Cours 5: Computational $\lambda$-calculus

Call-by-value + computational effects ⟶ evaluation strategy

In a cartesian closed category:

where

• $M,N$ are $\lambda$-terms
• ${⟶}_{\beta \eta }$ means $\beta$-reduction/$\eta$-expansion applied at any position / occurence of the $\lambda$-term

Keep in mind the specific monad:

Partiality monad:

Whenever we have a monad:

Kleisli category $Kl(T)$ of $T$:

• sets as objects

• functions $T:\{\begin{array}{l}A⟶TBA\mapsto 1+B\end{array}$ as morphisms: i.e. partial function from $A$ to $B$.

• Composition: $f:A\stackrel{Kl}{⟶}B$ and $g:B\stackrel{Kl}{⟶}C$ are composed in the following way: $f;Tg;{\mu }_T$

Does it corresponds to the way we compose partial functions ? Yes:

If $T$ is the partiality monad:

$$Kl(T)=pSet$$

where $pSet$ is the category of sets and partial functions.

If $P:A$, $[P]:TA$ ⟶ turn an implicit effect into an explicit one.

Strong monads

In the case of the partiality monad $T:Set⟶Set$:

where $1+B=\{\star \}\bigsqcup \{inrb\mid b\in B\}$

Cartesian category equipped with the identity strong monad is a cartesian closed category.

NB: The diagram going from $TA\times TB$ to $T(A\times B)$ doesn’t commute in general, but it does indeed if you have only ONE exception (partiality monad): it doesn’t matter when it is triggered.