Visualizing Logical Thinking using Homotopy A new learning method to survive in dynamically changing cyberworlds Kenji Ohmori 1, Tosiyasu L. Kunii 2 1 Computer and Information Sciences, Hosei University, Koganei, Tokyo, Japan 2 Morpho Inc, Bunkyo, Tokyo, Japan Abstract For logical thinking learners, it is important to learn problem solving formally and intuitively. Homotopy which is modern mathematics combines algebra and geometry together. Homotopy gives the most abstract level of explanation or the most basic invariant which is the most required characteristics for logical thinking. The homotopy lifting property gives good explanation of inductive reasoning. The homotopy extension property, which is dual to the homotopy lifting property, provides a good tool for deductive reasoning. As homotopy combining algebra with geometry, it provides intuitive explanation with informative figures. This paper shows how puzzle-based learning is explained mathematically and intuitively using these properties. Keywords: A maximum of 6 keywords logical thinking; puzzle-based learning; inductive and deductive reasoning; homotopy lifting and extension properties; problem-solving 1. Introduction Logical thinking [1], [2], [3] is an important method to solve inexperienced problems encountered in the complicated and sophisticated society of cyberworlds, where we are insisted to survive wisely in dynamically and rapidly changing surroundings. Logical thinking needs mathematics to solve problems formally and logically. However, learners are awkward to use classical mathematics which uses many expressions so that symbol manipulation is not intuitively understandable. Graphical user interface, which is improved with the development of Web systems, provides a visualized environment for logical thinking tools. Homotopy [4], [5], [6] in the field of topology of modern mathematics gives a better way for visualizing logical thinking since homotopy unifies algebra and geometry with informative illustrations of figures. In cyberworlds, homotopy has been applied to software development [7], [8], [9], where system requirements are formally transformed into concurrent processes described by the pi-calculus and then implemented as event-driven and multi-thread programs suitable for XMOS processors. The paper expresses that homotopy has two important properties: the homotopy lifting and extension properties. Furthermore, when applying homotopy into the field of cyberworlds, these two properties are used as tools for bottom-up and top-down methods, respectively. Logical thinking is mainly employed by deductive reasoning and inductive reasoning. Inductive reasoning is a thinking skill that is initiated by observing something, and then completed by a conclusion based on what has been observed. Inductive reasoning is thought as the process of inducing an abstract property from specific properties since it leads a common property or an invariant from specific facts. Deductive reasoning is a thinking skill that is started by forming a conclusion, and then achieved by showing that the conclusion follows from a set of premises. Deductive reasoning is considered as the process of deducting an abstract property into specific properties, so that inductive and deductive reasoning are equivalent to the bottom-up and top-down methods, respectively, used in the software development application. The homotopy lifting property and extension properties can be used as intuitive and informative skills since these properties are rich in graphical user interface. In this paper, these properties are applied to puzzle-based learning, which is a new teaching methodology focused on the development of problem-solving skills for logical thinking. Puzzle-based learning [10] is introduced to overcome the disadvantage of the conventional problem-solving where logical thinking skills are taught after learning professional knowledge. However, students want to improve their logical thinking skills before learning professional knowledge. As puzzles can be solved without professional knowledge, it is thought that puzzle-based learning is suitable for improving logical thinking skills by solving puzzles instead of professional problems. There may be objections to use homotopy when solving puzzles. As topology and homotopy are taught after classical mathematics, students tend to think that homotopy is a hard subject. However, as it is shown in this paper, homotopy is not a so difficult subject as compared with calculus which is taught in high schools.
