Post on 04-Jun-2018
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Foundations ofKnowledge-based Systems
Lecture 2
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Outline
Recap Knowledge Representation
Propositional Logic
Predicate Logic
Validity and Soundness
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Recap Definition of KBS A knowledge based system (KBS) is a software system
capable of supporting the explicit representation ofknowledge in some specific competence domain and ofexploiting it through appropriate reasoning mechanismsin order to provide high-level problem-solvingperformance.
KBS is a specific, dedicated, computer-based problem-solver, able to face complex problems, which, if solved byman, would require advanced reasoning capabilities, suchas deduction, abduction, hypothetical reasoning, model-
based reasoning, analogical reasoning, learning, etc.
Typical problems Diagnosis Scheduling Planning
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Recap Components of a KBS
Knowledge Base
Problem
Domain Knowledge
Reasoning Mechanism
Working Memory
Solution Knowledge-Based
System
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Knowledge Classification Knowledge can be classified into
Priori knowledge: Universally true and cannot bedenied without contradiction. Examples are mathematical laws, logical statements.
Posteriori knowledge: Represents information that is
verified using sensory experiences. This Knowledge can be denied based on new knowledge
without the need for contradictions.
Further classification includes Procedural knowledge: Knowing how to do something.
Declarative knowledge: Knowing that something istrue or false.
Tacit knowledge: Unconsciously knowing how to dosomething.
Explicit knowledge:
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Knowledge Representation
This is the way that knowledge is stored in aprogram. This implies that
There is a systematic way to store the information.
The knowledge is coded into the program.
Knowledge representation and reasoning - thestudy of formal ways of extracting informationfrom symbolically represented knowledge
Existing computer languages can be used and the
knowledge is stored in memory.
The stored knowledge and facts can be used inreasoning.
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Knowledge Representation
Knowledge can be represented in a varietyof ways.
The predominant knowledge
representation schemes are Frames and production rules.
Connections and weights.
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Knowledge Representation Some desirable features of any knowledge
representation scheme include: Completeness:Should support the acquisition of all
aspects of the knowledge.
Conciseness: Allow efficient acquisition so that
knowledge is stored compactly and is easily retrieved. Computational efficiency:It should be possible to
use the knowledge rapidly and without the need forexcessive computation.
Transparency:Should be such that it is possible to
understand its behaviour and how it arrives atconclusions.
Explicity:The important things should be explicit butthe details suppressed but available in case it is
required in future.
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Why use special tools
Traditional languages emphasize Efficiency
Maintainability
Portability
Not representational power
Traditional language control is
Primitive
Implicit in statement ordering
Pretty much fixed at compile time Good for algorithmic work, but knowledge is
implicit not explicit.
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Why use special tools
There is no single representation scheme that embodiesall the above characteristics.
Each of the representation schemes is suitable for certaintypes of application domain.
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Types of representation schemes
Some popular representation schemes include
Rule-based schemes:Information is stored asabstract rules that have general applicability.
Learning is explicit.
Instance based models:Do not operate on explicitrules. Exhibit rule-like behavior by being exposed to aseries of examples.
Learning is implicit.
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Logic and other schemes
Logic:Extensively used in Al programs.
Main purpose of logic: The soundness or unsoundness ofarguments.
Typically, an argument consists of statements calledpropositions, from which other statement(s) calledconclusion(s) are claimed to follow.
This is the basis of propositional logic.
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Propositional Logic
Proposition:A sentence that is either true orfalse.
Example: The following are propositions:
Sam is a happy man" (1)
"All cats are good pets" (2)
Propositions, because each is either true or false.
The following phrases are not propositions:
"Amy's pet" (3)
"Oh dear me!" (4)
Statements in propositional logic are usuallyexpressed symbolically.
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Propositional Logic
Example:
The following inference:
"If Sam is a happy man then Sam is a teacher"
Could be symbolically expressed as:
A: Sam is a happy man B: Sam is a teacher
This could be expressed in propositional logic as:
if A then B
Logic notation: A B (meaning proposition Aimplies proposition B).
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Propositional Logic
This is an example of a rule of inference called modus
ponens.
