# Mathematical equivalence of Account and eUTxO model

I've heard that the eUTxO model and the Account model are mathematically equivalent in terms of expressiveness in various YouTube videos. I've also heard that the eUTxO model is a strict superset of the Account model (ie: the eUTxO model can do everything the Account model can do, and more). Are these ideas/facts true? I've been trying to find a high quality source for this idea/fact but haven't been able to find one.

I tried reading the paper presenting the eUTxO model but my lack of formal math training made it a bit difficult and I didn't find any mentions of Account model equivalence.

Does anyone know of high quality sources which confirm or reject out these ideas?

The reason eUTXO is considered a superset of the account model is because it is possible to create constraint emitting machines (CEMs) on the eUTXO model, as mentioned in the paper you linked. To summarize:

Since "smart contract" logic is encapsulated entirely within individual UTXO(s), it is possible to perfectly capture logic of arbitrary complexity by aggregating and quarantining UTXOs into a sort of "black box" on-chain.

It is this CEM principle that forms the basis of the Hydra Head Protocol; allowing the creation of mini-ledgers that can be configured to behave in an account-style fashion, all while periodically committing snapshots to the overarching UTXO-style main-chain.

All of this is to say, that any and all UTXO logic can be expressed in an accounts-like fashion, but the reverse is not true. Therefore, the UTXO model is, expressively speaking, a superset of the account model.

• Thank you for the amazing answer! I know this is a big ask but do you mind elaborating? I don't have intuition for what the following means: `perfectly capture logic of arbitrary complexity by aggregating and quarantining UTXOs into a sort of "black box" on-chain`. Also, is there any chance you can elaborate on why the reverse isn't true? Is there a simple example of a UTXO smart contract that cannot be expressed in an account-like way? Commented Feb 16, 2022 at 21:43
• UTXOs are like little boxes that contain data (and logic) with zero external dependencies. If we aggregate a bunch of these little boxes into a bigger box (lets call the bigger box a "Head"), we can rigorously guarantee that this new Head is not logically dependent on anything outside of the UTXOs that went into its construction. However, we cannot make the same zero-external-dependency guarantees for an accounts-based ledger, because it is already, by its nature, one big box/state of constantly changing interdependencies (almost like one big head). Commented Feb 16, 2022 at 23:19

I stumbled across this paper by IOHK which seems to show this sort of equivalence.

I'm going to try to read it and see what I can learn. I suspect this might be the best resource for learning about this mathematical equivalence between those two models.

EDIT: this seems like the paragraph from the paper above which confirms the idea that eUTxO is a mathematical superset of the account model:

Despite the accounting equivalence implied by the translations, Proposition 2 and the translation of Sec. 5.2 show that it is generally necessary to create several account-based transactions to simulate the effect of a single UTXO-based transaction. This is important: while all the value transfers in a single UTXO-based transaction are processed atomically and simultaneously, the corresponding account-based transactions are not guaranteed to be so. Theoretically, there may be a wide time gap between their executions.