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Bridging

Rollup state updates facilitate interoperability between the layer 1 and the rollup. If the state of the rollup is verified and stored by the layer 1, then the layer 1 can also validate claims against that state, such as a claim that some tokens have been deposited into a bridge contract on the rollup. The L1 can also write to the state which is maintained by the L1, and the rollup can thus receive messages and tokens from the L1.
This is the idea used by the LX-to-LY bridge, which Polygon zkEVM uses to bridge ETH between the layer 1 and layer 2. In this design, part of the L1 state, a Merkle tree of messages to be sent to the L2, is represented directly in the L2 VM semantics. Since the canonical execution of L2 transactions happens in a smart contract on the L1, this executor is able to read from the appropriate L1 state when executing operations in the L2 VM.