For the given code block:

val io = maybeGivenUp.traverse_(task =>
  Logger[IO].warn(s"Give up move due to invalid move $response by $key for $task")
)

I was trying to use new function syntax with maybeGivenUp.traverse_: task =>, but got syntax error, strange

The only difference expect for naming, is the usage of traverse_ instead of traverse.

This is a good optimization 👍

Is there possibility and necessity to replace state.updated with state.add? Should be other methods be renamed?

They're the same but also not quite.
AppState is an opaque type with the underlying type is a Map. For object outside of AppState, they can't access to Map's functions they can only access to what We define in AppState object. So that's why we can use state.updated in AppState, but in Executor We need to use state.add

I was trying to use new function syntax with maybeGivenUp.traverse_: task =>, but got syntax error, strange

So, any function that ends with symbol cannot use fewer braces syntax.

So, any function that ends with symbol cannot use fewer braces syntax.

TIL, thanks!