an mcsat treatment of bit vectors work in progress
play

An MCSAT treatment of Bit-Vectors (work-in-progress) Stphane - PowerPoint PPT Presentation

Aug 13, 2022 •191 likes •996 views

An MCSAT treatment of Bit-Vectors (work-in-progress) Stphane Graham-Lengrand and Dejan Jovanovi CNRS - SRI International SMT workshop, 23rd July 2017 1/19 The model-constructing approach to SMT-solving MCSAT introduced in [dMJ13, JBdM13,

  1. An MCSAT treatment of Bit-Vectors (work-in-progress) Stéphane Graham-Lengrand and Dejan Jovanović CNRS - SRI International SMT workshop, 23rd July 2017 1/19

  2. The model-constructing approach to SMT-solving MCSAT introduced in [dMJ13, JBdM13, Jov17], following work on specific decision procedures for theories such as non-linear arithmetic [JdM12]. 2/19

  3. The model-constructing approach to SMT-solving MCSAT introduced in [dMJ13, JBdM13, Jov17], following work on specific decision procedures for theories such as non-linear arithmetic [JdM12]. MCSAT offers: ◮ a template for decision procedures ◮ an integration of such procedures with Boolean reasoning ◮ new possibilities for combining procedures [JBdM13, BGLS17] 2/19

  4. The model-constructing approach to SMT-solving MCSAT introduced in [dMJ13, JBdM13, Jov17], following work on specific decision procedures for theories such as non-linear arithmetic [JdM12]. MCSAT offers: ◮ a template for decision procedures ◮ an integration of such procedures with Boolean reasoning ◮ new possibilities for combining procedures [JBdM13, BGLS17] The template is a generalisation of how CDCL works. Run = alternation of search phases and conflict analysis phases Boolean theory can be given the same status as other theories. 2/19

  5. The model-constructing approach to SMT-solving MCSAT introduced in [dMJ13, JBdM13, Jov17], following work on specific decision procedures for theories such as non-linear arithmetic [JdM12]. MCSAT offers: ◮ a template for decision procedures ◮ an integration of such procedures with Boolean reasoning ◮ new possibilities for combining procedures [JBdM13, BGLS17] The template is a generalisation of how CDCL works. Run = alternation of search phases and conflict analysis phases Boolean theory can be given the same status as other theories. Terms and literals are created that do not belong to the input problem. 2/19

  6. Search phase 1. For each variable awaiting a value, track the set of feasible values (those not yet ruled out by the current constraints) 3/19

  7. Search phase 1. For each variable awaiting a value, track the set of feasible values (those not yet ruled out by the current constraints) 2. For one of the variables, pick a value out of that set. 3/19

  8. Search phase 1. For each variable awaiting a value, track the set of feasible values (those not yet ruled out by the current constraints) 2. For one of the variables, pick a value out of that set. 3. With that choice, some of the constraints may become unit : all of their free variables are assigned a value but one. 3/19

  9. Search phase 1. For each variable awaiting a value, track the set of feasible values (those not yet ruled out by the current constraints) 2. For one of the variables, pick a value out of that set. 3. With that choice, some of the constraints may become unit : all of their free variables are assigned a value but one. 4. See whether and how those new constraints unit in y restrict the feasible values for y , and update that set 3/19

  10. Search phase 1. For each variable awaiting a value, track the set of feasible values (those not yet ruled out by the current constraints) 2. For one of the variables, pick a value out of that set. 3. With that choice, some of the constraints may become unit : all of their free variables are assigned a value but one. 4. See whether and how those new constraints unit in y restrict the feasible values for y , and update that set 5. Repeat until all variables are assigned values & all constraints are satisfied or one of these sets becomes empty: there is a conflict 3/19

  11. Search phase 1. For each variable awaiting a value, track the set of feasible values (those not yet ruled out by the current constraints) 2. For one of the variables, pick a value out of that set. 3. With that choice, some of the constraints may become unit : all of their free variables are assigned a value but one. 4. See whether and how those new constraints unit in y restrict the feasible values for y , and update that set 5. Repeat until all variables are assigned values & all constraints are satisfied or one of these sets becomes empty: there is a conflict Simple process: we only look at what the constraints say once they become unit. Until then, we simply maintain for each constraint a watch list of variables, to detect when they become unit (as in CDCL). 3/19

