Predicate Logic or FOL

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Predicate Logic or FOL

• Chapter 8

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Propositional Logic cant say

• If X is married to Y, then Y is married to X.
• If X is west of Y, and Y is west of Z, then X is
  west of Z.
• And a million other simple things.
• Fix
• extend representation add predicates
• Extend operator(resolution) add unification

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Syntax

• See text for formal rules.
• All of propositional quantifiers, predicates,
  functions, and constants.
• Variables can take on values of constants or
  terms.
• Term reference to object
• Variables not allowed to be predicates.
• E.G. What is the relationship between Bill and
  Hillary?
• Text Notation variables lower case, constants
  upper
• Prolog Notation variables are upper case, etc

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Term

• A term with no variables is a ground term.
• Composite Objection function of terms or
  primitives
• Convenience we dont want to name all objects
• e.g. nounphrase(det(the),adj(tall),noun(tree)).
• E.g. leftLeg(John).
• Successor of 1 may be s(1), but we write 2.
• Successor of 2 s(s(1)), but we write 3.

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Goldbachs Conjecture

• For all n, if integer(n), even(n), greater(n,2)
  then there exists p1, p2, integer(p1),
  integer(p2), prime(p1),prime(p2), and
  equals(n,sum(p1,p2)).
• Quantifiers for all, there exists
• Predicates integer, greater, prime, even,
  equals.
• Constants 2
• Functions sum.

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Semantics

• Validity true in every model and every
  interpretation.
• Interpretation mapping of constants,
  predicates, functions into objects, relations,
  and functions.
• For Goldbach wrt to standard integer model
  interpretation mapping n to an even integer.
  (Context).

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Representing World in FOL

• All kings are persons.
• ? goes to?
• for all x, King(x) Person(x).
• for all x, King(x) gt Person(x).

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Representing World in FOL

• All kings are persons.
• for all x, King(x) gt Person(x). OK.
• for all x, King(x) Person(x). Not OK.
• this says every object is a king and a person.
• In Prolog person(X) - king(X).
• Everyone Likes icecream.
• for all x, Likes(x, icecream).

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Negating Quantifiers

• there exist x, P(x)
• for all x, P(x)
• For all x, Likes(x,Icecream)
• No one likes liver.
• For all x, not Likes(x,Liver)

• For all x, P(x)
• There exists x, P(x)
• There does not exist an x, not Likes(x,Icecream)
• Not there exists x, Iikes(x,Liver).

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More Translations

• Everyone loves someone.
• There is someone that everyone loves.
• Everyone loves their father.
• See text.

• For all x, there is a y(x) such that
  Loves(x,y(x)).
• There is an M such that for all x, Loves(x,M).
• M is skolem constant
• For all x, Loves(x,Father(x)).
• Father(x) is skolem function.

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Unification

• If p and q are logical expressions, then
  Unify(p,q) Substitution List S if using S makes
  p and q identical or fail.
• Standardize apart before unifying, make sure
  that p and q contain different variable names.

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Most General Unifier (MGU)

• f(X,g(Y)) unifies with f(g(Z),U) with
  substitutions X/g(a), Y/b, U/g(b), Z/b.
• But also if X/g(Z), U/g(Y).
• The MGU is unique up to renaming of variables.
• All other unifiers are unify with the MGU.
• Use Prolog with for unification.

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Occurs Checking

• When unifying a variable against a complex term,
  the complex term should not contain the same
  variable, else non-match.
• Prolog doesnt check this.
• Ex. f(X,X) and f(Y,g(Y)) should not unify.

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Modeling with Definite Clauses at most one
positive literal

• 1. It is a crime for an american to sell weapons
  to a hostile country.
• 1. American(x)Weapons(y)Hostile(z)
  Sell(x,y,z) gt Criminal (x).
• 2. The country Nono has some missiles.
• There exists x Owns(Nono,x)Missile(x).
• 2. Missile(M1). Skolem Constant
  introduction
• 2. Owns(Nono,M1).

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Prove West is a criminal

• 3. All of its missiles where sold to it by
  Colonel West.
• 3. Missile(x)Owns(Nono,x) gt Sells(West,x,Nono).
  
• 4. Missile(x) gt Weapon(x). .. common sense
• 5. Enemy(x,America) gt Hostile(x).
• 6. American(West).
• 7. Enemy(Nono,American).

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Forward Chaining

• Start with facts and apply rules until no new
  facts appear. Apply means use substitutions.
• Iteration 1 using facts.
• Missile(M1),American(West), Owns(Nono,M1),
  Enemy(Nono,America)
• Derive Hostile(Nono), Weapon(M1),
  Sells(West,M1,Nono).
• Next Iteration Criminal(West).
• Forward chaining ok if few facts and rules, but
  it is undirected.

