# hw4-soln - CISC 404/604 Homework 4 Solutions 1a. To show z...

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CISC 404/604 Homework 4 Solutions 1a. To show z xP ( x,z,f ( x,z )) ⇒ ∃ z x yP ( x,z,y ) is valid using resolution. .. Negate: ¬ ( z xP ( x,z,f ( x,z )) ⇒ ∃ z x yP ( x,z,y )) Convert to prenex: z 1 z 2 x 2 y x 1 ¬ ( P ( x 1 ,z 1 ,f ( x 1 ,z 1 )) P ( x 2 ,z 2 ,y )) Skolemize: z 2 y x 1 ¬ ( P ( x 1 ,a,f ( x 1 ,a )) P ( g ( z 2 ) ,z 2 ,y )) Drop quantiﬁers: ¬ ( P ( x 1 ,a,f ( x 1 ,a )) P ( g ( z 2 ) ,z 2 ,y )) Convert to CNF: { [ P ( x,a,f ( x,a ))] , [ ¬ P ( g ( z ) ,z,y )] } Resolution/Uniﬁcation: L = { [ P ( x,a,f ( x,a ))] , [ ¬ P ( g ( z ) ,z,y )] } sub = [ x | g ( z )] L sub = { [ P ( g ( z ) ,a,f ( g ( a ) ,a ))] , [ ¬ P ( g ( z ) ,z,y )] } sub = [ x | g ( z )] , [ z | a ] L sub = { [ P ( g ( a ) ,a,f ( g ( a ) ,a ))] , [ ¬ P ( g ( a ) ,a,y )] } sub = [ x | g ( z )] , [ z | a ] , [ y | f ( g ( a ) ,a )] L sub = { [ P ( g ( a ) ,a,f ( g ( a ) ,a ))] , [ ¬ P ( g ( a ) ,a,f ( g ( a ) ,a ))] } [ ] (by resolution)

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hw4-soln - CISC 404/604 Homework 4 Solutions 1a. To show z...

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