Problems with implication-scope-to-implication-rule

[Roman Treutlein]

I have been thinking about ImplicationScope to Implication conversion (and the reverse) and found some issues.

Let's say we have:

(ImplicationScopeLink (stv 1.0 1.0)
  (VariableList
    (VariableNode "$A")
    (VariableNode "$B"))
  (EvaluationLink
    (PredicateNode "does")
    (ListLink
      (VariableNode "$A")
      (VariableNode "$B")))
  (EvaluationLink
    (PredicateNode "isbeingdone")
    (VariableNode "$B")))

(ImplicationScopeLink (stv 1.0 1.0)
  (VariableList
    (VariableNode "$A"))
  (EvaluationLink
    (PredicateNode "isbeingdone")
    (VariableNode "$A"))
  (EvaluationLink
    (PredicateNode "willbedone")
    (VariableNode "$A")))

It seems reasonable to think that we can infer the following:

(ImplicationScopeLink (stv 1.0 1.0)
  (VariableList
    (VariableNode "$A")
    (VariableNode "$B"))
  (EvaluationLink
    (PredicateNode "does")
    (ListLink
      (VariableNode "$A")
      (VariableNode "$B")))
  (EvaluationLink
    (PredicateNode "willbedone")
    (VariableNode "$B")))

But atm this does not work. Let's try to do it and see where we fail. First we will have to use the implication-scope-to-implication-rule as the deduction-implication-rule we want to use only works on normal ImplicationLinks. This results in the following:

   (ImplicationLink (stv 1 1)
      (LambdaLink
         (VariableList
            (VariableNode "$A")
            (VariableNode "$B"))
         (EvaluationLink
            (PredicateNode "does")
            (ListLink
               (VariableNode "$A")
               (VariableNode "$B"))))
      (LambdaLink
         (VariableList
            (VariableNode "$A")
            (VariableNode "$B"))
         (EvaluationLink
            (PredicateNode "isbeingdone")
            (VariableNode "$B"))))

   (ImplicationLink (stv 1 1)
      (LambdaLink
         (VariableList
            (VariableNode "$A"))
         (EvaluationLink
            (PredicateNode "isbeingdone")
            (VariableNode "$A")))
      (LambdaLink
         (VariableList
            (VariableNode "$A"))
         (EvaluationLink
            (PredicateNode "willbedone")
            (VariableNode "$A"))))

And here we can see the issue. Namely that the LambdaLink wrapping the "isbeingdone" Predicate has 2 variables in one case and 1 in the other which stops the decuction-implication-rule from doing anything.
And if I understand this correctly can't change the implication-scope-to-implication-rule to just drop the extra Variable as that would break the equivalence between ImplicationScopeLink and the ImplicationLink.
So i think the correct solution would be to create a decuction-implication-scope-rule that can handle ImplicationScopeLinks by doing unification to get the mappings between variables and then using alpha-conversion to get 2 Links where the Variables Names line up correctly and then doing normal deduction.

[Nil Geisweiller]

I can see 3 ways

1. Introduce deduction-implication-scope-rule as you suggest
2. Add a rule that add extra variables to the declaration of the lambda (so that the deduction-implication-rule can then work).
3. Hack deduction-implication-rule to implicitly add missing variables when necessary.

I don't know which way is better, but I suppose it doesn't hurt to offer scoped variations of deduction rules, experiment and see, etc.

[Nil Geisweiller]

OK, I remember why the deduction rule is non-scoped, that is because the formula requires the probability of the predicates, and a mere ImplicationScope doesn't give that. However the correct thing to do is to add A, B and C as premises of the deduction rule, do you see what I mean?

[Roman Treutlein]

I am going to try to implement deduction-impliction-scope-rule with A,B and C as premises.

[Linas Vepštas]

Please excuse my random remarks. The bug report looks very valid. The intermdiate forms look broken:

 (ImplicationLink (stv 1 1)
      (LambdaLink
           (VariableList (VariableNode "$A")   (VariableNode "$B"))
           stuff)
      (LambdaLink
          (VariableList (VariableNode "$A")   (VariableNode "$B"))
           other-stuff)
)

There's nothing to connect the variables in the first clause to the variables in the second clause, except for an accidental name-collision. The above is the same as

 (ImplicationLink (stv 1 1)
      (LambdaLink
           (VariableList (VariableNode "$A")   (VariableNode "$B"))
           stuff)
      (LambdaLink
          (VariableList (VariableNode "$X")   (VariableNode "$Y"))
           other-stuff)
)

and who's to say that X and Y should be A and B? How could one ever know?

I think the rule that you want would be

(Bind
    (TypedVariable (Variable "$P") (Type 'PredicateNode))
    (ImplicationScopeLink 
        (VariableList (VariableNode "$A") (VariableNode "$B"))
        (EvaluationLink
              (PredicateNode "does")
              (ListLink    (VariableNode "$A") (VariableNode "$B")))
        (EvaluationLink
               (Variable "$P")
                (VariableNode "$B")))

        (ImplicationScopeLink
              (VariableList  (VariableNode "$X"))
             (EvaluationLink
                   (Variable "$P")
                  (VariableNode "$X"))
             (EvaluationLink
                   (PredicateNode "willbedone")
                   (VariableNode "$X")))

Note that $A, $B and $X are unrelated (and bound) while $P will be pattern matched to (PredicateNode "isbeingdone")

Now I really do not understand the internals of the backward-chainer, but above seems like a reasonable kind of rule ... yes?

[Nil Geisweiller]

The scoped implication deduction is reasonable, I agree, just want to point out that the non-scope implication rule using lambdas is also reasonable.

[Roman Treutlein]

(ImplicationLink (stv 1 1)
      (LambdaLink
           (VariableList (VariableNode "$A")   (VariableNode "$B"))
           stuff)
      (LambdaLink
          (VariableList (VariableNode "$X")   (VariableNode "$Y"))
           other-stuff)
)

and who's to say that X and Y should be A and B? How could one ever know?

I think what connects A and B with X and Y is the order. But i agree that this is a very flimsy representation.