PROPOSISTION LOGIC IN ARTIFICIAL INTELLIGENCE

Propositional logic is a branch of mathematics that studies the relationships between complete statements (propositions), which are either true or false. It uses symbols to represent these propositions and logical connectives (like AND, OR, NOT) to build complex statements, analyzing whether an argument is valid based on its logical structure. It is a foundational system for fields like mathematics, computer science, and philosophy, used to clarify reasoning and remove ambiguity from language.

1. Notes 8:
Predicate logic and inference
ICS 270a Spring 2003

2. Outline
New ontology

objects,relations,properties,functions.
New Syntax

Constants, predicates,properties,functions
New semantics

meaning of new syntax
Inference rules for Predicate Logic (FOL)

Resolution

Forward-chaining, Backword-chaining

unification
Readings: Nillson’s Chapters 15-16, Russel and Norvig Chapter
8, chapter 9

3. Propositional logic is not
expressive
Needs to refer to objects in the world,
Needs to express general rules

On(x,y)  ~ clear(y)

All man are mortal

Everyone who passed age 21 can drink

One student in this class got perfect score

Etc….
First order logic, also called Predicate calculus allows more
expressiveness

4. Syntax
Components

Infinite set of object constants, alphanumerc strings
Bb, 12345, Jerusalem, 270a, Irvine

Function constants of all aritys (convention: start with lower case)
motherOf, times, greaterThan, color

Relation constants, Predicates of all aritys (start with capital)
B21, Parent, multistoried, XYZ, prime
Terms

An object constant is a term

A function constant of arity n followed by n terms in parenthesis
is a term (called functional expression)
motherOf(Silvia), Sam, leftlegof(John), add1(5), times(5,plus(3,6))

5. Wffs sentences
An atomic sentence is formed from a predicate (relation)
symbol followed by a parenthesized list of symbols

Atoms: a relation constant (predicate) followed by n terms in
parenthesis seperated by commas.
 GreaterThan(6,3), Q, Q(A,B,C,D), Brother(Richard,John)
Married(FatherOf(Richard),MotherOf(John))
Complex propositional wffs:
Use logic connectives:

Brother(Richard,John) /\ Brother(John,Richard)

Older(John,30) V Younger(John,30)

Older(John) --> ~Younger(John,30) V P
 ,,,

6. Semantics: Worlds
The world consists of objects that has properties.

There are relations and functions between these objects

Objects in the world, individuals: people, houses, numbers,
colors, baseball games, wars, centuries
Clock A, John, 7, the-house in the corner, Tel-Aviv

Functions on individuals:
father-of, best friend, third inning of, one more than

Relations:
brother-of, bigger than, inside, part-of, has color, occurred after

Properties (a relation of arity 1):
red, round, bogus, prime, multistoried, beautiful

7. Semantics: Interpretation
An interpretation of a wff is an assignment that maps

object constants to objects in the worlds,

n-ary function symbols to n-ary functions in the world,

n-ary relation symbols to n-ary relations in the world
Given an interpretation, an atom has the value “true” in case
it denotes a relation that holds for those individuals denoted
in the terms. Otherwise it has the value “false”

Example: A,B,C,floor,
On, Clear

World:

On(A,B) is false, Clear(B) is true, On(C,F1) is true…

9. Semantics: Models
An interpretation satisfies a well-formed formula (wff)
(sentence) if the wff has the value “true” under the
interpretation.
An interpretation that satisfies a wff is a model of that
wff
Any wff that has the value “true” under all
interpretations is valid
Any wff that does not have a model is inconsistent or
unsatisfiable
If a wff w has a value true under all the models of a set
of sentences delta then delta logically entails w

10. Example of models
The formulas:
On(A,F1)  Clear(B)
Clear(B) and Clear(C)  On(A,F1)
Clear(B) or Clear(A)
Clear(B)
Clear(C)
Possible interpretations which are models:
On = {<B,A>,<A,floor>,<C,Floor>}
Clear = {<C>,<B>}

11. Quantification
Universal and existential quantifiers allow expressing
general rules with variables
Universal quantification

All cats are mammals


It is equivalent to the conjunction of all the sentences obtained
by substitution the name of an object for the variable x.
Syntax: if w is a wff then (forall x) w is a wff.
)( )( xMammalxCatx 
,,,,
)()(
)()(
)()(



FelixMammalFelixCat
RebbekaMammalRebbekaCat
SpotMammalSpotCat

12. Quantification: Existential
Existential quantification : an existentially quantified
sentence is true in case one of the disjunct is true
Equivalent to disjunction:

We can mix existential and universal quantification.

