Boolean Functions: Theory, Algorithms, and Applications

BOOLEAN FUNCTIONS

Theory, Algorithms, and

Applications

Yves CRAMA and Peter L. HAMMER

with contributions by

Claude Benzaken, Endre Boros,

Nadia Brauner, Martin C. Golumbic, Vladimir Gurvich,

Lisa Hellerstein, Toshihide Ibaraki, Alexander Kogan,

Kazuhisa Makino, and Bruno Simeone

HEC Management School of the University of Li`ege Belgium

Dedication...

Contents

I

FOUNDATIONS

13

1 Fundamental concepts and applications 15

1.1 Boolean functions: definitions and examples . . . 16

1.2 Boolean expressions . . . 21

1.3 Duality . . . 26

1.4 Normal forms . . . 28

1.5 Transforming an arbitrary expression into a DNF . . . 33

1.6 Orthogonal DNFs and number of true points . . . 38

1.7 Implicants and prime implicants . . . 41

1.8 Restrictions of functions, essential variables . . . 45

1.9 Geometric interpretation . . . 49

1.10 Monotone Boolean functions . . . 51

1.10.1 Definitions and examples . . . 52

1.10.2 DNFs and prime implicants of positive functions . . . 54

1.10.3 Minimal true points and maximal false points . . . 57

1.11 Recognition of functional and DNF properties . . . 59

1.12 Other representations of Boolean functions . . . 63

1.12.1 Representations over GF(2) . . . 63

1.12.2 Representations over the reals . . . 65

1.12.3 Binary decision diagrams and decision trees . . . 66

1.13 Applications . . . 70

1.13.1 Propositional logic and artificial intelligence . . . 71

1.13.2 Electrical and computer engineering . . . 74

1.13.3 Game theory . . . 77

1.13.4 Reliability theory . . . 80

1.13.5 Combinatorics . . . 82

1.13.6 Integer programming . . . 84 1

1.14 Exercises . . . 88

2 Boolean equations 91 2.1 Definitions and applications . . . 91

2.2 The complexity of Boolean equations: Cook’s theorem . . . 97

2.3 On the role of DNF equations . . . 99

2.4 What does it mean to “solve a Boolean equation”? . . . 104

2.5 Branching procedures . . . 106

2.5.1 Branching rules . . . 108

2.5.2 Preprocessing . . . 111

2.6 Variable elimination procedures . . . 114

2.7 The consensus procedure . . . 120

2.8 Mathematical programming approaches . . . 124

2.8.1 Integer linear programming . . . 124

2.8.2 Nonlinear programming . . . 129

2.8.3 Local search heuristics . . . 131

2.9 Recent trends and algorithmic performance . . . 132

2.10 More on the complexity of Boolean equations . . . 133

2.10.1 Complexity of equation solving procedures . . . 133

2.10.2 Random equations . . . 135

2.10.3 Constraint satisfaction problems and Schaefer’s theorem . . . 138

2.11 Generalizations of consistency testing . . . 141

2.11.1 Counting the number of solutions . . . 141

2.11.2 Generating all solutions . . . 142

2.11.3 Parametric solutions . . . 144

2.11.4 Maximum satisfiability . . . 146

2.12 Exercises . . . 153

3 Prime implicants and minimal DNFs 155 Peter L. Hammer and Alexander Kogan 3.1 Prime implicants . . . 156

3.1.1 Applications to propositional logic and artificial intelligence . . . . 156

3.1.2 Short prime implicants . . . 158

3.2 Generation of all prime implicants . . . 160