2. How to employ homotopy in cyberworlds Homotopy is defined in mathematics as follows. [Homotopy] A homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X [0, 1] Y from the product of the space X with the unit interval [0, 1] to Y such that, if x X then H(x, 0) = f(x) and H(x, 1) = g(x). If the second parameter of H is considered as time then H describes a continuous deformation of f into g where we have the function f at time 0 and the function g at time 1. A continuous function, which appears in the previous sentence, is important concept in mathematics. It is defined as follows. [Continuous function] A function f : X Y, where X and Y are topological spaces, is continuous if and only if the inverse image f ( 1) (V ) = {x X f(x) V } is open for every open set V Y. As cyberworlds are often employed in the set of discrete elements, mathematical terms have to be interpreted to adjust discrete characteristics of cyberworlds. The set of discrete elements is transformed into a topological space by introducing the trivial space or the discrete space. When the unit interval is discrete, the elements of the unit interval is treated as the components of a totally ordered set. Fig. 1 is an example of homotopy in cyberworlds. X and Y are the sets of discrete elements. Y is divided into the subset of {Y 0, Y 1, Y 2, Y 3, Y 4 }. It is considered that X is mapped to Y i at time t i. This example can be used to describe how a certain area has changed in history or how the financial state of a company has changed in recent years. Now, we will introduce topological spaces to X and Y and define a homotopy H. The topological space (X, T x ) is defined by introducing the subset T x = {φ, X}ofX. The interval is defined as follows. t i I, T I = {[t i, t j ] t i, t j I, i j}. I is a totally ordered set. The topological space (Y, T y ) is also defined by introducing the power set T y of {Y 0, Y 1, Y 2, Y 3, Y 4 }. Then, the homotopy H is defined by H(x, t i ) : x X, t i I y Y. Fig. 1: Homotopy employed in cyberworlds. 3. Homotopy lifting property in a discrete space The homotopy lifting property is used as a tool for deductive reasoning. [Homotopy lifting homotopy] The homotopy lifting property is defined as follows. Given any commutative diagram of continuous maps as shown in Fig. 2, the map p : E B has the homotopy lifting property if there is a continuous map Ĥ : Y I E such that p Ĥ = H. The homotopy Ĥ thus lifts H through p. Fig. 2 also includes an example of the homotopy lifting property. E is the surface of a grove and B is the projected space of B. X is a line which is mapped to E. However, the mapped lines are continuously changed along the time interval I as shown in Fig. 2. Fig. 2: Homotopy Lifting Property [Fiber bundle] A fiber bundle is a quadruple ξ = (E, B, F, p) consisting of a total space E, a base space B, a fiber F, and a bundle projection that is a continuous surjection called F bundle p : E B such that there exists an open covering U = {U} of B and, for each U U, a homeomorphism called a coordinate chart ϕ U : U F p 1 (U) exists such that the composite U F p 1 (U) U is the projection to the first factor U. Thus the bundle projection p : E B and the projection p B : B F B are locally equivalent. The fiber over b B is defined to be equal to p 1 (b), and it is noted that F is homeomorphic to p 1 (b) for every b B, namely b B, F = p 1 (b). A Mobius strip gives a good example to explain a fiber bundle as shown in Fig. 3. If a Mobius strip and its center circle described by the dot line are considered as E and B of a fiber bundle and any point of E is projected to the intersection of the center circle with the perpendicular line from the point of E to the center circle, then, F is a straight line segment vertical to the center circle.
Fig. 3: Mobius strip. Though a cylinder is different from a Mobius strip in shape, these objects are alike in homotopy. When constructing a fiber bundle for a cylinder, the cylinder has the same B and H as a Mobius strip. B of the cylinder is also a circle. The fibers are also straight line segments vertical to the center circle. However, the fibers of the Mobius strip is rotating along the center circle. On the other hand, the fibers of the cylinder are always straight, which causes the difference in shape. B is considered as a common property or invariant among similar objects in homotopy. A cylinder is deformed to the circle by moving any point perpendicularity to the point of the center circle. A Mobius strip is also deformed to the circle by the same moving. In the abstract level, a cylinder and a Mobius strip are the same. The invariance of two objects is explicitly represented by B. Google map is a good example to explain how a fiber bundle is applied in cyberworlds as shown in Fig. 4. E is a Google map which is constructed by a set of guide map layers, each of which corresponds to a business category such as restaurants, hospitals, and so on. B is the projected map from E. B is the common