• Proceedings of the Korean Information Science Society Conference
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• 2004.04b
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• pp.352-354
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• 2004

현재의 모델 체커는 모델이 속성을 만족하지 않을 경우 반례를 사용자에게 보여주어서 디버깅을 돕는다. 모델 체커에서 반례는 중요한 장점 중에 하나이지만 대부분의 모델 체커에서 반례로서 하나의 경로만을 보여주게 된다. 하지만 사용자가 원하는 것은 그 이상의 정보를 원할 수 있다. 따라서 반례에서 좀더 많은 정보를 보여줄 필요가 있다. 이런 종류의 연구로서 트리 형식의 반례 생성과 증명 형식의 반례생성이 있었다. 하지만 이 연구들은 시스템이 가질 수 있는 모든 경로를 알아낼 수는 없고 또한 증명 형식의 반례 생성의 경우 상태공간을 다른 형식으로 변경을 해야 한다. 본 논문에서는 반례로서 도달 가능한 모든 경로를 그래프 형식으로 보여줄 수 있는 그래프 형식의 반례를 정의하고 생성방법에 대해서 알아본다

• The KIPS Transactions:PartD
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• v.12D no.1 s.97
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• pp.81-90
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• 2005

Given a model and a property, model checking determines whether the model satisfies the property. In case the model does not satisfy the property model checking gives a counterexample which explains where the violation occurs. Since counterexamples are useful for model debugging as well as model understanding, counterexample generation is one of the indispensable components in the model checking tool. This paper presents efficient counterexample generation techniques when a safety property is falsified. These techniques are used to solve Push Push games which consist of 50 games. As a result, all the games are solved with the proposed techniques. However, with the original NuSMV, 42 games are solved but 8 failed. In addition, we obtain $86{\%}$ time improvement and $62{\%}$ space improvement compared to the original NuSMV in solving the game.

• Proceedings of the Korean Information Science Society Conference
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• 2003.10b
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• pp.358-360
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• 2003

모델검사는 시스템과 그 시스템이 만족해야할 속성을 입력받아서 시스템이 속성을 만족하는지를 자동으로 보여주는 기술이다. 시스템이 속성을 만족한다면 참을, 만족하지 않는다면 왜 만족하지 않는지에 대한 반례를 보여준다. 모델검사에서 반례는 시스템의 오류를 발견하는데 중요하게 사용된다. 또한 모델검사를 이용해서 게임시스템을 모델링하고 반례를 이용해서 게임에 대한 풀이를 알 수 있게 되었다. 하지만 이런 반례가 최적의 풀이인지는 알 수 없었다. 이 논문은 모델검사에서 나온 게임 풀이가 최단 풀이 경로인지를 살펴본다. 그리고 최단 풀이경로를 출력하도록 NuSMV를 수정하여 원래 NuSMV에서 생성된 게임 풀이와 길이를 비교해 본다.

• Journal of Educational Research in Mathematics
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• v.21 no.4
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• pp.379-398
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• 2011

In learning environment at mathematics education, prove and refute are essential abilities to demonstrate whether and why a statement is true or false. Learning proofs and counter examples within the domain of limit of sequence is important because preservice teacher encounter limit of sequence in many mathematics courses. Recently, a number of studies have showed evidence that pre service and students have problem with mathematical proofs but many research studies have focused on abilities to produce proofs and counter examples in domain of limit of sequence. The aim of this study is to contribute to research on preservice teachers' productions of proofs and counter examples, as participants showed difficulty in writing these proposition. More importantly, the analysis provides insight and understanding into the design of curriculum and instruction that may improve preservice teachers' learning in mathematics courses.

• Proceedings of the Korean Information Science Society Conference
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• 2006.06c
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• pp.142-144
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• 2006

모델 체킹(model checking)은 자동으로 소프트웨어의 속성을 검증하는 기법으로 그 필요성이 꾸준히 증가하고 있다. 시스템이 특정 속성(property)을 만족하지 않는 경우 모델 체커는 반례(Counterexample)를 생성하게 된다. 반례는 오류가 발생한 원인을 담고 있는 정보로서 오류를 이해하고 수정하는 작업에 많은 도움을 준다. 하지만 반례가 너무 길거나 이해하기 어려운 경우에는 분석에 많은 시간과 자원이 소요되기도 한다. 따라서 자동적으로 반례 안의 오류를 찾아내고 설명을 제공하는 기법의 필요성이 대두되고 있다. 본 논문에서는 추상모델(abstract model)에서 생성된 반례의 오류의 원인을 밝히는 자동화 기법을 제시한다.

