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Year 20 1 3 LECTURE 07 First Order Logic 2 Introduction to FirstOrder Logic Syntax and Semantics of FirstOrder Logic Using FirstOrder Logic Proof by Resolution Knowledge Engineering in FirstOrder Logic ID: 760356

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Course : T0264 – Artificial IntelligenceYear : 2013

LECTURE 07

First Order

Logic

Slide22

Introduction to First-Order LogicSyntax and Semantics of First-Order LogicUsing First-Order LogicProof by ResolutionKnowledge Engineering in First-Order LogicSummary

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Outline

Slide3Whereas propositional logic assumes the world contains facts, first-order logic (like natural language) assumes the world contains:Objects: people, houses, numbers, colors, baseball games, wars, …Relations: red, round, prime, brother of, bigger than, part of, comes between, …Functions: father of, best friend, one more than, plus, …

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T0264 - Artificial Intelligence

Introduction to

First-Order Logic

Slide4 = For all ; [e.g : every one, every body, any time, etc] = There exists ; [e.g : some one, some time, etc] = Implication ; [ if … then ….] = Equivalent ; biconditional [if … and … only … if …] = Not ; negation = OR ; disjunction = AND ; conjunction

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4

Standard Logic Symbols

Introduction to

First-Order Logic

Slide5Model for First-Order LogicSymbol and RepresentationsVariables x, y, a, b,...Connectives , , , , Equality = Quantifiers ,

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Syntax and Semantics

of First-Order Logic

Slide66

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Models for FOL: Example

Syntax and Semantics

of First-Order Logic

Slide7Atomic Sentences

Atomic sentence = predicate (term1,...,termn) or term1 = term2 “Richard the Lionheart is the brother of King John”. e.g., Brother(Richard, John) Complex terms as arguments in atomic sentences is : married (Father(Richard), Mother(John)) An atomic sentence is true in a given model if the relation referred to by the predicate symbol holds among the objects referred to by the arguments.

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T0264 - Artificial Intelligence

Syntax and Semantics

of First-Order Logic

Slide8Complex Sentences

Complex sentences are made from atomic sentences using connectives : “All King are person” written as : x : King(x) Person(x) “King John has a crown on his head” written as : y :Crown(y) OnHead(y, John)More complex sentences using multiple quantifier : p q : Brother (p, q) Sibling(p, q)Symmetric relationship : p q : Sibling(p, q) Sibling(q, p) e.g. Sibling(King John, Richard) Sibling(Richard, King John)

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T0264 - Artificial Intelligence

Syntax and Semantics

of First-Order Logic

Slide99

Sentences :Chandra is a student Chandra is student in computer scienceEach computer science student is member of school of computer Algorithm is difficult Each student in school of computer is like or hate the algorithm Each student is like one lesson The student which do not come to in difficult lesson, there are dislike to these lesson Chandra not come in algorithm

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Using

First-Order Logic

Slide1010

Change to First-Order Logic

Student (Chandra)Student (Chandra, computer_science)x : student(x) computer_science(x) member (x, school_of_computer)Difficult(Algorithm)x : student(x, computer_science) like(x, Algorithm) hate(x, Algorithm)x :y : like(x, y)x : y : student(x) difficult(y) come(x, y) like(x, y)8. come(Chandra, Algorithm)

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Using

First-Order Logic

Slide1111

Does Chandra like Algorithm?by backward chaining

like(Chandra, algorithm)

(8)

null

student(Chandra

) difficult(algorithm) come (Chandra, algorithm)

difficult(algorithm) come(Chandra, algorithm)

come(Chandra, algorithm)

(4)

(1)

(7)

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Using

First-Order Logic

Slide12T0264 - Artificial Intelligence

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Proof by Resolution

Inference procedure based on resolution work by using the principle of proof by

contradiction.

That is, to show that

KB

├

=

, we show that

(KB

)

is unsatisfied. We do this by contradiction.

See Resolution Algorithm

Therefore:

First : (KB

) is convert into CNF (normal clause form)

Second : show clauses obtained by resolving pairs in the row

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Algorithm : Resolution

Convert all the proposition logic to normal clause form (CNF).Negate P and convert the result to clause form. Add it to the set of clause in step Repeat until either a contradiction is found or no progress can be made, or a predetermined amount of effort has been expended. Select two clause. Call these the parent clauses.Resolve them together. The resolvent will be the disjunction of all of the literals of both parent clauses with appropriate substitutions performed and with the following exception: If there is one pairs of literals T1 and T2 such that one of the parent clauses contains T1 and the other contains T2 and if T1 and T2 are unifiable, then neither T1 nor T2 should appear in the resolvent. If there is more than one pair of complementary literals, only one pair should be omitted from the resolvent.

