Set-Blocked Clause and Extended Set-Blocked Clause in First-Order Logic

Due to scale and complexity of first-order formulas, simplifications play a very important role in first-order theorem proving, in which removal of clauses and literals identified as redundant is a significant component. In this paper, four types of clauses with the local redundancy property were proposed, separately called a set-blocked clause (SBC), extended set-blocked clause (E-SBC), equality-set-blocked clause (ESBC) and extended equality-set-blocked clause (E-ESBC). The former two are redundant clauses in first-order formulas without equality while the latter two are redundant clauses in first-order formulas with equality. In addition, to prove the correctness of the four proposals, the redundancy of the four kinds of clauses were proved. It was guaranteed eliminating clauses with the four forms has no effect on the satisfiability or the unsatisfiability of original formulas. In the end, effectiveness and confluence properties of corresponding clause elimination methods were analyzed and compared.