AndOr Graph

And/Or Graph:



Connecting Logic and

State Space

Motivations



• You have learned about FOL

– AND, OR, IF, etc.

• You have learned about state space

– tic-tac-toe game tree, Königsberg bridges,

etc.

• This lecture combines these two technologies

for logical inference using state space

representation

– And/Or Graph

Objectives



• State space representation of predicate

calculus

• What is And/Or graph representation?

• And/Or graph for car problem diagnostic

• And/Or graph for financial advisor

• And/Or graph for a simple English grammar

State space representation of predicate calculus





• Letters as nodes T

• Implications as links

•qp T T

•rp

•vq

•sr

•tr T T

•su Given: tr

•t t

By modus ponens

•s r



Propositions that are reachable from boxed

states are true propositions.

Summary: state space representation of

predicate calculus

T





T T

• Representation

– Logical expressions as states, p, q, r, …

– Inference rules as links, .

– Prolog facts and rules T T

• Correctness

– Soundness and completeness of predicate calculus

– Inference rules guarantee the correctness of conclusions

• Theorem proof

– State space search

– A Prolog query becomes a goal to be satisfied in a bottom up fashion.

Simple And/Or graph





• Or graph

•qrp









• And graph

•qr→p

And/or graph



a

b

c

aΛ bd

aΛ ce

bΛ df

f g

aΛ eh





• Is h true?

• Is h true if b is no longer true?

• What is the fastest way (CPU time) to show that X is true?

• Is q true? Closed world assumption.

Car diagnosis

Where is Fred?

7d location(fred,Z)







7a 7b 7c









7a 7b 7c 7d

Financial adviser knowledge base



1. savings_account (inadequate)

 investment(savings)

2. savings_account (adequate) 

income(adequate)  investment (stocks)

3. savings_account (adequate) 

income(inadequate) investment(combination)

4.  X amount_saved(X)   Y (dependents (Y) 

greater(X, minsavings(Y)))

 savings_account(adequate)

X is the dollar amount.

Y is the number of dependents.

minsavings is the number of dependents multiplied

by 5000.

Knowledge base cont’d



5.  X amount_saved(X)   Y (dependents (Y) 

 greater (X, minsavings(Y)))

 savings_account(inadequate)

6.  X earnings(X, steady)   Y (dependents (Y) 

greater (X, minincome(Y)))

 income(adequate)

7.  X earnings(X, steady)   Y (dependents (Y) 

 greater (X, minincome(Y)))

 income(inadequate)

Minimum income is

15,000 + (4000 * number of dependents)

8.  X earnings(X, unsteady)  income(inadequate)

Applying it to an individual



A person whose

9. amount_saved(20000)

10. earnings(30000, steady)

11. dependents (2)

What should he do about investment?

And/or graph searched by the financial advisor



A person whose

9. amount_saved(20000)

10. earnings(30000, steady)

11. dependents (2)

What should he do about

investment?

F T



T

A small English grammar

Summary



• And/Or graph combines first-order logic with

state space representation.

• Nodes are logical expressions.

• Arcs are logical inferences.

• Logical deduction is search in this space.

• Use this technology to engineer AI systems

for many applications such as diagnostics.