Introduction to Mathematics of Satisfiability

Introduction to Mathematics of Satisfiability

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  • Focuses on both theoretical and practical aspects of satisfiability
  • Discusses the important topic of clausal logic
  • Reduces the satisfiability of clausal theories to classical problems of integer programming and linear algebra
  • Offers shortcuts to programming with SAT, such as a variation of predicate logic without function symbols, cardinality constraints, and monotone constraints
  • Outlines the foundations of answer set programming and how it can be used for knowledge representation
  • Explains most mathematics of SAT from first principles
  • Provides additions, corrections, and improvements to the book on the author’s website


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.

Table of Contents


Sets, Lattices, and Boolean Algebras

Sets and Set-Theoretic Notation

Posets, Lattices, and Boolean Algebras

Well-Orderings and Ordinals

The Fixpoint Theorem

Introduction to Propositional Logic

Syntax of Propositional Logic

Semantics of Propositional Logic


Tautologies and Substitutions

Lindenbaum Algebra



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


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


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




Exercises appear at the end of each chapter.

Author Bio(s)

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

Editorial Reviews

This interesting book covers the satisfiability problem with a strong focus on its mathematical background. It includes the famous theorems on the problem as well as some exotic results. … To improve understanding, the book offers plenty of insightful examples, elegant proofs, and each chapter ends with about a dozen exercises. … What I like most about the book is the wide variety of ideas of which the usefulness to solve many problems is almost tangible … the book covers more potentially powerful techniques, such as the cutting plane rule and various autarky detection methods, than those used in the latest state-of-the-art solvers. … apart from the collection of elegant proofs — from major theorems to exotic lemmas — Introduction to Mathematics of Satisfiability is also a source of inspiration for students and researchers in the field of satisfiability.
Theory and Practice of Logic Programming, Vol. 11, Issue 1

… Through very current material at the heart of the book, the author presents and analyzes general algorithms that work better than exhaustive search … Marek also covers important special cases of the problem that turn out vulnerable to clever special attacks. … Summing Up: Recommended.
CHOICE, September 2010

… an invaluable reference for anyone who is interested in issues ranging from theoretical mathematical logic to computational logic. The book maintains a nice tradeoff between formalism and clarity. … The author excels at relating his expositions to the current state of the art, and he recognizes when his discussions are only the tip of the iceberg. … its most significant contribution is its accessible explanations of how and why algorithms and ideas expose work.
—Carlos Linares Lopez, Computing Reviews, March 2010

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