General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

A ∨ BT AT
BT A ⇒ BT AF
BT A ∨ B F
A F
B F
A ⇒ B F
A T
B F
A ⇔ BT AT BT
AF
BF A T
A ⇒ B T
B T
A ⇔ BF AT BF
AF
BT c○: Michael Kohlhase 205
A T
A ⇒ B T
¬A ∨ B T
¬(¬¬A ∧ ¬B) T
¬¬A ∧ ¬B F
¬¬A F
¬A T
A F
⊥
With these derived rules, theorem proving becomes quite efficient. With these rules, the tableau
(??) would have the following simpler form:
Tableaux with derived Rules (example)
Example 343
love(mary, bill) ∧ love(john, mary) ⇒ love(john, mary) F
love(mary, bill) ∧ love(john, mary) T
love(john, mary) F
love(mary, bill) T
love(john, mary) T
⊥
c○: Michael Kohlhase 206
Another thing that was awkward in (??) was that we used a proof for an implication to prove
logical consequence. Such tests are necessary for instance, if we want to check consistency or
informativity of new sentences 11 . Consider for instance a discourse ∆ = D 1 , . . . , D n , where n is EdNote:11
large. To test whether a hypothesis H is a consequence of ∆ (∆ |= H) we need to show that
C := (D 1 ∧ . . .) ∧ D n ⇒ H is valid, which is quite tedious, since C is a rather large formula, e.g.
if ∆ is a 300 page novel. Moreover, if we want to test entailment of the form (∆ |= H) often, –
for instance to test the informativity and consistency of every new sentence H, then successive
∆s will overlap quite significantly, and we will be doing the same inferences all over again; the
entailment check is not incremental.
Fortunately, it is very simple to get an incremental procedure for entailment checking in the
model-generation-based setting: To test whether ∆ |= H, where we have interpreted ∆ in a model
generation tableau T , just check whether the tableau closes, if we add ¬H to the open branches.
Indeed, if the tableau closes, then ∆ ∧ ¬H is unsatisfiable, so ¬((∆ ∧ ¬H)) is valid 12 , but this is EdNote:12
equivalent to ∆ ⇒ H, which is what we wanted to show.
¬B F
B T
Example 344 Consider for instance the following entailment in natural language.
Mary loves Bill. John loves Mary |= John loves Mary
13 We obtain the tableau EdNote:13
love(mary, bill) T
love(john, mary) T
¬(love(john, mary)) T
love(john, mary) F
⊥
11 EdNote: add reference to presupposition stuff
12 EdNote: Fix precedence of negation
13 EdNote: need to mark up the embedding of NL strings into Math
110