitment could be
   designed such that the delay could be reduced in a soft-fork.


## Further Work

While we've shown that TXO commitments certainly could be implemented without
increasing peak IO bandwidth/block validation latency significantly with the
delayed commitment approach, we're far from being certain that they should be
implemented this way (or at all).

1) Can a TXO commitment scheme be optimized sufficiently to be used directly
without a commitment delay? Obviously it'd be preferable to avoid all the above
complexity entirely.

2) Is it possible to use a metric other than age, e.g. priority? While this
complicates the pruning logic, it could use the UTXO set space more
efficiently, especially if your goal is to prioritise bitcoin value-transfer
over other uses (though if "normal" wallets nearly never need to use TXO
commitments proofs to spend outputs, the infrastructure to actually do this may
rot).

3) Should UTXO archiving be based on a fixed size UTXO set, rather than an
age/priority/etc. threshold?

4) By fixing the problem (or possibly just "fixing" the problem) are we
encouraging/legitimising blockchain use-cases other than BTC value transfer?
Should we?

5) Instead of TXO commitment proofs counting towards the blocksize limit, can
we use a different miner fairness/decentralization metric/incentive? For
instance it might be reasonable for the TXO commitment proof size to be
discounted, or ignored entirely, if a proof-of-propagation scheme (e.g.
thinblocks) is used to ensure all miners have received the proof in advance.

6) How does this interact with fraud proofs? Obviously furthering dependency on
non-cryptographically-committed STXO/UTXO databases is incompatible with the
modularized validation approach to implementing fraud proofs.


# References

1) "Merkle Mountain Ranges",
   Peter Todd, OpenTimestamps, Mar 18 2013,
   https://github.com/opentimestamps/opentimestamps-server/blob/master/doc/merkle-mountain-range.md

2) "Do we really need a mempool? (for relay nodes)",
   Peter Todd, bitcoin-dev mailing list, Jul 18th 2015,
   https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2015-July/009479.html

3) "Segregated witnesses and validationless mining",
   Peter Todd, bitcoin-dev mailing list, Dec 23rd 2015,
   https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2015-December/012103.html

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

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