Basis for Equalities Implied by a System of LRA Constraints - - PowerPoint PPT Presentation

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Basis for Equalities Implied by a System of LRA Constraints - - PowerPoint PPT Presentation

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Computing a Complete Basis for Equalities Implied by a System of LRA Constraints 7/2/2016 Linear Arithmetic / Linear Programming Input: coefficients ( ) bound ( ) variables Basis for Equalities Bromberger, Weidenbach

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SLIDE 1

Computing a Complete

Basis for Equalities

Implied by a System of LRA Constraints

7/2/2016

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SLIDE 2

2/15

Linear Arithmetic / Linear Programming Input:

7/2/2016 Basis for Equalities – Bromberger, Weidenbach

coefficients (∈ ℤ ) variables bound (∈ ℤ )

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SLIDE 3

Linear Arithmetic / Linear Programming Input: Goal: Example:

P NP

Complexity:

2/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

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SLIDE 4

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Largest Cube Test Fast Cube Tests for LIA constraint solving

(Bromberger & Weidenbach, IJCAR 2016)

  • largest cube inside the set of real solutions
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SLIDE 5

3/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

  • largest cube inside the set of real solutions
  • center point

Largest Cube Test Fast Cube Tests for LIA constraint solving

(Bromberger & Weidenbach, IJCAR 2016)

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SLIDE 6
  • largest cube inside the set of real solutions
  • center point → integer point
  • optimization LP (LRA) + evaluation

3/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

Largest Cube Test Fast Cube Tests for LIA constraint solving

(Bromberger & Weidenbach, IJCAR 2016)

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SLIDE 7

Equalities

substitute

5/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

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SLIDE 8

Equalities

substitute

5/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

Diophantine Equation Handler

A Practical Approach to Satisfiability Modulo Linear Integer Arithmetic

by A. Griggio. JSAT 2012

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SLIDE 9

Equalities

substitute

5/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

Diophantine Equation Handler

A Practical Approach to Satisfiability Modulo Linear Integer Arithmetic

by A. Griggio. JSAT 2012

Not Always Explicit!

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SLIDE 10

explicit implicit Implied Equalities 𝑦 𝑧 = 2 2

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 6/15

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SLIDE 11

explicit implicit Equality Basis

Basis for Equalities – Bromberger, Weidenbach 7/15 7/2/2016

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SLIDE 12

Equality Basis

Equality Basis: 1.set of linear independent equalities

  • 2. maximal

Implied Equalites: All linear combinations

  • f the

Equality Basis

+

Basis for Equalities – Bromberger, Weidenbach 7/15 7/2/2016

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SLIDE 13

− Equality Basis

Equality Basis: 1.set of linear independent equalities

  • 2. maximal

Implied Equalites: All linear combinations

  • f the

Equality Basis

Basis for Equalities – Bromberger, Weidenbach 7/15 7/2/2016

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SLIDE 14

Equality Basis

Simplification: Eliminating Equalities via Substitution

Basis for Equalities – Bromberger, Weidenbach 8/15 7/2/2016

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SLIDE 15

Equality Basis

Simplification: Eliminating Equalities via Substitution ? Verifying Implied Equalities: via Substitution (Nelson-Oppen)

Basis for Equalities – Bromberger, Weidenbach 8/15 7/2/2016

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SLIDE 16

Nelson-Oppen Combination

Basis for Equalities – Bromberger, Weidenbach 8/15 7/2/2016

𝜏 ≔ { } ,

pairs of equivalent variables

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SLIDE 17

Nelson-Oppen Combination

Basis for Equalities – Bromberger, Weidenbach 8/15 7/2/2016

semantic equivalence syntactic equivalence 𝜏 + normalizing Find equivalent variables with DAGs! pairs of equivalent variables

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SLIDE 18

Nelson-Oppen Combination

Basis for Equalities – Bromberger, Weidenbach 8/15 7/2/2016

semantic equivalence syntactic equivalence 𝜏 + normalizing pairs of equivalent variables

How do we find equalities?

