Semi-tensor product of matrices is a generalization of conventional matrix product for the case when the two factor matrices do not meet the dimension matching condition. It was firstly proposed about ten years ago. S...Semi-tensor product of matrices is a generalization of conventional matrix product for the case when the two factor matrices do not meet the dimension matching condition. It was firstly proposed about ten years ago. Since then it has been developed and applied to several different fields. In this paper we will first give a brief introduction. Then give a survey on its applications to dynamic systems, to logic, to differential geometry, to abstract algebra, respectively.展开更多
Refutation methods based on the resolution principle are generally applied to a (finite) set of sentences, which must have a series of pre-transformations (prenex normalization, Skolemization and conjunction normaliza...Refutation methods based on the resolution principle are generally applied to a (finite) set of sentences, which must have a series of pre-transformations (prenex normalization, Skolemization and conjunction normalization) before starting the refutation. In this paper, the authors first generalize the concept of abatract consistency class to the most general form-universal abstract consistency class, and prove its universal unifying principle. Then, based on the R-refutation, a universal refutation method is proposed and its soundness and completeness are proved by means of the universal unifying principle. This method can be applied directly to any finite set of wffs without preprocessing the wffs at all so that the refutation procedure is more natural.展开更多
基金Supported partly by National Natural Science Foundation of China under Grant No. 60221301 and 60334040 .Dedicated to Academician Han-Fu Chen on the occasion of his 70th birthday.
文摘Semi-tensor product of matrices is a generalization of conventional matrix product for the case when the two factor matrices do not meet the dimension matching condition. It was firstly proposed about ten years ago. Since then it has been developed and applied to several different fields. In this paper we will first give a brief introduction. Then give a survey on its applications to dynamic systems, to logic, to differential geometry, to abstract algebra, respectively.
文摘Refutation methods based on the resolution principle are generally applied to a (finite) set of sentences, which must have a series of pre-transformations (prenex normalization, Skolemization and conjunction normalization) before starting the refutation. In this paper, the authors first generalize the concept of abatract consistency class to the most general form-universal abstract consistency class, and prove its universal unifying principle. Then, based on the R-refutation, a universal refutation method is proposed and its soundness and completeness are proved by means of the universal unifying principle. This method can be applied directly to any finite set of wffs without preprocessing the wffs at all so that the refutation procedure is more natural.
