Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

=== Below is a minimal SDELang fragment showing Obatala constructs and a compact small-step semantics, then a compiler lowering into an intermediate representation (IR) that uses the Tree monad. ===

SDELang grammar (subset):

<syntaxhighlight>t ::= x
    | λx. t
    | t t
    | reflect t               -- promote: A -> O A
    | bind x <- t in u        -- monadic bind
    | merge t t               -- join two O A
    | sim t                   -- produce trajectory
    | leaf t                  -- syntactic sugar for reflect t

</syntaxhighlight>

Typing rules (informal):
* Γ ⊢ t : A ⇒ Γ ⊢ reflect t : O A.
* Γ ⊢ t : O A and Γ, x:A ⊢ u : O B ⇒ Γ ⊢ bind x <- t in u : O B.

Small-step semantics (selected, arrow →):
# bind x <- reflect v in u  →  u[x := v]  (left unit)
# bind x <- (bind y <- m in n) in u  →  bind y <- m in (bind x <- n in u)  (associativity / let-assoc)
# merge (reflect v) (reflect w)  →  reflect (merge_agent v w) (interpret merge at runtime or keep as Node)
# sim (reflect v) → base_sim(v) (simulate a leaf)
# sim (merge m n) → combine_sim(sim m, sim n) (organizational simulation)

Operational strategy:
* Evaluate inside-out (call-by-value for reflect). reflect v is a value when v is a value.
* bind reduces when the left side is a reflect or already reduced to a tree node; otherwise evaluate left expression.

Compiler lowering (SDELang → IR using Tree):
* reflect t  → Leaf(t') where t' is lowered expression
* merge a b  → Node(MergeOp, [a', b'])
* sim t      → Node(SimOp, [t'])  or produce a Traj call to a runtime sim function
* bind x <- t in u → lower t to a Tree T; then produce IR representing join (map (fun leaf -> lower(u) with x := leaf) T) — equivalently, join . fmap as in Coq.

Example compiler fragment (pseudo-code):

<syntaxhighlight lang="ocaml">('' Lowering: Expr -> Tree IR '')
let rec lower (e : expr) : tree_ir =
  match e with
  | Var x -> Leaf (IR_Var x)
  | Lam (x,b) -> Leaf (IR_Closure (x,lower b))
  | App (f,a) -> Leaf (IR_App (lower f, lower a))
  | Reflect t -> Leaf (IR_Value (lower t))  ('' leaf wraps the value '')
  | Merge (a,b) -> Node (MergeOp, [lower a; lower b])
  | Sim t -> Node (SimOp, [lower t])
  | Bind (x,t,u) ->
      let t' = lower t in
      ('' produce join (fmap (\leaf -> lower(u) [x:=leaf]) t') '')
      join (fmap (fun leaf => lower (subst u x (reify_leaf leaf))) t')

</syntaxhighlight>

reify_leaf extracts the payload from Leaf to substitute into u. At runtime, the Tree can be interpreted by EM-algebras (handlers).

Runtime execution model:
* The runtime holds: - Interpreter service that collapses Tree via an EM algebra (organization handler) — this is the implementation of α : Tree A → A. - Simulator service that given Node SimOp ... produces a Trajectory. - Coordination service that implements MergeOp semantics (e.g., conflict resolution, scheduling, resource allocation).