Tau Meta Language and the notion of Fixed Point

1. Tau Meta Language and the notion of Fixed Point

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Tau Meta Language is very unique in how it works. Each TML program is described as a loop without an arbitrary stopping condition. This is the case because the states of computation are finite. So a TML program merely defines what happens within a single iteration or "instance".

The loop continues until

S ∨ F

A) Two consecutive iterations return the same result (this means success)

i+(1).. i++(1).. S

B) Two non-consequetive iterations return the same result (this means fail)

i+(1).. i++(x).. i+++(1).. F

There can only ever be two possibilities so in essence this is boolean, an either or situation. A represents a Fixed Point which indicates the result is final because the state of the computation has not changed after two consequentive iterations.

To put it more precisely, if a loop ever has a length greater than 1 then it is a fail.

Partial fixed point requires ORDERED structures such as an ordered set. This allows for P-SPACE complexity to be recovered.