Model elimination

Abstract: I. Propositional Logic from the Viewpoint of Analytic Tableaux.- I. Preliminaries.- 0. Foreword on Trees.- 1. Formulas of Propositional Logic.- 2. Boolean Valuations and Truth Sets.- II. Analytic Tableaux.- 1. The Method of Tableaux.- 2. Consistency and Completeness of the System.- III. Compactness.- 1. Analytic Proofs of the Compactness Theorem.- 2. Maximal Consistency: Lindenbaum's Construction.- 3. An Analytic Modification of Lindenbaum's Proof.- 4. The Compactness Theorem for Deducibility.- II. First-Order Logic.- IV. First-Order Logic. Preliminaries.- 1. Formulas of Quantification Theory.- 2. First-Order Valuations and Models.- 3. Boolean Valuations vs. First-Order Valuations.- V. First-Order Analytic Tableaux.- 1. Extension of Our Unified Notation.- 2. Analytic Tableaux for Quantification Theory.- 3. The Completeness Theorem.- 4. The Skolem-Lowenheim and Compactness Theorems for First-Order Logic.- VI. A Unifying Principle.- 1. Analytic Consistency.- 2. Further Discussion of Analytic Consistency.- 3. Analytic Consistency Properties for Finite Sets.- VII. The Fundamental Theorem of Quantification Theory.- 1. Regular Sets.- 2. The Fundamental Theorem.- 3. Analytic Tableaux and Regular Sets.- 4. The Liberalized Rule D.- VIII. Axiom Systems for Quantification Theory.- 0. Foreward on Axiom Systems.- 1. The System Q1.- 2. The Systems Q2, Q*2.- IX. Magic Sets.- 1. Magic Sets.- 2. Applications of Magic Sets.- X. Analytic versus Synthetic Consistency Properties.- 1. Synthetic Consistency Properties.- 2. A More Direct Construction.- III. Further Topics in First-Order Logic.- XI. Gentzen Systems.- 1. Gentzen Systems for Propositional Logic.- 2. Block Tableaux and Gentzen Systems for First-Order Logic.- XII. Elimination Theorems.- 1. Gentzen's Hauptsatz.- 2. An Abstract Form of the Hauptsatz Ill.- 3. Some Applications of the Hauptsatz.- XIII. Prenex Tableaux.- 1. Prenex Formulas.- 2. Prenex Tableaux.- XIV. More on Gentzen Systems.- 1. Gentzen's Extended Hauptsatz.- 2. A New Form of the Extended Hauptsatz.- 3. Symmetric Gentzen Systems.- XV. Craig's Interpolation Lemma and Beth's Definability Theorem.- 1. Craig's Interpolation Lemma.- 2. Beth's Definability Theorem.- XVI. Symmetric Completeness Theorems.- 1. Clashing Tableaux.- 2. Clashing Prenex Tableaux.- 3. A Symmetric Form of the Fundamental Theorem.- XVII. Systems of Linear Reasoning.- 1. Configurations.- 2. Linear Reasoning.- 3. Linear Reasoning for Prenex Formulas.- 4. A System Based on the Strong Symmetrie Form of the Fundamental Theorem.- References.

Abstract: A sound and complete theorem prover for first-order logic is presented, which is based on the connection method. The inference machine is implemented using PROLOG technology, an approach taken also with other systems, most prominently with Stickel's PTTP. But SETHEO differs from those in essential characteristics, among which are the following ones. It incorporates a powerful preprocessing module for a reduction of the input formula. The main proof procedure is realized as a variant of Warren's abstract machine. For search pruning we perform subsumption and regular proofs. Factorization, lemma generation, and the application of proof schemata are offered as options. The entire system is implemented in C and is running on several machines. The most remarkable feature of SETHEO is its performance of up to 70 Klips on a SUN SPARC station 1 with 12 Mips. The paper comprises the theoretical background, the system architecture as well as details of the implementation.