On Fairness of Completion-Based Theorem Proving Strategies

Fairness is an important concept emerged in theorem proving recently, in particular in the area of completion-based theorem proving. Fairness is a required property for the search plan of the given strategy. Intuitively, fairness of a search plan guarantees the generation of a successful derivation if the inference mechanism of the strategy indicates that there is one. Thus, the completeness of the inference rules and the fairness of the search plan form the completeness of a theorem proving strategy. A search plan which exhausts the entire search space is obviously fair, albeit grossly inefficient. Therefore, the problem is to reconciliate fairness and efficiency. This problem becomes even more intricate in the presence of contraction inference rules rules that remove data from the data set. The known definitions of fairness for completion-based methods are designed to ensure the confluence of the resulting system. Thus, a search plan which is fair according to these definitions may force the prover to perform deductions completely irrelevant to prove the intended theorem. In a theorem proving strategy, on the other hand, one is usually only interested in proving a specific theorem. Therefore the notion of fairness should be defined accordingly. In this paper we present a target-oriented definition of fairness for completion, which takes into account the theorem to be proved and therefore does not require computing all the critical pairs. If the inference rules are complete and the search plan is fair with respect to our definition, then the strategy is complete. Our framework contains also notions of redundancy and contraction. We conclude by comparing our definition of fairness and the related concepts of redundancy and contraction with those in related works.