Logic and artificial intelligence Nils J. Nilsson

Logic and artificial
intelligence
Nils J. Nilsson
Presented by
Ida Sofie Gebhardt Stenerud
Ida Kokkersvold
Logical approach
In the future: successful AI systems
will be a combination of methods
Important to understand the different
approaches
This article: the logical approach to AI
The article contains 4 main
topics
Logical approach is based
on three theses
Relate to three theses about the role of
knowledge in intelligent systems
Theoretical foundation
Important observations
Challenging problems for AI and attempts to
solve them
Thesis 1 - Intelligent machines will have a knowledge
of their environment
Thesis 2 - The most versatile intelligent machines will
represent much of their knowledge about their
environment declaratively
Thesis 3 - For the most versatile machines, the
language in which declarative knowledge is
represented must be at least as expressive as firstorder
predicate calculus

Thesis 2
Thesis 3
Declarative knowledge: stored explicitly in form of
sentences in some language
The declarative knowledge and the component that
perform reasoning process is separate
Easier to modify
Natural language too ambiguous: depends
too much on the context
Many languages has limited expressive
power
Can be true in wider context than specific programs
The declarative knowledge must be context free
First-order predicate does have these
limitations and meet the minimal
representational requirements
With first-order predicate
you are able to
Meaning of sentences
say that one of two facts are true: disjunction
say that something is not true: negations
say that all the members of a class has a property
without explicitly listing them: universally quantifier
say that some of the member of a class has a property
without listing which one: existentially quantifier
The designer have to say something
about the meaning of the sentences
for the machine to be able to believe
something
(∀x) Bird(x) Fly(x)
(∀x) Fgkg(x) Dfk(x)

Model of how the machine
interacts with the real world
Logical approach and
functions
The state of the machines are sentences
M’ and W’ is a set of states
S’ is a set of inputs and A’ is a set of outputs
See: maps W’ into S’
Mem: maps S’ x M’ into M’
Act: maps S’ x M’ into A’
Effect: maps A’ x W’ into W’
Mem transforms the sentences into another
set of sentences
Mem(previous machine state + machine
input) = new machine state
Act is a function of sentences that produces
the machine’s actions
Act(previous machine state + machine
input) = machine actions
Logical approach and
functions
Effect(actions of machine + old world
state ) = new world state
Conceptualization
Suppose the designer think of the world a a
finite-state machine consisting of objects,
functions and relations
The designer have to guess what the
world’s objects, functions and relations are
Conceptualization describes this guess

Conceptualization
Logic and conceptualization
The designer need only to invent a
conceptualization that is good enough
The world is invented: can invent any
thing that makes the machine work
Conceptualization described as a set of sentences
World object: create an object constant
World relation: create a relation constant
World function: create a function constant
Modify conceptualization when
noticed deficient
Uses the syntax of predicate calculus
Example: (∀x) Bird(x) Fly(x)
Interpretation
Model
Use interpretation of sentences to represent
knowledge
Assignment of relation, object and function
to each relation, object and function
constant
Procedure for assigning values T and F to
each closed formula
Any interpretation for which all of the
sentences is evaluates to T is called a
model of the set of sentences
Compose a set of sentences in the
language such that the interpretation
of those sentences is a model of the
set of sentences.

Knowledge base
The sentences of the machine ∆
Initial state: ∆0
Mem produces ∆1, ∆2… ∆i,….
The designer must provide sufficient
sentences in the knowledge base such that
is model is limited
Procedurally via
declaratively
Procedurally: act and mem are highly
specified to the task the machine is
expected to perform
∆
Declaratively: act and mem are
general purpose and largely
independent of the application.
Logically entails
If all the models of ∆ are also models
of a sentence Φ, we say that ∆
logically entails Φ and write ∆╞ Φ.
Want mem to add sentences to ∆ that
are logically entailed by ∆
Rule of inference and
deduction
Rule of inference: any computation on a set
of sequences that produces new sentences.
Ex: modus ponens. Given AB, A will give
us B
If Ψ can be derived from ∆ by a sequence
of applications of rules of inference, we say
that can be deduced from ∆ and write ∆╞ Ψ

