There is no royal road... we have to spell out more details:
1. Consequence operator can be broken down to single steps:
Cn(F) = lim
n →∞
,
Stⁿ(F)
2. Each single step is the set union of the results of individual rules r ∈
R:
St(F) = ⋃ r(F)
3. A single rule is of the form:
head → tail
and it performs, for all facts f ∈ F:
r(F) = tail * { sub | head * sub = f }
where "x * sub" means substituting sub into formula x. The { | } is set
comprehension.
The last equation says to find all subs such that head * sub equals a fact
∈ F. That is unification.
What remains to do is to find a spatial interpretation of the act of
substitution, *. That depends on how to represent "formulas with holes",
which is predicate logic. For example, "father(X,Y)" is a formula with 2
holes.
If we can represent such "predicated" (or "quantified") formulas in space,
then we can spatialize substitution etc...
More later =)
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AGI
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