Exercise. [dt-001S]

Liam O'Connor

What is \textbf {fix}(\mathsf {id}) where \mathsf {id} is the identity function on \mathbb {B}_\bot ?
Solution.
- Liam O'Connor
$\bigsqcup \{ \bot , \mathsf {id}(\bot ), \mathsf {id}(\mathsf {id}(\bot )), \dots \} = \bot$
What is \textbf {fix}(\kappa _\mathit {F}) where \kappa _\mathit {F} is the constant function that always returns \mathit {F}?
Solution.
- Liam O'Connor
$\bigsqcup \{ \bot , \kappa _F(\bot ), \kappa _F(\kappa _F(\bot )), \dots \} = \mathit {F}$
What is \textbf {fix}(f) where \begin {array}{l} f : [\mathbb {N}_\bot ] \rightarrow [\mathbb {N}_\bot ]\\ f(x) = 1::x \end {array} (where [\mathbb {N}_\bot ] denotes Haskell-style lazy lists of \mathbb {N}_\bot )?
Solution.
- Liam O'Connor
$\bigsqcup \{ \bot , 1::\bot , 1::1::\bot , \dots \} = 1::1::1::\cdots$ Here $\textbf {fix}(f)$ would be the semantics of the recursive definition $\mathsf {ones} = 1 :: \mathsf {ones}$ .