The paper by Thomas J. Schaefer explores the complexity of satisfiability problems, particularly focusing on formulas with three literals per clause. It introduces an infinite class of satisfiability problems that includes the standard CNF satisfiability problem and shows that each member of this class is either polynomial-time decidable or NP-complete. The main result, the Dichotomy Theorem, characterizes the complexity of these problems based on the properties of the logical relations involved. Specifically, the satisfiability problem is polynomial-time decidable if the relations satisfy certain conditions (0-valid, 1-valid, weakly positive, weakly negative, affine, or bijunctive), and NP-complete otherwise. The paper also provides a classification of the polynomial-time decidable cases and discusses the complexity of satisfiability problems with constants and quantified formulas. Several specific completeness results are derived, including the NP-completeness of problems like ONE-IN-THREE SATISFIABILITY and NOT-ALL-EQUAL SATISFIABILITY. The work builds on earlier research and contributes to the understanding of the complexity landscape of satisfiability problems.The paper by Thomas J. Schaefer explores the complexity of satisfiability problems, particularly focusing on formulas with three literals per clause. It introduces an infinite class of satisfiability problems that includes the standard CNF satisfiability problem and shows that each member of this class is either polynomial-time decidable or NP-complete. The main result, the Dichotomy Theorem, characterizes the complexity of these problems based on the properties of the logical relations involved. Specifically, the satisfiability problem is polynomial-time decidable if the relations satisfy certain conditions (0-valid, 1-valid, weakly positive, weakly negative, affine, or bijunctive), and NP-complete otherwise. The paper also provides a classification of the polynomial-time decidable cases and discusses the complexity of satisfiability problems with constants and quantified formulas. Several specific completeness results are derived, including the NP-completeness of problems like ONE-IN-THREE SATISFIABILITY and NOT-ALL-EQUAL SATISFIABILITY. The work builds on earlier research and contributes to the understanding of the complexity landscape of satisfiability problems.