Hongbiao Gao

A systematic methodology for automated theorem finding

The problem of automated theorem finding is one of the 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, an approach of forward deduction based on the strong relevant logics was proposed. Following the approach, this paper presents a systematic methodology for automated theorem finding. To show the effectiveness of our methodology, the paper presents two case studies, one is automated theorem finding in NBG set theory and the other is automated theorem finding in Peanos arithmetic. Some known theorems have been found in our case studies.

Automated theorem finding by forward deduction based on strong relevant logic: 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, this paper presents a case study of automated theorem finding in NBG set theory by forward deduction based on the strong relevant logics. The ultimate goal of automated theorem finding in NBG set theory is to find new and interesting theorems. As the first step, this case study tries to do “rediscovery” in NBG set theory, i.e., to deduce already proved theorems from axioms, definitions and /or other theorems of NBG set theory. However, from the viewpoint of the mechanism of deducing theorems, “re-discovery” is as same as “discovery”. The paper shows several known theorems rediscovered successfully by the approach. The paper also shows issues of the approach for real “discovery”.

Finding Theorems in NBG Set Theory by Automated Forward Deduction Based on Strong Relevant Logic

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. For the problem, Cheng has proposed a forward deduction approach based on strong relevant logic. 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, the method of the rediscovery is ad hoc, but not systematic. This paper gives an analysis and discussion for our experiment method and results from the viewpoint of the systematic method. The paper also presents some issues and future research directions for a systematic method of automated theorem finding based on Cheng’s approach.

Research on Automated Theorem Finding: Current State and Future Directions

The problem of automated theorem finding is one of 33 basic research problems in automated reasoning which was originally proposed by Wos. The problem is still an open problem until now. This paper reviews the current state of the research on automated theorem finding and shows some future directions of automated theorem finding. In particular, we propose a systematic procedure for automated theorem finding based on Cheng’s approach, i.e., automated theorem finding by forward deduction based on strong relevant logics.

A Formal Analysis Method with Reasoning for Cryptographic Protocols

Formal analysis of protocols is to find out flaws in cryptographic protocols by formal method. In formal analysis methods with proving such as theorem proving and model checking, analysts must set possible attacks as targets, and then verify whether these attacks succeed or not. However, its impossible to enumerate all attacks so that flaws may be not detected. As an alternative way, a formal analysis method with reasoning has been proposed, but it can only be applied to key exchange protocols. This paper presents a formal analysis method with reasoning for various cryptographic protocols. By succeeding in detecting flaws of splitting protocols, we show that the proposed method is effective for various cryptographic protocols.

A Set of Metrics for Measuring Interestingness of Theorems in Automated Theorem Finding by Forward Reasoning: 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. The problem implicitly requires some metrics to be used for measuring interestingness of found theorems. However, no one addresses that requirement until now. This paper proposes the first set of metrics for measuring interestingness of theorems. The paper also presents a case study in NBG set theory, in which we use the proposed metrics to measure the interestingness of the theorems of NBG set theory obtained by using forward reasoning approach and confirms the effectiveness of the metrics.

A Bidirectional Transformation Supporting Tool for Formalization with Logical Formulas

In many applications in computer science and artificial intelligence, logical formulas are used as a formal representation to represent and/or specify various objects and relationships among them. However, transforming the informal propositional statements, e.g., declarative sentences and mathematical formulas, into logical formulas is not an easy task for most people. Moreover, when people obtain new logical formulas as results of deduction/reasoning based on logic, investigating the obtained formulas is also not an easy task for them. Although a tool to support transformation from the informal propositional statements of a target domain into logical formulas, and vice versa, is demanded, there is no such tool until now. This paper presents a bidirectional transformation method for formalization with logical formulas, and its supporting tool we are developing.

An epistemic programming approach for automated theorem finding

The problem of automated theorem finding is one of 33 basic research problems in automated reasoning which was originally proposed by Wos. The problem is still an open problem until now. Specific knowledge is the power of any scientist, therefore, if a scientist in a particular area takes part in the process of automated theorem finding, then the scientist should certainly make some contributions for automated theorem finding in the target area. Epistemic programming was proposed as a novel program paradigm to program epistemic processes of scientific discovery, which regards conditionals as the subject of computing, takes primary epistemic operations as basic operations of computing, and regards epistemic processes as the subject of programming. Epistemic programming provides not only programming means but also interactive means for scientists to control cognitive processes. This paper proposes an epistemic programming approach for automated theorem finding following the epistemic programming paradigm and shows some examples to do automated theorem finding by using the approach.

Explicitly Epistemic Contraction by Predicate Abstraction in Automated Theorem Finding: A Case Study in NBG Set Theory

In automated theorem finding by forward reasoning, there are many redundant theorems as intermediate results. This paper proposes an approach of explicitly epistemic contraction by predicate abstraction for automated theorem finding by forward reasoning in order to remove redundant theorems in a set of obtained theorems, and shows the effectiveness of the explicitly epistemic contraction by a case study of automated theorem finding in NBG set theory.

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.