diff --git a/bn.rkt b/bn.rkt
index df807cc..5e523e3 100644
--- a/bn.rkt
+++ b/bn.rkt
@@ -26,9 +26,10 @@
 
 ;;; Given a state s updates all the variables from xs.  This
 ;;; corresponds to a parallel mode.
-(define (update [bn : Network]            ; the Boolean network
-                [s : State]               ; the state to operate on
-                [xs : (Listof Variable)]) ; the variables to update
+(: update (-> Network State (Listof Variable) State))
+(define (update bn  ; the Boolean network
+                s   ; the state to operate on
+                xs) ; the variables to update
   (let ([new-s : State (hash-copy s)])
     (for ([x xs])
       (let ([f (hash-ref bn x)])
@@ -36,9 +37,11 @@
     new-s))
 
 ;;; A version of make-hash restricted to creating Boolean states.
-(define (make-state [mappings : (Listof (Pairof Variable Boolean))])
+(: make-state (-> (Listof (Pairof Variable Boolean)) State))
+(define (make-state mappings)
   (make-hash mappings))
 
 ;;; A version of make-hash restricted to creating Boolean networks.
-(define (make-bn [funcs : (Listof (Pairof Variable UpdateFunc))])
+(: make-bn (-> (Listof (Pairof Variable UpdateFunc)) Network))
+(define (make-bn funcs)
   (make-hash funcs))