In Swift, how to cast to protocol with associated type?

[Edited to fix: : SpecialValue, not = SpecialValue]

This is not possible. SpecialValueController is an "incomplete type" conceptually so the compiler cannot know. SpecialValueType, although it is constrained by SpecialValue, it is not known until it is determined by any adopting class. So it is a really placeholder with inadequate information. as?-ness cannot be checked.

You could have a base class that adopts SpecialController with a concrete type for SpecialValueController, and have multiple child classes that inherit from the adopting class, if you're still seeking a degree of polymorphism.

Unfortunately, Swift doesn't currently support the use of protocols with associated types as actual types. This however is technically possible for the compiler to do; and it may well be implemented in a future version of the language.

A simple solution in your case is to define a 'shadow protocol' that SpecialController derives from, and allows you to access currentValue through a protocol requirement that type erases it:

// This assumes SpecialValue doesn't have associated types – if it does, you can
// repeat the same logic by adding TypeErasedSpecialValue, and then using that.
protocol SpecialValue {
  // ...
}

protocol TypeErasedSpecialController {
  var typeErasedCurrentValue: SpecialValue? { get }
}

protocol SpecialController : TypeErasedSpecialController {
  associatedtype SpecialValueType : SpecialValue
  var currentValue: SpecialValueType? { get }
}

extension SpecialController {
  var typeErasedCurrentValue: SpecialValue? { return currentValue }
}

extension String : SpecialValue {}

struct S : SpecialController {
  var currentValue: String?
}

var x: Any = S(currentValue: "Hello World!")
if let sc = x as? TypeErasedSpecialController {
  print(sc.typeErasedCurrentValue as Any) // Optional("Hello World!")
}