FUZZY LOGIC Felix Distel Dresden, WS 2012/13 About the Course - - PowerPoint PPT Presentation

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FUZZY LOGIC Felix Distel Dresden, WS 2012/13 About the Course - - PowerPoint PPT Presentation

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Faculty of Computer Science Chair of Automata Theory FUZZY LOGIC Felix Distel Dresden, WS 2012/13 About the Course Course Material Metamathematics of Fuzzy Logic by Petr Hjek available on course website: Slides Lecture Notes

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Faculty of Computer Science Chair of Automata Theory

FUZZY LOGIC

Felix Distel

Dresden, WS 2012/13

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About the Course

Course Material

  • Metamathematics of Fuzzy Logic by Petr Hájek
  • available on course website:

– Slides – Lecture Notes (from a previous semester) – Exercise Sheets

Contact Information

  • felix@tcs.inf.tu-dresden.de
  • lat.inf.tu-dresden.de/teaching/ws2012-2012/FL/

Exams

Oral exams at the end of the semester or during semester break

TU Dresden, WS 2012/13 Fuzzy Logic Slide 2

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Classical Logic

  • suited for properties that are

– identifiable, – distinct, – clear-cut.

  • Examples:

– days of the week, – marital status, – . . .

TU Dresden, WS 2012/13 Fuzzy Logic Slide 3

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Imprecise Knowledge

Is Italy a small country?

TU Dresden, WS 2012/13 Fuzzy Logic Slide 4

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Imprecise Knowledge

Is Italy a small country? Depends.

TU Dresden, WS 2012/13 Fuzzy Logic Slide 4

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Imprecise Knowledge

Is Italy a small country? Depends. Other examples for fuzzy proper- ties

  • old
  • warm
  • tall
  • . . .

TU Dresden, WS 2012/13 Fuzzy Logic Slide 4

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Degrees of Membership

large

2 4 6 8 10 12 14 16 0.2 0.4 0.6 0.8 1 area in 106 km2 truth degree

TU Dresden, WS 2012/13 Fuzzy Logic Slide 5

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Degrees of Membership

warm

5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 temperature in ◦C truth degree

TU Dresden, WS 2012/13 Fuzzy Logic Slide 6

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Crisp vs. Fuzzy Logics

  • Crisp Logics: Only truth values 1 and 0.

= ⇒ characteristic function

  • Fuzzy Logics: Truth values from the interval [0, 1].

= ⇒ membership function

TU Dresden, WS 2012/13 Fuzzy Logic Slide 7

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Fuzzy vs. Probabilistic Logics

Both use truth values

  • Fuzzy Logics: vagueness

– statement is neither completely true nor false – e.g. “The Dresden TV Tower is a tall building. ”

  • Probabilistic Logics: belief or uncertainty

– statement is either true nor false, but outcome unknown – e.g. “Tomorrow it will rain. ”

TU Dresden, WS 2012/13 Fuzzy Logic Slide 8

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Question

How to interpret conjunction?

For the country of Turkey we might have:

  • membership in Huge: 0.046,
  • membership in Asian: 0.969

What is the membership degree of Turkey in Huge ⊓ Asian?

TU Dresden, WS 2012/13 Fuzzy Logic Slide 9

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Question

How to interpret conjunction?

For the country of Turkey we might have:

  • membership in Huge: 0.046,
  • membership in Asian: 0.969

What is the membership degree of Turkey in Huge ⊓ Asian?

Possible choices

  • Minimum of 0.046 and 0.969
  • Product of 0.046 and 0.969
  • . . .

= ⇒ There is not just one fuzzy logic!

TU Dresden, WS 2012/13 Fuzzy Logic Slide 9

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Generalize Operators

Classical logical operators, such as

  • conjunction,
  • disjunction,
  • negation, and
  • implication

need to be generalized. Generalizations should be

  • truth functional
  • “behave well” logically (e.g. conjunction should be associative, commutative,

etc.)

TU Dresden, WS 2012/13 Fuzzy Logic Slide 10

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t-Norms

Definition

Binary operator ⊗: [0, 1] × [0, 1] → [0, 1]

  • associative,
  • commutative,
  • monotone, and
  • has unit 1.

