Artificial Intelligence
CS 165A
Thursday, October 25, 2007
 Knowledge and reasoning (Ch 7)
• Propositional logic
1
Who is this key historical AI figure?
• Built a calculating machine that could
add and subtract (which Pascal’s
couldn’t)
• But his dream was much grander – to
reduce human reasoning to a kind of
calculation and to ultimately build a
machine capable of carrying out such
calculations
Gottfried Leibniz (1646-1716)
• Co-inventor of the calculus
“For it is unworthy of excellent men to lose hours like slaves in the labor of
calculation which could safely be relegated to anyone else if the machine
were used.”
2
Notes
• HW#2 posted, due Tuesday
– 5x5 Tic Tac Toe game
– Can work in teams of two, but different
partner than in HW#1
– Can use your (or your team’s) code from
HW#1 as starting point
– You’ll need to use a heuristic to evaluate
positions and go down N ply in your search
so that the program returns an answer in a
“short” amount of time
 About 10 seconds maximum....
X : 2 points
O : 3 points
___________
|X|O|X|X|X|
| |O|O|X| |
| | |O|X| |
| | | |O| |
| | | | |O|
¯¯¯¯¯¯¯¯¯¯¯
3 in a row : 1 point
4 in a row : 3 points
5 in a row : 5 points
3
KB Agents
True sentences
TELL
Knowledge Base
Domain specific content; facts
ASK
Inference engine
Domain independent algorithms; can
deduce new facts from the KB
4
Thursday Quiz
1. If KB is the knowledge base and  is a logical sentence,
what is the meaning of this statement?
KB

2. Propositions P and Q are both true. Is this sentence true or
false:
P  Q
Briefly explain.
5
Syntax and semantics example
• The PQ system – Syntax
– A correct sentence must have the form
 {0 or more dashes} p {0 or more dashes} q {0 or more dashes}
– For example
--p---q-----
--p---q----
--p---q--p--
-p-q-
-p-q--
-q-p--
pq
p-q
-p--
--p-q-
--pq--
--p-p-q----
• The PQ system – Semantics
Not allowable
– # plus # equals #
– Which of the above sentence are TRUE?
LEGAL sentences and TRUE sentences are not the same thing!
6
Semantics
--p---q----Sentences
-pq-
--p--q-----
World
Semantics
Representation (abstraction)
2+3=5
Facts
1+0=1
2+2=5
Note: Facts may or may not be true
If sentence P is false, then sentence P is true
7
Inference and Entailment
• Given a set of (true) sentences, logical inference generates
new sentences
– Sentence  follows from sentences { i }
– Sentences { i } entail sentence 
– The classic example is modus ponens: P  Q and P entail what?
• A knowledge base (KB) entails sentences 

KB
• An inference procedure i can derive  from KB
KB
i

8
Inference and Entailment (cont.)
Sentences
Sentence
ENTAILS
Semantics
World
Semantics
Representation
Facts
Fact
FOLLOWS
9
Using propositional logic: rules of inference
• Inference (n.):
a. The act or process of deriving logical conclusions from premises
known or assumed to be true.
b. The act of reasoning from factual knowledge or evidence.
• Inference rules capture patterns of sound inference
– Once established, we don’t need to show the truth table every time
• Examples of inference rules:
– ((P  H)  H) P
 I.e., if ((P  H)  H) is in the KB, then we can conclude P
– ((P  Q)  P) Q
 I.e., if ((P  Q)  P) is in the KB, then we can conclude Q
10
Inference engine
• An inference engine is a program that applies inference
rules to knowledge
– Goal: To infer new (and useful) knowledge
• Separation of
– Knowledge
– Rules
Inference engine
– Control
Which rules should we apply when?
11
Inference procedures
• An inference procedure
– Generates new sentences  that purport to be entailed by the
knowledge base
…or...
– Reports whether or not a sentence  is entailed by the knowledge
base
• Not every inference procedure can derive all sentences that
are entailed by the KB
• A sound or truth-preserving inference procedure generates
only entailed sentences
• Inference derives valid conclusions independent of the
semantics (i.e., independent of the interpretation)
12
Inference procedures (cont.)
• Soundness of an inference procedure
– i is sound if whenever KB i , it is also true that KB
– I.e., the procedure only generates entailed sentences