property of the layers and invariant. Fig. 4: Google map. 4. Inductive reasoning using homotopy lifting property The homotopy lifting property explains mathematically and intuitively how a puzzle is solved by inductive reasoning. Let us consider to solve the following puzzle. [Gas station puzzle] There are n gas stations located on a circular track. The total gas available in these gas stations is just enough for one car to complete the loop. Show that there is a gas station from which a car (initially with an empty tank) can start and complete the loop by collecting gas from the gas station on its way around. (Z. Michalewicz, M. Michalewicz, Puzzle-Based Learning: An introduction to critical thinking, mathematics, and problem solving, p64) The puzzle is specified by providing 12 gas stations, each of which is given a name, distance from station A and available gas as shown in Table 1. Table 1: An Example of a Table St A B C D E F G H I J K L Dis 0 30 80 115 150 170 205 215 245 295 305 325 Qt 9 5 10 8 4 1 7 4 4 12 3 5 The commutative diagram of the homotopy lifting property is constructed as shown in Fig. 5. The information of gas stations provided in Table 1 is assigned in X. X is the list of stations with name, distance and available gas. That is, x = ((A, 0, 9), (B, 30, 5),...) X. I is a series of the starting points, from each of which the car starts to go around the circular track. That is I = (t 0, t 1,..., t 11 ), where t i means that the car starts from the (i + 1) th station. The element of X I is a pair of the gas station list and a starting point. H(x, t) maps the station list from starting point t to a remaining gas graph. E is constructed to show how much gas remains at the tank along the circular track starting from the gas station designated by I. When gas is exhausted, E shows how much gas should be borrowed. In Fig. 5, the graph for t i shows the change of remaining gas along the track when the car starts from the (i + 1) th station. If remaining gas is negative, the same amount of gas must be borrowed. The graphs on E are different in quantity. But, these graphs have the same shape, which is a common property of these graphs. By removing the values of remaining gas and connecting the starting station with the ending so that a graph becomes a loop, each graphs on E is projected to B as shown in Fig. 5. As B is invariant in this puzzle, we can obtain the solution of the puzzle from B. Since B is a loop, it has at least one lowest point as an invariant. If the car starts from the lowest position, the car can run the circular track without running out of gas. There is another puzzle, which is solved using mathematical induction. [Strange country] There is a strange country where every road is a one way road. Moreover, every pair of cities is connected by only one directed road. Show that there exists
capital at L 0, L 1 and the new city becomes the new capital at L 2. The countries in E is projected to the graph of B, which is invariant. The graph of B is the same as one of X. Therefore, the puzzle has been proved. Fig. 5: Homotopy lifting property for gas station puzzle. a city (possibly the capital of this strange country?) that can be reached from every city directly or via at most one other city. (Z. Michalewicz, M. Michalewicz, Puzzle-Based Learning: An introduction to critical thinking, mathematics, and problem solving, p66) By assuming that the capital exists for n cities, let s try to prove that the capital exists for n + 1 cities. This puzzle is usually proved by mathematical induction. Mathematical induction can be visualized using the homotopy lifting property. At first, let s define X and I. A new country with n + 1 city is constructed by providing the new roads that connect a new city with the old cities. As a road is one way, the number of countries with n + 1 cities becomes the combination of one way roads. The one way roads connecting the new city with the old cities are described by (c 1, c 2,..., c n ) where c i = 0 if the road is directed from the i th city to the new city, otherwise c i = 1. I is defined by the combination of directions and is sorted as Huffman codes. The set of directions is described by I = (t 0, t 1,..., t 2 n 1) and t 0 = (0, 0...0, 0), t 1 = (0, 0...0, 1), t 2 = (0, 0...1, 1),... t 2n 1 = (1, 1...1, 1). X is constructed as the country with n cities. By assumption, n cities are classified into three groups as shown in Fig. 6: the capital, the directly reachable cities from which a one-way road to the capital is provided, and others, which are named indirectly reachable cities. From an indirectly reachable city, a one-way road to a directly reachable city is provided. Homotopy Ĥ is defined by Ĥ(x, t) = y where x X, t = (c t 1, c t 2,..., c t n) I, y E. The countries with n + 1 cities can be divided into three layers: L 0, L 1, L 2 according to how new roads connected to the new added city are provided. If the roads are provided so that the capital is directly reached