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Are there any approaches to or any research on having decentralized applications in such a way that not every full node has to execute them?

I'm not asking specifically about the Ethereum Project but more generally whether there is anything in this regard.

Are there any other approaches which take away the gossip behavior from decentralized applications, enabling them to scale?

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Yes. A method using sharding is one approach which can be used is proposed for Ethereum. In a sharded system, work is distributed so that workers only process a subset of all the available work. In Ethereum, because the workers are not trusted, work done in different shards can be checked by randomly sampling transactions from the shards.

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