In computability theory, a system of data-manipulation rules […] is said to be Turing complete or computationally universal if it can be used to simulate any single-taped Turing machine
However, my question is about the formal definition of Turing completeness. I've searched around, but never really found any good definitions of what is a _"a system of data manipulatio rules", or even "simulation".
I suspect that one way to define such notions is to use dynamical systems (maybe creating an equivalence relation on the state-space for representing the states of a Turing machine or equivalent system). On the other side, I know that logic has connections to the theory of automatas. Can logic be viewed as a dynamical system? Does that make sense? For which mathematical structures does it make sense to ask about turing completeness or even computability? Does it have to be a dynamical system?
Anyway, given those considerations, my question is "what is the formal definition of Turing completeness?"