Course : T 0264 – Artificial Intelligence - PowerPoint Presentation

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Course : T0264 – Artificial IntelligenceYear : 2013

LECTURE 07

First Order

Logic

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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

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Whereas 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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Introduction to

First-Order Logic

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 = 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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Standard Logic Symbols

Introduction to

First-Order Logic

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Model for First-Order LogicSymbol and RepresentationsVariables x, y, a, b,...Connectives , , , , Equality = Quantifiers , 

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Syntax and Semantics

of First-Order Logic

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Models for FOL: Example

Syntax and Semantics

of First-Order Logic

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Atomic 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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Syntax and Semantics

of First-Order Logic

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Complex 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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Syntax and Semantics

of First-Order Logic

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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

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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

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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

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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

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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

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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 (ab) = a  b (ab) = 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

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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

)

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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

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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

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computer_science(Chandra) v

like

(Chandra,Algorithm)

school_of_computer(Chandra) v

like

(Chandra,Algorithm)

3

hate(Chandra,

Algorithm)

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Chandra/x2

Chandra/x1

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Proof by Resolution

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The 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

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Seven Basic Gate

ABANDORNANDNORXORXNORNOT BBufferB = X0000110110010110100-10011010- -11110001 -1

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Knowledge

Engineering

i

n First-Order Logic

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A 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

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In Boolean representation as : Or Logic Circuit

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The Electronic Circuit Domain

Knowledge

Engineering

i

n First-Order Logic

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Integrated Circuit (IC)

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Knowledge

Engineering

i

n First-Order Logic

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Resolution 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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Summary

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Next 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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