Predicate Logic or FOL

Chapter 8

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.

– that’s 2nd order logic

• Notation: unlike Prolog, variables are lower case, etc.

Term• A term with no variables is a ground term.

• Composite Objection: function of terms or primitives– Convenience: we don’t want to give names to

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.

Goldbach’s 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.

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).

Representing World in FOL

• All kings are persons.

• for all x, King(x) => 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).

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).

More Translations

• Everyone loves someone.

• There is someone that everyone loves.

• Everyone loves their father.

• See text.

• For all x, there is a y such that Loves(x,y).

• There is an M such that for all y, Loves(x,M).

• M is skolem constant• For all x,

Loves(x,Father(x)).• Father(x) is skolem

function.

Inference: almost complete• Propositionalization: Grounding out the variables

– For x, King(x) and Greedy(x) => Evil(x).– King(John). Greedy(John).– Replace all variables by given constants =>

• Evil(John) and replace fully instantiate predicates by propositional variables.

• If process finite, then propositional completeness => predicate completeness.

• Function symbols yield infinite number of terms, but theorem almost true.

• Semidecidable. If entailed, eventually find a proof, but if not entailed, proof search is unbounded.

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.

Most General Unifier (MGU)

• f(X,g(Y)) unifies with f(g(Z),U) with substitutions {X/g(a), Y/b, U/g{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.

Occurs Checking

• When unifying a variable against a complex term, the complex term should not contain the same variable, else non-match.

• Prolog doesn’t check this.

• Ex. f(X,X) and f(Y,g(Y)) should not unify.

Modeling with Definite Clauses: 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) => Criminal (x).

2. The country Nono has some missiles. There exists x Owns(Nono,x)&Missile(x). 2’. Missile(M1). … Constant introduction2’’. Owns(Nono,M1).

Prove: West is a criminal3. All of its missiles where sold to it by Colonel

West.

3’. Missile(x)&Owns(Nono,x) => Sells(West,x,Nono).

4’. Missile(x) => Weapon(x). .. “common sense”

5’. Enemy(x,America) => Hostile(x).

6’. American(West).

7’. Enemy(Nono,American).

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.

Resolution gives forward chaining

• Enemy(x,America) =>Hostile(x)

• Enemy(Nono,America)• |- Hostile(Nono)

• Not Enemy(x,America) or Hostile(x)

• Enemy(Nono,America)• Resolve by {x/Nono}• To Hostile(Nono)

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.

Resolution yeilds Backward Chaining

• A(x) &W(y)&H(z)& S(x,y,z) =>C(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.

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 =>d or e and

• a,b,c, -d => e etc.

• Prolog forces directionality and results in an incomplete theorem prover.

FOL -> 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.

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.

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.

• Let’s take in on faith!

Results• Proof of theorems in:

– Lattice Theory– Group theory– Logic

• But didn’t generate the theorem.

• Lenat’s phd thesis AM generated mathematical theorems, but none of interest.

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

Monotonicity• Unluckily, the pilot’s plane caught on fire.

• Luckily, he jumped out with his parachute.

• Unluckily the parachute didn’t work.

• Luckily he landed in a swimming pool.

• Unluckily it had no water.

• Luckily is was all a dream.

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

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.

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 what’s in story– Represent what’s not in story but relevant.– Inferencing

Example Questions

• Was Goldilocks hungry?

• Was Goldilocks tired?

• Why did the bed break?

• Could the baby bear say “Papa, don’t talk unless you are spoken too”.

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.

Mycin: by Shortliffe

• 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.

Soy Bean 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.