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Proof by Resolution

Slide1414

Algorithm : Resolution

If the resolvent is the empty clause, then a contradiction has been found. If it is not, then add it to the set of clause available to the procedure.A simple pseudo code resolution algorithm for proportional logic

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Proof by Resolution

Slide1515

Conversion to Normal Clause Form (CNF Algorithm)

Eliminate , using : a b = a b.Reduce the scope of each to a single term, using de Morgan’s laws: (p) = p (ab) = a b (ab) = a b x : P(x) =x : P(x) x : P(x) = x : P(x)3. Standardize variables.For sentences like (∀x P(x)) ∨ (∃x Q(x)) which use the same variable name twice, change the name of one of the variables. This avoids confusion later when we drop the quantifiers. For example, from ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃y Loves(y, x)]. we obtain: ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃z Loves(z, x)].

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Proof by Resolution

Slide1616

Convert to Normal Clause Form (cont’d)

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Proof by Resolution

Move all quantifiers to the left of the formula without changing their relative order.

Eliminate

existential quantifiers by

inserting

Skolem

functions.

∃x P(x) into P(A), where A is a new constant

Drop universal quantifiers

Convert

the matrix into a

conjunction

of

disjoints,

using

associativity

and

distributivity

(

d

istribute ORs over ANDs

)

Slide1717

Resolution Example

Representation in CNF: student(Chandra)computer_science(Chandra)computer_science(x1) v school_of_computer(x1)difficult(Algorithm)school_of_computer(x2) v like(x2,Algorithm) v hate(x2,Algorithm)like(x3,f(x3))student(x4) v difficult(y1) v come(x4,y1) v like(x4,y1)come(Chandra, Algorithm)

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Proof by Resolution

Slide1818

Show that:Chandra hate Algorithmby Resolution

difficult

(Algoritma) v come(Chandra,Algorithm)

come(Chandra, Algorithm)

null

2

4

8

1

student(Chandra) v

difficult

(Algorithm) v come(Chandra,Algorithm)

like(Chandra,Algorithm)

Chandra/x4 ; Algoritma/y1

7

computer_science(Chandra) v

like

(Chandra,Algorithm)

school_of_computer(Chandra) v

like

(Chandra,Algorithm)

3

hate(Chandra,

Algorithm)

5

Chandra/x2

Chandra/x1

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Proof by Resolution

Slide19The Knowledge Engineering Process Identify the taskAssemble the relevant knowledgeDecide on a vocabulary of predicates, functions, and constantsEncode general knowledge about a domainEncode a description of the specific problem instancePose queries to the inference procedure and get answersDebug the knowledge base

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Knowledge

Engineering

i

n First-Order Logic

Slide20Seven Basic Gate

ABANDORNANDNORXORXNORNOT BBufferB = X0000110110010110100-10011010- -11110001 -1

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Knowledge

Engineering

i

n First-Order Logic

Slide21A digital circuit C1, purporting to be a one-bit full adder.

The first two inputs are the two bit to be added, and the third input is a carry bit.The first output is the sum, and the second output is a carry bit for the next adder.

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The Electronic Circuit Domain

Knowledge

Engineering

i

n First-Order Logic

Slide22In Boolean representation as : Or Logic Circuit

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The Electronic Circuit Domain

Knowledge

Engineering

i

n First-Order Logic

Slide23Integrated Circuit (IC)

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Knowledge

Engineering

i

n First-Order Logic

Slide24Resolution is complete for propositional logicForward, backward chaining are linear-time, complete for Horn clausesPropositional logic lacks expressive powerFirst-order logic:objects and relations are semantic primitivessyntax: constants, functions, predicates, equality, quantifiersIncreased expressive power: sufficient to define wumpus world

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T0264 - Artificial Intelligence

Summary

Slide25Next Presentations(1 group-6 persons)

Adversarial SearchClassical PlanningPlanning and Acting in the Real WorldProbabilistic Reasoning and Over TimeMaking simple and Complex DecisionLearning from ExamplesReinforcement LearningNatural Language ProcessingDeep LearningPerception and Robotics

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