Examples

1
Examples
2
Example
Everybody loves somebody. Everybody loves a
lover. Show that everybody loves
everybody. "x.y.loves( x, y) "u."v."w.(loves(v,
w) Þ loves(u, v)) Ø"x."y.loves( x, y) negated
conclusion In clausal form they
become loves(x, f( x)) Øloves(v, w), loves(u,
v) Øloves(jack, jill)
3

• loves(x, f(x)) Premise
• Øloves(v, w), loves(u, v) Premise
• Øloves(jack, jill) Goal
• loves(u, x) 1,2
• 4,3

4
Horses, dogs, and rabbits
Every horse can outrun every dog. Some
greyhounds can outrun every rabbit. Show that
every horse can outrun every rabbit. "x."y.(Horse
( x) Ù Dog( y) Þ Faster( x, y))
y.(Greyhound(y) Ù "z.(Rabbit( z) Þ Faster( y,
z))) "y.(Greyhound( y) Þ Dog( y)) (background
knowledge) "x."y."z.(Faster(x, y) Ù Faster( y, z)
Þ Faster( x, z)) (background knowledge) Ø"x."y.(H
orse(x) Ù Rabbit(y) Þ Faster(x, y)) negated
conclusion
5
"x."y.(Horse( x) Ù Dog( y) Þ Faster( x,
y)) y.(Greyhound(y) Ù "z.(Rabbit( z) Þ Faster(
y, z))) "y.(Greyhound( y) Þ Dog(
y)) "x."y."z.(Faster(x, y) Ù Faster( y, z) Þ
Faster( x, z)) Ø"x."y.(Horse(x) Ù Rabbit(y) Þ
Faster(x, y)) In clausal form they
become ØHorse( x), ØDog(y), Faster( x,
y) Greyhound(Greg) ØRabbit(z), Faster(Greg,
z) ØGreyhound(y), Dog(y) ØFaster(x, y),
ØFaster(y, z), Faster(x, z) continued
6
Lets transform the goal into clausal
form Ø"x."y.(Horse(x) Ù Rabbit(y) Þ Faster(x,
y)) I Ø"x."y.(Ø(Horse(x) Ù Rabbit(y)) Ú Faster(x,
y)) N Ø"x."y.(ØHorse(x) Ú ØRabbit(y) Ú Faster(x,
y)) ?x. ?y.(Horse(x) Ù Rabbit(y) Ù ØFaster(x,
y)) S ?x. ?y.(Horse(x) Ù Rabbit(y) Ù ØFaster(x,
y)) E Horse(Jack) Ù Rabbit(Smith) Ù ØFaster(Jack,
Smith)) A Horse(Jack) Ù Rabbit(Smith) Ù
ØFaster(Jack, Smith)) D Horse(Jack) Ù
Rabbit(Smith) Ù ØFaster(Jack, Smith)) O
Horse(Jack) Rabbit(Smith) ØFaster(Jack,
Smith)
7

• Lets try to infer the using resolution
• ØHorse(x), ØDog(y), Faster(x, y)
• Greyhound(Greg)
• ØRabbit(z), Faster(Greg, z)
• ØGreyhound(y), Dog(y)
• ØFaster(x, y), ØFaster(y, z), Faster(x, z)
• Horse(Jack)
• Rabbit(Smith)
• ØFaster(Jack, Smith)
• ØDog(y), Faster(Jack, y) 1,6
• Faster(Jack, y), ØGreyhound(y) 4,9
• Faster(Jack, Greg) 2,10
• Faster(Greg, Smith) 3,7
• ØFaster(Greg, z), Faster(x, z) 5,11
• Faster(x, Smith) 12,13
• 8,14

8
Resolution Strategies
9
Linear Restriction Strategy

• A linear resolution is one in which one of the
  parents is in the initial database or an ancestor
  of the other parent.
• Example.
• 1. p, q
• 2. p,Øq
• 3. Øp, q
• 4. Øp,Øq
• 5. p 1,2
• 6. q 3,5
• 7. Øp 4,6
• 8. 5,7
• Linear Resolution is
• refutation complete!!

10
Theo

• Theo is theorem prover based on linear resolution
  (as well as other resolution strategies besides
  linearity)
• Accepts input in clausal form.
• Uses
• for ?
• for ? (doesnt use the set notation for
  clauses, but the OR, i.e. notation, e.g. p(x)
  q(y) instead of p(x), q(y))
• Distinguish variables from constants, by adding
  parenthesis. E.g. tuna().
• If you omit parenthesis and write instead just
  tuna, it will be considered a variable.