  12. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 4/19

  13. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 4/19

  14. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 4/19

  15. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 4/19

  16. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 4/19

  17. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i SAT 4/19

  18. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 5/19

  19. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 5/19

  20. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 5/19

  21. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 5/19

  22. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } x 1 j , . . . C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } x 2 j , . . . C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } x 3 j , . . . . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i Conflict 5/19

  23. Implementing the set of feasible values for y This has to be a data-structure with operations for 1. updating the set whenever a new constraint becomes unit in y , 2. detecting when the set becomes empty, and 3. proposing a value from the feasible set. 6/19

  24. Implementing the set of feasible values for y This has to be a data-structure with operations for 1. updating the set whenever a new constraint becomes unit in y , 2. detecting when the set becomes empty, and 3. proposing a value from the feasible set. This is theory-dependent: ◮ For LRA, this can be an interval 6/19

  25. Implementing the set of feasible values for y This has to be a data-structure with operations for 1. updating the set whenever a new constraint becomes unit in y , 2. detecting when the set becomes empty, and 3. proposing a value from the feasible set. This is theory-dependent: ◮ For LRA, this can be an interval ◮ For bit-vectors, [ZWR16] use the combination of ◮ an interval, e.g. [0000 , 0010] (understanding bitvectors in arithmetic modulo) ◮ and a pattern imposing the value of some of the bits, e.g. ???1 6/19

  26. Alternative proposition: Use a Binary Decision Diagram (BDD) over the bits of y . 7/19

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

Solving bitvectors with MCSAT: explanations from bits and pieces Stphane Graham-Lengrand, Dejan

Solving bitvectors with MCSAT: explanations from bits and pieces Stphane Graham-Lengrand, Dejan

Solving bitvectors with MCSAT: explanations from bits and pieces Stphane Graham-Lengrand, Dejan Jovanovi, Bruno Dutertre SRI International IJCAR, July 2020 1/32 tl;dl (Too Long; Didnt Listen) MCSAT (Model-Constructing

1.27k views • 98 slides
Evaluating Treatment Effects and Replicability - work in progress - Victor Gonzalez-Jimenez

Evaluating Treatment Effects and Replicability - work in progress - Victor Gonzalez-Jimenez

Evaluating Treatment Effects and Replicability - work in progress - Victor Gonzalez-Jimenez Karl H. Schlag University of Vienna University of Vienna November 4, 2019 1 Motivation Replication: some want to emphasize validity of experiments,

309 views • 16 slides
Vectors Vectors and Scalars Properties of Vectors Components of a Vector and Unit

Vectors Vectors and Scalars Properties of Vectors Components of a Vector and Unit

Vectors Vectors and Scalars Properties of Vectors Components of a Vector and Unit Vectors Homework 1 Vectors and Scalars Vector - quantity that has magnitude and direction e.g. displacement, velocity, acceleration, force

560 views • 16 slides
TRADITION PRIDE PROGRESS WATER TREATMENT PLANT WASTEWATER TREATMENT PLANT PARKS DEPARTMENT

TRADITION PRIDE PROGRESS WATER TREATMENT PLANT WASTEWATER TREATMENT PLANT PARKS DEPARTMENT

TRADITION PRIDE PROGRESS WATER TREATMENT PLANT WASTEWATER TREATMENT PLANT PARKS DEPARTMENT COMMUNITY GARDEN GLOW IN THE DARK DISC GOLF COURSE PAVEMENT MANAGEMENT PLAN CITY ZONING MAP COMFORT INN & SUITES HOTEL GARVIN PARK PLAYGROUND

366 views • 17 slides
Fault Tree Represent at ion of Securit y Requirement s 0 Work in progress Work in progress

Fault Tree Represent at ion of Securit y Requirement s 0 Work in progress Work in progress

Fault Tree Represent at ion of Securit y Requirement s 0 Work in progress Work in progress One Picture is Worth a Thousand Words Couple Dozen Connectives I liano Cer vesat o iliano@itd.nrl.navy.mil I TT I ndust r ies, inc @ NRL Washingt

