9 unicaon example xknows john x loves john x knows

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: ersonlovedby ( x )) € 7 Inference Rules •  Existen<al Introduc<on α (g) SUBST ({v / g}, ∃v : α (v)) •  How to read this: € –  We know that the sentence α is true –  Can subs<tute variable v for any constant g in α and (w/existen<al quan<fier) and α will s<ll be true –  Why is this OK? Generalized Modus Ponens Example •  If has_US_birth_cer<ficate(X) then natural_US_ci<zen(X) •  has_US_birth_cer<ficate(Obama) •  Conclude SUBST({Obama/X},natural_US_ci<zen(X)) •  i.e., natural_US_ci<zen(Obama) 8 Generalized Modus Ponens SUBST (θ , pi ' ) = SUBST (θ , pi )∀i p1 ', p2 ',… pn ', ( p1 ∧ p2 ∧ … ∧ pn ⇒ q) SUBST (θ , q) € •  How to read this: € –  We have an implica<on which implies q –  Any consistent subs<tu<on of variables on the LHS must yield a valid conclusion on the RHS Unifica<on •  Subs<tu<on is a non ­trivial maMer •  We need an algorithm unify: Unify( p, q) = θ : Subst(θ , p) = Subst(θ , q) •  Important: Unifica<on replaces variables: ∃xLoves( John, x ) ∃xHates( John, x ) € •  Are these the same x? € 9 Unifica<on Example ∀xKnows( John, x ) ⇒ Loves( John, x ) Knows( John, Jane) ∀yKnows( y, Leonid ) ∀yKnows( y, Mother ( y )) ∀xKnows( x, Elizabeth) Note: All unquan<fied variables are assumed universal from here on. € Unify(Knows( John, x ), Knows( John, Jane)) = Unify(Knows( John, x ), Knows( y, Leonid )) = Unify(Knows( John, x ), Knows( y, Mother ( y ))) = € Unify(Knows( John, x ), Knows( x, Elizabeth)) = € {x / Jane} {x / Leonid, y / John} {y / John, x / Mother ( John)} {x1 / Elizabeth, x2 / John} € € € Most General Unifier •  Unify(Knows(John,x),Knows(y,z)) –  {y/John,x/z} –  {y/John,x/z,w/Freda} –  {y/John,x/John,z/John) •  When in doubt, we should always return the most general unifier (MGU) –  MGU...
View Full Document

Ask a homework question - tutors are online