Introduction to Mathematics of Satisfiability

Although this area has a history of over 80 years, it was not until the creation of efficient SAT solvers in the mid-1990s that it became practically important, finding applications in electronic design automation, hardware and software verification, combinatorial optimization, and more. Exploring the theoretical and practical aspects of satisfiability, Introduction to Mathematics of Satisfiability focuses on the satisfiability of theories consisting of propositional logic formulas. It describes how SAT solvers and techniques are applied to problems in mathematics and computer science as well as important applications in computer engineering.

The book first deals with logic fundamentals, including the syntax of propositional logic, complete sets of functors, normal forms, the Craig lemma, and compactness. It then examines clauses, their proof theory and semantics, and basic complexity issues of propositional logic. The final chapters on knowledge representation cover finite runs of Turing machines and encodings into SAT. One of the pioneers of answer set programming, the author shows how constraint satisfaction systems can be worked out by satisfiability solvers and how answer set programming can be used for knowledge representation.

Victor W. Marek is a professor in the Department of Computer Science at the University of Kentucky.

Preface

Sets, Lattices, and Boolean Algebras
Sets and Set-Theoretic Notation
Posets, Lattices, and Boolean Algebras
Well-Orderings and Ordinals
The Fixpoint Theorem

Introduction to PropositionalLogic
Syntax of Propositional Logic
Semantics of Propositional Logic
Autarkies
Tautologies and Substitutions
Lindenbaum Algebra
Permutations
Duality
Semantical Consequence

Normal Forms
Canonical Negation-Normal Form
Occurrences of Variables and Three-Valued Logic
Canonical Forms
Reduced Normal Forms
Complete Normal Forms
Lindenbaum Algebra Revisited
Other Normal Forms

The Craig Lemma
Craig Lemma
Strong Craig Lemma
Tying up Loose Ends

Complete Sets of Functors
Beyond De Morgan Functors
Tables
Field Structure in Bool
Incomplete Sets of Functors, Post Classes
Post Criterion for Completeness
If-Then-Else Functor

Compactness Theorem
König Lemma
Compactness, Denumerable Case
Continuity of the Operator Cn

Clausal Logic and Resolution
Clausal Logic
Resolution Rule
Completeness Theorem
Query Answering with Resolution
Davis–Putnam Lemma
Semantic Resolution
Autark and Lean Sets

Algorithms for SAT
Table Method
Hintikka Sets
Tableaux
Davis–Putnam Algorithm
Boolean Constraint Propagation
The DPLL Algorithm
Improvements to DPLL?
Reduction of the Search SAT to Decision SAT

Easy Cases of SAT
Positive and Negative Formulas
Horn Formulas
Autarkies for Horn Theories
Dual Horn Formulas
Krom Formulas and 2-SAT
Renameable Classes of Formulas
XOR Formulas

SAT, Integer Programming, and Matrix Algebra
Encoding of SAT as Inequalities
Resolution and Other Rules of Proof
Pigeon-Hole Principle and the Cutting Plane Rule
Satisfiability and {-1, 1}-Integer Programming
Embedding SAT into Matrix Algebra

Coding Runs of Turing Machine, and "Mix-and-Match"
Turing Machines
The Language
Coding the Runs
Correctness of Our Coding
Reduction to 3-Clauses
Coding Formulas as Clauses and Circuits
Decision Problem for Autarkies
Search Problem for Autarkies
Either-Or CNFs
Other Cases

Computational Knowledge Representation with SAT
Encoding into SAT, DIMACS Format
Knowledge Representation over Finite Domains
Cardinality Constraints, the Language Lcc
Weight Constraints
Monotone Constraints

Knowledge Representation and Constraint Satisfaction
Extensional Relations, CWA
Constraint Satisfaction and SAT
Satisfiability as Constraint Satisfaction
Polynomial Cases of Boolean CSP
Schaefer Dichotomy Theorem

Answer Set Programming
Horn Logic Revisited
Models of Programs
Supported Models
Stable Models
Answer Set Programming and SAT
Knowledge Representation and ASP
Complexity Issues for ASP

Conclusions

References

Index

Exercises appear at the end of each chapter.