11
Clausal Curiosity (Theo format)

• animal(f(x1)), loves(g(x1),x1)
• ?loves(x2,f(x2)), loves(g(x2),x2)
• ?animal(y1), ?kills(x3,y1), ?loves(z,x3)
• ?animal(y2), loves(jack,y2)
• kills(jack,tuna), kills(curiosity,tuna)
• ?cat(x4), animal(x4)
• cat(tuna)
• ?kills(curiosity,tuna)

animal(f(x1)) loves(g(x1),x1) loves(x2,f(x2))
loves(g(x2),x2) animal(y1) kills(x3,y1)
loves(z,x3) animal(y2) loves(jack,y2) kills(
jack(),tuna()) kills(curiosity(),tuna()) cat(x4
) animal(x4) cat(tuna()) negated_conclusion ki
lls(curiosity(),tuna())
12
Theo output

• Proof Found!
• Axioms
• 1 gtanimalfx lovesgxx
• 2gtlovesxfx lovesgxx
• 3 gtanimalx killsyx loveszy
• 4 gtanimalx lovesyx
• 5gtkillsjacktuna killscuriositytuna
• 6 gtcatx animalx
• 7 gtcattuna
• Negated conclusion
• 8Sgtkillscuriositytuna

Phase 0 clauses used in proof 9Sgt(8a5b)
killsjacktuna 10Sgt(9a3b) animaltuna
lovesxjack 11gt(10a6b) lovesxjack
cattuna 12Sgt(11b7a) lovesxjack 15 gt(4a1a)
lovesxfy lovesgyy 16 gt(15a2a) lovesgxx Phases
1 and 2 clauses used in proof 17Sgt(16a,12a)
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Compile

• You can create well formed formulas (WFF) i.e.
  FOL sentences and then convert into clausal form
  using compile
• _at_ is forall
• ! is exists
• Variables and constants distinguished by
  quantification. Variables are quantified,
  constants arent.
• Use ( and ) for functions and relations.
• Use and for grouping sentences.
• is AND, is OR, gt is IMPLIES
• Example
• _at_x met_with_at_sea(x) noticed_thing(x) gt
  entered_in_log(x).

14
Books

• The only books in this library, that I do not
  recommend for reading, are unhealthy in tone
• The bound books are all well-written
• All the romances are healthy in tone
• I do not recommend that you read any of the
  unbound books.
• All the romances in this library are
  well-written.
• _at_x book(x) recommend(x) gt healthy(x).
• _at_x book(x) healthy(x) gt recommend(x).
• _at_x book(x) bound(x) gt wellwritten(x).
• _at_x romance(x) gt healthy(x).
• _at_x romance(x) gt book(x).
• _at_x book(x) bound(x) gt recommend(x).
• theorem (negated)
• _at_x romance(x) gt wellwritten(x) .

15
Songs

• Things sold in the street are of no great
  value
• Nothing but rubbish can be sold for a song
• Eggs of the Great Auk are very valuable
• It is only what is sold in the streets which is
  really rubbish.
• Conclusion An egg of the Great Auk is not to
  be had for a song.
• _at_x soldInStreet(x) gt ofGreatValue(x).
• _at_x_at_y exchange(y,x) song(x) gt rubbish(y).
• _at_x greatAukEgg(x) gt ofGreatValue(x).
• _at_x rubbish(x) gt soldInStreet(x).
• _at_x soldInStreet(x) gt rubbish(x).
• theorem (negated)
• !x!y greatAukEgg(y) exchange(y,x)
  song(x) .

16
Birds

• No birds, except ostriches, are 9 feet high
• There are no birds in this aviary that belong
  to anyone but me
• No ostrich lives on mince-pies
• I have no birds less than nine-feet high.
• Conclusion No bird in this aviary lives on
  mince-pies.
• _at_x bird(x) height(x,Nine) gt ostrich(x).
• _at_x ostrich(x) gt bird(x).
• _at_x bird(x) gt belongs(x,Me).
• _at_x ostrich(x) gt lives(x,MincePies).
• _at_x bird(x) belongs(x,Me) gt height(x,Nine).
• theorem (negated)
• _at_x bird(x) gt lives(x,MincePies) .

17
British Lion

• No discussions in our Debating-Club can rouse
  the British Lion, so long as they are checked
  when they become too noisy
• Discussions, unwisely conducted, endanger the
  peacefulness of our Debating-Club
• Discussions, that go on while Tomkins is in the
  Chair, rouse the British Lion
• Discussions in our Debating-Club, when wisely
  conducted, are always checked when they become
  too noisy.
• !x discussion(x) in(x, OurDebatingClub)
  noisy(x) gt checked(x) rouse(x,
  BritishLion) .
• _at_x discussion(x) checked(x) gt rouse(x,
  BritishLion). background
• _at_x_at_y discussion(x) wiselyConducted(x)
  in(x,y) gt endanger(x, Peacefulness, y).
• _at_x_at_y discussion(x) in(x,y) chair(Tomkins, y)
  gt rouse(x, BritishLion).
• _at_x discussion(x) in(x, OurDebatingClub)
  wiselyConducted(x) gt noisy(x) gt checked(x).

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• Conclusion Discussions, that go on while
  Tomkins is in the chair, endanger the
  peacefulness of our Debating-Club..
• theorem (negated)
• _at_x discussion(x) in(x,OurDebatingClub)
  chair(Tomkins, OurDebatingClub) gt endanger(x,
  Peacefulness, OurDebatingClub) .

19
Algebra

• In an associative system with left and right
  solutions,
• there is a right identity element.
• P(x,y,z) means xy z
• (xy)z x(yz)
• xy u yz v
• _at_x_at_y_at_z_at_u_at_v_at_w p(x,y,u) p(y,z,v) p(u,z,w) gt
  p(x,v,w).
• _at_x_at_y_at_z_at_u_at_v_at_w p(x,y,u) p(y,z,v) p(x,v,w) gt
  p(u,z,w).
• _at_x_at_y!s p(x,s,y).
• _at_x_at_y!s p(s,x,y).
• theorem (negated)
• !e_at_x p(x,e,x).