Inference in rst order logic is semidecidable compare

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online