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Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Fuzzy Sets and Fuzzy TechniquesLecture 10 Fuzzy Logic and Approximate

Reasoning

Nataa Sladoje

Centre for Image AnalysisUppsala University

February 22, 2007

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Outline

1 Introduction

2 Fuzzy Logic

3 Fuzzy Implications

4 Binary Fuzzy Relations

5 Approximate Reasoning

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Introduction

Classical logic - a very brief overview Chapter 8.1 Multivalued logic Chapter 8.2

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewPropositional logic

Logic is the study of methods and principles ofreasoning in all its possible forms.

Propositions - statements that are required to be trueor false.

The truth value of a proposition is the opposite of thetruth value of its negation.

Instead of propositions, we use logic variables. Logicvariable may asses one of the two truth values, if it issubstituted by a particular proposition.

Propositional logic studies the rules by which newlogic variables can be produced from some given logicvariables. The internal structure of the propositionsbehind the variables does not matter!

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewLogic functions

Logic function assigns a truth value to a combination oftruth values of its variables:

f : {true, false}n {true, false}

2n choices of n arguments 22n logic functions of nvariables.

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewLogic functions of two variables

v2 1 1 0 0 Function Adoptedv1 1 0 1 0 name symbol1 0 0 0 0 Zero function 02 0 0 0 1 NOR function v1 v23 0 0 1 0 Inhibition v1 > v24 0 0 1 1 Negation v25 0 1 0 0 Inhibition v1 < v26 0 1 0 1 Negation v17 0 1 1 0 Exclusive OR v1 v28 0 1 1 1 NAND function v1|v29 1 0 0 0 Conjunction v1 v210 1 0 0 1 Equivalence v1 v211 1 0 1 0 Assertion v112 1 0 1 1 Implication v1 v213 1 1 0 0 Assertion v214 1 1 0 1 Implication v1 v215 1 1 1 0 Disjunction v1 v216 1 1 1 1 One function 1

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewLogic primitives

Observe, e.g.:

14(v1, v2) = 15(6(v1, v2), v2)10(v1, v2) = 9(14(v1, v2), 12(v1, v2))

A task: Express all the logic functions of n variables byusing only a small number of simple logic functions,preferably of one or two variables.

Such a set is a complete set of logic primitives. Examples:

{negation, conjunction, disjunction} = {6, 9, 15},{negation, implication} = {6, 14}.

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewLogic formulae

Definition1. If v is a logic variable, then v and v are logic formulae;2. If v1 and v2 are logic formulae, then v1 v2 and v1 v2

are also logic formulae;3. Logic formulae are only those defined (obtained) by the

two previous rules.

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewLogic formulae

Each logic formula generates a unique logic function.Different logic formulae may generate the same logicfunction. Such are called equivalent.Examples:

(v1 v2) (v1 v2)(v1 v2) ((v1 v2) (v1 v2))

Tautology is (any) logic formula that corresponds to a logicfunction one.Contradiction is (any) logic formula that corresponds to alogic function zero.

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewInference rules

Inference rules are tautologies used for making deductiveinferences.Examples: (a (a b)) b modus ponens (b (a b)) a modus tollens (a b) (b c)) (a c) hypothetical

syllogism

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewPredicate logic

There are situations when the internal structure ofpropositions cannot be ignored in deductive reasoning.

Propositions are, in general, of the form

x is P

where x is a symbol of a subject and P is a predicatethat characterizes a property.

x is any element of universal set X , while P is a functionon X , which for each value of x forms a proposition.

P(x) is called predicate; it becomes true or false forany particular value of x .

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Classical logic: A brief overviewPredicate logic-extensions

n-ary predicates P(x1, x2, . . . , xn) Quantification of applicability of a predicate with respect

to the domain of its variablesExistential quantification: (x)P(x)Universal quantification: (x)P(x)

It holds:

(x)P(x) =

xXP(x) (x)P(x) =

xXP(x)

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Multivalued LogicsThree-valued logic

Third truth value is allowed:truth: 1, false: 0, intermediate: 12 .

While it is accepted to have p = 1 p, the definitions ofother primitives differ in different three-valued logics.

For the best known three-valued logics (ukasiewicz,Bochvar, Kleene, Heyting, Reichenbach), primitivescoincide with two valued counterparts for the variableshaving values 0 or 1 (see Table 8.4, p.218).

None of the mentioned logics satisfies law of excludedmiddle, or law of contradiction.

quasi-tautology is a logic formula that never assumestruth value 0;

quasi-contradiction is a logic formula that neverassumes truth value 1.

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Multivalued Logicsn-valued logic

The set of truth values is

Tn = {0 =0

n 1 ,1

n 1 ,2

n 1 , . . . ,n 2n 1 ,

n 1n 1 = 1}.

Truth values are interpreted as degrees of truth.Primitives in n-valued logics of ukasiewicz, denoted by Ln,are:

p = 1 pp q = min[p,q]p q = max[p,q]

p q = min[1,1 + q p]p q = 1 |p q|

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Multivalued Logicsukasiewicz n-valued logic

L2 is a classical two valued logic. L takes truth values to be all rational numbers in [0,1]. By L1, the logic with truth values being all real numbers

in [0,1] is denoted.It is called the standard ukasiewicz logic.It is isomorphic with a fuzzy set theory basedon standard operations.

There exists no finite complete set of logic primitives for anyinfinite-valued logic. Using a finite set of primitives, only asubset of all logic functions can be defined.

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Fuzzy logic

Fuzzy propositions Chapter 8.3 Linguistic hedges Chapter 8.5 Fuzzy quantifiers Chapter 8.4

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Fuzzy propositions

The range of truth values of fuzzy propositions is not only{0,1}, but [0,1].The truth of a fuzzy proposition is a matter of degree.Classification of fuzzy propositions: Unconditional and unqualified propositions

The temperature is high. Unconditional and qualified propositions

The temperature is high is very true." Conditional and unqualified propositions

If the temperature is high, then it is hot. Conditional and qualified propositions

If the temperature is high, then it is hot is true.

Fuzzy Setsand FuzzyTechniques

NataaSladoje

Introduction

Fuzzy Logic

FuzzyImplications

Binary FuzzyRelations

ApproximateReasoning

Linguistic hedges

Linguistic hedges are linguistic terms by which otherlinguistic terms are modified.Tina is young is true.Tina is very young is true.Tina is young is very true.Tina is very young is very true.

Fuzzy predicates and fuzzy truth values can bemodified.Crisp predicates cannot be modified.

Examples of hedges: very, fairly, extremely.

Fuzzy