• Journal of KIISE:Software and Applications
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• v.30 no.7_8
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• pp.642-648
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• 2003

In this paper, we are interested in checking equivalence of FSMs(finite state machines). Two FSMs are equivalent if and only if their responses are always equal with each other with respect to the same external stimuli. Equivalence checking FSMs makes complicated FSM be substituted for simpler one, if they are equivalent. We can also determine the system satisfies the requirements, if they are all written in FSMs. In this paper, we regard equivalence checking problem as model checking one. For doing so, we construct the product model $M ={M_A} {\beta}{M_B} from two FSMs ${M_A} and {M_B}$. And we also get the temporal logic formula ${\Phi}$ from the equivalence checking definition. Then, we can check with model checker whether if satisfies ${\Phi}$, written $M= {.\Phi}$. Two FSMs are equivalent, if $M= {.\Phi}$ Otherwise, it is not equivalent. In that case, model checker generates counterexamples which explain why FSMs are not equivalent. In summary, we solve the equivalence checking problem with model checking techniques. As a result of applying to several examples, we have many satisfiable results.

• Journal of Korea Game Society
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• v.4 no.1
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• pp.58-66
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• 2004

This paper uses counterexamples for solving reachability games. An objective. of the game we consider here is to find out a minimal path from an initial state to the goal state. We represent initial states and game rules as finite state model and the goal state as temporal logic formula. Then, model checking is used to determine whether the model satisfies the formula. In case the model does not satisfy the formula, model checking generates a counterexample that shows how to reach the goal state from an initial state. In this way, we solve many of small-sized Push Push games. However, we cannot handle larger-sized games due to the state explosion problem. To mitigate the problem, abstraction is used to reduce the state space to be che cked. As a result, unsolved games are solved with the abstraction technique we propose inthis paper.

• 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.

• School Mathematics
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• v.11 no.1
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• pp.55-70
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• 2009

The purpose of this study is to implement Lakatos method in the teaching and learning of geometry for middle school students. In his landmark book , Lakatos suggested the following instructional approach: an initial conjecture was produced, attempts were made to prove the conjecture, the proofs were repeatedly refuted by counterexamples, and finally more improved conjectures and refined proofs were suggested. In the study, students were selected from the high achieving students who participated in the special mathematics and science program offered by the city council of Seoul. The students were given a contradictory geometric proposition, and expected to find the cause of the fallacy. The students successfully identified the fallacy following the Lakatos method. In this process they also set up a primitive conjecture and this conjecture was justified by the proof and refutation method. Some implications were drawn from the result of the study.

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1351-1365
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• 2020

Let S and T be w-linked extension domains of a domain R with S ⊆ T. In this paper, we define what satisfying the w_R-GD property for S ⊆ T means and what being w_R- or w-GD domains for T means. Then some sufficient conditions are given for the w_R-GD property and w_R-GD domains. For example, if T is w_R-integral over S and S is integrally closed, then the w_R-GD property holds. It is also given that S is a w_R-GD domain if and only if S ⊆ T satisfies the w_R-GD property for each w_R-linked valuation overring T of S, if and only if S ⊆ (S[u])_w satisfies the w_R-GD property for each element u in the quotient field of S, if and only if S_𝔪 is a GD domain for each maximal w_R-ideal 𝔪 of S. Then we focus on discussing the relationship among GD domains, w-GD domains, w_R-GD domains, Prüfer domains, PνMDs and Pw_RMDs, and also provide some relevant counterexamples. As an application, we give a new characterization of Pw_RMDs. We show that S is a Pw_RMD if and only if S is a w_R-GD domain and every w_R-linked overring of S that satisfies the w_R-GD property is w_R-flat over S. Furthermore, examples are provided to show these two conditions are necessary for Pw_RMDs.