PLANNING AND MACHINE LEARNING
Reasoning is the act of deriving a
conclusion from certain premises using a given methodology.
Reasoning is a process of thinking;
reasoning is logically arguing; reasoning is drawing inference.
When a system is required to do
something, that it has not been explicitly told how to do, it must reason. It
must figure out what it needs to know from what it already knows.
Many types of Reasoning have long
been identified and recognized, but many questions regarding their logical and
computational properties still remain controversial.
The popular methods of Reasoning
include abduction, induction, model-based, explanation and confirmation. All of
them are intimately related to problems of belief revision and theory
development, knowledge assimilation, discovery and learning.
1. Reasoning
Any
knowledge system to do
something, if
it has not been
explicitly told
how to do it then it must reason.
The system must figure out what it
needs to know from what it already knows. Example
If we know : Robins are birds. All
birds have wings. Then if we ask : Do robins have wings?
Some reasoning (although very
simple) has to go on answering the
question.
Reasoning is the act of deriving a conclusion from certain premises using a given methodology.
Any knowledge
system must reason, if it is required to do something which has not been told
explicitly .
For
reasoning, the system must find out what it needs to know from what it already
knows.
Example :
If we know : Robins are birds.
All birds have wings
Then if we ask: Do robins have wings?
To answer this question - some
reasoning must go.
areas:
Mathematical Reasoning – axioms, definitions, theorems, proofs
Logical Reasoning – deductive, inductive, abductive
Non-Logical Reasoning – linguistic , language
These three areas of reasoning, are
in every human being, but the ability level depends on education, environment
and genetics.
The IQ (Intelligence quotient) is the summation of mathematical
reasoning skill and the logical reasoning.
The EQ (Emotional Quotient) depends mostly on
non-logical reasoning capabilities.
Note : The Logical Reasoning is
of our concern in AI
Logical Reasoning
Logic is a language
for reasoning. It is a collection of
rules called Logic arguments, we use when doing logical reasoning.
Logic reasoning is the process of drawing conclusions from premises using rules of inference.
The study of logic is divided into
formal and informal logic. The formal logic is
sometimes called
symbolic logic.
Symbolic logic is the study of |
symbolic abstractions |
(construct) that |
capture the formal features of |
logical inference by a |
formal system. |
Formal system consists of two components, a
formal language plus
a set of inference rules. The formal system has axioms.
Axiom is
a sentence that is always true within the system.
Sentences are derived using the system's axioms and rules of derivation are called theorems.
Formal Logic
The Formal logic is the study of
inference with purely formal content, ie. where content is made explicit.
Examples - Propositional logic and
Predicate logic.
Here the logical arguments are a
set of rules for manipulating symbols. The rules are of two types
◊ Syntax rules : say how to build meaningful
expressions.
Inference
rules : say how to obtain true formulas from other true formulas.
Logic also needs semantics, which says how to assign meaning
to expressions.
Informal Logic
The Informal logic is the study of
natural language arguments.
The analysis
of the argument structures in ordinary language is part of informal logic.
The focus
lies in distinguishing good arguments (valid) from bad arguments or fallacies
(invalid).
■ Formal Systems
Formal systems can have following
three properties :
‡ Consistency : System's theorems do
not contradict.
‡ Soundness : System's rules of derivation will never
infer anything false, so long as
start is with only true premises.
Completeness : There are no true sentences in the system that cannot be proved using the derivation rules of the system.
System Elements
Formal systems consist of following
elements :
A finite set of symbols for constructing formulae.
A grammar, is a way of constructing well-formed formulae (wff).
A set of axioms; each axiom has to be a wff.
A set of inference rules.
A set of theorems.
A well-formed formulae, wff,
is any string generated by a grammar. e.g., the sequence of symbols ((α → β
)
→ (¬ β → ¬
α )) is a WFF because it is grammatically correct in
propositional logic.
■ Formal Language
A formal language may be viewed as
being analogous to a collection of words
or a collection of sentences.
In computer science, a formal
language is defined by precise mathematical or machine process able formulas.
A formal language L
is characterized as a set F of finite-length sequences of elements drawn from a
specified finite set A of symbols.
Mathematically, it is an unordered
pair L = { A, F }
If
A
is words
then the set A is called alphabet
of L, and the elements of F are called words.
‡ If A is sentence
then the set A is called the
lexicon or vocabulary of F, and the elements of F are then called sentences.
‡ The mathematical theory that
treats formal languages in general is known as formal language theory.
Uncertainty in Reasoning
■
The world is an uncertain place;
often the Knowledge is imperfect which causes uncertainty. Therefore reasoning must be able
to operate under uncertainty.
