1. Inference in first-order logic

2. Outline Reducing first-order inference to propositional inference
Unification
Generalized Modus Ponens
Forward chaining
Backward chaining
Resolution

3. Universal instantiation (UI)
Every instantiation of a universally quantified sentence is entailed by it:
v α Subst({v/g}, α)
for any variable v and ground term g(without any variable)
E.g., x King(x)  Greedy(x)  Evil(x) yields:
King(John)  Greedy(John)  Evil(John)
King(Richard)  Greedy(Richard)  Evil(Richard)
King(Father(John))  Greedy(Father(John))  Evil(Father(John)) .

4. Existential instantiation (EI)
For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base:
v α
Subst({v/k}, α)
E.g., x Crown(x)  OnHead(x,John) yields:
Crown(C1)  OnHead(C1,John)
provided C1 is a new constant symbol, called a Skolem constant

5. Reduction to propositional inference
Suppose the KB contains just the following:
x King(x)  Greedy(x)  Evil(x)
King(John)
Greedy(John)
Brother(Richard,John)
Instantiating the universal sentence in all possible ways, we have:
King(John)  Greedy(John)  Evil(John)
King(Richard)  Greedy(Richard)  Evil(Richard)
The new KB is propositionalized: proposition symbols are
King(John), Greedy(John), Evil(John), King(Richard), etc.

6. Reduction contd.
Every FOL KB can be propositionalized so as to preserve entailment
A ground sentence is entailed by new KB iff entailed by original KB
Idea: propositionalize KB and query, apply resolution, return result
Problem: with function symbols, there are infinitely many ground terms,
e.g., Father(Father(Father(John)))

7. Reduction contd.
Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositionalized KB
Idea: For n = 0 to ∞ do
create a propositional KB by instantiating with depth-$n$ terms
see if α is entailed by this KB
Problem: works if α is entailed, loops if α is not entailed
Theorem: Turing (1936), Church (1936) Entailment for FOL is semidecidable
algorithms exist that say yes to every entailed sentence
no algorithm exists that also says no to every nonentailed sentence.

8. Problems with propositionalization
Propositionalization seems to generate lots of irrelevant sentences.
Example
from:
x King(x)  Greedy(x)  Evil(x)
King(John)
y Greedy(y)
Brother(Richard,John)
it seems obvious that Evil(John), but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant
With p k-ary predicates and n constants, there are p·nk instantiations.

9. Unification
We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y)
θ = {x/John,y/John} works
Unify(α,β) = θ if αθ = βθ
p q θ
Knows(John,x) Knows(John,Jane)
Knows(John,x) Knows(y,OJ)
Knows(John,x) Knows(y,Mother(y))
Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of variables, e.g., Knows(John,z27) Knows(z17,OJ)
{x/Jane}
{x/OJ, y/John}
{x/Mother(John),y/John}
No substitution possible yet.

10. Unification
We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y)
θ = {x/John,y/John} works
Unification finds substitutions that make different logical expressions look identical
UNIFY takes two sentences and returns a unifier for them, if one exists
UNIFY(p,q) =  where SUBST(,p) = SUBST (,q)
Basically, find a  that makes the two clauses look alike

11. Unification Examples UNIFY(Knows(John,x), Knows(John,Jane)) = {x/Jane}
UNIFY(Knows(John,x), Knows(y,Bill)) = {x/Bill, y/John}
UNIFY(Knows(John,x), Knows(y,Mother(y))= {y/John, x/Mother(John)
UNIFY(Knows(John,x), Knows(x,Elizabeth)) = fail
Last example fails because x would have to be both John and Elizabeth
We can avoid this problem by standardizing:
The two statements now read
UNIFY(Knows(John,x), Knows(z,Elizabeth))
This is solvable:
UNIFY(Knows(John,x), Knows(z,Elizabeth)) = {x/Elizabeth,z/John}

12. Unification To unify Knows(John,x) and Knows(y,z)
Can use θ = {y/John, x/z }
Or θ = {y/John, x/John, z/John}
The first unifier is more general than the second.
There is a single most general unifier (MGU) that is unique up to renaming of variables.
MGU = { y/John, x/z }

13. Unification Unification algorithm:
Recursively explore the two expressions side by side
Build up a unifier along the way
Fail if two corresponding points do not match

