NbE for sums and S4

Kripke model for lambda calculus

$(K, \leq)$ — preorder of possible worlds

Relation of forcing: for world w in K and n-ary predicate X, $w \Vdash X$ — n-ary relation on w, such that

For $w' \geq w$ , $(w \Vdash X)(d_1, \ldots, d_n) \to (w', \Vdash X)(d_1, \ldots, d_n)$

For compound propositions (= types):

$w \Vdash A \to B$ means for all $w' \ge w$ $w' \Vdash A$ implies $w' \Vdash B$

$w \Vdash <A, B>$ means $w \Vdash A$ AND $w \Vdash B$

$w \Vdash \text{Either}(A, B)$ means $w \Vdash A$ OR $w \Vdash B$

Again, meta language must have functions, pairs and sums

Universal model

$K$ (possible worlds) will be a set of different contexts $\Gamma$

$\Gamma \leq \Gamma'$ if $\Gamma \subseteq \Gamma'$

Forcing: $\Gamma \Vdash A$ — a set of normal form derivations $\Gamma \vdash^{nf} A$ , where $A$ — closed and atomic

Then, for example, by definition

$\Gamma \Vdash A \to B$ when for all $\Gamma' \geq \Gamma$ $\Gamma' \vdash^{nf} A$ implies $\Gamma' \vdash^{nf} B$ — a function in meta-language

Kripke model for logic with sums

New binary relation: $w \Vdash_\bot^C$ — world is exploding at formula $C$

Strong forcing $w \Vdash_s A$ defined as before for atomic formulas

Non-strong forcing: $w \Vdash A$ if

$\forall C\, \forall w' \geq w \left( \forall w'' \geq w'\, w'' \Vdash_s A \to w'' \Vdash_\bot^C \right) \to w' \Vdash_\bot^C$

Extend strong forcing to all formulas:

$w \Vdash_s A \to B$ if for all $w' \geq w$ $w' \Vdash A$ implies $w' \Vdash B$
$w \Vdash_s <A, B>$ if $w \Vdash A$ AND $w \Vdash B$
...

Universal model for Kripke model with sums

K is the set of contexts again
Strong forcing: $\Gamma \Vdash_s A$ if there is a derivation in normal form $\Gamma \vdash^{nf} A$ , for atomic formula $A$
Exploding: $\Gamma \Vdash_\bot^C$ if there is a derivation in normal form $\Gamma \vdash^{nf} A$

Forcing:

$\forall C\, \forall \Gamma' \geq \Gamma \left( \forall \Gamma'' \geq \Gamma'\, \Gamma'' \Vdash_s A \to \Gamma'' \vdash^{nf} C \right) \to \Gamma' \vdash^{nf} C$

In Haskell:

data Sem
  = Sem ((StrongForcing -> Term) -> Term)

data StrongForcing
  = FInt Term
  | FPair Sem Sem
  | FSum (Either Sem Sem)
  | FFun (Sem -> Sem)

Reification

"unit": $\eta: w \Vdash_s A \to w \Vdash A$ , $\eta(a) = \lambda k.\, k a$

"bind": $* : (\forall w' \geq w\, w' \Vdash_s A \to w' \Vdash B) \to w \Vdash A \to w \Vdash B$ ,

$\phi*a = \lambda k.\, a\ (\lambda b.\, \phi\ b\ k)$

Forcing ( $w \Vdash A$ ) is kinda "Cont" monad storing strong forcing ( $w \Vdash_s A$ )

"run": $\mu : w \Vdash X \to w \Vdash_s X$ for atomic $X$ , by definition of $\Vdash$

Reification

reify_X (a) = $\mu(a)$ , $X$ — atomic

reify_(A->B) (f) = $\eta\left(\lambda s.\, \lambda x.\, \text{reify}_B (s (\text{reflect}_A x))\right)$

reify_(Either A B) (a) =

$\eta\left( \lambda s.\, (\text{Left}\ l \to \text{reify}_A l; \text{Right}\ r \to \text{reify}_B r)\right)$

reflect_X (a) = $\eta(a)$ , $X$ — atomic

reflect_(A->B) (f) = $\eta\left( \lambda x.\, \text{reflect}_B (f (\text{reify}_A x)) \right)$

reflect_(Either A B) (t) =

$\lambda k.\, \text{case}\ t\ \text{of}\ \Bigl[\text{Left}\ l \to k (\text{Left}\ (\text{reflect}_A l));$

$\text{Right}\ r \to k (\text{Right}\ (\text{reflect}_B r)) \Bigr]$

Modal logic

New type: $\Box A$

New terms:

quot t : $\Box A$ if $t : A$

let box u = t in e: A if $t : \Box A$ and $\Gamma; \Delta, u:A \vdash e : B$

Two contexts: $\Gamma; \Delta \vdash t : A$

exp :: Int -> [] (Int -> Int)
exp b =
  if zero b
    then box (\_ -> 1)
    else
      let box u = exp (b - 1)
      in box (\x -> x * (u x))
      
\x -> let box u = exp 3 in u x :: Int -> Int

Kripke CPS model

Strong forcing:

$\Gamma; \Delta \Vdash_s \text{box}\ A$ is a pair of $\Gamma; \emptyset \vdash t$ and $\Gamma; \Delta \Vdash A$

eval(box t) = $\eta( <\text{graft}\ t, \text{eval}\ t>)$

eval(let box u = t in e) =

$\lambda k.\, (\text{eval}\ t) (<t, s> \to (\text{eval}\ b) k)$

reify([] A)(t) = $t (<t, s> \to \text{box}\ t)$

reflect([] A)(t) =

$\lambda k.\, \text{let box}\ u = t\ \text{in}\ k <u, \text{reflect}_A u>$

NbE for lambda calculus with sums and for modal logic