)(),( xCatspotxxSister 
d)...Cat(Richarhard,Spot)Sister(Ric
Cat(Felix)ix,Spot)Sister(Fel
a)Cat(Rebeccecca,Spot)Sister(Reb
Cat(Spot)Spot)tSister(Spo



,

13. Useful Equivalences
De Morgan laws
Inference rules:

Universal instantiation: foall x f(x) |-- f(alpha)

Existential generalization: from f(alpha) |-- exist x f(x)

14. Modeling a domain;
Conceptualization
The kinship domain:

object are people

Properties include gender and they are related by relations
such as parenthood, brotherhood,marriage

predicates: Male, Female (unary) Parent,Sibling,Daughter,Son...

Function:Mother Father

15. Resolution in the predicate calculus
Unification

Algorithm unify
Using unification in predicate calculus resolution
Completeness and soundness
Converting a wff to clause form
The mechanics of resolution
Answer extraction

16. Inference Rules in FOL
All the inference rules for propositional logic applies and additional
three rules that use substitution; assigning constants to variables

Subst(x/constant)

Subst({x/Sam,y/Pam},Likes(x,y)) = Likes(Sam,Pam)
Universal elimination

From fromall x Likes(x,IceCream) we can substitute {x/Ben} and infer
Likes(Ben,IceCream)
Existential Elimination

For any sentence and for any symbol k that does not appear elsewhere
in the knowledge-base
exists x Kill(x,Victim) we can infer Kill(Murderer,victim)

As long as Murdered soes not appear anywhere
Existential introduction

From Likes(Jerry,IceCream) we can infer exists x Likes(x,IceCream)

17. Predicate calculus resolution
Suppose the 
1 and 
2 are two clauses represented as set of
literals. If there is an atom  in 
1
and a literal ~  in 
2
such
as  and  have a common unifier 
Then these two clauses have a resolvent row. It is obtained
by applying the substitution  to 
1
 
2
leaving out
complementary literals
Example:
xyfsubst
CxRAxPCBQAyfP


)(
)},(),({)},,()),(({

18. Converting to clause form
),(
)27,(
)28,()27,(
)(),(
),()28,()27,()()( ,
ABS
BI
AIAI
BPAP
yxSyIxIyPxPyx




19. Example: Answer Extraction

20. Example: Resoluction Refutation
Prove I(A,27)

21. Rule-Based Systems
Rule-based Knowledge Representation and Inference

combines rules
if DOG(x) then TAIL(X)

with working memory,
DOG(SPOT)
DOG(LUNA)

is not a formal knowledge representation scheme like logic
somewhat like a combination of propositional and FOPL
uses simple methods for inference
often combined with other mechanisms such as inheritance
motivated by practical use rather than theoretical properties
also has some basis in cognitive models

22. Forward Chaining Procedure
While (rules can still fire or goal is not reached)
for each rule in the rulebase which has not fired
1. try to bind premises by matching to
assertions in WM
2. if all premises in a rule are supported then assert
the conclusion and add to WM
Repeat 1. and 2. for all matching/ instantiation alternatives
end
continue

23. Example: Zoo Rulebase
Want to identify animals based on their characteristics
could have rules like

if A1 and A2 and A3.....and A27 then animal is a monkey

i.e., one very specific rule per animal
Better to use intermediate concepts

e.g., the intermediate concept of a mammal

if ?x is a mammal and B1...and B3 then ?x is a monkey
Why are intermediate concepts useful?

rules are easier to create

rulebase is easier to maintain

easier to perform inference

however: finding good intermediate concepts is non-trivial!

24. Where do the rules come from ?
Manual Knowledge Acquisition

interview experts

expert describes the rules

process of translating expert knowledge into a formal representation is also
known as knowledge engineering

this is notoriously difficult
experts are often very poor at explicitly describing what they do
e.g., ask Michael Jordan how he plays basketball or ask Tiger Woods how he plays golf
Automated Knowledge Acquisition

derive rules automatically from formal specifications
e.g., by automatic analysis of solution manuals

machine learning
learn rules from data

25. Inference Network Representation
We can represent forward chaining with rules as a network:
has hair
eats meat
tawny color
black stripes
has a tail
is a tiger
is a carnivore
is a mammal