14. The unification algorithm

15. The unification algorithm

16. Simple Example Brother(x,John)Father(Henry,y)Mother(z,John)
Brother(Richard,x)Father(y,Richard)Mother(Eleanore,x)

17. Generalized Modus Ponens (GMP)
p1', p2', … , pn', ( p1  p2  …  pn q) qθ
p1' is King(John) p1 is King(x)
p2' is Greedy(y) p2 is Greedy(x)
θ is {x/John,y/John} q is Evil(x)
q θ is Evil(John)
GMP used with KB of definite clauses (exactly one positive literal)
All variables assumed universally quantified
where pi'θ = pi θ for all i

18. Soundness of GMP Need to show that p1', …, pn', (p1  …  pn  q) ╞ qθ
provided that pi'θ = piθ for all I
Lemma: For any sentence p, we have p ╞ pθ by UI
(p1  …  pn  q) ╞ (p1  …  pn  q)θ = (p1θ  …  pnθ  qθ)
p1', \; …, \;pn' ╞ p1'  …  pn' ╞ p1'θ  …  pn'θ
From 1 and 2, qθ follows by ordinary Modus Ponens

19. Storage and Retrieval
Use TELL and ASK to interact with Inference Engine
Implemented with STORE and FETCH
STORE(s) stores sentence s
FETCH(q) returns all unifiers that the query q unifies with
Example:
q = Knows(John,x)
KB is:
Knows(John,Jane), Knows(y,Bill), Knows(y,Mother(y))
Result is
1={x/Jane}, 2=, 3= {John/y,x/Mother(y)}

20. Storage and Retrieval First approach:
Create a long list of all propositions in Knowledge Base
Attempt unification with all propositions in KB
Works, but is inefficient
Need to restrict unification attempts to sentences that have some chance of unifying
Index facts in KB
Predicate Indexing
Index predicates:
All “Knows” sentences in one bucket
All “Loves” sentences in another
Use Subsumption Lattice (see below)

21. Storing and Retrieval Subsumption Lattice
Child is obtained from parent through a single substitution
Lattice contains all possible queries that can be unified with it.
Works well for small lattices
Predicate with n arguments has a 2n lattice
Structure of lattice depends on whether the base contains repeated variables
Knows(x,y)
Knows(x,John)
Knows(x,x)
Knows(John,x)
Knows(John,John)

22. Forward Chaining Forward Chaining Idea:
Start with atomic sentences in the KB
Apply Modus Ponens
Add new atomic sentences until no further inferences can be made
Works well for a KB consisting of Situation  Response clauses when processing newly arrived data

23. Forward Chaining First Order Definite Clauses
Disjunctions of literals of which exactly one is positive:
Example:
King(x)  Greedy(x)  Evil(x)
King(John)
Greedy(y)
First Order Definite Clauses can include variables
Variables are assumed to be universally quantified
Greedy(y) means y Greedy(y)
Not every KB can be converted into first definite clauses

24. Example knowledge base
The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.
Prove that Col. West is a criminal

25. Example knowledge base contd.
... it is a crime for an American to sell weapons to hostile nations:
American(x)  Weapon(y)  Sells(x,y,z)  Hostile(z)  Criminal(x)
Nono … has some missiles, i.e., x Owns(Nono,x)  Missile(x):
Owns(Nono,M1) and Missile(M1)
… all of its missiles were sold to it by Colonel West
Missile(x)  Owns(Nono,x)  Sells(West,x,Nono)
Missiles are weapons:
Missile(x)  Weapon(x)
An enemy of America counts as "hostile“:
Enemy(x,America)  Hostile(x)
West, who is American …
American(West)
The country Nono, an enemy of America …
Enemy(Nono,America)

26. Forward chaining algorithm

27. Forward chaining proof

28. Forward chaining proof

29. Forward chaining proof

30. Properties of forward chaining
Sound and complete for first-order definite clauses
Datalog = first-order definite clauses + no functions
FC terminates for Datalog in finite number of iterations
May not terminate in general if α is not entailed
This is unavoidable: entailment with definite clauses is semidecidable

31. Efficiency of forward chaining
Incremental forward chaining: no need to match a rule on iteration k if a premise wasn't added on iteration k-1
 match each rule whose premise contains a newly added positive literal
Matching itself can be expensive:
Database indexing allows O(1) retrieval of known facts
e.g., query Missile(x) retrieves Missile(M1)
Forward chaining is widely used in deductive databases

