BOOLEAN FUNCTIONS Theory, Algorithms, and Applications

Transcription

1 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 September 23, 2010

2 Copyright Yves Crama HEC Management School of the University of Liège Belgium Dedication... Acknowledgments...

3 Contents I FOUNDATIONS 13 1 Fundamental concepts and applications Boolean functions: definitions and examples Boolean expressions Duality Normal forms Transforming an arbitrary expression into a DNF Orthogonal DNFs and number of true points Implicants and prime implicants Restrictions of functions, essential variables Geometric interpretation Monotone Boolean functions Definitions and examples DNFs and prime implicants of positive functions Minimal true points and maximal false points Recognition of functional and DNF properties Other representations of Boolean functions Representations over GF(2) Representations over the reals Binary decision diagrams and decision trees Applications Propositional logic and artificial intelligence Electrical and computer engineering Game theory Reliability theory Combinatorics Integer programming

4 2 CONTENTS 1.14 Exercises Boolean equations Definitions and applications The complexity of Boolean equations: Cook s theorem On the role of DNF equations What does it mean to solve a Boolean equation? Branching procedures Branching rules Preprocessing Variable elimination procedures The consensus procedure Mathematical programming approaches Integer linear programming Nonlinear programming Local search heuristics Recent trends and algorithmic performance More on the complexity of Boolean equations Complexity of equation solving procedures Random equations Constraint satisfaction problems and Schaefer s theorem Generalizations of consistency testing Counting the number of solutions Generating all solutions Parametric solutions Maximum satisfiability Exercises Prime implicants and minimal DNFs 155 Peter L. Hammer and Alexander Kogan 3.1 Prime implicants Applications to propositional logic and artificial intelligence Short prime implicants Generation of all prime implicants Generation from the set of true points Generation from a DNF representation: The consensus method.. 163

5 CONTENTS Generation from a DNF representation: Complexity Generation from a CNF representation Logic minimization Quine-McCluskey approach: Logic minimization as set covering Local simplifications of DNFs Computational complexity of logic minimization Efficient approximation algorithms for logic minimization Extremal and typical parameter values Number of prime implicants Extremal parameters of minimal DNFs Typical parameters of Boolean functions and their DNFs Exercises Duality theory 207 Yves Crama and Kazuhisa Makino 4.1 Basic properties and applications Dual functions and expressions Normal forms and implicants of dual functions Dual-comparable functions Applications Duality properties of positive functions Normal forms and implicants of dual functions Dual-comparable functions Applications Algorithmic aspects: The general case Definitions and complexity results Dualization by sequential distributivity Algorithmic aspects: Positive functions Some complexity results A quasi-polynomial dualization algorithm Additional results Exercises II SPECIAL CLASSES Quadratic functions 249

6 4 CONTENTS Bruno Simeone 5.1 Basic definitions and properties Why are quadratic Boolean functions important? Special classes of quadratic functions Classes Functional characterizations Quadratic Boolean functions and graphs Graph models of quadratic functions The matched graph The implication graph Conflict codes and quadratic graphs Reducibility of combinatorial problems to quadratic equations Introduction Bipartite graphs Balance in signed graphs Split graphs Forbidden-color graph bipartition Totally unimodular matrices with two nonzero entries per column The Kőnig-Egerváry property for graphs Single-bend wiring Max-quadratic functions and VLSI design A level graph drawing problem A final look into complexity Efficient graph-theoretic algorithms for quadratic equations Introduction Labeling algorithm (L) Alternative Labeling algorithm (AL) Switching algorithm (S) Strong Components algorithm (SC) An experimental comparison of algorithms for quadratic equations Quadratic equations: Special topics The set of solutions of a quadratic equation Parametric solutions Maximum 2-satisfiability On-line quadratic equations

7 CONTENTS Prime implicants and irredundant forms Introduction A transitive closure algorithm for finding all prime implicants A restricted consensus method and its application to computing the transitive closure of a digraph Irredundant normal forms and transitive reductions Dualization of quadratic functions Introduction The dualization algorithm Exercises Horn functions 327 Endre Boros 6.1 Basic definitions and properties Applications of Horn functions Propositional rule bases Functional dependencies in data bases Directed graphs, hypergraphs and Petri nets Integer programming and polyhedral combinatorics False points of Horn functions Deduction in AI Horn equations Horn equations and the unit literal rule Pure Horn equations and forward chaining More on Horn equations Prime implicants of Horn functions Properties of the set of prime implicants Minimization of Horn DNFs Minimizing the number of terms Minimizing the number of literals Minimization of the number of source sides Dualization of Horn functions Special classes Submodular functions Bidual Horn functions Double Horn functions

