Automated Theorem Finding by Forward Deduction Based on the Semi-lattice Model of Formal Theory: A Case Study in NBG Set Theory

The problem of automated theorem finding is one of 33 basic research problems in automated reasoning which was originally proposed by Wos in 1988, and it is still an open problem. To solve the problem, a forward deduction approach based on the strong relevant logics was proposed. To verify the effectiveness of the approach, we tried to rediscover already known theorems in NBG set theory by using the approach, and succeeded in rediscovery of several known theorems. However, from the viewpoint of automated theorem finding, our method of the rediscovery is ad hoc, but not systematic. This paper proposes a systematic method based on the semi-lattice model of formal theory for deducing theorems and finding theorems which are two key phases of automated theorem finding. The paper presents a case study for automated theorem finding in NBG set theory and also shows some future research directions for automated theorem finding.

[1]  Larry Wos.  The problem of automated theorem finding , 2004, Journal of Automated Reasoning.

[2]  Jingde Cheng,et al.  EPLAS: An Epistemic Programming Language for All Scientists , 2007, International Conference on Computational Science.

[3]  Peng Lu,et al.  Multi-Intelligent-Agent Supported Workflow Management System , 2007 .

[4]  Jingde Cheng.  Entailment Calculus as the Logical Basis of Automated Theorem Finding in Scientific Discovery , 1995 .

[5]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[6]  Jingde Cheng,et al.  A Semilattice Model for the Theory Grid , 2007, Third International Conference on Semantics, Knowledge and Grid (SKG 2007).

[7]  Jingde Cheng,et al.  Automated theorem finding by forward deduction based on strong relevant logic: A case study in NBG set theory , 2012, 2012 International Conference on Machine Learning and Cybernetics.

[8]  Larry Wos,et al.  Automated reasoning - 33 basic research problems , 1988 .

[9]  Jingde Cheng,et al.  Finding Theorems in NBG Set Theory by Automated Forward Deduction Based on Strong Relevant Logic , 2013, COCOON.

[10]  Jingde Cheng,et al.  A Strong Relevant Logic Model of Epistemic Processes in Scientific Discovery , 1998, EJC.

[11]  J. W. Addison,et al.  Automated development of fundamental mathematical theories , 1992 .

[12]  Jingde Cheng,et al.  A Cooperative Grid Computing Approach to Automated Theorem Finding and Automated Problem Proposing , 2007, KES.

[13]  Jingde Cheng,et al.  FreeEnCal: A Forward Reasoning Engine with General-Purpose , 2007, KES.

[14]  Jingde Cheng,et al.  Practical implementation of EPLAS: An Epistemic Programming Language for all scientists , 2011, 2011 International Conference on Machine Learning and Cybernetics.