32. Hard matching example
Diff(wa,nt)  Diff(wa,sa)  Diff(nt,q)  Diff(nt,sa)  Diff(q,nsw)  Diff(q,sa)  Diff(nsw,v)  Diff(nsw,sa)  Diff(v,sa)  Colorable()
Diff(Red,Blue) Diff (Red,Green) Diff(Green,Red) Diff(Green,Blue) Diff(Blue,Red) Diff(Blue,Green)
Colorable() is inferred iff the CSP has a solution
CSPs include 3SAT as a special case, hence matching is NP-hard

33. Backward Chaining
Improves on Forward Chaining by not making irrelevant conclusions
Alternatives to backward chaining:
restrict forward chaining to a relevant set of forward rules
Rewrite rules so that only relevant variable bindings are made:
Use a magic set
Example:
Rewrite rule: Magic(x)American(x) Weapon(x) Sells(x,y,z) Hostile(z)Criminal(x)
Add fact: Magic(West)

34. Backward Chaining Idea:
Given a query, find all substitutions that satisfy the query.
Algorithm:
Work on lists of goals, starting with original query
Algo finds every clause in the KB that unifies with the positive literal (head) and adds remainder (body) to list of goals

35. Backward chaining algorithm
SUBST(COMPOSE(θ1, θ2), p) = SUBST(θ2, SUBST(θ1, p))

36. Backward chaining example

37. Backward chaining example

38. Backward chaining example

39. Backward chaining example

40. Backward chaining example

41. Backward chaining example

42. Backward chaining example

43. Backward chaining example

44. Properties of backward chaining
Depth-first recursive proof search: space is linear in size of proof
Incomplete due to infinite loops
fix by checking current goal against every goal on stack
Inefficient due to repeated subgoals (both success and failure)
fix using caching of previous results (extra space)
Widely used for logic programming

45. Logic programming: Prolog
Algorithm = Logic + Control
Basis: backward chaining with Horn clauses + bells & whistles
Widely used in Europe, Japan (basis of 5th Generation project)
Program = set of clauses:
head :- literal1, … literaln.
criminal(X) :- american(X), weapon(Y), sells(X,Y,Z), hostile(Z).
Depth-first, left-to-right backward chaining
Built-in predicates for arithmetic etc., e.g., X is Y*Z+3
Built-in predicates that have side effects (e.g., input and output predicates, assert/retract predicates)
Closed-world assumption ("negation as failure")
e.g., given alive(X) :- not dead(X).
alive(joe) succeeds if dead(joe) fails

46. Prolog Appending two lists to produce a third: append([],Y,Y).
append([X|L],Y,[X|Z]) :- append(L,Y,Z).
query: append(A,B,[1,2]) ?
answers: A=[] B=[1,2]
A=[1] B=[2]
A=[1,2] B=[]

47. Prolog Has problems with repeated states and infinite paths
Example: Path finding in graphs
path(X,Z) :- link(X,Z)
path(X,Z) :- path(X,Y),link(Y,Z)
path(a,c) A B C
link(a,c)
link(Y,c)
fail
path(a,Y)
{ Y/b}
link(a,b)
{Y/b }

48. Prolog Has problems with repeated states and infinite paths
Example: Path finding in graphs
path(X,Z) :- path(X,Y),link(Y,Z)
path(X,Z) :- link(X,Z)
path(a,c) A B C
path(a,Y)
link(Y,b)
fail
path(a,Y’)
link(Y’,Y)
path(a,Y)
link(Y,b)

49. Resolution
Resolution for propositional logic is a complete inference procedure
Existence of complete proof procedures in Mathematics would entail:
All conjectures can be established mechanically
All mathematics can be established as the logical consequence of a set of fundamental axioms
Gődel 1930: Completeness Theorem for first order logic
Any entailed sentence has a finite proof
No algorithm given until J.A. Robinson’s resolution algorithm in 1965
Gődel 1931: Incompleteness Theorem:
Any logical system with induction is necessarily incomplete
There are sentences that are entailed, but not proof can be given

50. Resolution First order logic requires sentences in CNF
Conjunctive Normal Form: Each clause is a disjunction of literals, but literals can contain variables, which are assumed to be universally quantified
Example: Convert
x American(x)  Weapon(y)  Sells(x,y,z)  Hostile(z)  Criminal(x)
American(x)   Weapon(y)  Sells(x,y,z)   Hostile(z) Criminal(x)

51. Conversion to CNF Everyone who loves all animals is loved by someone:
x [y Animal(y)  Loves(x,y)]  [y Loves(y,x)]
1. Eliminate biconditionals and implications
x [y Animal(y)  Loves(x,y)]  [y Loves(y,x)]
2. Move  inwards: x p ≡ x p,  x p ≡ x p
x [y (Animal(y)  Loves(x,y))]  [y Loves(y,x)]
x [y Animal(y)  Loves(x,y)]  [y Loves(y,x)]
x [y Animal(y)  Loves(x,y)]  [y Loves(y,x)]

