Decision Procedures and Model Building or How to Improve Logical Information in Automated Deduction

Abstract

The field of automated theorem proving is about 40 years old. During this time many new logic calculi were developed which thoroughly reshaped the discipline of deduction. The key feature of these calculi, in contrast to the “traditional” logic calculi, is efficient mechanizability. Instead of proof transformation (as in classical proof theory) proof search became the main issue. Programs searching for proofs of theorems formalized in some logical syntax are commonly called theorem provers. Thus most of the existing theorem provers can be considered as (deterministic) implementations of (nondeterministic) calculi: in fact their activity essentially consists in production of deductions till a proof (or a refutation) of the theorem under consideration is eventually found. In order to be useful theorem provers must (at least) be efficient, sound and complete. While soundness is absolutely mandatory, completeness may (in specific circumstances) sacrificed for higher efficiency. The first calculus which fulfilled all three requirements defined above was Robinson’s resolution [21]. For a long time, particularly in the seventies and eighties, increasing efficiency under preservation of soundness and completeness was virtually the only goal in the field of automated deduction. This was the time where most of the refinements of resolution, tableaux- and connection type calculi and equational calculi were developed. Only few papers, in particular those of S.Y. Maslov and W.J. Joyner, addressed the logical quality of theorem provers, i.e. the amount of logical information they are capable to produce.KeywordsDecision ProcedureHorn ClauseGround InstanceUnit ClauseAutomate DeductionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.