basis for equalities
play

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

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


  1. Computing a Complete Basis for Equalities Implied by a System of LRA Constraints 7/2/2016

  2. Linear Arithmetic / Linear Programming Input: coefficients (∈ ℤ ) bound (∈ ℤ ) variables Basis for Equalities – Bromberger, Weidenbach 7/2/2016 2/15

  3. Linear Arithmetic / Linear Programming Input: Goal: P NP Complexity: Example: Basis for Equalities – Bromberger, Weidenbach 7/2/2016 2/15

  4. Fast Cube Tests for LIA constraint solving (Bromberger & Weidenbach, IJCAR 2016) Largest Cube Test • largest cube inside the set of real solutions Basis for Equalities – Bromberger, Weidenbach 7/2/2016 3/15

  5. Fast Cube Tests for LIA constraint solving (Bromberger & Weidenbach, IJCAR 2016) Largest Cube Test • largest cube inside the set of real solutions • center point Basis for Equalities – Bromberger, Weidenbach 7/2/2016 3/15

  6. Fast Cube Tests for LIA constraint solving (Bromberger & Weidenbach, IJCAR 2016) Largest Cube Test • largest cube inside the set of real solutions • center point → integer point • optimization LP (LRA) + evaluation Basis for Equalities – Bromberger, Weidenbach 7/2/2016 3/15

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

  8. Equalities substitute Diophantine Equation Handler A Practical Approach to Satisfiability Modulo Linear Integer Arithmetic by A. Griggio. JSAT 2012 Basis for Equalities – Bromberger, Weidenbach 7/2/2016 5/15

  9. Equalities substitute Diophantine Equation Handler A Practical Approach to Satisfiability Modulo Linear Integer Arithmetic by A. Griggio. JSAT 2012 Not Always Explicit! Basis for Equalities – Bromberger, Weidenbach 7/2/2016 5/15

  10. Implied Equalities 𝑦 𝑧 = 2 2 explicit implicit Basis for Equalities – Bromberger, Weidenbach 7/2/2016 6/15

  11. Equality Basis explicit implicit Basis for Equalities – Bromberger, Weidenbach 7/2/2016 7/15

  12. Equality Basis Equality Basis: Implied Equalites: 1.set of All linear independent linear combinations equalities of the 2. maximal Equality Basis + Basis for Equalities – Bromberger, Weidenbach 7/2/2016 7/15

  13. Equality Basis Equality Basis: Implied Equalites: 1.set of All linear independent linear combinations equalities of the 2. maximal Equality Basis − Basis for Equalities – Bromberger, Weidenbach 7/2/2016 7/15

  14. Equality Basis Simplification: Eliminating Equalities via Substitution Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15

  15. Equality Basis Simplification: Verifying Eliminating Implied Equalities: Equalities via via Substitution Substitution (Nelson-Oppen) ? Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15

  16. Nelson-Oppen Combination pairs of equivalent variables 𝜏 ≔ { } , Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15

  17. Nelson-Oppen Combination pairs of equivalent variables 𝜏 + normalizing semantic syntactic equivalence equivalence Find equivalent variables with DAGs! Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15

  18. Nelson-Oppen Combination pairs of How do we find equalities? equivalent variables 𝜏 + normalizing semantic syntactic equivalence equivalence Find equivalent variables with DAGs! Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15

  19. Largest Cube Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 9/15

  20. Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15

  21. Equality Test Interior & Surface Interior Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15

  22. Equality Test Interior & Surface Interior Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15

  23. Equality Test sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15

  24. Equality Test sat unsat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15

  25. Equality Explanation original Positive Linear Combinations Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15

  26. Equality Explanation original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15

  27. Equality Explanation strict original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15

  28. Equality Explanation strict Multiply original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15

  29. Equality Explanation strict Multiply original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15

  30. Equality Explanation strict original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15

  31. Equality Test Original System of Constraints actual implying strict inequalities inequalities inequalities Equality Explanation Minimal Conflict System for Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15

  32. Equality Test Original System of Constraints actual implying strict inequalities inequalities inequalities Equality Explanation Minimal Conflict System for Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15

  33. Equality Test Original System of Constraints strict non-strict inequalities inequalities System for Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15

  34. Equality Test Original System of Constraints strict non-strict inequalities inequalities Satisfiable Assignment for Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15

  35. Equality Test Original System of Constraints strict non-strict inequalities inequalities System for Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15

  36. Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15

  37. Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15

  38. Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15

  39. Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15

  40. Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15

  41. Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15

  42. Computing an Equality Basis strict equalities original substitutions sat sat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15

  43. Computing an Equality Basis simplified equalities original substitutions sat sat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 14/15

  44. Computing an Equality Basis simplified equalities original substitutions sat sat ? Nelson- Oppen Basis for Equalities – Bromberger, Weidenbach 7/2/2016 14/15

  45. Conclusions Finding Equalities: A Basis for Equalities: Applications: Thank you for your attention! Basis for Equalities – Bromberger, Weidenbach 7/2/2016 15/15

  46. Inequality Tableau & Bounds Representation Representation Equalitiy Test substitution Equalitiy Basis Basis for Equalities – Bromberger, Weidenbach 7/2/2016 16/17

  47. Inequality Tableau & Bounds Representation Representation Equalitiy Test pivoting substitution Equalitiy Basis Basis for Equalities – Bromberger, Weidenbach 7/2/2016 16/17

Recommend


More recommend