Inference in FOL Compared to predicate logic, more abstract reasoning and specific conclusions

D. Goforth, COSC 4117, fall 2006 FOL knowledge bases  Facts about environment involve statements about specific objects E.g., Dentist(Bill), Likes(Mary, Candy)  General knowledge is mainly statements about sets of objects involving quantifiers E.g.,  x Dentist(x) ⇒ Likes(x, Candy)

D. Goforth, COSC 4117, fall 2006 Deductive reasoning from general to specific  how do quantified sentences get applied to facts? universal quantifier existential quantifier instantiation: substituting a reference to an object for a variable inference: conclusions entailed in KB

D. Goforth, COSC 4117, fall 2006 Instantiating Universal quantifier (UI)  x p(x)  statement is always true  any substitution makes a legitimate statement  format:  x p(x) subst( {x/k}, p(x) ) (K is any constant or function from KB) p(K)

D. Goforth, COSC 4117, fall 2006 Instantiating Existential quantifier (UI)  x p(x)  statement is true for some object  name the object for which it is true  format:  x p(x) subst( {x/k}, p(x) ) (k is a new constant, never used before Skolem constant) p(k)

D. Goforth, COSC 4117, fall 2006 Brute force reasoning  use instantiation to create a ‘propositional’ logic KB  complete BUT...  presence of functions causes infinitely large set of sentences (Father(Al), Father(Father(Al))  semi-decidable (disproofs never end)

D. Goforth, COSC 4117, fall 2006 Direct reasoning  x man(x)  mortal(x)  man(Socrates) 1.Substitute for instantiation: subst( {x/Socrates}, man(x)  mortal(x)) man(Socrates)  mortal(Socrates) 2.modus ponens mortal(Socrates)

D. Goforth, COSC 4117, fall 2006 Substitutions for reasoning  generalized modus ponens p 1, p 2, p 3, (p 1 ^ p 2 ^ p 3 )=> q subst( {x 1 /k 1, x 2 /k 2..}, q) Unification: substitutions so that the sentences are consistently instantiated

D. Goforth, COSC 4117, fall 2006 Substitutions for reasoning  generalized modus ponens example Parent(Art,Barb), Parent(Barb,Carl), (Parent(x,y) ^ Parent(y,z ) ⇒ Grandparent(x,z) subst( {x/Art, y/Barb,z/Carl}, q) (Parent(Art,Barb) ^ Parent(Barb,Carl ) ⇒ Grandparent(Art,Carl) Grandparent(Art,Carl)

D. Goforth, COSC 4117, fall 2006 Consistent substitutions  unification algorithm – p.278 or variant here example  x likes(Bill, x)(Bill likes everyone)  y likes(y, Mary) (everyone likes Mary) subst( {Bill/y, Mary/x}, likes(Bill, Mary)) makes two predicates identical

D. Goforth, COSC 4117, fall 2006 Application example  x likes(Bill, x)  y likes(y, Mary) => ~trusts(y,Father(Mary)) subst( {Bill/y, Mary/x}, likes(Bill, Mary)) makes two predicates identical likes(Bill, Mary), likes(Bill, Mary) => ~trusts(Bill,Father(Mary))  ~trusts(Bill,Father(Mary))

D. Goforth, COSC 4117, fall 2006 Examples  unify: Likes(x,Art), Likes(Father(y), y)  {Art/y} Likes(x,Art), Likes(Father(Art), Art)  {Art/y, Father(Art)/x}  unify: Likes(x,Art), Likes(Bart, x)  fails, can’t subst x for Art and Bart

D. Goforth, COSC 4117, fall 2006 Examples  unify: Likes(x,Art), Likes(Bart, x)  fails, can’t subst x for Art and Bart BUT where did ‘x’ come from?  Art likes everybody:  x Likes(x, Art)  Everybody likes Bart:  x Likes(Bart, x) standardize apart:  z 0 Likes(Bart, z 0 ) then Likes(Bart, Art) is OK with subst ( {Bart/x, Art/z 0 } )

Unify(L 1, L 2 ) // L 1, L 2 are both predicates or both objects 1.If (L 1 or L 2 is variable or constant) i.if (L 1 ==L 2 ) return {} (no subst required) ii.if (L 1 is variable) – if L 1 in L 2 return fail else return {L 2 /L 1 } iii.if (L 2 is variable) – if L 2 in L 1 return fail else return {L 1 /L 2 } iv.return fail // both constants or functions // L 1,L 2 are predicates if we get to here 2.If predicate symbols of L 1,L 2 not identical, return fail 3.If L 1,L 2 have different number of arguments, return fail 4.Subst = {} 5.For (i = 1 to number of arguments in L 1,L 2 ) i.S = Unify(L 1.argument[i],L 2.argument[i]) ii.if (S==fail) return fail iii.if (S!={}) apply S to remainder of L 1,L 2 Subst = Subst U S 6.Return Subst Unification algorithm

Unify(L 1, L 2 ) // L 1, L 2 are predicates or objects 1.If (L1 or L2 is variable or constant) i.if (L 1 ==L 2 ) Art, Art x,x ii.if (L 1 is variable) – if L 1 in L 2 return fail else return {L 2 /L 1 } x, Father(x) x, Mother(y) i.if (L 2 is variable) – if L 2 in L 1 return fail else return {L 1 /L 2 } ii.return fail Art, Bart // L 1,L 2 are predicates if we get to here 2.If predicates of L 1,L 2 not identical Likes(x,y) Brother(z,w) 3.If L 1,L 2 have different # of arguments Band(x,y,z), Band(t,v) 4.Subst = {} 5.For (i = 1 to # of args in L 1,L 2 ) i.S = Unify(L 1.arg[i],L 2.arg[i]) Likes(Bill,x) Likes(y,Father(y)) ii.if (S==fail) return fail iii.if (S!={}) apply S to remainder of L 1,L 2 Likes(Bill,x) Likes(Bill,Father(Bill)) Subst = Subst U S 6.Return Subst Unification algorithm - examples

D. Goforth, COSC 4117, fall 2006 Inference: Reasoning methods  Forward chaining  Backward chaining  Resolution

D. Goforth, COSC 4117, fall 2006 Resolution 1.convert sentences to equivalent conjunctive normal form (CNF) 2.apply resolution refutation