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What if i make a smart contract that does modify the state, but the all its functions are constant.(it will still compile) It means that i can use my smart contract for free without paying for the gas. Wouldn't this be a lose to the miners? How does Ethereum handles this situation?

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Yes, you can. Calling constant function doesn't create transaction. Your client just computes result locally.

However, constant functions can not change state (because they're constant)

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You cannot change contract state without a blockchain transaction.

The reason it still compiles is because the Solidity compiler does not yet figure out with 100% accuracy which functions modify contract state. When I try to compile it, I do get this compiler warning:

Warning: Function declared as view, but this expression (potentially) modifies the state (...)

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