■
AI systems must have ability to
reason under conditions of uncertainty.
Monotonic Logic
Formal logic is a set of rules for
making deductions that seem self evident. A Mathematical logic formalizes such
deductions with rules precise enough to program a computer to decide if an
argument is valid, representing objects and relationships symbolically.
Examples
Predicate logic and the
inferences we perform on it.
All humans are mortal. Socrates
is a human. Therefore Socrates is mortal.
In monotonic reasoning if we
enlarge at set of axioms we cannot retract any existing assertions or axioms.
Most formal logics have a monotonic consequence relation, meaning that adding a formula to a
theory never produces a reduction of its set of consequences. In other words, a
logic is monotonic if the truth of a proposition does not change when new
information (axioms) are added. The traditional logic is monotonic.
‡ In mid 1970s, Marvin Minsky and
John McCarthy pointed out that pure classical logic is not adequate to
represent the commonsense nature of human reasoning. The reason is, the human
reasoning is non-monotonic in nature. This means, we reach to conclusions from
certain premises that we would not reach if certain other sentences are included
in our premises.
‡ The non-monotonic human reasoning
is caused by the fact that our knowledge about the world is always
incomplete and
therefore we are fo ed to reason in the absence of
complete information. Therefore we often revise our conclusions, when new
information becomes available.
Thus, the need for non-monotonic reasoning in
AI was recognized, and several formalizations of
non-monotonic reasoning.
Only the non-monotonic logic
reasoning is presented in next few slides.
Non-Monotonic Logic
Inadequacy of monotonic logic for
reasoning is said in the previous slide. A monotonic logic cannot handle :
Reasoning by default : because
consequences may be derived only because of lack of evidence of the contrary.
Abductive reasoning : because consequences
are only deduced as most likely explanations.
Belief revision : because new
knowledge may contradict old beliefs.
A non-monotonic logic is a formal
logic whose consequence relation is not monotonic. A logic is non-monotonic if
the truth of a proposition may change when new information (axioms) are added.
‡ Allows a statement to be
retracted.
‡ Used to formalize plausible
(believable) reasoning.
Example 1 :
Birds typically fly.
Tweety is a bird.
--------------------------
Tweety (presumably) flies.
‡
Conclusion of non-monotonic argument may not be correct.
Example-2 : (Ref. Example-1)
If Tweety is a penguin, it is
incorrect to conclude that Tweety flies. (Incorrect because, in example-1,
default rules were applied when case-specific information was not available.)
All non-monotonic reasoning are
concerned with consistency. Inconsistency is resolved, by removing the relevant
conclusion(s)
derived by default rules, as shown
in the example below.
Example -3 :
The truth value (true or false), of propositions such as "Tweety is a bird" accepts default that is normally true, such as "Birds typically fly". Conclusions derived was "Tweety flies". When an inconsistency is recognized, only the truth value of the last type is changed.
2. Different Methods of Reasoning
Mostly three kinds of logical
reasoning: Deduction,
Induction, Abduction.
Deduction
Example:
"When it rains, the grass gets wet. It rains. Thus, the grass is
wet."
This means in determining the
conclusion; it is using rule and its precondition to make a conclusion.
Applying
a general principle to a special case.
Using
theory to make predictions
Usage:
Inference engines, Theorem provers, Planning.
Induction
Example:
"The grass has been wet every time it has rained. Thus, when it rains, the
grass gets wet."
This means in determining the
rule; it is learning the rule after numerous examples of conclusion following
the precondition.
Deriving
a general principle from special cases
From
observations to generalizations to knowledge
Usage:
Neural nets, Bayesian nets, Pattern recognition
■ Abduction
.
Example: "When it rains, the
grass gets wet. The grass is wet, it must have rained."
Means determining the
precondition; it is using the conclusion and the rule to support that the
precondition could explain the conclusion.
Guessing that some general
principle can relate a given pattern of cases
Extract hypotheses to form a
tentative theory
Usage: Knowledge discovery,
Statistical methods, Data mining.
Analogy
Example:
"An atom, with its nucleus and electrons, is like the solar system, with
its sun and planets."
Means analogous; it is
illustration of an idea by means of a more familiar idea that is similar to it
in some significant features. and thus said to be analogous to it.
finding a
common pattern in different cases
usage:
Matching labels, Matching sub-graphs, Matching transformations.
Note: Deductive reasoning and
Inductive reasoning are the two most commonly used explicit methods of
reasoning to reach a conclusion.
More about different methods of
Reasoning
■ Deduction Example
Reason from facts and general
principles to other facts.
