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A chain-complete partial order (equivalently, a pointed dcpo) is a set $D$ with a partial order $\leq$ such that all chains of $D$ have a supremum. The least upper bound ($\bigsqcup$) of the empty chain is the least element $\bot$ of the CCPO.

A function $f\colon M \to N$ is monotone if for all $a, b \in M$, the following holds: $$a \leq b \implies f(a) \leq f(b)$$

A function $f\colon M \to N$ between two CCPOs is Scott-continuous if it is monotone and for every chain $C$ of $M$, we have

$$f(\bigsqcup_{c \in C} c) = \bigsqcup_{m \in C} f(c)\,.$$

Scott-continuous functions play an important part in defining denotational semantics of programs, and as is well-known in computing science, every Turing-computable function is Scott-continuous$^0$. Is the converse true? Is every Scott-continuous function computable?

  1. See this question, from which I took and edited some definitions.
Jozef Mikušinec
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1 Answers1

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$\newcommand{\TM}{T\!M}$

Let $\TM$ be the set of Turing machines and $2 = \{0,1\}$.

The powerset of any set with $\leq$ equal to $\subseteq$ is a CCPO, its least upper bounds being the unions of the chains.

Take the function $f\colon P(\TM) \to P(\TM \times 2)$ that maps a set of Turing machines $S$ to $$\{ (t, h) \mid t \in S \text{ and $h$ equals whether $t$ halts on empty input } \}$$

The function $f$ is Scott-continuous $-$ it is defined pointwise, so it's monotonous and $$f(\bigcup_{c \in C} c) = \bigcup_{c \in C} f(c)\,,$$ and therefore $$f(\bigsqcup_{c \in C} c) = \bigsqcup_{c \in C} f(c)\,.$$

It is also obviously not computable.

Jozef Mikušinec
  • 145
  • 7