Avoiding fake boundaries in interval analysis

(1)

Avoiding fake boundaries in interval analysis

Guilherme Schvarcz Franco and Luc Jaulin

(2)

Guilherme Schvarcz Franco

Presentation

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Guilherme Schvarcz Franco

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Resume:

– Graduate in Systems Analysis – UNIFRA - Brazil

– Graduate in Geomatics – UFSM - Brazil

– Master degree in computer sciences at UFRGS – Brazil

– PhD at ENSTA Bretagne - Brest

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Advisor: Luc Jaulin

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Localization Problem

A B∪A B∪AB∪((A∪B∪C)∩C)

A B C ∪ A B C

Inner contractor Outer contractor

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Fake boundary

A B∪A B∪AB∪((A∪B∪C)∩C)

A B C ∪ A B C

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( A ∪ B )∪(( A ∪ B )∩C )

Simplification

A B∪ AB ∪ A B∪(( A ∪ B∪ C )∩C ) A ∩( B ∪ B )∪ A B ∪(( A ∪ B ∪C )∩C ) A ∪ A B∪(( A ∪ B∪ C )∩C )

(( A ∪ A )∩( A ∪ B ))∪(( A ∪ B ∪C)∩C)

A ∪ B

By distributive law over union

By complement law and identity law

( A ∪ B )∪(( A ∪ B ∪C )∩C )

By distributive law over union

By complement law and identity law

( A ∪ B )∪(( A ∪ B )∩C )∪C C

By distributive law over union By complement law

By absorption law

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Expected set

A ∪ B A B

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Karnaugh Map

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Karnaugh Map

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A B∪ A B ∪ AB∪(( A ∪ B∪ C )∩C ) ^{A B C} ^∪ ^{A B C}

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A ∪ B A B

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Solution

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Two parts:

– Build Karnaugh Map

– Extract the Minimal Equation

( A C ∪ A B )∩(C ∪ D)

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Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

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Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

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Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

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Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

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Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

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Extracting the Minimal Equation

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Extracting the Minimal Equation

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Extracting the Minimal Equation

(20)

Extracting the Minimal Equation

(21)

Extracting the Minimal Equation

(22)

Extracting the Minimal Equation

A C D ∪ A C D∪ A B C ∪ A B C

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Extracting the Minimal Equation

(24)

Extracting the Minimal Equation

(25)

Extracting the Minimal Equation

A C ∪ A D B

(26)

Extracting the Minimal Equation

A C ∪ A D B

A C D ∪ A C D∪ A B C ∪ A B C ∪ A D B

(27)

Questions?

[email protected] [email protected]

Schvarcz Guilherme Schvarcz Franco

Avoiding fake boundaries in interval analysis

Avoiding fake boundaries in interval analysis

Guilherme Schvarcz Franco and Luc Jaulin

Presentation

Guilherme Schvarcz Franco

Resume:

– Graduate in Systems Analysis – UNIFRA - Brazil

– Graduate in Geomatics – UFSM - Brazil

– Master degree in computer sciences at UFRGS – Brazil

– PhD at ENSTA Bretagne - Brest

Advisor: Luc Jaulin

Localization Problem

A B C ∪ A B C

Fake boundary

A B C ∪ A B C

( A ∪ B )∪(( A ∪ B )∩C )

Simplification

A B∪ AB ∪ A B∪(( A ∪ B∪ C )∩C ) A ∩( B ∪ B )∪ A B ∪(( A ∪ B ∪C )∩C ) A ∪ A B∪(( A ∪ B∪ C )∩C )

(( A ∪ A )∩( A ∪ B ))∪(( A ∪ B ∪C)∩C)

A ∪ B

( A ∪ B )∪(( A ∪ B ∪C )∩C )

( A ∪ B )∪(( A ∪ B )∩C )∪C C

Expected set

A ∪ B A B

Karnaugh Map

Karnaugh Map

A B∪ A B ∪ AB∪(( A ∪ B∪ C )∩C ) A B C ∪ A B C

A ∪ B A B

Solution

Two parts:

– Build Karnaugh Map

– Extract the Minimal Equation

( A C ∪ A B )∩(C ∪ D)

Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

Building Karnaugh Map

( A C ∪ A B )∩(C ∪ D)

Extracting the Minimal Equation

Extracting the Minimal Equation

Extracting the Minimal Equation

Extracting the Minimal Equation

Extracting the Minimal Equation

Extracting the Minimal Equation

A C D ∪ A C D∪ A B C ∪ A B C

Extracting the Minimal Equation

Extracting the Minimal Equation

Extracting the Minimal Equation

A C ∪ A D B

Extracting the Minimal Equation

A C ∪ A D B

A C D ∪ A C D∪ A B C ∪ A B C ∪ A D B

Questions?

[email protected] [email protected]

A B∪ A B ∪ AB∪(( A ∪ B∪ C )∩C ) ^{A B C} ^∪ ^{A B C}