394 Analytic and Non-analytic Proofs

hi this paper we present an algorithm for translating from a particular non-anMytic proof system to analytic proofs. Moreover, some results about the translation in the other direction are refornmlated and known algorithms improved, hnplementation of the algorithms presented for use in research and teaching logic is under way at Carnegie-Mellon University in the framework of TPS and its educational counterpart ETPS. Finally we show how to obtain non-analytic proofs from resolution refutations. As an application, resolution refutations can be translated into comprehensible natural deduction proofs. 1. I n t r o d u c t i o n In automated theorem proving different kinds of proof systems have been used. Traditional proof systems, such as Hilbert-style proofs or natural deduction we call non-analytic, while resolution or mating proof systems we call analytic. There are many good reasons to study the connections between analytic and non-analytic proofs. We would like a theorem prover to make efficient use of both analytic and non-analytic methods to get the best of both worlds. The advantages of analytic proofs are well known. One of the most important advantage is that they seem to be ideally suited for an efficient automatic search for a proof on the computer. On the other hand there is much to gain from the use of non-anMytic proof systems in addition to analytic methods. Non-analytic proofs can be presented in a comprehensible and pleasing format. If we can translate, say, resolution refutations into legible non-anMytic proofs, we can help the mathematician understand the atttomatically generated proof. Valuable work here has been done by Miller [10]. The natural deduction proofs obtained from mating refutations are often elegant and easy to understand and use such mathematically common concepts as proof by contradiction and case-analysis, and make use of intuitive operations such as backchaining. Better translations which arc the object of current research would make this even more useful for a wider class of theorems. The ability to freely translate between analytic and non-analytic proofs also gives us a tool for creating a more elegant natural deduction style proof from a given one. We would