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

=== We sketch a small-step operational semantics for an SDELang fragment with Obatala effects. Terms: ===

<syntaxhighlight>t ::= x | λx.t | t t | reflect t | bind x ← t in u | merge(t,t) | sim t | ... 

</syntaxhighlight>

Typing (informal): if Γ ⊢ t : A then Γ ⊢ reflect t : O A. Bind types: if Γ ⊢ t : O A and Γ, x:A ⊢ u : O B then Γ ⊢ (bind x ← t in u) : O B.

Small-step rules (selected):
# Reflect intro (value): reflect v is a value of type O A if v is a value.
# Bind-reflect: </code>`<code> bind x ← (reflect v) in u  -->  u[x := v] </code>`<code> (left unit law operationalized)
# Bind-join (flatten): </code>`<code> bind x ← (bind y ← m in n) in u  -->  bind y ← m in (bind x ← n in u) </code>`<code> (associativity operationalization)
# Flatten rule (join): </code>`<code> flatten (reflect v) --> reflect v flatten (flatten m) --> flatten m  (idempotence in particular contexts) <syntaxhighlight>5. Simulate: </syntaxhighlight> sim (reflect v) --> sim_leaf(v)  -- base simulation of single agent sim (merge m n) --> combine_sim(sim m, sim n) </code>`<code> combine_sim depends on organization coordination semantics.

Handler semantics: Handler α : O A → A reduces bind x ← t in α by interpreting t via α.

This operational semantics respects the monad laws (subject to congruence and beta rules).