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?

2 Answers 2


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? Feb 16 at 21:43
  • 1
    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).
    – zhekson
    Feb 16 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.

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