8 6 CONTENTS Acyclic Horn functions Generalizations Renamable Horn expressions and functions Q-Horn functions Extended Horn expressions Polynomial hierarchies built on Horn expressions Exercises Orthogonal forms and shellability Computation of orthogonal DNFs Shellings and shellability Definition Orthogonalization of shellable DNFs Shellable DNFs vs. shellable functions Dualization of shellable DNFs The lexico-exchange property Definition LE property and leaders Recognizing the LE property Dualization of functions having the LE property Shellable quadratic DNFs and graphs Applications Exercises Regular functions Relative strength of variables and regularity Basic properties Regularity and left-shifts Recognition of regular functions Dualization of regular functions Regular set covering problems Regular minorants and majorants Largest regular minorant with respect to a given order Smallest regular majorant with respect to a given order Regular minorization and set covering problems Higher-order monotonicity Generalizations of regularity

9 CONTENTS Weakly regular functions Aligned functions Ideal functions Relations among classes Exercises Threshold functions Definitions and applications Basic properties of threshold functions Characterizations of threshold functions Recognition of threshold functions A polynomial-time algorithm for positive DNFs A compact formulation The general case Prime implicants of threshold functions Chow parameters of threshold functions Chow functions Chow parameters and separating structures Computing the Chow parameters Threshold graphs Exercises Read-once functions 535 Martin C. Golumbic and Vladimir Gurvich 10.1 Introduction Dual implicants Implicants and dual implicants The dual subimplicant theorem Characterizing read-once functions The properties of P 4 -free graphs and cographs Recognizing read-once functions Learning read-once functions Related topics and applications of read-once functions The readability of a Boolean function Factoring general Boolean functions Positional games

10 8 CONTENTS 10.8 Historical notes Exercises Characterizations of special classes by functional equations 581 Lisa Hellerstein 11.1 Characterizations of positive functions Functional equations Characterizations of particular classes Horn functions Linear functions and related classes Quadratic and degree k functions Conditions for characterization Finite characterizations by functional equations Exercises III GENERALIZATIONS Partially defined Boolean functions 607 Toshihide Ibaraki 12.1 Introduction Extensions of pdbfs and their representations Definitions Support sets of variables Patterns and theories of pdbfs Roles of theories and co-theories Decision trees Extensions within given function classes Positive extensions Monotone (unate) extensions Degree-k extensions Horn extensions Threshold extensions Decomposable extensions k-convex extensions Best-fit extensions of pdbfs containing errors

11 CONTENTS Extensions of pdbfs with missing bits Three types of extensions Complexity results Minimization with don t cares Conclusion Exercises Pseudo-Boolean functions Definitions and examples Representations Polynomial expressions, pseudo-boolean normal forms and posiforms Piecewise linear representations Disjunctive and conjunctive normal forms Extensions of pseudo-boolean functions The polynomial extension Concave and convex extensions The Lovász extension Pseudo-Boolean optimization Local optima An elimination algorithm for global optimization Extensions and relaxations Posiform transformations and conflict graphs Approximations Special classes of pseudo-boolean functions Quadratic functions and quadratic 0-1 optimization Monotone functions Supermodular and submodular functions Unimodular functions Threshold and unimodal functions Exercises A Graphs and hypergraphs 721 A.1 Undirected graphs A.1.1 Subgraphs A.1.2 Paths and connectivity A.1.3 Special classes of graphs

12 10 CONTENTS A.2 Directed graphs A.2.1 Directed paths and connectivity A.2.2 Special classes of digraphs A.2.3 Transitive closure and transitive reduction A.3 Hypergraphs B Algorithmic complexity 729 B.1 Decision problems B.2 Algorithms B.3 Running time, polynomial-time algorithms and the class P B.4 The class NP B.5 Polynomial-time reductions and NP-completeness B.6 The class conp B.7 Cook s theorem B.8 Complexity of list-generation and counting algorithms C JBool: A software tool 743 Claude Benzaken and Nadia Brauner C.1 Introduction C.2 Work interface C.2.1 Menu bar C.2.2 Function windows C.2.3 Text zone C.3 Creating a Boolean function C.3.1 Function syntax and presentation C.3.2 Creation modes C.3.3 Saving a function C.4 Editing a function C.4.1 Changing the normal form C.4.2 Modifying the variable set C.5 Operations on Boolean functions C.5.1 Equivalent presentations of a Boolean function C.5.2 Constructions C.5.3 Operations on two Boolean functions C.5.4 Testing properties of a function Bibliography 751