Find equivalent variables with DAGs!

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SLIDE 19

Largest Cube Test

9/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

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SLIDE 20

Equality Test

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Equality Test Interior & Surface Interior

10/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

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SLIDE 22

Equality Test Interior & Surface Interior

10/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

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SLIDE 23

Equality Test unsat sat

10/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

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SLIDE 24

Equality Test unsat sat unsat

10/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

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SLIDE 25

Equality Explanation

11/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

Positive Linear Combinations

  • riginal
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SLIDE 26

Equality Explanation

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 11/15

  • riginal
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SLIDE 27

Equality Explanation

7/2/2016 Basis for Equalities – Bromberger, Weidenbach

strict

  • riginal

11/15

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SLIDE 28

Equality Explanation

7/2/2016 Basis for Equalities – Bromberger, Weidenbach

strict

  • riginal

Multiply

11/15

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SLIDE 29

Equality Explanation

7/2/2016 Basis for Equalities – Bromberger, Weidenbach

strict

  • riginal

11/15

Multiply

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Equality Explanation

7/2/2016 Basis for Equalities – Bromberger, Weidenbach

  • riginal

11/15

strict

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Equality Test

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 12/15

Original System of Constraints System for Equality Test

strict inequalities implying inequalities actual inequalities Equality Explanation Minimal Conflict

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SLIDE 32

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 12/15

Original System of Constraints System for Equality Test

strict inequalities implying inequalities actual inequalities Equality Explanation Minimal Conflict

Equality Test

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SLIDE 33

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 12/15

Original System of Constraints System for Equality Test

strict inequalities non-strict inequalities

Equality Test

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7/2/2016 Basis for Equalities – Bromberger, Weidenbach 12/15

Original System of Constraints

strict inequalities non-strict inequalities

Satisfiable Assignment for Equality Test

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SLIDE 35

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 12/15

Original System of Constraints

strict inequalities non-strict inequalities

System for Equality Test Equality Test

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SLIDE 36

Computing an Equality Basis

  • riginal

strict equalities substitutions sat unsat

13/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

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SLIDE 37

Computing an Equality Basis

  • riginal

strict equalities substitutions

7/2/2016

sat unsat

Basis for Equalities – Bromberger, Weidenbach 13/15

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SLIDE 38

Computing an Equality Basis

  • riginal

strict equalities substitutions

7/2/2016

sat unsat

Basis for Equalities – Bromberger, Weidenbach 13/15

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SLIDE 39

Computing an Equality Basis

  • riginal

strict equalities substitutions

7/2/2016

sat unsat

Basis for Equalities – Bromberger, Weidenbach 13/15

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SLIDE 40

Computing an Equality Basis

  • riginal

strict equalities substitutions sat unsat

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 13/15

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SLIDE 41

Computing an Equality Basis

  • riginal

strict equalities substitutions sat unsat

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 13/15

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SLIDE 42

Computing an Equality Basis

  • riginal

strict equalities substitutions sat sat

7/2/2016 Basis for Equalities – Bromberger, Weidenbach 13/15

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SLIDE 43

Computing an Equality Basis

  • riginal

simplified equalities substitutions

14/15 7/2/2016

sat sat

Basis for Equalities – Bromberger, Weidenbach

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SLIDE 44

Computing an Equality Basis

  • riginal

simplified equalities substitutions

Nelson- Oppen

?

14/15 7/2/2016

sat sat

Basis for Equalities – Bromberger, Weidenbach

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SLIDE 45

Conclusions A Basis for Equalities:

Thank you for your attention!

15/15 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

Finding Equalities: Applications:

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Inequality Representation

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Tableau & Bounds Representation Equalitiy Test Equalitiy Basis substitution

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SLIDE 47

Inequality Representation

16/17 7/2/2016 Basis for Equalities – Bromberger, Weidenbach

Tableau & Bounds Representation Equalitiy Test Equalitiy Basis substitution pivoting