Completeness

• Is there a sound and complete inference procedure for FOPL? Yes

• Shown to be true in Godel's completeness theorem. There are inference rules that allow a complete proof procedure.
  • Don’t confuse this with Godel's incompleteness theorem discussed on page 288

• The completeness theorem says that there is an inference procedure that will return true if a sentence is true. What if the sentence is false?

Semidecidable

• The completeness theorem does not guarantee that the inference procedure will return false if the sentence is false. There is no inference procedure guaranteed to do that. FOPL is only semidecidable (related to the Turing Machine halting problem).

Resolution-Refutation Proofs

• Resolution is sound and refutation is complete; if a sentence is unsatisfiable, resolution will derive a contradiction (proof is in the text).

• Resolution can be used to establish that a sentence is entailed by the KB, but cannot be used to generate all logical consequences of a set of WFF (the KB).

Overview of resolution refutation proof strategy

Resolution won't always yield an answer

• Entailment is only semidecidable; example
  KB: brother(carl,deb), mother(deb,dave),
  mother(sue,deb), father(dick,jon),
  male(dave), male(dick), male(jon)
  brother(x,y) ^ mother(y,z) -> uncle(x,z)
  brother(x,y) -> male(x)
  mother(x,y) -> female(x)
  male(x) -> ~female(x)
  can prove uncle(dick,dave)?