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Jurusan Teknik Informatika Universitas Widyatama

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Reasoning (Proposional Logic)

Pertemuan : 5

Dosen Pembina :

Sriyani Violina

Danang Junaedi

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Deskripsi

Reasoning

Propositional logic

Problems with propositional logic

Overview

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Pertemuan ini mempelajari bagaimana

memecahkan suatu masalah dengan teknik

reasoning.

Metode reasoning yang dibahas pada

pertemuan ini adalah propositional logic

Deskripsi

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Representations are Models of the world

If we are going to create programs that are intelligent, then we need to figure out how to represent models of the world

An important aspect of an AI agent is its model of the world

As humans, we carry sophisticated models of the world in our heads

They allow us to predict certain things about the future, play scenarios though in our heads to see what might happen, and not be constantly surprised by everything we see

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The Role of a Model

Represent the environment

Provide a structure for the assimilation of new knowledge

Be able to change in the light of new evidence, but not too readily

Dry run without the need for sensor input or actual actuator output

Be able to generate new facts that have not been sensed from those that have

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Reasoning

Definisi : Teknik menyelesaikan masalah dengan cara merepresentasikan masalah ke dalam basis pengetahuan (knowledge base) menggunakan logic atau bahasa formal

Metode

Untuk masalah yang memiliki kepastian

Proportional Logic

First Order Logic/Predicate Calculus

Untuk masalah yang tidak memiliki kepastian

Fuzzy Logic

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Perbedaan Reasoning dengan

Searching

Searching

Masalah direpresentasikan dalam state dan ruang masalah,

Solusi ditentukan / dihasilkan dengan menggunakan strategi searching

Reasoning

Masalah direpresentasikan dalam basis pengetahuan dan

Solusi ditentukan / dihasilkan dengan melakukan proses penalaran

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Logic

Contrariwise, continued Tweedledee, if it

was so, it might be, and if it were so, it would

be; but as it isnt, it aint. Thats logic!

Lewis Carroll

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What is a logic?

A formal language Syntax what expressions are legal (well-formed)

Semantics what legal expressions mean

in logic the truth of each sentence with respect to each possible

world.

E.g the language of arithmetic X+2 >= y is a sentence, x2+y is not a sentence

X+2 >= y is true in a world where x=7 and y =1

X+2 >= y is false in a world where x=0 and y =6

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Logic consists of a

representation

Logical constants: true, false

Proposition symbols: P, Q, R, ...

Logical connectives: & (or ^), , , ,

Parentheses: ( )

Propositional logic is an extremely simple

representation

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Models

Logicians typically think in terms of models,

which are formally structured worlds with

respect to which truth can be evaluated.

m is a model of a sentence if is true in m

M() is the set of all models of

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Propositional Logic

One way to represent or model the world is propositional logic.

It can reason about such logical implications, we introduce Propositional Logic (PL).

It allows us

to express simple truth facts like I have money or I have an iPod,

to express logical statements like If I have money then I have an iPod, and

to draw conclusions like If I have money and (if I have money then I have an iPod) then I have an iPod.

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Propositional logic (PL)

A simple language useful for showing key ideas and

definitions

User defines a set of propositional symbols, like P and Q.

User defines the semantics of each propositional symbol:

P means It is hot

Q means It is humid

R means It is raining

A sentence (well formed formula) is defined as follows:

A symbol is a sentence

If S is a sentence, then S is a sentence

If S is a sentence, then (S) is a sentence

If S and T are sentences, then (S T), (S T), (S T), and (S T) are sentences

A sentence results from a finite number of applications of the above rules

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Semantics

A proposition symbol can mean anything at all. The logical system has no understanding of its meaning.

The semantics of propositional logic specifies the meaning of the logical connectives and how they combine propositions.

A truth table can be used to specify the semantics of the logical

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Propositional Logic

Represents facts as being either true or false

Formally represented by a letter, e.g. P or Q These symbols can represent whole propositions such as Elvis is alive.

Actually refer to facts about the environment, e.g. the speed limit in town is 30mph

Single facts can be combined into sentences using Boolean operators such as (^,v )

These sentences, if true, can become facts in the Knowledge Base (KB).

A KB is said to entail a sentence if it is true in the KB

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Propositional logic

Logical constants: true, false

Propositional symbols: P, Q, S, ... (atomic sentences)

Wrapping parentheses: ( )

Sentences are combined by connectives:

...and [conjunction]

...or [disjunction]

...implies [implication / conditional]

..is equivalent [biconditional]

...not [negation] Literal: atomic sentence or negated atomic sentence

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Syntax rules for propositional

logic The constants true and false are propositions by

themselves.

A proposition symbol such as P or Q is a proposition by itself.

Wrapping parentheses around a proposition produces proposition.

A proposition can be formed by using a logical connective to combine two propositions.

Constants or proposition symbols by themselves make atomic propositions. Propositions containing a connective are called complex propositions.

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Propositional Logic - Syntax

Sentences in Propositional Logic are defined in Backus-Naur Form (BNF):

A variable symbol (P,Q,R,), and the constantsTrue, False are correct sentences.

Given correct sentences a, b: , , , ( a) is a correct sentence. (negation)

(a ^ b) is a correct sentence. (conjunction)

(a b) is a correct sentence. (disjunction)

(a b) is a correct sentence. (implication)

(a b) is a correct sentence. (equivalence)

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A BNF grammar of sentences in

propositional logic

S := ;

:= |

;

:= "TRUE" | "FALSE" |

"P" | "Q" | "S" ;

:= "(" ")" |

|

"NOT" ;

:= "AND" | "OR" | "IMPLIES" |

"EQUIVALENT" ;

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Propositional Logic - Syntax

To avoid the excessive use of parentheses, we agree to use

the following convention on the order of precedence of

operators:

(highest), ^ , V , , (lowest)

Now, we can write

((((P) V (Q ^ R)) True) S)

as

P V Q ^ R True S

Even though these sentences are ambiguous in syntax

because of their unique semantics (to be defined next) we

allow sentences like P V Q V R, P ^ Q ^ R, and P Q

R. Sentences like P Q R are not allowed.

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Propositional Logic - Syntax

A model or instantiation to a sentence in

propositional logic is an assignment of truth

values to each variable:

For the sentence P Q ^ R True S

potential models are:

m1 = { P=true, Q=true, R=false, S=false }

m2 = { P=false, Q=true, R=false, S=true}

We write a|m to denote the evaluation of a

under model m.

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Examples of PL sentences

(P Q) R

If it is hot and humid, then it is raining

Q P

If it is humid, then it is hot

Q

It is humid.

A better way:

Ho = It is hot

Hu = It is humid

R = It is raining

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Truth tables

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Truth tables II

The five logical connectives:

A complex sentence:

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Models of complex sentences

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Inference rules

Logical inference is used to create new sentences that logically follow from a given set of predicate calculus sentences (KB).

An inference rule is sound if every sentence X produced by an inference rule operating on a KB logically follows from the KB. (That is, the inference rule does not create any contradictions)

An inference rule is complete if it is able to produce every expression that logically follows from (is entailed by) the KB. (Note the analogy to complete search algorithms.)

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Sound rules of inference

Here are some examples of sound rules of inference

A rule is sound if its conclusion is true whenever the premise is true

Each can be shown to be sound using a truth table

RULE PREMISE CONCLUSION

Modus Ponens A, A B B

And Introduction A, B A B

And Elimination A B A

Double Negation A A

Unit