Consolution as a Framework for Comparing Calculi

In (Eder, 1991) the consolution method is introduced as a bridge between the idea of matrix methods (see (Bibel, 1987)) and resolution. Eder defines a consolution calculus together with a construction which yields a consolution refutation for every resolution refutation. This gives both a demonstration that resolution can be seen as a special strategy within consolution and a completeness proof for consolution. However, Eder did not investigate a formal treatment of matrix methods with consolution; he only gave an informal description of matrix methods as a starting point for consolution. It appeared fascinating to us to see how easily the two ideas of a matrix method and a resolution calculus can be combined in the consolutionmethod. We tried to followthis line to use consolution as a framework to relate matrix methods and resolution by defining special consolution calculi. It turned out that this is not as straightforward as we expected. However we can reach this goal by making numerous changes to the definition of consolution. The changes are harmless in the sense that the inference rules of the modified calculus subsume that of the original calculus. The documentation of these changes is one purpose of this paper. Another purpose is to demonstrate that consolution is well suited for comparing calculi. This is a topic of significant importance for anyone trying to understand the essence of deduction or to implement a deduction system. In particular we show that when defined appropriately, consolution can simulate step by step a certain model elimination calculus (model elimination as defined by Loveland), a connection method calculus and a tableau model elimination calculus on which proof procedures like PTTP (Stickel, 1989; Stickel, 1988) and SETHEO (Letz et al., 1992) are based. The result for resolution and consolution was shown in (Eder, 1991). Altogether we will prove the diagram from figure 1, where the means that calculus can be simulated stepwise by calculus ; dashed arrows indicate a “weaker” simulatability relation after certain calculus restrictions.