3.2.1 Generation from the set of true points . . . 161

CONTENTS 3

3.2.3 Generation from a DNF representation: Complexity . . . 172

3.2.4 Generation from a CNF representation . . . 176

3.3 Logic minimization . . . 177

3.3.1 Quine-McCluskey approach: Logic minimization as set covering . . 179

3.3.2 Local simplifications of DNFs . . . 182

3.3.3 Computational complexity of logic minimization . . . 187

3.3.4 Efficient approximation algorithms for logic minimization . . . 194

3.4 Extremal and typical parameter values . . . 198

3.4.1 Number of prime implicants . . . 198

3.4.2 Extremal parameters of minimal DNFs . . . 200

3.4.3 Typical parameters of Boolean functions and their DNFs . . . 201

3.5 Exercises . . . 204

4 Duality theory 207 Yves Crama and Kazuhisa Makino 4.1 Basic properties and applications . . . 207

4.1.1 Dual functions and expressions . . . 207

4.1.2 Normal forms and implicants of dual functions . . . 209

4.1.3 Dual-comparable functions . . . 211

4.1.4 Applications . . . 215

4.2 Duality properties of positive functions . . . 218

4.2.1 Normal forms and implicants of dual functions . . . 218

4.2.2 Dual-comparable functions . . . 220

4.2.3 Applications . . . 221

4.3 Algorithmic aspects: The general case . . . 225

4.3.1 Definitions and complexity results . . . 225

4.3.2 Dualization by sequential distributivity . . . 229

4.4 Algorithmic aspects: Positive functions . . . 233

4.4.1 Some complexity results . . . 233

4.4.2 A quasi-polynomial dualization algorithm . . . 236

4.4.3 Additional results . . . 241

4.5 Exercises . . . 243

II

SPECIAL CLASSES

247

Bruno Simeone

5.1 Basic definitions and properties . . . 249

5.2 Why are quadratic Boolean functions important? . . . 251

5.3 Special classes of quadratic functions . . . 254

5.3.1 Classes . . . 254

5.3.2 Functional characterizations . . . 255

5.4 Quadratic Boolean functions and graphs . . . 256

5.4.1 Graph models of quadratic functions . . . 256

5.4.2 The matched graph . . . 257

5.4.3 The implication graph . . . 259

5.4.4 Conflict codes and quadratic graphs . . . 264

5.5 Reducibility of combinatorial problems to quadratic equations . . . 267

5.5.1 Introduction . . . 267

5.5.2 Bipartite graphs . . . 267

5.5.3 Balance in signed graphs . . . 269

5.5.4 Split graphs . . . 269

5.5.5 Forbidden-color graph bipartition . . . 270

5.5.6 Totally unimodular matrices with two nonzero entries per column . 271 5.5.7 The K˝onig-Egerv´ary property for graphs . . . 272

5.5.8 Single-bend wiring . . . 274

5.5.9 Max-quadratic functions and VLSI design . . . 275

5.5.10 A level graph drawing problem . . . 277

5.5.11 A final look into complexity . . . 281

5.6 Efficient graph-theoretic algorithms for quadratic equations . . . 281

5.6.1 Introduction . . . 281

5.6.2 Labeling algorithm (L) . . . 283

5.6.3 Alternative Labeling algorithm (AL) . . . 287

5.6.4 Switching algorithm (S) . . . 288

5.6.5 Strong Components algorithm (SC) . . . 292

5.6.6 An experimental comparison of algorithms for quadratic equations . 294 5.7 Quadratic equations: Special topics . . . 296

