Chapter 4 - The Inverse Method

Abstract

There are several methods of theorem proving. The best known methods are resolution-based and tableau-based ones. Resolution-based methods convert the goal formula into clausal norm form and perform derivation on the set of clauses using a saturation algorithm. Tableau-based methods operate directly on formulas by reducing goals to subgoals and trying to solve the subgoals, either by reduction to new subgoals or by unification-based methods. The inverse method is much less known. Like the tableau method, it operates on formulas, but the main operation is not reduction of goals to subgoals, but rather construction of goals from previously proved subgoals. Like resolution-based methods, the inverse method uses a saturation algorithm. The inverse method is unusual compared to semantic tableaux and related methods. Instead of searching for a derivation in a goal-directed manner, by reducing the goal to subgoals until all subgoals reduce to axioms, it tries to prove the goal in the inverse direction: from axioms. Since in most calculi the number of axioms is infinite, proving theorems in the inverse direction requires one to use some properties of the calculi.