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On termination, an mgu for a unifiable partition can be found by comparing the terminal graph with the initial graph.
Once we can compute an mgu for any unifiable partition of a set of expressions (or show the partition not to be unifiable, if that is the case), we are ready to make inferences by resolution.
From [right arrow] P(G(r s) r s) and P(x y u)P(y z v)P(x v w) [right arrow] P(u z w) we infer P(r z r) [right arrow P(s z s) by a resolution in which M + {P(G(r s) r s)} and N + {P(x y u),P(x v w)}, since {M [union] N} is unifiable with mgu {x = G(r s), y = v + r, u + w + s}.
From P(x y u)P(y z v)P(x v w) [right arrow] P(u z w) and P(a b c)P(b d e)P(c d f) [right arrow] P(a e f) we infer P(x y a)P(y b v)P(y b v)P(x v c)P(b d e)P(c d f) [right arrow] P(a e f) by a resolution in which M = {P(u z w)} and N = {P(a b c)}, since {M [union] N} is unifiable with mgu {u = a, z = b, w = c}.
From two given clauses, only a finite number of clauses can be inferred by resolution--one for each choice of the 'cut' sets M and N for which the partition {M [union] N} is unifiable.
The unifiable p-part partition that is the essential ingredient of a hyperresolution is called its kernel.
from A [right arrow] B and C infer [right arrow]B[delta] if there is a cover of A [right arrow] B by C whose kernel is unifiable with mgu [delta].
If the kernel of the cover is unifiable with mgu [sigma], or it guarantees that we can easily relabel the tree so it turns into a hyperresolution deduction, from these clauses as premises, of the same unconditional clause [right arrow] B [sigma] that the ultraresolution inference obtains directly from them in one step.
The thick lines in Figure 9 show the pairs of the unifiable kernel of the cover.