<p>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:</p>
<p><strong>Since &quot;smart contract&quot; logic is encapsulated <em>entirely</em> within individual UTXO(s), it is possible to <em>perfectly</em> capture logic of arbitrary complexity by aggregating and quarantining UTXOs into a sort of &quot;black box&quot; on-chain.</strong></p>
<p>It is this CEM principle that forms the basis of the <a href="https://eprint.iacr.org/2020/299.pdf" rel="nofollow noreferrer">Hydra Head Protocol</a>; 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.</p>
<p>All of this is to say, that <strong>any and all UTXO logic can be expressed in an accounts-like fashion, but <em>the reverse is not true</em></strong>. Therefore, the UTXO model is, expressively speaking, a superset of the account model.</p>

CC BY-SA 4.0
2022-02-16T21:43:32.617Z
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?
Paymahn Moghadasian
CC BY-SA 4.0
2022-02-16T23:19:10.610Z
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
CC BY-SA 4.0
2022-02-16T18:14:12.500Z
2022-02-16T18:14:12.500Z
zhekson
<p>I stumbled across <a href="https://iohk.io/en/research/library/papers/chimeric-ledgerstranslating-and-unifying-utxo-based-and-account-based-cryptocurrencies/" rel="nofollow noreferrer">this paper</a> by IOHK which seems to show this sort of equivalence.</p>
<p>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.</p>
<p>EDIT: this seems like the paragraph from the paper above which confirms the idea that eUTxO is a mathematical superset of the account model:</p>
<blockquote>
<p>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.</p>
</blockquote>

CC BY-SA 4.0
2022-02-16T22:34:22.440Z
2022-02-17T17:12:45.720Z
2022-02-17T17:12:45.720Z
Paymahn Moghadasian
<p>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.</p>
<p>I tried reading the <a href="https://iohk.io/en/research/library/papers/the-extended-utxo-model/" rel="nofollow noreferrer">paper presenting the eUTxO model</a> but my lack of formal math training made it a bit difficult and I didn't find any mentions of Account model equivalence.</p>
<p>Does anyone know of high quality sources which confirm or reject out these ideas?</p>

<eutxo>
Mathematical equivalence of Account and eUTxO model
# Mathematical equivalence of Account and eUTxO model

(Asked by: username_0 on 2022-02-16)

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](https://iohk.io/en/research/library/papers/the-extended-utxo-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?

**Tags:** eutxo

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## 2 answers

### Answer by username_1 on 2022-02-16. Score: 4 | Accepted answer

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](https://eprint.iacr.org/2020/299.pdf); 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.

> **Comments:**
> **username_0**: 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?  
> **username_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).  

---

### Answer by username_0 on 2022-02-16. Score: 1

I stumbled across [this paper](https://iohk.io/en/research/library/papers/chimeric-ledgerstranslating-and-unifying-utxo-based-and-account-based-cryptocurrencies/) 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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