Artificial Intelligence
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1 Artificial Intelligence SLD Resolution [1]
2 Back to Propositional Logic SLD Resolution [2]
3 Horn Clauses (in L P ) wff {B, D, A, C}, {A, B} (D A C) B, B A {B}, {A} {B}, {A, B} SLD Resolution [3]
4 Lost in Translation wff (A B) C (A B) C A B C A (B C) A (B C) (A B) (A C) (A B), (A C) (A B) C (A B) C (A B) C (A C) (B C) (A C), (B C) (A B) C (A B) C A B C A (B C) A B C SLD Resolution [4]
5 SLD Resolution { } {A} {A, B, C} {B, C} {B, D} {A} {C}, {D}, {B, D}, {A, B, C} {D, C} {C} {D} {C} { } SLD Resolution [5]
6 SLD trees {A} {C}, {D}, {B, D}, {A, B, C} {A} Sequence {B, C} {D, C} {C} {C}, {D}, {B, F}, {B, E}, {B, D}, {A, B, C} {A} { } {A} {B, C} {F, C} {E, C} {D, C} {C} { } SLD Resolution [6]
7 SLD Resolution L P SLD Resolution [7]
8 SLD resolution in First-Order Logic SLD Resolution [8]
9 Horn Clauses in L FO {Greek(socrates)}, {Pyramid(x)}, {Sister(sally, motherof(paul))} {Human(x), Greek(x)}, x (Greek(x) Human(x)) {Female(x), Parent(k(x),x), Parent(k(y),y), Sister(x,y)} xy ((Female(x) z (Parent(z,x) Parent(z,y))) Sister(x,y)) {Above(x,y), On(x k(x))}, {Above(x,y), On(j(y),y)} xy (Above(x,y) (z On(x,z) v On(v,y))) {Mortal(socrates)} {Sister(sally,x), Sister(x,paul)} x (Sister(sally,x) Sister(x,paul)) SLD Resolution [9]
10 SLD Resolution in L FO 1 2 std( ) std( ) std( i )[] [] std( i ) SLD Resolution [10]
11 SLD Resolution in L FO SLD Resolution [11]
12 SLD Trees {{Human(x), Greek(x)}, {Mortal(y), Human(y)}, {Greek(socrates)}, {Greek(plato)}, {Greek(aristotle)}} goal {Mortal(x), Human(x)} 1: {Mortal(x)} [] {Mortal(x)}, {Mortal(y 1 ), Human(y 1 ),} [] 2: {Human(y 1 )} [x/y 1 ] {Human(y 1 )}, {Human(x 1 ), Greek(x 1 )} [x/y 1 ] 3: {Greek(x 1 )} [x/y 1 ][y 1 /x 1 ] {Greek(x 1 )} {Greek(socrates)} [x/y 1 ][y 1 /x 1 ] 4: {} [x/y 1 ][y 1 /x 1 ][x 1 /socrates] SLD Resolution [12]
13 SLD Trees {{Human(x), Greek(x)}, {Mortal(y), Human(y)}, {Greek(socrates)}, {Greek(plato)}, {Greek(aristotle)}} goal {Mortal(x), Human(x)} {Greek(x 1 )} {Greek(socrates)} [x/y 1 ][y 1 /x 1 ] 1: {Mortal(x)} [] {Mortal(x)}, {Mortal(y 1 ), Human(y 1 ),} [] 2: {Human(y 1 )} [x/y 1 ] {Human(y 1 )}, {Human(x 1 ), Greek(x 1 )} [x/y 1 ] 3: {Greek(x 1 )} [x/y 1 ][y 1 /x 1 ] {Greek(x 1 )} {Greek(plato)} [x/y 1 ][y 1 /x 1 ] {Greek(x 1 )} {Greek(aristotle)} [x/y 1 ] [y 1 /x 1 ] 4: {} [x/y 1 ][y 1 /x 1 ][x 1 /socrates] 5: {} [x/y 1 ][y 1 /x 1 ][x 1 /plato] 6: {} [x/y 1 ][y 1 /x 1 ][x 1 /aristotle] SLD Resolution [13]