5.7.1 The set of solutions of a quadratic equation . . . 296

5.7.2 Parametric solutions . . . 300

5.7.3 Maximum 2-satisfiability . . . 303

CONTENTS 5

5.8 Prime implicants and irredundant forms . . . 305

5.8.1 Introduction . . . 305

5.8.2 A transitive closure algorithm for finding all prime implicants . . . 306

5.8.3 A restricted consensus method and its application to computing the transitive closure of a digraph . . . 310

5.8.4 Irredundant normal forms and transitive reductions . . . 319

5.9 Dualization of quadratic functions . . . 320

5.9.1 Introduction . . . 320

5.9.2 The dualization algorithm . . . 321

5.10 Exercises . . . 324

6 Horn functions 327 Endre Boros 6.1 Basic definitions and properties . . . 327

6.2 Applications of Horn functions . . . 331

6.2.1 Propositional rule bases . . . 331

6.2.2 Functional dependencies in data bases . . . 332

6.2.3 Directed graphs, hypergraphs and Petri nets . . . 333

6.2.4 Integer programming and polyhedral combinatorics . . . 334

6.3 False points of Horn functions . . . 336

6.3.1 Deduction in AI . . . 338

6.4 Horn equations . . . 340

6.4.1 Horn equations and the unit literal rule . . . 340

6.4.2 Pure Horn equations and forward chaining . . . 343

6.4.3 More on Horn equations . . . 346

6.5 Prime implicants of Horn functions . . . 346

6.6 Properties of the set of prime implicants . . . 354

6.7 Minimization of Horn DNFs . . . 359

6.7.1 Minimizing the number of terms . . . 360

6.7.2 Minimizing the number of literals . . . 364

6.7.3 Minimization of the number of source sides . . . 365

6.8 Dualization of Horn functions . . . 369

6.9 Special classes . . . 373

6.9.1 Submodular functions . . . 373

6.9.2 Bidual Horn functions . . . 374

6.9.4 Acyclic Horn functions . . . 377

6.10 Generalizations . . . 379

6.10.1 Renamable Horn expressions and functions . . . 379

6.10.2 Q-Horn functions . . . 381

6.10.3 Extended Horn expressions . . . 384

6.10.4 Polynomial hierarchies built on Horn expressions . . . 385

6.11 Exercises . . . 386

7 Orthogonal forms and shellability 393 7.1 Computation of orthogonal DNFs . . . 394

7.2 Shellings and shellability . . . 398

7.2.1 Definition . . . 398

7.2.2 Orthogonalization of shellable DNFs . . . 400

7.2.3 Shellable DNFs vs. shellable functions . . . 403

7.3 Dualization of shellable DNFs . . . 405

7.4 The lexico-exchange property . . . 407

7.4.1 Definition . . . 407

7.4.2 LE property and leaders . . . 408

7.4.3 Recognizing the LE property . . . 409

7.4.4 Dualization of functions having the LE property . . . 416

7.5 Shellable quadratic DNFs and graphs . . . 417

7.6 Applications . . . 419

7.7 Exercises . . . 420

8 Regular functions 423 8.1 Relative strength of variables and regularity . . . 423

8.2 Basic properties . . . 428

8.3 Regularity and left-shifts . . . 436

8.4 Recognition of regular functions . . . 439

8.5 Dualization of regular functions . . . 445

8.6 Regular set–covering problems . . . 453

8.7 Regular minorants and majorants . . . 457

8.7.1 Largest regular minorant with respect to a given order . . . 460

8.7.2 Smallest regular majorant with respect to a given order . . . 465

8.7.3 Regular minorization and set–covering problems . . . 466

8.8 Higher-order monotonicity . . . 469

CONTENTS 7

8.9.1 Weakly regular functions . . . 477

8.9.2 Aligned functions . . . 478

8.9.3 Ideal functions . . . 479

8.9.4 Relations among classes . . . 480

8.10 Exercises . . . 481

9 Threshold functions 485 9.1 Definitions and applications . . . 485

9.2 Basic properties of threshold functions . . . 489

9.3 Characterizations of threshold functions . . . 495

9.4 Recognition of threshold functions . . . 500

9.4.1 A polynomial-time algorithm for positive DNFs . . . 500

9.4.2 A compact formulation . . . 502

9.4.3 The general case . . . 504

9.5 Prime implicants of threshold functions . . . 506

9.6 Chow parameters of threshold functions . . . 512

9.6.1 Chow functions . . . 513

9.6.2 Chow parameters and separating structures . . . 515

9.6.3 Computing the Chow parameters . . . 521

9.7 Threshold graphs . . . 523

9.8 Exercises . . . 531

10 Read-once functions 535 Martin C. Golumbic and Vladimir Gurvich 10.1 Introduction . . . 535

10.2 Dual implicants . . . 538

10.2.1 Implicants and dual implicants . . . 538

10.2.2 The dual subimplicant theorem . . . 539

10.3 Characterizing read-once functions . . . 544

10.4 The properties of P4-free graphs and cographs . . . 552

10.5 Recognizing read-once functions . . . 556

10.6 Learning read-once functions . . . 564

10.7 Related topics and applications of read-once functions . . . 567

10.7.1 The readability of a Boolean function . . . 567

10.7.2 Factoring general Boolean functions . . . 568

10.8 Historical notes . . . 572

10.9 Exercises . . . 574

11 Characterizations of special classes by functional equations 581 Lisa Hellerstein 11.1 Characterizations of positive functions . . . 582

