A Graph-based Interpretation for Finding Solution Strategies of Contradiction Problems in the Butterfly Diagram

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1 , pp A Graph-based Interpretation for Finding Solution Strategies of Contradiction Problems in the Butterfly Diagram Jung Suk Hyun 1 and Chan Jung Park 2,1 1 Department of Management Information Systems, Jeju National University, Jejudaehak-ro 102, Jeju-do, 63243,Republic of Korea 2 Department of Computer Education, Jeju National University, Jejudaehak-ro 102, Jeju-do, 63243,Republic of Korea {jsyun,cjpark}@jejunu.ac.kr Abstract. The Butterfly diagram, a component of the Butterfly model, has been developed for resolving contradiction problems creatively. It contains a tradeoff relation between two main functions of a given problem. Also, it represents a contradiction relationship between two conflicting states generated from the two functions achievements. By showing the relationships with a diagram, problem solvers understand the contradiction of the given problem easily, and can derive the right solution strategy for the given problem. In this paper, we devise the graph-based representation method to find out the solution strategies for a given problem automatically based on the Graph theory. We handle three different types of contradiction problems in this paper. Keywords: Software Automation, the Butterfly Diagram, Contradiction Problem Solving, Graph Theory, Algorithm Representation 1 Introduction The Butterfly model is a model for resolving contradictions. In many academic areas such as Mathematics and Logic, a contradiction has been treated as false or wrong. Thus, when two contradictory requirements conflict with each other, people usually throw up one requirement in order to achieve the other. However, the Butterfly model takes the both-and strategy by analyzing a problem with the Butterfly diagram and finding conflicting relationship resides in the problem. In other words, the Butterfly model includes algorithms that solve problems by determining the conflicting types between given functions and the contradictory relations between two system states [1][2][3]. In the model, in order to define a given contradiction problem visually and to understand easily, the Butterfly diagram has been introduced [2][4][5]. The Butterfly diagram contains a trade-off relation between two functions of a given problem. Also, it represents a contradiction relationship between two conflicting states generated 1 The corresponding author is Chan Jung Park(cjpark@jejunu.ac.kr) ISSN: ASTL Copyright 2016 SERSC

2 Advanced Science and Technology Letters after performing the two functions. By showing the relationship with a diagram, it helps to understand the contradiction of the given problem clearly, and to derive the right solution strategy for the given problem. The Hyun et al. s research [1] defined a system with two conflicting functions and two contradictory states. And then, it classified the contradiction types and their solution strategies based on propositional logic theory. At this time, based on the research results described in [1], we adopt graph representation in order to find out the solution strategies in an algorithmic way in this paper. Among 9 types of contradiction problems proposed in [1], we firstly choose 3 different types of the problems. And then, we represent them with adjacent lists [6] to find out possible solution strategies. Finally, we explain the method that derives right solution strategies from the conflicting relations of the problem. 2 Contradiction Types and their Solution Dimension One of the purpose of drawing a Butterfly diagram for a given contradiction problem is to define the problem formally and to explain the relationship between the contradiction easily. In order to execute automatically on the computer, we should change the Butterfly diagram to a machine readable architecture. We can represent the Butterfly diagram with digraph, one of the data structures frequently used in computer programming languages [6]. Before we explain the graph-based representation, we summarize the contradiction types and their solution dimensions described in [1]. First of all, we define some terminologies used in this paper. A system S is composed of three components, w, s, and u, i.e. S = {w, ~w, u, ~u, s, ~s}, where w is a function S wants to perform, u is a function S does not want to occur, and s as a system state belongs to one of the sufficient condition, the necessary condition, and the necessary and sufficient condition for w(u) [1][5]. Assume that w be a wanted function and u be an unwanted function. Then, we define that w and ~u has a trade-off relation. Let s and ~s be states. Then, s and ~s has a contradiction relationship [5]. In Hyun et al. s research [1], w, u, s, are assumed to be propositions. Then, there are 9 possible conditional proposition relationships according to their conditions. Table I shows the 9 types made of the combinations of the conditional propositions between the wanted function w in a system and the system state s for performing w and the conditional propositions between the unwanted function u in a system and the s for performing u. Table 1. 9 Types of Conditional Proposition Relationships that occur Contradictions [1] s and u s u w and s s u s u w s (w s) (s u) (w s) (s u) (w s) (s u) w s (w s) (s u) (w s) (s u) (w s) (s u) w s (w s) (s u) (w s) (s u) (w s) (s u) Copyright 2016 SERSC 221