• Annual Conference of KIPS
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• 2012.11a
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• pp.1578-1580
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• 2012

모델 검증은 시스템이 만족해야 하는 속성을 자동으로 검사하는 정형 검증 기법으로써, 많은 도메인에서 활용되고 있다. 특히 모델 검증 도구들에 따라 상태 공간 탐색 방식이 다르고, 상태 공간 탐색 방식에 따라서 생성되는 반례도 달라진다. 본 논문에서는 모델 검증의 대표적인 도구인 SPIN과 SMV에서 생성하는 반례를 상호 비교한다.

• The KIPS Transactions:PartD
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• v.10D no.5
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• pp.813-820
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• 2003

This paper solves Push-Push game with the model checker NuSMY which exhaustively explores all search space to determine whether a model satisfies a property. In case a model doesn't satisfy properties to be checked, NuSMV generates a counterexample which tells where this unsatisfaction occurs. However, the algorithm for generating counterexample in NuSMV traverses a search space twice so that it is inefficient for solving the game we consider here. To save the time to be required to complete the game, we revise the part of counterexample generation so that it traverses a search space once. As a result, we obtain 62% time improvement and 11% space improvement in solving the game with modified NuSMV.

• Communications of Mathematical Education
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• v.30 no.3
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• pp.393-418
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• 2016

Theory and fundamentals of mathematics consist mostly of proposition form. Activities by research of the proposition which leads to determine the true or false, justify the true propositions and refute with counterexample improve logical reasoning skills of students in emphases on mathematics education. Also, utilizing of counterexamples in school mathematics combines mathematical knowledge through the process of finding a counterexample, help the concept study and increase the critical thinking. These effects have been found through previous research. But many studies say that the learners have difficulty in generating counterexamples for false propositions and materials have not been developed a lot for the counterexample utilizing that can be applied in schools. So, this study analyzed the current textbook and examined the use of counterexamples and developed educational materials for counterexamples that can be applied at schools. That materials consisted of making true & false propositions and students was divided into three groups of academic achievement level. And then this study looked at the change of the students' thinking after counterexample classes. As a study result, in all three groups was showed a positive change in the cognitive domain and affective domain. Especially, in top-level group was mainly showed a positive change in the cognitive domain, in upper-middle group was mainly showed in the cognitive and the affective domain, in the sub-group was mainly found a positive change in the affective domain. Also in this study shows that the class that makes true or false propositions in education of utilizing counterexample, made students understand a given proposition, pay attention to easily overlooked condition, carefully observe symbol sign and change thinking of cognitive domain helping concept learning regardless of academic achievement levels of learners. Also, that class gave positive affect to affective domain that increase interest in the proposition and gain confidence about proposition.

• Journal for History of Mathematics
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• v.26 no.5_6
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• pp.401-416
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• 2013

There has been increasing interest in recent years in the pedagogical importance of counter-examples that focuses on pedagogical perspectives. But there is no research that undergraduate students' generating counter-examples and proposing the true statements. This study analyze 6 undergraduate students' response to interview tasks and the process of their generating counter-examples and proposing true statements. The results of interviews are that the more undergraduate students generate various counter-examples, the more valid they propose true statements. If undergraduate students have invalid understanding of logical implication and generate only one counter-example, they would not propose true statements that modify the given statement, preserving the antecedent. In pre-service teacher's education and school mathematics class, we need to develop materials and textbooks about counter-examples and false statements.

• Journal of KIISE:Software and Applications
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• v.36 no.7
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• pp.523-530
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• 2009

Despite the growing need for customized operating system kernels for embedded devices, kernel development continues to suffer from insufficient reliability and high testing cost for several reasons such as the high complexity of the kernel code. To alleviate these difficulties, this study proposes the MOdel-based KERnel Testing (MOKERT) framework for detection of concurrency bugs in the kernel. MOKERT translates a given C program into a corresponding Promela model, and then tries to find a counter example with regard to a given requirement property, If found, MOKERT executes that counter example on the real kernel code to check whether the counter example is a false alarm or not, The MOKERT framework was applied to the Linux proc file system and confirmed that the bug reported in a ChangeLog actually caused a data race problem, In addition, a new data race bug in the Linux proc file system was found, which causes kernel panic.