basis for equalities
play

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

Mar 21, 2023 •126 likes •598 views

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

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

2010 Equalities Act Since the 1970s there have been lots of different laws passed in Parliament

2010 Equalities Act Since the 1970s there have been lots of different laws passed in Parliament

2010 Equalities Act Since the 1970s there have been lots of different laws passed in Parliament to stop different groups of people being treated unfairly In 2010 the Government brought all these together in the Equalities Act In March this

427 views • 10 slides
Decision Procedures An Algorithmic Point of View Equalities and Uninterpreted Functions D.

Decision Procedures An Algorithmic Point of View Equalities and Uninterpreted Functions D.

Decision Procedures An Algorithmic Point of View Equalities and Uninterpreted Functions D. Kroening O. Strichman ETH/Technion Version 1.0, 2007 Part III Equalities and Uninterpreted Functions Outline 1 Introduction to Equality Logic

1.03k views • 90 slides
Equalities Act 2010 and School Accessibility Plans This is a reminder to Governors about their

Equalities Act 2010 and School Accessibility Plans This is a reminder to Governors about their

Equalities Act 2010 and School Accessibility Plans This is a reminder to Governors about their statutory duty under the Equalities Act 2010 to ensure that their School has a current Accessibility Plan in place. The Plan must identify the

278 views • 4 slides
Unranked Tree Automata with Sibling Equalities and Disequalities Wong Karianto Christof Lding

Unranked Tree Automata with Sibling Equalities and Disequalities Wong Karianto Christof Lding

Unranked Tree Automata with Sibling Equalities and Disequalities Wong Karianto Christof Lding Lehrstuhl fr Informatik 7 RWTH Aachen ICALP 2007 Wrocaw, 913 July 2007 Motivations(1): Tree Automata with Subtree Equalities Finite

834 views • 35 slides
Equalities Annual Review 12/13 and Forward Planning 13/14 and Forward Planning 13/14 Page 5

Equalities Annual Review 12/13 and Forward Planning 13/14 and Forward Planning 13/14 Page 5

Equalities Annual Review 12/13 and Forward Planning 13/14 and Forward Planning 13/14 Page 5 Victoria Lawson and Shama Sarwar Agenda Item 6 Agenda Item 6 Annual Review of Equalities 12/13 Sutton will be including a range of equality

109 views • 6 slides
Including Intersectional Identities Sam Rankin Intersectional Equalities Coordinator Grounds

Including Intersectional Identities Sam Rankin Intersectional Equalities Coordinator Grounds

Including Intersectional Identities Sam Rankin Intersectional Equalities Coordinator Grounds covered Race / Ethnicity Faith / Belief Disability Sex Sexual Orientation Characteristics Gender Identity Some labels people use for themselves

265 views • 12 slides
Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil

Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil

Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil Ghorbal 1 Andrew Sogokon 2 e Platzer 1 Andr 1. Carnegie Mellon University 2. University of Edinburgh SAS, Munich, Germany September 11th, 2014

1.02k views • 53 slides
Quiz Suppose u 1 , . . . , u n is a basis for U and v 1 , . . . , v k is a basis for V . Prove that

Quiz Suppose u 1 , . . . , u n is a basis for U and v 1 , . . . , v k is a basis for V . Prove that

Quiz Suppose u 1 , . . . , u n is a basis for U and v 1 , . . . , v k is a basis for V . Prove that u 1 , . . . , u n , v 1 , . . . , v k is a basis for U V . Two parts to the proof: 1. Show that Span { u 1 , . . . , u n , v 1 , . . . , v k }

712 views • 26 slides
Computational Higher Inductive Types Computing with Custom Equalities Jason Gross jgross@mit.edu

Computational Higher Inductive Types Computing with Custom Equalities Jason Gross jgross@mit.edu

Computational Higher Inductive Types Computing with Custom Equalities Jason Gross jgross@mit.edu MIT CSAIL Student Workshop January 22, 2014 Properties of Equality Warm Up: Linked Lists Example: Unordered Sets Canonical Inhabitants Higher

482 views • 46 slides
Computational Higher Inductive Types Computing with Custom Equalities Jason Gross jgross@mit.edu

Computational Higher Inductive Types Computing with Custom Equalities Jason Gross jgross@mit.edu

Computational Higher Inductive Types Computing with Custom Equalities Jason Gross jgross@mit.edu MIT CSAIL Student Workshop April 10, 2014 Properties of Equality Warm Up: Linked Lists Example: Unordered Sets Canonical Inhabitants Higher

362 views • 10 slides
On a new orthonormal basis for RBF native spaces and its fast computation Stefano De Marchi and

On a new orthonormal basis for RBF native spaces and its fast computation Stefano De Marchi and

On a new orthonormal basis for RBF native spaces and its fast computation Stefano De Marchi and Gabriele Santin Torino: June 11, 2014 Outline Introduction 1 Change of basis 2 3 WSVD Basis The new basis 4 5 Numerical Results Further

813 views • 42 slides
ASSESSING RISK TO BAME STAFF A COVID -19 AND EQUALITIES FORUM Wednesday 17th June 2020

ASSESSING RISK TO BAME STAFF A COVID -19 AND EQUALITIES FORUM Wednesday 17th June 2020

ASSESSING RISK TO BAME STAFF A COVID -19 AND EQUALITIES FORUM Wednesday 17th June 2020 7pm - 8.30pm A virtual meeting between 4 expert panellists on a NHS experienced Panel: Roger Kline OBE * Dr. Joan Myers OBE Dr. Satheesh