3 In the Table 1, represents a sufficient condition, whereas represents a necessary and sufficient condition. For example, w s means that if w then s (or in order to do w, s is required). means the exclusive-or. In other words, when p and q are propositions, if p is true and q is false, then, p q is true, and vice versa. When p and q are all true or p and q are all false then, p q is false. 3 Graph Representations for 3 Types of Problems In a graph theory, a directed graph G is an ordered pair of (V, E) where V is a set of vertices and E is a set of directed edges, which are made of ordered pairs of vertices V [6]. A directed graph G can be represented with n lists of n vertices. The i th list contains vertex j if there exists an edge from vertex i to vertex j [6]. In the following, we represent an adjacent list corresponding to a given Butterfly diagram. Type 1: The graph G1 = (V, E), where V={w, ~w, u, ~u, s, ~s} and E={(w,s), (s,u), (~u, ~s), (~s, ~w)}. In this case, the problem solving objectives is w ~u. In the graph, when we follow the vertex w and ~u, we meet s and ~s. Thus, in order to achieve the problem solving objectives w ~u, we define the solution strategy for this type as s ~s. We can find the solution strategy by following the link from w and ~u respectively. w is linked to s and ~u is linked to ~s. Thus, the ultimate solution to solve w ~u is s ~s. (w s) (s u) w ~u s ~s (b) The adjacent list of G1 Fig. 1. The graph representation for the Type 1 contradiction problem Type 2: The graph G2 = (V, E), where V={w, ~w, u, ~u, s, ~s} and E={(s,w), (s,u), (~u, ~s), (~w, ~s)}. In this case, we have no adjacent vertex from w. Thus, the problem solving objective w should be remain in the solution strategy. Similarly, ~u reaches to ~s in Fig. 2. Thus, the solution strategy becomes w ~s. 222 Copyright 2016 SERSC

4 (w s) (s u) w ~u w ~s (b) The adjacent list of G2 Fig. 2. The graph representation for the Type 2 contradiction problem Type 3: The graph G3 = (V, E), where V={w, ~w, u, ~u, s, ~s} and E={(w,s), (u,s), (~s, ~u), (~s, ~w)}. We apply the same rule to this case. The objectives is w ~u. w goes to s, but there is no vertex from ~u. Thus, the solution strategy became s ~u. (w s) (s u) w ~u s ~u (b) The adjacent list of G3 Fig. 3. The graph representation for the Type 3 contradiction problem 4 Conclusions The Butterfly diagram helps to find out a solution by defining a solution strategy based on the analysis of the trade-off relation and the contradiction relation hidden in a given problem [1][2][4]. In this paper, in order to find a solution strategy for a given Copyright 2016 SERSC 223

5 contradiction problem automatically, we apply the representation method for a directed graph. Since the Butterfly diagram consists of a set of nodes and a set of directed edges, it can be mapped into the corresponding adjacent list. After we draw the corresponding list, we firstly determine the problem-solving objectives. The problem-solving objectives were proved in the Hyun et al. s previous research [1]. By using the given problem solving objectives, we traverse the adjacent list until there is no more adjacent vertex. Then, we can find the solution strategy of the objectives. In this paper, we covered only 3 types. As an example case of the Type 1, we describe bi-focal lens. When the elderly people read a book closely (w), they need concave lens (s). However, with concave lens, they do not look farther (u). Thus, convex lens (~s) are needed for looking farther (~u). w and ~u forms a trade-off relation and s and ~s forms a contradiction relation. As an example case of the Type 2, we describe the safety bike. If the size of the wheels of a bike is big (s), then it increases the speed of a bike (w). Also, the big wheels cause unsafety (u). We want a fast but safe bike. As an example case of the Type 3, we describe a shopping card problem. in order to increase sales amount (w), a big bag is necessary (s). On the contrary, if a bag is small (~s), the shopping is convenient (~u). In other words, the shopping is inconvenient (u) when the shopping bag is big (s). In this case, the objectives are big sales amount and the convenient shopping at the same time. In the future, we will cover the rest of the 6 contradiction problem types and prove its correctness totally. Then, we can achieve an automatic method to solve contradiction problems more easily. References 1. Hyun, J. S., Park, C. J.: Logical interpretation about problem types and solution strategies of the butterfly model for the automation of contradiction-based provlem solving. Teaching, Assessment and Learning (TALE), 2014 International Conference on. IEEE, pp (2014) 2. Hyun, J. S., Park, C. J.: The butterfly model for supporting creative problem solving. Knowledge, Information and Creativity Support Systems (KICSS), 2012 Seventh International Conference on IEEE, pp (2012) 3. Hyun, J. S., Park, C. J.: Learning effects of divide-and-combine principles and state models on contradiction problem solving and growth mindset. Knowledge Management, vol. 14, no. 5, pp , (2014) 4. Hyun, J. S., Park, C. J.: A conflict-based model for problem-oriented software engineering and its applications solved by dimension change and use of intermediary, in CCIS, vol. 59, pp D. Slezak et al., Eds. Spinger-Verlag: Berlin (2009) 5. Hyun, J. S., Park, C. J.: Effects of the Butterfly diagram education in primary and secondary schools on contradiction problem solving ability. In Frontiers in Education Conference (FIE), IEEE, pp (2015) 6. Horowitz, E., and Sahni, S.: Fundamentals of data structures. No. 04; QA76. D35, H6., Pitman, (1983) 224 Copyright 2016 SERSC