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Resolution gives forward chaining

• Enemy(x,America) gtHostile(x)
• Enemy(Nono,America)
• - Hostile(Nono)

• Not Enemy(x,America) or Hostile(x)
• Enemy(Nono,America)
• Resolve by x/Nono
• To Hostile(Nono)

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Backward Chaining

• Start with goal, Criminal(West) and set up
  subgoals. This ends when all subgoals are
  validated.
• Iteration 1 subgoals American(x), Weapons(y) and
  Hostile(z).
• Etc. Eventually all subgoals unify with facts.

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Resolution yeilds Backward Chaining

• A(x) W(y)H(z) S(x,y,z) gtC(x)

• -A(x) or W(y) or
• H(z) or S(x,y,z) or C(x).
• Add goal C(West).
• Yields A(West) or
• -W(y) or H(z) or
• -S(West,y,z). Etc.

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Resolution is non-directional

• Both a power (inference representation) and a
  weakness (no guidance in search)
• -a or b or c or d or e equals
• a,b,c gtd or e and
• a,b,c, -d gt e etc.
• Prolog forces directionality and results in an
  incomplete theorem prover.

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FOL -gt Conjuctive Normal Form

• Similar to process for propositional logic, but
• Use negations rules for quantifiers
• Standarize variables apart
• Universal quantification is implicit.
• Skolemization introduction of constants and
  functions to remove existential quantifiers.

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Skolemization

• Introduction of constants or functions when
  removing existential quantifier.
• There exists an x such that P(x) becomes P(A)
  for some new constant symbol A.
• Everyone has someone who loves him
• For all x, Loves(F(x),x) where F(x) is a new
  function.

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Resolution in CNF

• Just like propositional case, but now
  complimentary first order literals unify.
• Theorem (skipping proof)FOL with resolution is
  refutation complete, i.e. if S is a set of
  unsatisfiable clauses, then a contradiction
  arises after a finite number of resolutions.
• Lets take in on faith!

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Results

• Proof of theorems in
• Lattice Theory
• Group theory
• Logic
• But didnt generate the theorem.
• Lenats phd thesis AM generated mathematical
  theorems, but none of interest.

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Limitations

• 2nd order What is the relationship between Bush
  and Clinton?
• Brittle If knowledge base has contradiction,
  then anything derivable. (false P)
• Scaleability
• Expensive to compute
• Difficult to write down large number of
  statements that are logically correct.
• Changing World (monotoncity) what was true, is
  not longer.
• Likelihoods What is likelihood that patient has
  appendicitis given high temp.
• Combining Evidence

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Situation Calculus/Planning

• The world changes and actions change it.
• What to do?
• Early approach Define Actions via
• Preconditions conjunctions of predicates
• Effects changes to world if operator applied
• Delete conditions predicates to delete
• Add conditions predicates to add

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Blocks World Example

• Action Move(b,x,y)move b from x to y
• Preconditions
• On(b,x)Clear(y)Block(b) Clear(b)
• Careful and b \ y else problems ( b to b)
• Postconditions
• On(b,y) Clear(x) not On(b,x) not Clear(y)
• Similar for other operators/actions.
• Now search better plan searchers possible.

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More Extensions

• Special axioms for time, space, events,
  processes, beliefs, goals
• Try to do any simple story, e.g. Goldilocks and
  three bears.
• How would you know you did it?
• Problems
• Represent whats in story
• Represent whats not in story but relevant.
• Inferencing

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Time

• Before (x,y) implies After(y,x)
• After(x,y) imples Before(y,x)
• Before(x,y) and Before(y,z) implies Before(x,z)
  etc.
• When are you done?
• What about during?

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Space and more

• In(x,y) and In(y,z) implies in(x,z).
• Infront, behind, etc
• Frame problem you turn, some predicates change
  and some dont.
• etc.
• And lots more heat, wind, hitting, physical
  objects versus thoughts, knowing,

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Example Questions

• Was Goldilocks hungry?
• Was Goldilocks tired?
• Why did the bed break?
• Could the baby bear say Papa, dont talk unless
  you are spoken too.

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Expert Systems Engineering Approach

• We can keep the representation language of FOL,
  but do not adopt the semantics.
• Attach to each fact and rule a belief ()
• Provide an ad hoc calculus for combining beliefs.
• Now multiple proofs valuable since they will add
  evidence.
• This worked, if domain picked carefully. The hard
  part getting the rules or knowledge.

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Mycin by Shortliffe 1976

• First rule based system that performed better
  than average physician at blood disease
  diagnosis.
• Required 500 rules that were painful to capture.
  (Knowledge Acquisition)
• Used ad hoc calculus to combine confidences in
  rules and facts.

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SoyBean Disease Diagnosis

• Expert diagnostician built a rule-based expert
  system for the task.
• System worked, but not as good as he was.
• Some knowledge was not captured.
• Using Machine Learning, rules were create from a
  large data base.
• The ML rules did better than the expert rules,
  but did not perform as well as the expert.