Sound and complete
A rule of inference is called sound if rules of
inference have the property of that if ∆╞ Φ,
then ∆╞ Φ.
A set of inference rules that has the
property that if ∆╞ Φ then the rules will
eventually produce Φ, is called complete.
An important component of mem is sound
inference
Comments on the logical
approach
How to model the physical world Objects,
functions, relations
How to model
• My intentions
• My plans
• Change
• Ect
Comments on the logical
approach
Sound and unsound
inference
The hard problem is to express
commonsense knowledge.
Deciding the language is easier than
deciding what to say!
Example
• A = ”It is raining outside”
• B = ”I will become wet”
Logic inference
1) A
2) A B
3) B conclusion

Sound and unsound
inference
Nilssons argument:
Human reasoning
• It is raining..
• So it seems like I will get wet
• But what if I use my umbrella?
• And what if i stay inside?
How can a computer know infere this using
logic??
We can use more than one model
• We can add a model that knows about
umbrellas, staying inside…
We can use circumscription
• will be explained later..
Generalization
Generalization using logic –
Sound inference!
Cat(Cat_1) ^ Eyes(Cat_1 , yellow)
Cat(Cat_2) ^ Eyes(Cat_2 , yellow)
Cat(Cat_3) ^ Eyes(Cat_3 , yellow)
Cat(Cat_4) ^ Eyes(Cat_4 , yellow)
Human generalization:
”All cats have yellow eyes”
Add the following statements:
(Ax)(Ey) (Cat(X) Eyes (x, y))
• All cats have some color on their eyes
(Ax)(Ay)(Az) (Eyes(x,y) ^ Eyes(x,z) (y=z))
• All cats have the same color on their eyes

Generalization using logic –
Sound inference!
Logic and eficciency
Critisism: Logic is to inefficient to use in AI
Now we can soudly conclude that
(Ax) (Cat(x) Eyes(x, yellow))
Nilssons answer:
1) Several programs do efficient theory
proval
2) For other purposes, convert to for
example LISP
Logic in LISP
Logic in LISP
Example: All cats has fur.
Logic
• (Ax) ( Cat(x) Fur(x) )
• Cat(myCat)
In LISP:
• (Cat myCat)
• (or (not(Cat myCat)) (Fur myCat) )
How to deduce Fur(myCat) in LISP?
1) Logical deduction rules
2) Semantic attachment
Remember: (a b) equivalent with (~a v b)

Meta-knowledge
Exceptions in logic
We can reason about knowledge, as
well as reason with it
• Can this be done in logic?
Solution:
• Use more than one model
• Also done in science!
• Use meta-theories to decide which
model to use!
” If the fuel tank is empty and I turn the
key then my car will start“
• What if the tank is full of water?
• What if the battery is empty?
We want a simple conceptualisation,
even though we know that the world is
detailed
What to do about
exceptions?
Solution 1: Ignore them
• A scientific model is never accurate
Solution 2: Non-monotonic reasoning
Solution 3: Curcumscription
Non-monotonic reasoning
Regular logic
1. ~ Empty-tank(X) ^ Turn_key(X) Start(x)
2. ~ Empty_tank(myCar)
3. Turn_key(myCar)
4. start(myCar) conclusion
Non-monotonic reasoning:
introduce Full_of_water(myCar) will give ~ Start(x)

Circumscription
Challenges for logic
Introduce the AB(X) predicate
~ Empty_tank(x) ^ Turn_key(x) ^ ~AB(x) Start(x)
(Ax) AB(x) = [Full_of_water(x), Battery_empty(x)]
Change: transition between states
• At (Person, Place, Time)
• At (Ida, Home, 08.00) At (Ida, School, 10.00)
• How does states change by action?
• The frame problem: How to specify
aspects of the world that does not
change?
Other challenges
Fuzzy logic
• At( Ida, School, 10.00, 97%)
• How to use this in reasoning?
Any questions?
Embedded systems:
• Reacting to stimuli
• Reasoning
• Responding to the world
• Solution:
• Action networks and different levels of
reasoning