# Notation for a Kleene fixed point with a starting element

Assume a CPO $Q,\leq$ and a Scott-continuous function $f : Q \rightarrow Q$. As it is known, the chain $\bot \leq f(\bot) \leq \ldots \leq f^n(\bot)$ (where $f^n$ denotes the function $n-1$-times composed with itself) leads to a fixed point of $f$.

It is also clear, if $f$ is inflationary, that for any $q \in Q$, the chain $q \leq f(q) \leq \ldots \leq f^m(q)$ leads to a fixed point. (This follows immediately by just considering the subset $\{\,p\,|\, p \in Q \wedge q \leq p\}$, which is also a CPO.)

This second form is often more useful in practice. But how to denote it? While the fixed point of $f$ is easily written $\text{fix} f$, is there a standard convention for the second form? Writing $\text{fix} f(q)$ seems intuitive but is - strictly speaking - wrong, since $f(q)$ is not a function (it might also be confusing in the case of higher-order functions).

edit: Added note, that $f$ must be inflationary.

Suppose $Q$ is a join semilattice. Define $g:Q \to Q$ by

$$g(x) = f(x) \sqcup q.$$

Then $\text{fix} \; g$ is equal to the least fixpoint of $q,f(q),f^2(q),\dots$. So, at least for cpo's that form a join semilattice, you can indicate it pretty naturally using existing notation -- there's not a strong need to invent a new notation (though you could of course introduce one if you wanted).

If $Q$ is not a join semilattice, I think you can still define

$$g(x) = \sup \{ f(x), q \}$$

and then do the same thing. The least fixpoint of $q,f(q),f^2(q),\dots$ will be $\text{fix} \; g$.

Note that, in general, there might not be such a fixed point.

E.g. $$\begin{array}{l} f : \mathbb{N}_\bot \to \mathbb{N}_\bot\\ f(\bot) = \bot\\ f(\lfloor n\rfloor) = \lfloor n + 1\rfloor \end{array}$$ is the standard denotation of the lazy lambda calculus function $\lambda n.\ n+1$. It is Scott-continuous.

Its least fixed point is $\bot$ -- actually, there are no other fixed points. So, if we consider the "chain" starting from $q = \lfloor 42 \rfloor$, which is $\lfloor 42 \rfloor, f(\lfloor 42 \rfloor), f^2(\lfloor 42 \rfloor), \ldots$ we discover that it's not even a chain, and as a set has no supremum in $\mathbb{N}_\bot$.

Hence, one should be careful here:

It is also clear, that for any $q \in Q$, the chain $q \leq f(q) \leq \ldots \leq f^m(q)$ leads to a fixed point. (This follows immediately by just considering the subset $\{\,p\,|\, p \in Q \wedge q \leq p\}$, which is also a CPO.)

If that is a chain, then it leads to a fixed point -- but it might not be a chain. Also, the subset you mention is a CPO: in the previous example it is the singleton CPO $\{q\}=\{\lfloor 42 \rfloor\}$. However, in general, $f$ is not an internal operation on that CPO. Indeed, in the example we immediately escape that.

• You are right, I forgot to mention that the function also needs to be inflationary. Do I understand your answer correctly in that only $\bot < \lfloor n \rfloor$? – choeger Feb 8 '17 at 18:57
• @choeger Yes, $\mathbb{N}_\bot$ only has $\bot < \lfloor n \rfloor$ as a strict inequality. (and $\lfloor - \rfloor : \mathbb{N}\to\mathbb{N}_\bot$ denotes the lifting injection -- I am probably being overly pedantic in using that explicitly here) – chi Feb 8 '17 at 19:32