TU Dresden, WS 2012/13 Fuzzy Logic Slide 11

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Continuous t-norms

Fundamental continuous t-norms

Gödel: x ⊗ y = min(x, y)

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 x ⊗ y x y TU Dresden, WS 2012/13 Fuzzy Logic Slide 12

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Continuous t-norms

Fundamental continuous t-norms

Gödel: x ⊗ y = min(x, y) Product: x ⊗ y = x · y

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 x ⊗ y x y TU Dresden, WS 2012/13 Fuzzy Logic Slide 12

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Continuous t-norms

Fundamental continuous t-norms

Gödel: x ⊗ y = min(x, y) Product: x ⊗ y = x · y Łukasiewicz: x ⊗ y = max(0, x + y − 1)

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 x y x ⊗ y TU Dresden, WS 2012/13 Fuzzy Logic Slide 12

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Truth Functions for Boolean Connectives

Connective Truth Function Definition conjunction (&) t-norm (⊗) associative, commutative, monotone, unit 1, (usually also continuous) implication (→) ? negation (¬) ? disjunction (∨) ?

TU Dresden, WS 2012/13 Fuzzy Logic Slide 13

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Generalizing Modus Ponens

Modus Ponens in the Crisp Case

φ ∧ (φ → ψ) then ψ.

Fuzzy Generalization of Modus Ponens

x ⊗ (x ⇒ y)

  • z

≤ y

Residuum

Choose z maximal with this property: x ⇒ y = max{z | x ⊗ z ≤ y}

TU Dresden, WS 2012/13 Fuzzy Logic Slide 14

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Uniqueness of Residuum

Lemma 1.2

For every continous t-norm ⊗ x ⇒ y = max{z | x ⊗ z ≤ y} is the unique operator satisfying z ≤ x ⇒ y iff x ⊗ z ≤ y

TU Dresden, WS 2012/13 Fuzzy Logic Slide 15

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Truth Functions for Boolean Connectives

Connective Truth Function Definition conjunction (&) t-norm (⊗) associative, commutative, monotone, unit 1, (usually also continuous) implication (→) residuum (⇒) x ⊗ y ≤ z iff y ≤ x ⇒ z negation (¬) precomplement ⊖ x ⇒ 0 disjunction (∨) ?

TU Dresden, WS 2012/13 Fuzzy Logic Slide 16

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Ordinal Sums

Definition

Given (ai, bi), i ∈ I family disjoint open intervals, ⊗i, i ∈ I family of t-norms x ⊗ y =

  • s−1

i

  • si(x) ⊗i si(y)
  • if x, y ∈ (ai, bi)

min{x, y}

  • therwise

where si(x) = x − ai bi − ai is the ordinal sum

i∈I(⊗i, ai, bi). TU Dresden, WS 2012/13 Fuzzy Logic Slide 17

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Plots of Ordinal Sums

x 0.3 0.7 1

TU Dresden, WS 2012/13 Fuzzy Logic Slide 18

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Plots of Ordinal Sums

x 0.3 0.7 1 0.3 0.7 1 y

TU Dresden, WS 2012/13 Fuzzy Logic Slide 18

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Plots of Ordinal Sums

x 0.3 0.7 1 0.3 0.7 1 y

TU Dresden, WS 2012/13 Fuzzy Logic Slide 18

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Plots of Ordinal Sums

x 0.3 0.7 1 0.3 0.7 1 y Product Łukasiewicz Gödel

TU Dresden, WS 2012/13 Fuzzy Logic Slide 18

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Plots of Ordinal Sums

x 0.3 0.7 1 0.3 0.7 1 y Product Łukasiewicz Gödel Gödel Gödel

TU Dresden, WS 2012/13 Fuzzy Logic Slide 18

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Plots of Ordinal Sums

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 x y x ⊗ y TU Dresden, WS 2012/13 Fuzzy Logic Slide 18

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Isomorphisms between t-norms

Isomorphic t-norms

If there is s is a bijective, monotone function s: [0, 1] → [0, 1] satisfying x ⊗1 y = s−1 s(x) ⊗2 s(y)

  • then ⊗1 and ⊗2 are called isomorphic.

TU Dresden, WS 2012/13 Fuzzy Logic Slide 19

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Isomorphic t-norms

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 x y x ⊗ y

Łukasiewicz t-norm (aka 1st Schweizer-Sklar t-norm) x ⊗ y = max{x + y − 1, 0}

TU Dresden, WS 2012/13 Fuzzy Logic Slide 20

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Isomorphic t-norms

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 x ⊗ y x y

2nd Schweizer-Sklar t-norm x ⊗ y =

  • max{x2 + y2 − 1, 0}

TU Dresden, WS 2012/13 Fuzzy Logic Slide 20

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Basic Logic

Syntax

P countable set of propositional variables, ⊗ continuous t-norm. Formulas of PC(⊗) are

  • 0,
  • p,
  • f1 & f2, and
  • f1 → f2.

Semantics

Valuation V : P → [0, 1] Zero V(0) = 0, Strong Conjunction V(φ & ψ) = V(φ) ⊗ V(ψ), Implication V(φ → ψ) = V(φ) ⇒ V(ψ).

