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# Inference in rst order logic is semidecidable compare

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Unformatted text preview: B,[q],{}) Func<on Back_Chain_List(KB,qlist,q) If qlist= then return q q< ­head(qlist) For each qi’ in KB such that qi< ­Unify(q,qi’) succeeds do Answers < ­ Answers + [q,qi] For each (pi^…^pn=>qi’)in KB: qi< ­Unify(q,qi’) succeeds do Answers< ­ Answers+ Back_Chain_List(KB,Subst(qi,[pi…pn]),[q,qi])) return union of Back_Chain_List(KB,Tail(qlist),q) for each q in answers 13 Completeness ∀xP( X ) ⇒ Q( x ) ∀x¬P( X ) ⇒ R( x ) ∀xQ(x ) ⇒ S ( x ) ∀xR( x ) ⇒ S ( x ) S ( A) ??? •  Problem: Generalized Modus Ponens not complete €•  Goal: A sound and complete inference procedure for ﬁrst order logic Generalized Resolu<on θ = Unify( p j, ¬qk ) ( p1 ∨ … p j … ∨ pm ), (q1 ∨ … qk … ∨ qn ) SUBST (θ , ( p1 ∨ … p j−1 ∨ p j+1 … ∨ pm ∨ q1 ∨ … qk −1 ∨ qk +1 … ∨ qn )) € € •  If the same term appears in both posi<ve and nega<ve form in two disjunc<ons, they cancel out when disjunc<ons are combined 14 Generalized Resolu<on Example (¬P( x ) ∨ Q( x )) (P( x ) ∨ R( x )) (¬Q(x ) ∨ S ( x )) (¬R(x ) ∨ S ( x )) S ( A) ??? Example on board/tablet… € Resolu<on Proper<es •  Proof by refuta<on (asser<ng nega<on and resolving to nil) is sound and complete (NB: We did not do this in the previous example) •  Resolu<on is not complete in a genera<ve sense, only in a tes<ng sense •  This is only part of the job •  To use resolu<on, we must convert everything to a canonical form 15 Canonical Form •  •  •  •  •  •  •  •  Eliminate Implica<ons Move nega<on inwards Standardize (apart) variables Move quan<ﬁers Le_ Skolemize Drop universal quan<ﬁers Distribute AND over OR FlaMen nested conjunc<ons and disjunc<ons Computa<onal Proper<es •  We can enumerate the set of all proofs •  We can check if a proof is valid •  First order logic is complete (Gödel) •  What if no valid proof exists? •  Inference in ﬁrst order logic is semi ­decidable •  Compare with hal<ng problem (hal<ng problem is semi ­decidable) 16 Gödel’s Incompleteness Result •  Gödel’s incompleteness result is, perhaps, beM...
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