The Blossom of Finite Semantic Trees

Automated deduction in first-order logic finds almost all its roots in Herbrand’s work, starting with Herbrand’s interpretations, a clausal calculus, and rules for unification. J.A. Robinson’s key contribution was the formulation of resolution and its completeness proof, in which semantic trees were semi-apparent. Robinson and Wos introduced the specific treatment of equality commonly called paramodulation. The systematic introduction of orderings to cut the search space is due to Lankford. Kowalski studied in more details the case of Horn clauses, while Peterson gave the first proof that paramodulation inside variables was superfluous, assuming a term ordering order-isomorphic to the natural numbers. Knuth studied the case of equality unit clauses, under the name of completion. All these works were done by using standard proof techniques, including semantic trees [Kow69]. Further progress required more powerful proof techniques.