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Proof Of Work

Short story: Assign Onward does not depend on proof of work to function.

Trust

If you are not going to trust the largest consensus of transaction processors to record your transaction, who will you trust? That all depends on what you are trusting them with... If you are pre-purchasing coffees from a particular coffee shop, you are already implicitly trusting that coffee shop to honor those pre-purchases in the future. You might also trust an established coupon processor to provide reliable recording and exchange services.

Bridges

Although Assign Onward as implemented does not use Proof Of Work, it makes sense to build automated exchanges with existing blockchains that do use Proof Of Work (such as Bitcoin...)

Open Source

Of course, being open source, modular, and simple, it would not be difficult to add a Bitcoin-like Proof Of Work requirement into the Assign Onward protocol, but would it make any sense to do so? I think not... Bitcoin is established, and small-time Proof Of Work based blockchains have already been proven vulnerable to double-spend attack. Unless a Proof Of Work blockchain enters the market with substantial miner share, competitive with Bitcoin and Ethereum, it will be vulnerable in the way the Bitcoin Gold and many other smaller cryptocurrencies were demonstrated to be vulnerable to 51% attacks in mid 2018.

Bottom line, Assign Onward protocols could be extended/modified to use Proof Of Work as the chain extending block selection mechanism, but then they wouldn't be compliant with the Assign Onward core principles anymore. Something like the Stellar Consensus Protocol would be more in-line with Assign Onward's philosophy.