618 views • 23 slides
OCCUPATIONAL HEALTH CONSIDERATIONS FOR WORK WITH VIRAL VECTORS, RNA INTERFERENCE AND GENE

OCCUPATIONAL HEALTH CONSIDERATIONS FOR WORK WITH VIRAL VECTORS, RNA INTERFERENCE AND GENE

OCCUPATIONAL HEALTH CONSIDERATIONS FOR WORK WITH VIRAL VECTORS, RNA INTERFERENCE AND GENE EDITING GARY R. FUJIMOTO, M.D. OCCUPATIONAL AND INTERNAL MEDICINE CONSULTANT OCTOBER 25, 2016 VIRAL VECTORS Definition: Viruses engineered to

548 views • 30 slides
Work Work Done by a Constant Force The Scalar (or Dot) Product of Two Vectors Work Done

Work Work Done by a Constant Force The Scalar (or Dot) Product of Two Vectors Work Done

Work Work Done by a Constant Force The Scalar (or Dot) Product of Two Vectors Work Done by a Variable Force Homework 1 Work Done by a Constant Force W = F r cos The unit of work is the joule (J) (1 J = 1 N m) 2 Forces

556 views • 12 slides
Deciding Bit-Vector Formulas with mcSAT Aleksandar Zelji c Uppsala University Philipp R

Deciding Bit-Vector Formulas with mcSAT Aleksandar Zelji c Uppsala University Philipp R

Deciding Bit-Vector Formulas with mcSAT Aleksandar Zelji c Uppsala University Philipp R ummer Christoph Wintersteiger Uppsala University Microsoft Research SMT Workshop July 1 st , 2016 We present... A novel decision procedure for the

821 views • 39 slides
Motivation The big picture: CDCL, arithmetic, MCSAT The CDSAT approach Discussion Maria Paola

Motivation The big picture: CDCL, arithmetic, MCSAT The CDSAT approach Discussion Maria Paola

Outline Motivation The big picture: CDCL, arithmetic, MCSAT The CDSAT approach Discussion Conflict-driven reasoning 1 Maria Paola Bonacina Dipartimento di Informatica, Universit` a degli Studi di Verona, Verona, Italy, EU Invited talk 24th

876 views • 51 slides
Methods of Adding Vectors Geometrically MCV4U: Calculus & Vectors Recall that two vectors are

Methods of Adding Vectors Geometrically MCV4U: Calculus & Vectors Recall that two vectors are

g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Methods of Adding Vectors Geometrically MCV4U: Calculus & Vectors Recall that two vectors are equivalent if they have the same magnitude and direction. This means that vectors

281 views • 4 slides
Geometric Vectors A geometric vector is a representation of a vector using an arrow diagram, or

Geometric Vectors A geometric vector is a representation of a vector using an arrow diagram, or

Culminating Review for Vectors An Introduction to Vectors Applications of Vectors 4 2 1 Equations of Lines and Planes 0011 0010 1010 1101 0001 0100 1011 Relationships between Points, Lines and Planes 5 An Introduction to Vectors Geometric

341 views • 31 slides
Geometric Vectors A geometric vector is a representation of a vector using an arrow diagram, or

Geometric Vectors A geometric vector is a representation of a vector using an arrow diagram, or

Culminating Review for Vectors An Introduction to Vectors Applications of Vectors 4 2 1 Equations of Lines and Planes 0011 0010 1010 1101 0001 0100 1011 Relationships between Points, Lines and Planes 5 An Introduction to Vectors Geometric

737 views • 18 slides
Work MCV4U: Calculus & Vectors Work is when a force is applied to an object, causing it to

Work MCV4U: Calculus & Vectors Work is when a force is applied to an object, causing it to

a l g e b r a i c v e c t o r s a l g e b r a i c v e c t o r s Work MCV4U: Calculus & Vectors Work is when a force is applied to an object, causing it to move. It is defined as the the product of the objects displacement and the

810 views • 3 slides
CS 162 Intro to Programming II Vectors 1 Vectors A

CS 162 Intro to Programming II Vectors 1 Vectors A

CS 162 Intro to Programming II Vectors 1 Vectors A container from the Standard Template Library Holds a set of elements, like an array Flexible number of elements - can grow and shrink

511 views • 10 slides
Vectors MA1S1 Tristan McLoughlin tristan@maths.tcd.ie Vectors Some quantities (which we will

Vectors MA1S1 Tristan McLoughlin tristan@maths.tcd.ie Vectors Some quantities (which we will