14 SLD Trees {{Mortal(felix), Cat(felix)}, {Human(x), Greek(x)}, {Mortal(y), Human(y)}, {Greek(socrates)}, {Greek(plato)}, {Greek(aristotle)}} goal {Mortal(x), Human(x)} 1: {Mortal(x)} [] {Mortal(x)}, {Mortal(felix), Cat(felix)} [] 2: Cat(felix) [x/felix] goal 2: cannot be resolved {Mortal(x)}, {Mortal(y 1 ), Human(y 1 ),} [] 3: {Human(y 1 )} [x/y 1 ] {Human(y 1 )}, {Human(x 1 ), Greek(x 1 )} [x/y 1 ] 4: {Greek(x 1 )} [x/y 1 ][y 1 /x 1 ] {Greek(x 1 )} {Greek(socrates)} [x/y 1 ][y 1 /x 1 ] {} [x/y 1 ][y 1 /x 1 ][x 1 /socrates] SLD Resolution [14]
15 *The discreet charme of functions [a, b, c, ] cons/2 a [b, c] cons(a, cons(b, cons(c, nil))) [a, b, c] Append/3 x y z nil. AL x Append(nil, x, x) x y z (Append(x, y, z) s Append(cons(s, x), y, cons(s, z))) {AL + z Append(cons(a, nil), cons(b, cons(c, nil), z) } Append(cons(a, nil), cons(b, cons(c, nil)), cons(a, cons(b, cons(c, nil)))) {AL + x y Append(x, y, cons(a, cons(b, nil)))} Append(cons(a, nil), cons(b, nil), cons(a, cons(b, nil))) Append(nil, cons(a, cons(b, nil)), cons(a, cons(b, nil))) Append(cons(a, cons(b, nil)),nil, cons(a, cons(b, nil))) SLD Resolution [15]
16 The world of lists [a, b, c, ] cons/2 a [b, c] cons(a,cons(b,cons(c,nil))) [a, b, c] Append/3 x y z nil. [] nil [a] cons(a,nil) [a,b] cons(a,cons(b,nil)) [a [b,c]] cons(a,[b,c]) AL x Append(nil,x,x) xyz (Append(x,y,z) s Append([s,x],y,[s,z])) SLD Resolution [16]
17 The world of lists x Append(nil, x, x) y x Append(nil, cons(y, x), cons(a, x)) x Append(nil, x, x), y x Append(nil, cons(y, x), cons(a, x)) x Append(nil, x, x), y x Append(nil, cons(y, x), cons(a, x)) {Append(nil, x, x)}, {Append(nil, cons(y, k(y)), cons(a, k(y)))} k/1 Append(nil, x, x), Append(nil, cons(y, k(y)), cons(a, k(y))))... Append/3 = [x/cons(a, k(a)), y/a] {Append(nil, cons(a,k(a)), cons(a,k(a)))}, {Append(nil, cons(a, k(a)), cons(a, k(a)))} SLD Resolution [17]
18 The world of lists in Prolog % Identical to built-in predicate append/3, although it uses "cons" % as a defined predicate, thus allowing trace-ability. append(cons(s,x),y,cons(s,z)) :- append(x,y,z). append(nil,x,x). % WARNING: express your queries with cons. Examples: %?- append(cons(a,nil), cons(b,cons(c, nil)),cons(a,cons(b,cons(c, nil)))). %?- append(x,y,cons(a,cons(b,cons(c, nil)))). SLD Resolution [18]
