Chapter 9

Logical Inference

Remember that the purpose of logical inference is to generate new sentences from old sentences where the new sentences are entailed by the old sentences:
- KB |= P

entailment means that the new sentences are necessarily true given that the old sentences are true. The new sentences must be true in all world/models in which the old sentences are true.

An inference procedure that generates only entailed sentences is called sound

Inference Rules

The Knowledge Base

The axioms in a knowledge base are satisfiable and they are true for the world/model being represented

Inference rules are axioms that can be added to any knowledge base because they are valid
- KB |= P
- KB |- P

Unsound inference rules

Are there sensible unsound inference rules?

Best done in uncertain reasoning systems based on probability theory

Puzzle K.B. with ground terms only

1. contains(b,apples)
2. label(a,apple)
3.label(b,oranges)
4. label(c,bananas)
5.contains(a,apples) | contains(a,oranges) | contains(a,bananas)
6.contains(b,apples) | contains(b,oranges) | contains(b,bananas)
7.contains(c,apples) | contains(c,oranges) | contains(c,bananas)

Puzzle continued

8. contains(b,apples) => ~contains(a,apples) ^ ~contains(c,apples)
9. contains(c,oranges) => ~contains(a,oranges) ^ ~contains(b,oranges)
10. contains(a,bananas) => ~contains(b,bananas) ^ ~contains(c,bananas)
11. label(a,apples) => ~contains(a,apples)
19. label(c,bananas) => ~contains(c,bananas)

First Puzzle proof

20. (1, 8, MP) ~contains(a,apples) ^ ~contains(c,apples)
21. (20, AE) ~contains(c,apples)
22. (4,19,MP) ~contains(c,bananas)
23. (7, 21, 22,UR) contains(c,oranges)
(really 2 steps needed)
24. (23, 9, MP) ~contains(a,oranges)
25. (2, 11, MP) ~contains(a,apples)
26. (5, 24, 25, UR) contains(a,bananas)
(really 2 steps needed)

Puzzle KB with variables

1. contains(b,apples)
2. label(a,apple)
3.label(b,oranges)
4. label(c,bananas)
5. x contains(x,apples) | contains(x,oranges) | contains(x,bananas)
6. x contains(b, x) => ~contains(a, x) ^ ~contains(c, x)
7. x contains(c, x) => ~contains(a, x) ^ ~contains(b, x)
8. x, y label(x,y) => ~contains(x,y)

Second Puzzle Proof

9. (1,6, MP, [x/apples]) ~contains(a,apples) ^ ~contains(c,apples)
10. (20,AE) ~contains(c,apples)
11. (4, 8, MP, [x/c, y/bananas]) ~contains(c,bananas)
12. (5,10,11,UR,[x/c]) contains(c,bananas)
.
.
.

Could I prove sentences that aren't true in the puzzle world, e.g., contains(apples,oranges)?

New inference rules for FOPL

New inference rules

Automated proof procedure

related to search
- start state = KB
- operators = inference rules
- goal state = theorem required

problem: big branching factor; lots of rules and lots of ways to apply some rules especially universal elimination

Generalized Modus Ponens

Example

1. x tall(x) ^ heavy(x) -> big(x)
2. tall(john)
3. heavy(john)
4. (1,2,3,GMP,[x/john]) big(john)

Unification

The process of finding the substitution is called unification; the substitution itself is a unifier; Generalized Modus Ponens helps focus search on the substitutions that matter

Unification takes 2 atomic sentences and returns a substitution that makes them look the same.
- unify(p,q) = theta, where subst(theta,p) = subst(theta,q)

Unification Examples

unify(P(a,x), P(a,b)) = [x/b]
unify(P(a,x), P(y,b)) = [x/b, y/a]
unify(P(a,x), P(y,f(y)) = [y/a, x/f(a)]
unify(P(a,x), P(x,b)) = ?
unify(P(a,b), P(x,x)) = failure
unify(P(a,f(a)), P(x,x)) = failure

We always want the most general unifier (mgu); this is the one that makes the least commitment about the bindings of the variables

e.g.: unify(P(x,f(y),b), P(x,f(b),b)) = [y/b] OR [x/a,y/b]; which one is the mgu?

Constraints on Unification

1. Each occurrence of a variable has the same term substituted for it
2. a variable can never be replaced by a term containing that variable, e.g., can’t use [x/f(x)]