26. Inference using Backward Chaining
Establish a hypothesis

e.g., Tony is a tiger

a hypothesis is as assertion whose truth we wish to test
i.e., assert the hypothesis
see if that is consistent with other data
Work backwards, i.e.,

match the hypothesis to the conclusion of the rules

look at the premise conditions of the rule
if these are unknown,
declare these as intermediate hypotheses

backward chain recursively (depth-first manner)
Can use search algorithms to control how backward chaining proceeds

27. Inference Network Representation of Backward
Chaining
•We can represent backward chaining as a network:
has hair
eats meat
tawny color
black stripes
is a tiger
is a carnivore
is a mammal
forward pointing eyes
has claws
has teeth

28. Backward Chaining Procedure
While (unsupported hypotheses exist or goal is not reached)
for each hypothesis H
for each rule R whose conclusion matches with H
try to support each rule’s premises by
1. matching assertions in WM
2. recursively applying backward chaining
if rule’s premises are supported
conclude that H is true
end
end
continue

29. Forward Chaining or Backward
Chaining:
which is better?
Which direction is better depends on the circumstances
4 main factors of relevance

the number of possible start states and number of possible goal
states
smaller is better

the branching factor in each direction
smaller is better

what type of explanation the user wants
more natural explanations may be important

the typical action which triggers a rule
i.e., data-driven or hypothesis-driven?

30. What is a rule-based expert system
(ES)?
ES performs a task in limited domains that require human expertise

medical diagnosis; fault diagnosis; status monitoring; data
interpretation; mineral exploration; credit checking; tutoring; computer
configuration
ES solves problems by application of domain specific knowledge
rather than weak methods.
Domain knowledge is acquired by interviewing human experts
ES can explain a problem solution in human-understandable terms
ES cannot operate in situations that call for common sense

31. Examples of Expert Systems
1965 DENDRAL Stanfordanalyze mass spectrometry data
1965 MACSYMA MIT symbolic mathematics problems
1972 MYCIN Stanforddiagnosis of blood diseases
1972 Prospector SRI Mineral Exploration
1975 Cadeceus U. of PITTInternal Medicine
1978 DIGITALIS MIT digitalis therapy advise
1979 PUFF Stanfordobstructive airway diseases
1980 R1 CMU computer configuration
1982 XCON DEC computer configuration
1983 KNOBS MITRE mission planning
1983 ACE AT&T diagnosis faults in telephone cables
1984 FAITH JPL spacecraft diagnosis
1986 ACES Aerospacesatellite anomaly diagnosis
1986 Cambpelldiagnose cooker malfunctions
1986 DELTA/CATS GE diagnosis of diesel locomotives
1987 AMEX credit authorization
1992 MAX NYNEX telephone network troubleshooting
1995 CaltechPacBell network management
1997 UCI planning drug treatment for HIV

32. Mycin: Diagnosis and treatment of
blood diseases
RULES
PREMISE ($AND (SAME CNTXT GRAM GRAMNEG)
(SAME CNTXT MORH ROD)
(SAME CNTXT AIR AEROBIC))
ACTION: (CONCLUDE CNTXT CLASS ENTEROBACTERIACEAE .8)
If the stain of the organism is gramneg, and
the morphology of the organism is rod, and
the aerobicity of the organism is aerobic
THEN there is strongly suggestive evidence (.8) that
the class of organism is enterocabateriaceae
DATA
ORGANISM-1:
GRAM = (GRAMNEG 1.0)
MORP = (ROD .8) (COCCUS .2)
AIR = (AEROBIC .6)(FACUL .4)

33. The XCON Application
A rule-based expert system

expert in the sense that the rules capture expert design knowledge

addresses a design/configuration problem
Digital Equipment Corporation (or Compaq by now!)

XCON = rule-based expert system for computer configuration

decides how peripherals are configured on new orders

has 10,000 rules

developed in early 1980’s

based on “reactive rule-based systems”
if antecedents then action
forward chaining in style

estimated to have saved DEC several hundred million $’s

34. Limitations of Rule-Based
Representations
Can be difficult to create

the “knowledge engineering” problem
Can be difficult to maintain

in large rule-bases, adding a rule can cause many unforeseen
interactions and effects => difficult to debug
Many types of knowledge are not easily represented by rules

uncertain knowledge: “if it is cold it will probably rain”

information which changes over time

procedural information (e.g. a sequence of tests to diagnose a
disease)

35. Summary
Knowledge Representation methods

formal logic

rule-based systems (less formal)
Rule-based representations are composed of

Working memory:

Rules
Inference occurs by

forward or backward chaining
Rule-based expert systems

useful for certain classes of problems which do not have direct algorithmic solutions

have their limitations