How to delay reduction until bindings are ready?

[ngeiswei]

Describe the bug

The backward chainer requires the interpreter to be able to delay reduction under certain conditions, such as the arguments are fully grounded.

To Reproduce

Run the following MeTTa script

;; Define Nat
(: Nat Type)
(: Z Nat)
(: S (-> Nat Nat))

;; Define greater than zero
(: 0< (-> Number Bool))
(= (0< $x) (< 0 $x))

;; Define backward chainer
(: bc (-> $a                            ; Knowledge base space
          $b                            ; Query
          Nat                           ; Maximum depth
          $b))                          ; Result
;; Base cases
(= (bc $kb (: $prf $ccln) $_) ; Kownledge base look-up
   (match $kb (: $prf $ccln) (: $prf $ccln)))
(= (bc $kb (: CPU (0⍃ $x)) $_) ; CPU check
   (if (0< $x) (: CPU (0⍃ $x)) (empty)))
;; Recursive step
(= (bc $kb (: ($prfabs $prfarg) $ccln) (S $k))
   (let* (((: $prfabs (-> (: $prfarg $prms) $ccln)) ; Recurse on proof abstraction
           (bc $kb (: $prfabs (-> (: $prfarg $prms) $ccln)) $k))
          ((: $prfarg $prms) ; Recurse on proof argument
           (bc $kb (: $prfarg $prms) $k)))
     (: ($prfabs $prfarg) $ccln)))

;; Define knowledge base
!(bind! &kb (new-space))
!(add-atom &kb (: 2 Prime)) ; 2 is a prime number
!(add-atom &kb (: Rule (-> (: $_ (0⍃ $x))   ; If $x is greater than 0
                           (-> (: $x Prime) ; and is a prime number, then
                               (0⍃' $x))))) ; $x is a prime number greater than zero

;; Test backward chainer
!(bc &kb (: $prf (0⍃' $x)) (S (S Z)))

I tried to provide the simplest possible knowledge base, thus the moronic theory used here.

Expected behavior

It should output

[(: ((Rule CPU) 2) (0⍃' 2))]

Actual behavior

It outputs instead

[]

Additional context

• MeTTa version: 0.1.8.dev36+g0e4b76e.d20240411, compiled with "minimal".

• Without "minimal", the output is (let* (((: $prfabs#34 (-> (: $prfarg#35 $prms#38) (0⍃' $x))) (bc GroundingSpace-0x2b472b8 (: $prfabs#34 (-> (: $prfarg#35 $prms#38) (0⍃' $x))) (S Z))) ((: $prfarg#35 $prms#38) (bc GroundingSpace-0x2b472b8 (: $prfarg#35 $prms#38) (S Z)))) (: ($prfabs#34 $prfarg#35) (0⍃' $x))).

  • It should be noted that if the query is replaced by !(bc &kb (: $prf (0⍃' 2)) (S (S Z))), meaning that $x is already grounded, then the backward chainer is able to find the proof.
  • It should also be noted that if the order of the premises is swapped, as in

    !(add-atom &kb (: Rule (-> (: $x Prime)       ; If $x is a prime number
                         (-> (: $_ (0⍃ $x)) ; and is greater than 0, then
                             (0⍃' $x)))))   ; $x is a prime number greater than zero

    then it also find the proof. Unfortunately this ordering trick can not always be used for more intricate theories.

  • Additionally I tried to wrap a conditional using is-closed in the body of 0<, as follows

    
    ;; Return True iff $x is closed
    (: is-closed (-> Atom Bool))
    (= (is-closed $x) (case (get-metatype $x)
                  ((Symbol True)
                   (Grounded True)
                   (Variable False)
                   (Expression (if (== $x ())
                                   True
                                   (and (let $head (car-atom $x) (is-closed $head))
                                        (let $tail (cdr-atom $x) (is-closed $tail))))))))

  ;; Greater than zero (: 0< (-> Number Bool)) (= (0< $x) (if (is-closed $x) (< 0 $x) (empty)))

  
  but without success.  I suppose a clever use of `NotReducible` may help but I did not succeed so far.