11.2 Functional equations . . . 583

11.3 Characterizations of particular classes . . . 586

11.3.1 Horn functions . . . 586

11.3.2 Linear functions and related classes . . . 587

11.3.3 Quadratic and degree k functions . . . 588

11.4 Conditions for characterization . . . 591

11.5 Finite characterizations by functional equations . . . 597

11.6 Exercises . . . 602

III

GENERALIZATIONS

605

12.2 Extensions of pdBfs and their representations . . . 611

12.2.1 Definitions . . . 611

12.2.2 Support sets of variables . . . 612

12.2.3 Patterns and theories of pdBfs . . . 615

12.2.4 Roles of theories and co-theories . . . 620

12.2.5 Decision trees . . . 625

12.3 Extensions within given function classes . . . 630

12.3.1 Positive extensions . . . 630

12.3.2 Monotone (unate) extensions . . . 632

12.3.3 Degree-k extensions . . . 634

12.3.4 Horn extensions . . . 635

12.3.5 Threshold extensions . . . 638

12.3.6 Decomposable extensions . . . 640

12.3.7 k-convex extensions . . . 646

CONTENTS 9

12.5 Extensions of pdBfs with missing bits . . . 653

12.5.1 Three types of extensions . . . 653

12.5.2 Complexity results . . . 656

12.6 Minimization with don’t cares . . . 661

12.7 Conclusion . . . 664

12.8 Exercises . . . 665

13 Pseudo-Boolean functions 669 13.1 Definitions and examples . . . 669

13.2 Representations . . . 676

13.2.1 Polynomial expressions, pseudo-Boolean normal forms and posiforms . . . 676

13.2.2 Piecewise linear representations . . . 680

13.2.3 Disjunctive and conjunctive normal forms . . . 680

13.3 Extensions of pseudo-Boolean functions . . . 684

13.3.1 The polynomial extension . . . 686

13.3.2 Concave and convex extensions . . . 687

13.3.3 The Lov´asz extension . . . 691

13.4 Pseudo-Boolean optimization . . . 692

13.4.1 Local optima . . . 693

13.4.2 An elimination algorithm for global optimization . . . 695

13.4.3 Extensions and relaxations . . . 696

13.4.4 Posiform transformations and conflict graphs . . . 698

13.5 Approximations . . . 701

13.6 Special classes of pseudo-Boolean functions . . . 702

13.6.1 Quadratic functions and quadratic 0-1 optimization . . . 702

13.6.2 Monotone functions . . . 705

13.6.3 Supermodular and submodular functions . . . 708

13.6.4 Unimodular functions . . . 714

13.6.5 Threshold and unimodal functions . . . 716

13.7 Exercises . . . 717

A Graphs and hypergraphs 721 A.1 Undirected graphs . . . 721

A.1.1 Subgraphs . . . 722

A.1.2 Paths and connectivity . . . 722

A.2 Directed graphs . . . 725

A.2.1 Directed paths and connectivity . . . 725

A.2.2 Special classes of digraphs . . . 725

A.2.3 Transitive closure and transitive reduction . . . 726

A.3 Hypergraphs . . . 727

B Algorithmic complexity 729 B.1 Decision problems . . . 729

B.2 Algorithms . . . 731

B.3 Running time, polynomial-time algorithms and the class P . . . 733

B.4 The class NP . . . 733

B.5 Polynomial-time reductions and NP-completeness . . . 735

B.6 The class coNP . . . 736

B.7 Cook’s theorem . . . 737

B.8 Complexity of list-generation and counting algorithms . . . 739

C JBool: A software tool 743 Claude Benzaken and Nadia Brauner C.1 Introduction . . . 743

C.2 Work interface . . . 744

C.2.1 Menu bar . . . 745

C.2.2 Function windows . . . 745

C.2.3 Text zone . . . 746

C.3 Creating a Boolean function . . . 746

C.3.1 Function syntax and presentation . . . 746

C.3.2 Creation modes . . . 747

C.3.3 Saving a function . . . 748

C.4 Editing a function . . . 749

C.4.1 Changing the normal form . . . 749

C.4.2 Modifying the variable set . . . 749

C.5 Operations on Boolean functions . . . 749

C.5.1 Equivalent presentations of a Boolean function . . . 750

C.5.2 Constructions . . . 750

C.5.3 Operations on two Boolean functions . . . 750

C.5.4 Testing properties of a function . . . 750