• Completeness of an inference procedure
– i is complete if whenever KB , it is also true that KB i 
– I.e., the procedure can find a proof for any sentence that is entailed
• The derivation of a sentence by a sound inference
procedure is called a proof
– Hence, the proof theory of a logical language specifies the
reasoning steps that are sound
13
Logics
• We will soon define a logic which is expressive enough to
say most things of interest, and for which there exists a
sound and complete inference procedure
– I.e., the procedure will be able to derive anything that is derivable
from the KB
– This is first-order logic, a.k.a. first-order predicate calculus
– But first, we need to define propositional logic
14
Propositional (Boolean) Logic
• Symbols represent propositions (statements of fact,
sentences)
– P means “San Francisco is the capital of California”
– Q means “It is raining in Seattle”
• Sentences are generated by combining proposition symbols
with Boolean (logical) connectives
15
Propositional Logic
• Syntax
– True, false, propositional symbols
– ( ) ,  (not),  (and),  (or),  (implies),  (equivalent)
• Examples of sentences in propositional logic
P1, P2, etc. (propositions)
S1  S2
( S1 )
S1  S2
 S1
true
S1  S2
P1  true  ( P2  false )
S1  S2
PQ  QP
16
Propositional (Boolean) Logic (cont.)
• Semantics
– Defined by clearly interpreted symbols and straightforward
application of truth tables
– Rules for evaluating truth: Boolean algebra
– Simple method: truth tables
2N rows for N propositions
17
Propositional (Boolean) Logic (cont.)
• Make sure you know simple Boolean logic
Associative, commutative, and distributive laws
PQ  QP
(P  Q)  R  P  (Q  R)
P  (Q  R)  (P  Q)  (P  R)
P  P
(P  Q)  P  Q
(P  Q)  P  Q
DeMorgan’s Laws
P  Q  P  Q
Important!
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Basic logical equivalences
Equivalent Sentences
P  Q
PQ
P
True  P
PQ
True  P  Q
P  False
P
P
P  False
19
Satisfiability and Validity
• Is this true: ( P  Q ) ?
– It depends on the values of P and Q
– This is a satisfiable sentence – there are some interpretations for
which it is true
– In other words, it depends: it could be true or false
• Is this true: ( P  P ) ?
– No, it is never true
– This is an unsatisfiable sentence (self-contradictory) – there is no
interpretation for which it is true
• Is this true: ( ((P  Q)  Q)  P ) ?
– Yes, independent of the values of P and Q
– This is a valid sentence – it is true under all possible
interpretations (a.k.a. a tautology)
– Truth tables can test for validity
20
Propositional logic (cont.)
• AC
– The premise implies the conclusion ... or ...
 The antecedent implies the consequent
– What if A  C is always true? Example:
Then we can say that the antecedent entails the consequent.
In other words, P and H can be seen as variables – this is true for any
statements P and Q.
21
• One can say that the premise entails the conclusion
– ((P  H)  H) entails P
– ((P  H)  H) P
• Have we also shown that ((P  H)  H)
P ?
Well, yes and no....
We still don’t have a method to derive this.
But if we add this inference rule explicitly to our
system, with P and H as variables representing any two
propositions, then we do have such a procedure.
22
Using propositional logic: rules of inference
• Inference rules capture patterns of sound inference
– Once established, don’t need to show the truth table every time
– E.g., we can define an inference rule: ((P  H)  H) P for
variables P and H
• Alternate notation for inference rule   :


“If we know , then we can conclude ”
(where  and  are propositional logic sentences)
23
Inference
• We’re particularly interested in
KB

or
1,  2, …

• Inference steps
KB
1

KB, 1
2

KB, 1, 2
3

…
So we need a mechanism to do this!
Inference rules that can be applied to sentences in our KB
24
Important Inference Rules for Propositional Logic
25
Resolution Rule: one rule for all inferences
p  q, q  r
p r
Propositional calculus resolution
Remember: p  q  p  q, so let’s rewrite it as:
p  q, q  r
p  r
or
a  b, b  c
ac
Resolution is really the “chaining” of implications.
Would like to show that resolution is sound and (essentially)
complete (use Deduction Rule for proof)
26
Inference in Propositional Logic
• Three ways to answer: “Is Mary the grandparent of Ann?”
– One way:
 Find sentence P (“Mary is the grandparent of Ann”) in KB
– Another way:
 Put sentence P in the KB, and use inference rules to prove a
contradiction (false)
– Yet another way:
 Find sentence Q (“Mary is parent of Frank”), sentence R (“Frank
is the parent of Ann”), and sentence Q  R  S, where S is the
proposition “Mary is grandparent of Ann”
 Then apply modus ponens: Q  R  S, Q  R
to conclude S (“Mary is the grandparent of Ann”)
“If Mary is the parent of Frank and Frank is the parent of Ann, then Mary
is the grandparent of Ann”
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