Guarantees that the conclusion is
true.
Modus Ponens : a valid form of
argument affirming the antecedent.
If it is rainy, John carries an
umbrella
It is rainy / John carries an
umbrella.
◊ If p then q
p/q
Modus Tollens : a valid form of argument denying the consequent.
If it
is rainy, John carries an umbrella
John does not carry an umbrella /
It is not rainy
◊ If p then q
not q / not p
Induction Example
Reasoning from many instances to
all instances.
‡ Good Movie
Fact : You have liked all movies starring Mery.
Inference You will like
her next movie.
‡ Birds
Facts: Woodpeckers,
swifts, eagles, finches have four toes on each foot.
Inductive Inference All
birds have 4 toes on each foot.
(Note: partridges have only 3).
‡ Objects
Facts : Cars, bottles, blocks fall if not held up.
Inductive Inference If
not supported, an object will fall.
(Note: an unsupported helium balloon will rise.)
‡ Medicine
Noted : People who had cowpox did not get smallpox.
Induction : Cowpox prevents smallpox.
Problem : Sometime inference is
correct, sometimes not correct.
Advantage : Inductive inference may
be useful even if not correct. It generates a proposition which may be
validated deductively.
Abduction Example
Common form of human reasoning–
"Inference to the best explanation".
In Abductive reasoning you make an
assumption which, if true, together with your general knowledge, will explain
the facts.
‡
Dating
Fact: Mary asks John
to a party.
Abductive Inferences : Mary likes John.
John is Mary's last choice.
Mary wants to make someone else jealous.
‡ Smoking
house
Fact: A large
amount of black
smoke is coming from a home.
Abduction1: the house
is on fire.
Abduction2: bad cook.
‡ Diagnosis
Facts: A thirteen
year-old boy has a sharp
pain in his right side, a fever, and a high white blood count.
Abductive : inference Appendicitis.
Problem: Not always correct; many explanations possible.
Advantage : Understandable conclusions.
Analogy Example
Analogical Reasoning yields
conjectures, possibilities.
If A is like B in some ways, then
infer A is like B in other ways.
‡ Atom and Solar System
Statements: An atom, with its nucleus and
electrons, is like the
solar system, with its sun and
planets.
Inferences: Electrons travel around the
nucleus.
Orbits are ci ular.
Orbits are all in one plane.
Electrons have little people
living on them.
Idea: Transfer information from known
(sou e)
to unknown (target).
‡ Sun and Girl
Statement: She is like the sun to me.
Inferences: She lights up my life.
She gives me warmth.
? She is gaseous.
? She is spherical.
‡ Sale man Logic
Statement: John has a fancy car and a pretty
girlfriend.
Inferences: If Peter buys a fancy car,
Then Peter will have a pretty
girlfriend.
Problems : Few analogical inferences are
correct
Advantage : Suggests novel possibilities. Helps to
organize information.
In many problem domains it is not
possible to create complete, consistent models of the world. Therefore agents
(and people) must act in uncertain worlds (which the real world is). We want an
agent to make rational decisions even when there is not enough information to
prove that an action will work.
Uncertainty is omnipresent because
of
Incompleteness
Incorrectness
Uncertainty in Data or Expert
Knowledge
Data
derived from defaults/assumptions
Inconsistency
between knowledge from different experts.
“Best
Guesses”
Uncertainty in Knowledge
Representation
Restricted
model of the real system.
Limited
expressiveness of the representation mechanism.
Uncertainty in Rules or Inference
Process
Incomplete
because too many conditions to be explicitly enumerated
Incomplete
because some conditions are unknown
Conflict
Resolution
4. Reasoning and KR
To certain extent, the reasoning
depends on the way the knowledge is represented or chosen.
A good
knowledge representation scheme allows easy, natural and plausible (credible)
reasoning.
Reasoning methods are broadly identified as :
Formal reasoning: Using basic
rules of inference with logic knowledge representations.
Procedural reasoning: Uses procedures that specify how to perhaps
solve sub problems.
Reasoning by analogy : This is as
Human do, but more difficult for AI systems.
Generalization and abstraction:
This is also as Human do; are basically learning and understanding methods.
Meta-level reasoning : Uses
knowledge about what we know and ordering them as per importance.
Note : What ever may be the
reasoning method, the AI model must be able to reason under conditions of
uncertainty mentioned before.
5. Approaches to Reasoning
There are three different
approaches to reasoning under uncertainties.
Symbolic reasoning
Statistical reasoning
Fuzzy logic reasoning
The first two approaches are
presented in the subsequent slides.
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