Yes, how about m_{X} : (A × (A × X)^{A})^{A} → (A × X)^{A}?
We have another collection of state machines, m_{X} : (A × (A × X)^{A})^{A}, which has A as a state set, and which outputs to another collection of state machines (this one too has A as a state set, and X as output set).How do we find for such a compound state machine a match in a state machine on A?
Write it in Lisp:
(define (mx f)
(let (tr1 out1)
((car f)(cadr f)));two components of f:A → (A × (A × X)^{A}
(lambda (a)
(let a1 (tr1 a));state after first transition
(let f2 (out1 a)) ;mapped a state to another state machine
(let (tr2 out2) ((car f2) (cadr f2))) ;second machine components
(list (tr2 a1) (out2 a1))))
See what happens here: we have a function from A to A×(A×X)^{A}, which consists of transition A → A and output A → (A×X)^{A}, that is, for each a we have another state a1 and an output function; the resulting function from A to A×X should just apply that output function to the new state.
Boring Exercise: Prove that this is actually a monad. |