TU Dresden, WS 2012/13 Fuzzy Logic Slide 21

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Abbreviations

Weak Conjunction φ ∧ ψ := φ &(φ → ψ), Weak Disjunction φ ∨ ψ :=

  • (φ → ψ) → ψ
  • ∧
  • (ψ → φ) → φ
  • Negation

¬φ := φ → 0 Equivalence φ ≡ ψ := (φ → ψ) &(ψ → φ) One 1 := 0 → 0.

TU Dresden, WS 2012/13 Fuzzy Logic Slide 22

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1-Tautologies

Formula φ such that V(φ) = 1 for every valuation V.

TU Dresden, WS 2012/13 Fuzzy Logic Slide 23

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Different t-Norms, Different 1-Tautologies

¬¬φ → φ

TU Dresden, WS 2012/13 Fuzzy Logic Slide 24

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Different t-Norms, Different 1-Tautologies

¬¬φ → φ

  • Łukasiewicz:

V(¬¬φ → φ) = ⊖ ⊖ V(φ) ⇒ V(φ) = 1 −

  • 1 − V(φ)
  • ⇒ V(φ)

= V(φ) ⇒ V(φ) = 1 1-tautology for Łukasiewicz

TU Dresden, WS 2012/13 Fuzzy Logic Slide 24

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Different t-Norms, Different 1-Tautologies

¬¬φ → φ

  • Łukasiewicz:

V(¬¬φ → φ) = ⊖ ⊖ V(φ) ⇒ V(φ) = 1 −

  • 1 − V(φ)
  • ⇒ V(φ)

= V(φ) ⇒ V(φ) = 1 1-tautology for Łukasiewicz

  • Gödel or Product: Not a 1-tautology! Assume V(φ) = 0.5, then

V(¬¬φ → φ) = ⊖ ⊖ V(φ) ⇒ V(φ) = ⊖0 ⇒ V(φ) = 1 ⇒ 0.5 = 0.5

TU Dresden, WS 2012/13 Fuzzy Logic Slide 24

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1-Tautologies for all t-Norms

1-tautologies for ⊗Ł 1-tautologies for ⊗min 1-tautologies for ⊗Π

TU Dresden, WS 2012/13 Fuzzy Logic Slide 25

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1-Tautologies for all t-Norms

1-tautologies for ⊗Ł 1-tautologies for ⊗min 1-tautologies for ⊗Π We are interested in 1-tautologies for all t-norms.

TU Dresden, WS 2012/13 Fuzzy Logic Slide 25

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Axioms of Basic Logic

(A1) (φ → ψ) →

  • (ψ → χ) → (φ → χ)
  • (A2) φ & ψ → φ

(A3) φ & ψ → ψ & φ (A4) φ &(φ → ψ) → ψ &(ψ → φ) (A5)

  • φ → (ψ → χ)
  • → (φ & ψ → χ)

(A6) (φ & ψ → χ) →

  • φ → (ψ → χ)
  • (A7)
  • (φ → ψ) → χ
  • →
  • (ψ → φ) → χ
  • → χ
  • (A8) 0 → φ
  • Deduction Rule: Modus Ponens

TU Dresden, WS 2012/13 Fuzzy Logic Slide 26

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Proofs in Basic Logic

Instantiation

Substitution arbitrary formulae for variable names, e.g. (ρ → σ) & ψ → (ρ → σ) is obtained from φ & ψ → φ by substituting ρ → σ for φ.

TU Dresden, WS 2012/13 Fuzzy Logic Slide 27

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Proofs in Basic Logic

Instantiation

Substitution arbitrary formulae for variable names, e.g. (ρ → σ) & ψ → (ρ → σ) is obtained from φ & ψ → φ by substituting ρ → σ for φ.

Application of Modus Ponens

BL ⊢ ρ & σ → σ & ρ Instance of (A3) BL ⊢ (ρ & σ → σ & ρ) →

  • (σ & ρ → σ) → (ρ & σ → σ)
  • Instance of (A1)

BL ⊢ (σ & ρ → σ) → (ρ & σ → σ) modus ponens BL ⊢ σ & ρ → σ Instance of (A2) BL ⊢ ρ & σ → σ modus ponens

TU Dresden, WS 2012/13 Fuzzy Logic Slide 27

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Goal

1-tautologies for ⊗Ł 1-tautologies for ⊗min 1-tautologies for ⊗Π

TU Dresden, WS 2012/13 Fuzzy Logic Slide 28

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Goal

1-tautologies for ⊗Ł 1-tautologies for ⊗min 1-tautologies for ⊗Π = BL?

TU Dresden, WS 2012/13 Fuzzy Logic Slide 28