52. Conversion to CNF
Standardize variables: each quantifier should use a different one
x [y Animal(y)  Loves(x,y)]  [z Loves(z,x)]
Skolemize: a more general form of existential instantiation.
Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables:
x [Animal(F(x))  Loves(x,F(x))]  Loves(G(x),x)
Drop universal quantifiers:
[Animal(F(x))  Loves(x,F(x))]  Loves(G(x),x)
Distribute  over  :
[Animal(F(x))  Loves(G(x),x)]  [Loves(x,F(x))  Loves(G(x),x)]

53. Resolution Inference Rule
Full first-order version:
l1  ···  lk, m1  ···  mn
Subst(θ ,l1  ···  li-1  li+1  ···  lk  m1  ···  mj-1  mj+1  ···  mn)θ
where Unify(li, mj) = θ.
The two clauses are assumed to be standardized apart so that they share no variables.
For example,
Rich(x)  Unhappy(x)
Rich(Ken)
Unhappy(Ken)
with θ = {x/Ken}
Apply resolution steps to CNF(KB  α); complete for FOL

54. Resolution
Show KB ⊢ α by showing that KB   α is unsatisfyable

55. Resolution Example Everyone who loves all animals is loved by someone
Anyone who kills an animal is loved by no one.
Jack loves all animals.
Either Jack or Curiosity killed the cat, who is named Tuna
All dogs kill a cats
Rintintin is a dog
Question: Did Curiosity kill the cat?

56. Resolution Example Everyone who loves all animals is loved by someone.
Formulate in FOL
x [y [Animal(y)  Loves(x,y)]]  [z Loves(z,x)]
Remove Implications
x [[y Animal(y)  Loves(x,y)]]  [z Loves(z,x)]
x [[y  Animal(y)  Loves(x,y)]]  [z Loves(z,x)]
Move negation inward
x [y   Animal(y)   Loves(x,y)]  [z Loves(z,x)]
x [y Animal(y)   Loves(x,y)]  [z Loves(z,x)]
Skolemize
x [ Animal(F(x))   Loves(x,F(x))]  [Loves(G(x),x)]
N.B.: Argument of Skolemization function are all universally qualified variables
Drop universal quantifier
[ Animal(F(x))   Loves(x,F(x))]  [Loves(G(x),x)]
Use distributive law (and get two clauses)
Animal(F(x))  Loves(G(x),x);  Loves(x,F(x))  Loves(G(x),x)

57. Resolution Example Anyone who kills an animal is loved by no one.
Transfer to FOL
x [y (Animal(y)  Kills(x,y)] (z Loves(z,x)
Remove Implications
x [y (Animal(y)  Kills(x,y)]  (z Loves(z,x)
Move negations inwards
x [ y Animal(y)   Kills(x,y)]  (z  Loves(z,x))
Remove quantifiers
Animal(y)   Kills(x,y)   Loves(z,x)

58. Resolution Example Jack loves all animals. FOL form
x [Animal(x)  Loves(Jack, x)]
Remove implications
x [Animal(x)  Loves(Jack, x)]
Remove quantifier
Animal(x)  Loves(Jack, x)

59. Resolution Example
Either Jack or Curiosity killed the cat, who is named Tuna.
FOL form
Kills(Jack,Tuna)  Kills(Curiosity,Tuna); Cat(Tuna)

60. Resolution Example All cats are animals FOL form Remove implications
x [Cat(x)  Animal(x)
Remove implications
x [Cat(x)  Animal(x)]
Remove quantifier
Cat(x)  Animal(x)

61. Resolution Example All dogs kill a cats FOL form Remove implications
x [Dog(x)  y[Cat(y)  Kills(x,y)]]
Remove implications
x [Dog(x)  y[Cat(y)  Kills(x,y)]]
Skolemize
x [Dog(x)  [Cat(H(x))  Kills(x,H(x))]]
Drop universal quantifiers
Dog(x)  [Cat(H(x))  Kills(x,H(x))]
Distribute (and obtain two clauses)
Dog(x)  Cat(H(x); Dog(x)  Kills(x,H(x))]

62. Resolution Example
Rintintin is a dog
FOL form
Dog(Rintintin)

63. Resolution Example Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)

64. Resolution Example Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)
Question: Did Curiosity kill the cat?
Kills(Curiosity,Tuna)]