339 views • 21 slides
Linear programming Input: System of inequalities or equalities over the reals R A linear cost

Linear programming Input: System of inequalities or equalities over the reals R A linear cost

Linear programming Input: System of inequalities or equalities over the reals R A linear cost function Output: Value for variables that minimizes cost function Example: Minimize 6x+4y Subject to 3x + 2y + 5z 1 z = 2 Note:

551 views • 30 slides
Private inductive types July 2013 Introduction Higher Inductive types: adding equalities

Private inductive types July 2013 Introduction Higher Inductive types: adding equalities

Private inductive types July 2013 Introduction Higher Inductive types: adding equalities Preventing inconsistencies Preserving convertibility Simulating with private types What is this thing called Equality A family of

421 views • 16 slides
South Gloucestershire Equalities Forum 9 October 2013 2011 Census Overview Focus on: Ethnicity,

South Gloucestershire Equalities Forum 9 October 2013 2011 Census Overview Focus on: Ethnicity,

South Gloucestershire Equalities Forum 9 October 2013 2011 Census Overview Focus on: Ethnicity, limiting long-term illness and care Andy Cornelius Principal Research & Intelligence Officer South Gloucestershire Council 2011 Census for

2.29k views • 192 slides
Eta-equalities in Martin-L of type theory Ansten Klev Czech Academy of Sciences Mathematical

Eta-equalities in Martin-L of type theory Ansten Klev Czech Academy of Sciences Mathematical

Eta-equalities in Martin-L of type theory Ansten Klev Czech Academy of Sciences Mathematical Logic and its Applications Kanazawa, 9 March 2018 : 1 Martin-L of type theory I Forms of judgement : type = : type a : a = b :

915 views • 13 slides
Natural Language Processing The Turing Test Eliza State of the art Conversational Agent

Natural Language Processing The Turing Test Eliza State of the art Conversational Agent

Natural Language Processing The Turing Test Eliza State of the art Conversational Agent The Turing Test: Background Participating in a natural language conversation is hard: ambiguity vagueness context real world

129 views • 10 slides
W e can alw a ys rename v ariables whatev er w e lik e, so this step is

W e can alw a ys rename v ariables whatev er w e lik e, so this step is

Closure Prop erties of CFL's | Substitut i on If a substitution assigns a CFL to ev ery sym b ol s in the alphab et of a CFL L , then s ( L ) is a CFL. Pro of T ak e a grammar for L and a gramma r

171 views • 4 slides
Symbolwise MAP Estimation for Multiple-Trace Insertion/Deletion/Substitution Channels Ryo

Symbolwise MAP Estimation for Multiple-Trace Insertion/Deletion/Substitution Channels Ryo

Symbolwise MAP Estimation for Multiple-Trace Insertion/Deletion/Substitution Channels Ryo Sakogawa and Haruhiko Kaneko Tokyo Institute of Technology ISIT2020 R. Sakogawa, H. Kaneko (TokyoTech) MAP Estimation for Multiple-Trace IDS ISIT2020 1

625 views • 25 slides
Parallel Numerical Algorithms Chapter 3 Dense Linear Systems Section 3.3 Triangular

Parallel Numerical Algorithms Chapter 3 Dense Linear Systems Section 3.3 Triangular

Triangular Systems 1D Algorithms Wavefront Algorithms 2D Algorithms and TRSM Parallel Numerical Algorithms Chapter 3 Dense Linear Systems Section 3.3 Triangular Linear Systems Michael T. Heath and Edgar Solomonik Department of

514 views • 40 slides
Update on Alternatives Assessment Under Reach M A R C H 1 4 , 2 0 1 6 F A C I L I T A T E D B

Update on Alternatives Assessment Under Reach M A R C H 1 4 , 2 0 1 6 F A C I L I T A T E D B

Update on Alternatives Assessment Under Reach M A R C H 1 4 , 2 0 1 6 F A C I L I T A T E D B Y : J O E L T I C K N E R , S C D J O E L _ T I C K N E R @ U M L . E D U L O W E L L C E N T E R F O R S U S T A I N A B L E P R O D U C T

419 views • 28 slides
Innovation Priorities for UK Bioenergy: Technological Expectations within Path Dependence Les

Innovation Priorities for UK Bioenergy: Technological Expectations within Path Dependence Les

Innovation Priorities for UK Bioenergy: Technological Expectations within Path Dependence Les Levidow, Theo Papaioannou and Alexander Borda-Rodriguez Open University OU-MCT Energy Research Day Conference 6 March 2013 Acknowledgements

716 views • 20 slides
Linear Systems Linear Systems Transform Ax = b into an equivalent but Transform Ax b into

Linear Systems Linear Systems Transform Ax = b into an equivalent but Transform Ax b into

Solving Linear Systems g y Linear Systems Linear Systems Transform Ax = b into an equivalent but Transform Ax b into an equivalent but simpler system. Gaussian Elimination Multiply on the left by a nonsingular Multiply on the

189 views • 6 slides
Royal Economic Society Gary Beckers "A Theory of the Allocation of Time" Royal

Royal Economic Society Gary Beckers "A Theory of the Allocation of Time" Royal

Royal Economic Society Gary Beckers "A Theory of the Allocation of Time" Royal Economic Society Arthur Lewbel Boston College March 2015 Lewbel (Boston College) 03/15 1 / 11 Setting the table: Before "Treatise on the

1.34k views • 113 slides

More recommend