Oct 24 - Lesson Plans - 473 Intro to AI

From Resolution to Prolog

Today:
1. Syntactic Reasoning
 	Frege Systems
	Modus Ponens
	Propositional resolution
4. Resolution proofs
5. Unification
Monday:
6. Answering Queries with Resolution
7. Resolution strategies
	Linear resolution
8. Prolog

---------------------------------------------------
[1] Syntactic Reasoning

Starting today we will look at a very different way of doing
inference.

Instead of thinking about TRUTH or the MEANING of formulas,
logic allows us to reason by simple syntactic manipulation of
sentences!

We will be covering a lot of fairly deep material quickly.
Why?  Because a major goal of this course is to introduce you
to many important tools from AI, that you may use in your future
career.

One of these is Prolog, a powerful, logic-based language that is
widely used in expert systems, natural language understanding,
databases, and other areas.

But we have to cover some ground before Prolog really makes sense.

For the assigned readings I selected what I thought were the most
critical passages from R&N -- we could easy spend the entire quarter
on just the chapters on logical reasoning alone!  But if you find the
material in this class (and possibly next!) either confusing or
interesting, I encourage you to go back and re-read the chapters
straight through.

FREGE SYSTEM

1. A logic
2. A set of rules for deriving new sentences from old

MODUS PONENS

Propositional logic

P, P=>Q
-------
   Q

same as:

P, ~P v Q
-------
   Q

RESOLUTION

P v R, ~P v Q
-------------
     Q

Example:
logic is cool or confusing; logic is not confusing or I am a liar;
THEREFORE logic is cool or I am a liar

IN GENERAL

P v X1 v...v Xj, ~P v Y1 v...v Yk
---------------------------------
X1 v...v Xj v Y1 v...v Yk

---------------------------------------------------
[2] Resolution Proof:

Completeness of Resolution:
If a formula is UNSATISFIABLE, there is a resolution proof of the
empty clause (false).

Want to prove F is VALID (true in all truth assignments)
Can do this with a resolution proof that ~F entails FALSE.

A proof is often written as A DIRECTED GRAPH that ends in FALSE.

Example: 
If times are good tuition goes up.          ~G v U
If times are not good tuition goes up.       G v U
Conclusion: tuition goes up.             negate conclusion: ~U

DO PROOF.

---------------------------------------------------
[3] Resolution with Variables

DPPL and Walksat only work on fully-instantiated formulas - no
variables!
These are also called GROUND formulas.

THE ADVANTAGE OF RESOLUTION IS THAT IT CAN WORK DIRECTLY ON FORMULAS
WITH VARIABLES!

TRUE FIRST-ORDER REASONING!

The classical syllogism:

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.

(all x).(man(x) => mortal(x))
CLAUSAL FORM:

~man(x) v mortal(x)        man(Socrates)
              DERIVE     
            mortal(Socrates)

The algorithm for matching terms and variables is UNIFICATION

Examples:
Unification of         and                is                where

mortal(x)          mortal(SOCRATES)    mortal(SOCRATES)     x/SOCRATES
mortal(SOCRATES)   mortal(HENRY)       does not unify!
mortal(x)          mortal(y)           mortal(x)            y/x
child(x,BILL)      child(JOHN,y)       child(JOHN,BILL)     x/JOHN, y/BILL

Resolution with variables:
1. Unify COMPLEMENTARY propositions
3. Apply substitutions from unification THROUGHOUT clauses
2. Join clauses - complementary propositions

Technical note: must always make sure clauses use different
names for their variables - rename if necessary!

Example:

All men are mortal.
The child of a mortal is mortal.
The chidren of men are mortal.

(all y,z) ((mortal(y) & child(z,y))->mortal(z))

~man(x) v mortal(x)
~mortal(y) v ~child(z,y) v mortal(z)
THEREFORE CAN CONCLUDE
~man(x) v ~child(z,x) v mortal(z)

ADD: mortal(JOHN), mortal(FRED), child(ANDY,JOHN), child(TOM,FRED).
WANT TO PROVE:  mortal(FRED)

DO REFUTATION PROOF.
NOTE: DID NOT NEED TO REASON ABOUT ANDY OR JOHN!

Unification algorithm - simple recursive algorithm in book.
Also handles complex terms with function symbols!

---------------------------------------------------

IF TIME: In class exercise:
Using the predicates:

happy(x)
person(x)
loves(x,y)
rich(x)
mother(x,y)
spouse(x,y)

People who love happy people are happy.
People who are rich are happy.
People love their mother.
People love their spouse.

Prove: Bill Gates mother-in-law is happy.