This says that if propositionAis true, and the rule ofinferenceA Bis true, then Bwill also be true.
Propositions can be combined using logical connectivese.g. "If I listen to music and the room is warm then I fall asleep
Rewriting this symbolically: PropositionA: I listen to music
Proposition B: The room is warm
Proposition C: I fall asleep
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Propositional Logic
Then this can be written in logic notation as:
A B C
Connective symbols
The symbols shown in the table are used to denote someof the most common connectives used in propositionallogic.
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Propositional Logic
Symbol Meaning Interpretation-A
A B
A B
A B
Not A
A and B
A or B
A implies B
Negation. Negation of propositionAis true ifAis false and vice versaConjunction. Aand Bonly true ifA
and Bare both true, otherwise falseDisjunction. Aor Bis true ifAistrue or Bis true.Implication. IfAis true andAimplies Bis true, then Bis true. IfA
is false andAimplies Bis true thenanything goes. That is, Bcould betrue or false, since implication saysnothing about case whenAis false
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Propositional Logic
Truth table
The meanings of the connectives and their results aresummarized in the table
A B A A B A B A B1100
1010
0011
1000
1110
1011
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Predicate Logic
Propositional logic is inadequate for solving some
problems because a proposition has to be treated as asingle entity that is either true or false.
Predicate logic overcomes this by allowing a proposition
to be broken down into two components. Arguments
Predicates.
It allows the use of variables, in addition to supporting
the rules of inference derived from propositional logic(i.e. modus ponens etc.).
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Predicate Logic
Example
Consider the proposition:
Kamau has brown hair.
This could be written in predicate logic notation
as: HAS (Kamau, short hair)
In the example,
Predicate: HAS Arguments: Kamau and brown hair.
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Predicate Logic
Quantifiers in predicate logic
Predicate logic also allows for the use ofquantifiers.
This means that the language can be extended to
propositions that refer to a range of a variable. For example, consider the proposition:
Every man loves a woman.
This can be expressed in predicate logic using
quantifiers as:
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Predicate Logic
x, Man(x) y, s.t. Woman(y) Loves(x, y)
Which reads: For any object xin the world if xis a Man, then there exists an
object y, such that yis a woman and xLoves y.
Quantifiers
: The universal quantifier since it refers to all objects inthe (male) population.
: The existential quantifier since it refers to at least oneobject in the (female) population.
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Predicate Logic
Now consider the proposition: every Welshman is a Man.
This would be expressed in formal logic as:
x, Welshman(x) Man(x)
Which reads: for any object x, if xis a Welshman, then xis a man.
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Predicate Logic
Then from the two facts it can be concluded, using the
rules of inference, that the following fact must be true:
x, Welshman(x) y, s.t. Woman(y) Loves(x, y)
That is, every Welshman loves a woman.
The example seems to lead to an obvious conclusion.However, for other examples such intuitive conclusions
would be less obvious.
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Logic: Validity and Soundness
Consider the following deductive argument:
If you are in Chiromo, then you are in Nairobi
If you are in Nairobi, then you are in Kenya
Therefore if you are in Chiromo you are in Kenya
Both premises and conclusion happen to be true
statements, But if you substitute Kampalafor Nairobi theargument will have false premises.
Therefore, there are arguments that intuitively seem tobe valid in the sense that the conclusions somehow
follow from the premises, but which still have somethingmissing.
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Logic: Validity and Soundness
Validity:
A deductive argument (or argument form) is valid ifand only if it is impossible for its conclusion to be falsewhen its premises are true.
Soundness:
A deductive argument is sound if it is valid and hastrue premises.
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Advantages of formal logic
There is a set of rules called rules of inferenceby which
facts that are known to be true can be used to deriveother facts, which must also be true.
The truth of any new proposition can be checked, in a
well-specified manner, against the facts that are alreadyknown to be true. Logical inferences will only guarantee the truth of a conclusion if
the premises leading to the conclusion are also true.
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Summary
Knowledge Base Systems
Components
Types
Knowledge Classification
Knowledge Representation Propositional Logic
Predicate Logic