Short answers, pulled from the story.
John Alan Robinson introduced the resolution rule in 1965. This breakthrough arrived when he published a syntactical unification algorithm that allowed mathematicians to instantiate formulas on demand during the proof process rather than beforehand.
The resolution rule operates as a single valid inference rule within propositional logic systems. It produces a new clause implied by two clauses containing complementary literals and yields a sound and complete decision procedure for formula satisfiability when coupled with a complete search algorithm.
Paramodulation serves as a specialized technique for handling equality predicates within logical clauses. The operation generates all equal versions of clauses except reflexive identities and searches another clause for a subterm that unifies with one side of the equality to replace it with the other side.
Several software packages implement resolution-based theorem proving techniques today including Otter, Prover9, Vampire, SNARK, SPASS, CARINE, GKC, and Logictools online prover. These programs catalog diverse strategies for handling propositional and first-order logic while serving as practical tools for verifying mathematical theorems and solving logical puzzles.