This is based off Boboris02's MBOT.
Now, what I will try to do in this blog post, is to try and make phi systems well-defined. However, this will be only limited to Φ(x), ⇒, ∪, and Φy(x), which means uncomputables will not be included.
Part 1 - The basics
Now, the function Φ(x) alone does not have a definition. To give Φ(x) a definition, we need to create a phi system, then put Φ(x) inside that phi system. A phi system is represented by an equation that must contain at least one instance of Φ(x) (or Φy(x)).
A phi system can contain either Φ(x) or another phi system. The definition of Φ(x) must be in the equation that describes the phi system that contains Φ(x).
Part 2 - Some definitions
⇒ alone does not have a definition, like Φ(x). The definition of ⇒ must be included in the equation of a phi system.
∪ alone does not have a definition, again, but there is a difference: the definition of ∪ must involve chains of ⇒ (which means, umm, how do i explain this without using ellipses, just take ⇒n, then add ⇒n at the end to get ⇒n⇒n, then repeat the process m times and you get a chain of ⇒).
The definition of ∪ must be included in the equation of a phi system.
Part 3 - But wait, there's more!
If ⇒ and ∪ aren't enough, we can have Φy(x)! It doesn't have a definition alone, and for a specific y (or for all y), it can have a definition in the equation of a phi system.
Part 4 - Invalid expressions
What makes an invalid expression?
An expression is invalid if Φ(x)'s definition makes Φ(x) end up without a value. For example, if Φ(x)'s definition ends in an infinite loop, or if Φ(x)'s definition doesn't exist.
Part 5 - How powerful are phi systems?