65. Resolution Example Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)
Kills(Curiosity,Tuna)]
Cat(Tuna) , Cat(x)  Animal(x)
Unify(Cat(Tuna), Cat(x)) = { x/Tuna }
First line thus resolves to:
Animal(Tuna)

66. Resolution Example Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)
Kills(Curiosity,Tuna)
Animal(Tuna)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna), Kills(Curiosity,Tuna)
Resolves to:
Kills(Jack,Tuna)

67. Resolution Example
Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)
Kills(Curiosity,Tuna)
Animal(Tuna)
Kills(Jack,Tuna)
Animal(y)   Kills(x,y)   Loves(z,x), Animal(Tuna)
Unify(Animal(Tuna), Animal(y)) = {y/Tuna}
Resolves to:
 Kills(x,Tuna)   Loves(z,x),

68. Resolution Example
Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)
Kills(Curiosity,Tuna)
Animal(Tuna)
Kills(Jack,Tuna)
 Kills(x,Tuna)   Loves(z,x),
 Loves(x,F(x))  Loves(G(x),x), Animal(z)  Loves(Jack, z)
Unify( Loves(x,F(x)) , Loves(Jack, z)) = { x / Jack, z / F(x)}
Resolvent clause is obtained by substituting the unification rule
Loves(G(Jack),Jack)  Animal(F(Jack))

69. Resolution Example
Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)
Kills(Curiosity,Tuna)
Animal(Tuna)
Kills(Jack,Tuna)
 Kills(x,Tuna)   Loves(z,x)
Loves(G(Jack),Jack)  Animal(F(Jack))
Animal(F(x))  Loves(G(x),x), Loves(G(Jack),Jack)  Animal(F(Jack))
Unify(Animal(F(x)) , Animal(F(Jack)))= { x / Jack}
Resolvent clause is obtained by substituting the unification rule
Loves(G(Jack),Jack)

70. Resolution Example  Kills(x,Tuna)   Loves(z,x), Loves(G(Jack),Jack)
Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)
Kills(Curiosity,Tuna)
Animal(Tuna)
Kills(Jack,Tuna)
 Kills(x,Tuna)   Loves(z,x)
Loves(G(Jack),Jack)  Animal(F(Jack))
Loves(G(Jack),Jack)
 Kills(x,Tuna)   Loves(z,x), Loves(G(Jack),Jack)
Unify( Loves(z,x), Loves(G(Jack),Jack) ) = { x / Jack, z / G(Jack)}
Resolvent clause is obtained by substituting the unification rule
 Loves(G(Jack),Jack)

71. Resolution Example  Loves(G(Jack),Jack), Loves(G(Jack),Jack)
Animal(F(x))  Loves(G(x),x)
 Loves(x,F(x))  Loves(G(x),x)
Animal(y)   Kills(x,y)   Loves(z,x)
Animal(x)  Loves(Jack, x)
Kills(Jack,Tuna)  Kills(Curiosity,Tuna)
Cat(Tuna)
Cat(x)  Animal(x)
Dog(x)  Cat(H(x)
Dog(x)  Kills(x,H(x))]
Dog(Rintintin)
Kills(Curiosity,Tuna)
Animal(Tuna)
Kills(Jack,Tuna)
 Kills(x,Tuna)   Loves(z,x)
Loves(G(Jack),Jack)  Animal(F(Jack))
Loves(G(Jack),Jack)
 Loves(G(Jack),Jack)
 Loves(G(Jack),Jack), Loves(G(Jack),Jack)
Resolvent clause is empty. Proof succeeded

72. Resolution Resolution is refutation-complete Theorem provers
If a set of sentences is unsatisfiable, then resolution will be able to produce a contradiction
Theorem provers
Use control in order to be more efficient
Focus of most research effort
Separate control from knowledge base
Example: Otter (Organized Technique for Theorem proving and Effective Research)
A set of clauses known as the SoS - Set of Support
The important facts about a problem
Search if focused on resolving SoS with another axiom
A set of usable axioms
Background knowledge about problem field
Rewrites / demodulators
Rules to transform expressions into a canonical form
Set of parameters or clauses that defines the control strategy to allow user to control search and filtering functions to eliminate useless subgoals

73. Theorem Prover Successes
First formal proof of Gődel’s Incompleteness Theorem (1986)
Robbins algebra (a simple set of axioms) is Boolean algebra (1996)
Software verification:
Remote agent spacecraft control program (2000)

74. Resolution proof: definite clauses