In this paper, we propose a new lemma facility, called the strong contraction rule, for model elimination calculus, and we study some implementation issues of strong contraction in a PTTP-based theorem prover. Strong contraction has the ability to shorten possible proofs and prevent some irrelevant computation to a target goal. Strong contraction never broadens the search space, in principle, and thus, has a stable effect on the acceleration of top-down proof search. However, it is difficult to embed the strong contraction into a PTTP-based theorem prover, because strong contraction imposes a truncation operation of center chains in model elimination calculus. We show a self-truncation-style Prolog code, which makes it possible to retain the high performance of a PTTP-based prover with strong contraction. Finally, we evaluate the performance and effect of strong contraction by performing some experiments.

Predicate Circumscription is a fundamental formalization of common sense reasoning. In this paper, we study a new approximation formula of it. In our previous works, we investigated Lifschitz's pointwise circumscription and its generalization, which functions as a finite approximation to predicate circumscription in the first-order framework. In this paper, at first, we study the ability of the generalized pointwise circumscription more closely, and give a simple example which shows that it cannot be complete even when a minimized predicate has only finite extension on the minimal models. Next, we introduce a new approximation formula, called finite constructive circumscription, in order to overcome that limitation. Finally, we compare expressive power of the two approximation methods with of predicate circumscription schema, and propose a open problem that should be solved to clarify that the completeness of predicate circumscription schema with respect to minimal model semantics.