Vectors MA1S1 Tristan McLoughlin tristan@maths.tcd.ie Vectors Some quantities (which we will call scalars) have purely numerical values (like mass, volume, temperature) while others (which are called vectors) also have a direction associated

368 views • 22 slides
Are we making any progress at all in the treatment of Advanced hepatocellular carcinoma? Bert

Are we making any progress at all in the treatment of Advanced hepatocellular carcinoma? Bert

Are we making any progress at all in the treatment of Advanced hepatocellular carcinoma? Bert H. ONeil, MD Cusick Professor of Oncology Director, GI and Phase I Clinical Trials IU Simon Cancer Center Disclosures Amgen Inc, Bayer

792 views • 20 slides
Network Flow IV - Given information about flights that an airline needs to Applications II

Network Flow IV - Given information about flights that an airline needs to Applications II

CS 573: Algorithms, Fall 2014 Airline Scheduling Problem Network Flow IV - Given information about flights that an airline needs to Applications II provide, generate a profitable schedule. 1. Input: detailed information about legs of

252 views • 13 slides
E ffi cient Bregman Projections Onto the Simplex Walid Krichene Syrine Krichene Alexandre Bayen

E ffi cient Bregman Projections Onto the Simplex Walid Krichene Syrine Krichene Alexandre Bayen

Introduction Projection Algorithms Numerical experiments E ffi cient Bregman Projections Onto the Simplex Walid Krichene Syrine Krichene Alexandre Bayen Electrical Engineering and Computer Sciences, UC Berkeley ENSIMAG and Criteo Labs, France

577 views • 38 slides
More Realistic Power Grid Verification Based on Hierarchical Current and Power constraints 2

More Realistic Power Grid Verification Based on Hierarchical Current and Power constraints 2

More Realistic Power Grid Verification Based on Hierarchical Current and Power constraints 2 Chung-Kuan Cheng, 2 Peng Du, 2 Andrew B. Kahng, 1 Grantham K. H. Pang, 1 Yuanzhe Wang, 1 Ngai Wong Grantham K. H. Pang, Yuanzhe Wang, Ngai Wong 1.

489 views • 30 slides
Constrained optimization Problem in standard form minimize f ( x ) subject to a i ( x ) = 0, for i

Constrained optimization Problem in standard form minimize f ( x ) subject to a i ( x ) = 0, for i

Constrained optimization Problem in standard form minimize f ( x ) subject to a i ( x ) = 0, for i = 1 , 2 , p c j ( x ) 0 for j = 1 , 2 , , q a i : R n R c j : R n R f : R n R f ( x ) = , if problem is

492 views • 24 slides
Policy & Financing Executive Directors Office & Behavioral Health Hearing Kim

Policy & Financing Executive Directors Office & Behavioral Health Hearing Kim

JOINT BUDGET COMMITTEE Colorado Department of Health Care Policy & Financing Executive Directors Office & Behavioral Health Hearing Kim Bimestefer, Executive Director Tom Leahey, Interim Pharmacy Office Director Craig Domeracki,

609 views • 48 slides
reflect those of anyone else in the Federal Reserve System.

reflect those of anyone else in the Federal Reserve System.

Hutchins Center on Fiscal and Monetary Policy Event Should the Fed stick with the 2 percent inflation target or rethink it? John Williams Federal Reserve Bank of San Francisco January 8, 2018 The views expressed here are those of the

374 views • 3 slides
Financial Assets and Liabilities, U.S. Nonfinancial Corporations 32 (dollars; natural log scale)

Financial Assets and Liabilities, U.S. Nonfinancial Corporations 32 (dollars; natural log scale)

Financial Assets, U.S. Nonfinancial Corporations as percentage of total (fixed + financial) assets 70% 65% 60% 55% 50% 45% 40% 35% 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Financial Assets and

264 views • 4 slides
Financial capability and saving behaviour Rio de Janeiro December 4th 2014 Ellen K. Nyhus

Financial capability and saving behaviour Rio de Janeiro December 4th 2014 Ellen K. Nyhus

Financial capability and saving behaviour Rio de Janeiro December 4th 2014 Ellen K. Nyhus Professor, School of Business and Law University of Agder, Kristiansand, Norway President International Association for Research in Economic

530 views • 15 slides

More recommend