19 Infinite SLD Trees (fairness of SLD) {{S(a,b)}, {S(b,c)}, {S(x,z), S(x,y), S(y,z)}} {S(a,x)} goal: S(a,x) [] {S(a,x)}, {S(a,b)} [] {} [x/b] SLD Resolution [19]
20 Infinite SLD Trees (fairness of SLD) {{S(a,b)}, {S(b,c)}, {S(x,z), S(x,y), S(y,z)}} {S(a,x)} goal: S(a,x) [] {S(a,x)}, {S(a,b)} [] {} [x/b] {S(a,x)}, {S(x 3,z 3 ), S(x 3,y 3 ), S(y 3,z 3 )} [] {S(a,y 3 ), S(y 3,z 3 )} [x 3 /a, x/z 3 ] {S(a,y 3 ), S(y 3,z 3 )}, {S(a,b)} [x/z 3, x 3 /a] {S(b,z 3 )} [x/z 3, x 3 /a] {S(b,z 3 )}, {S(b,c)} [x/z 3, x 3 /a] {} [x/z 3, x 3 /a, z 3 /c] ( [x/c]) SLD Resolution [20]
21 Infinite SLD Trees (fairness of SLD) {{S(a,b)}, {S(b,c)}, {S(x,z), S(x,y), S(y,z)}} {S(a,x)} goal: S(a,x) [] [ ] {S(a,x)}, {S(x 3,z 3 ), S(x 3,y 3 ), S(y 3,z 3 )} [] {S(a,y 3 ), S(y 3,z 3 )} [x 3 /a, x/z 3 ] {S(a,y 3 ), S(y 3,z 3 )}, {S(a,b)} [x/z 3, x 3 /a] {S(b,z 3 )} [x/z 3, x 3 /a] {S(b,z 3 )}, {S(b,c)} [x/z 3, x 3 /a] {} [x/z 3, x 3 /a, z 3 /c] ( [x/c]) {S(b,z 3 )}, {S(x 4,z 4 ), S(x 4,y 4 ), S(y 4,z 4 )} [x/z 3, x 3 /a] {S(b,y 4 ), S(y 4,z 4 )} [x/z 3, x 3 /a, z 3 /z 4, x 4 /b] {S(b,y 4 ), S(y 4,z 4 )}, {S(x 5,z 5 ), S(x 5,y 5 ), S(y 5,z 5 )} [x/z 3, x 3 /a, z 3 /z 4, x 4 /b] {S(b,y 5 ), S(y 5,z 5 ), S(z 5,z 4 )} [x/z 3, x 3 /a, z 3 /z 4, x 4 /b, y 4 /z 5, x 5 /b] [ ] SLD Resolution [21]
22 Infinite SLD Trees (fairness of SLD) {{S(x,z), S(x,y), S(y,z)},{S(a,b)}, {S(b,c)}} {S(a,x)} goal: S(a,x) [] {S(a,x)}, {S(x 1,z 1 ), S(x 1,y 1 ), S(y 1,z 1 )} [] {S(a,y 1 ), S(y 1,z 1 )} [x 1 /a, x/z 1 ] {S(a,y 1 ), S(y 1,z 1 )}, {S(x 2,z 2 ), S(x 2,y 2 ), S(y 2,z 2 )} [x 1 /a, x/z 1 ] {S(z 2,z 1 ), S(x 2,y 2 ), S(y 2,z 2 )} [x 1 /a, x/z 1, x 2 /a, y 1 /z 2 ] [ ] SLD Resolution [22]
23 Infinite SLD Trees (fairness of SLD) {{S(x,z), S(x,y), S(y,z)},{S(a,b)}, {S(b,c)}} {S(a,x)} goal: S(a,x) [] {S(a,x)}, {S(x 1,z 1 ), S(x 1,y 1 ), S(y 1,z 1 )} [] {S(a,y 1 ), S(y 1,z 1 )} [x 1 /a, x/z 1 ] {S(a,y 1 ), S(y 1,z 1 )}, {S(x 2,z 2 ), S(x 2,y 2 ), S(y 2,z 2 )} [x 1 /a, x/z 1 ] {S(z 2,z 1 ), S(x 2,y 2 ), S(y 2,z 2 )} [x 1 /a, x/z 1, x 2 /a, y 1 /z 2 ] [ ] {S(a,x)}, {S(a,b)} [] {} [x/b] {S(a,x)}, {S(x 3,z 3 ), S(x 3,y 3 ), S(y 3,z 3 )} [] {S(a,y 3 ), S(y 3,z 3 )} [x 3 /a, x/z 3 ] {S(a,y 3 ), S(y 3,z 3 )}, {S(a,b)} [x/z 3, x 3 /a] {S(b,z 3 )} [x/z 3, x 3 /a] {S(b,z 3 )}, {S(b,c)} [x/z 3, x 3 /a] {} [x/z 3, x 3 /a, z 3 /c] ( [x/c]) SLD Resolution [23]
24 Infinite SLD Trees (fairness of SLD) SLD Resolution [24]
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