from the new city, these countries belong to L 0. If the roads are provided so that the new city is reached from the capital and at least one of the directly reachable cities are reached from the new city, these countries become L 1. Otherwise, the countries belong to L 2 where an old indirectly reachable cities become a directly or indirectly reachable city. The old capital remains as the Fig. 6: Strange country 5. Homotopy extension property in a discrete space The homotopy extension property is dual to the homotopy lifting property, which is defined as follows. [Homotopy extension property] Given any commutative diagram of continuous maps as shown in Fig. 7, there is a continuous map ˆK : X Y I such that p 0 ˆK = k and ˆK i = K. The homotopy ˆK thus extends K over i and lifts k through p 0 where p 0 (λ) = λ(0). In the above definition, λ is called a path and is defined as follows. [path] A continuous map λ : I X yields a path. λ(0) = x and λ(1) = y are called the initial and terminal points. The path is denoted by w = (W ; λ) where W = λ(i). Y I is called a path space and is defined as follows. [Path space] The path space on Y, denoted Y I, is the space {λ : I Y continuous}. In Fig. 7, an example of the homotopy extension property in cyberworlds is depicted. The example shows a Google map application where a Google map provided for each customer is generated from the Google map. X is the Google map consisting of business category layers, each of which shows the places of buildings for a given business category, such as restaurants, hospitals and so on. A is the map for those layers which is included in X. Y I is a set of Google maps for each customer. I is a series of categories. A set of continuous functions {λ} is assigned for each customer such as Peter, Betty and so on. λ p (I) is a path for Peter, which describes a Google map for Peter.
Fig. 7: Homotopy Extension Property 6. Deductive reasoning using homotopy extension property The homotopy extension property explains intuitively how a puzzle is solved by deductive reasoning. Let us consider to solve the following puzzle. [Chess board] There is a chess board with a dimension of 2 m (i.e., it consists of 2 m squares) with a hole, i.e., one arbitrary square is removed. There are number of L shaped tiles and your task is cover the board with these tiles (the orientation of a tile is not important). How can you do it? Where should you start? (Z. Michalewicz, M. Michalewicz, Puzzle-Based Learning: An introduction to critical thinking, mathematics, and problem solving, p68) chess board where a hole is arbitrary provided. A is part of the chess board. The chess board is divided into four equal rectangular blocks and the block which has the hole is detached from the other blocks. The other blocks form an L shaped board. Therefore, A consists of the rectangular block with the hole and the L shaped board without the hole. I is a sequence of times at which an L shaped tile is placed on the chess board. That is, I = (t 0, t 1,..., t 85 ) At t i, the i th tile is placed. Y I is a path space that describes the changes of the chess board by placing L shaped tiles. Now, let us consider to place the first L shaped tile on the chess board. The rectangular square is possible to be covered by L shaped tiles since the number of squares that have to be covered by tiles can be divided by three. (Three is the number of squares that a tile needs to occupy.) Other blocks that does not have the hole become possible to be covered by tiles if one square is removed from each block. It is achieved by placing an L shaped tile on the three blocks, so that it surrounds the corner where these blocks meet together. As Y I is a path space, paths must be defined. Let s define the elements of X and Y I as follows. X = {b i,j i, j = 1..16} where i and j are a row and column number, respectively. Y is the same as X. Y I = {λ i,j (t) i, j = 1..16, t I} where λ i,j starts from b i,j Y. Homotopy ˆK is defined by ˆK(b i,j, t) : b i,j X, t I λ i,j (t) Y I. When the first tile is placed, path λ 9,9 (t) starts from b 9,9 at time t 0, circles around the tile and ends at b 9,9 at time t 1. At the same time, λ 8,9 (t) and λ 9,8 (t) circulate around the tile. This process is repeated to each divided four blocks. Fig. 9 shows how a new L shaped tile is placed in the divided block with the hole. For constructing Y I, other pathes λ ( 5, 5), λ ( 4, 5) and λ ( 5, 4) starting at t 1 and ending at t 2 are provided by circling around the new tile. Fig. 8: Chess board puzzle - Divide and Conquer The puzzle is solved using the homotopy extension property as shown in Fig. 8. To simplify the puzzle, a chess board with 16 16 squares is provided, so that 85 L shaped tiles have to be placed on the chess board. The commutative diagram of the homotopy extension property is constructed as follows. X is assigned as the Fig. 9: Chess board puzzle - Divide and Conquer 7. Conclusions The paper describes how homotopy is applied to logical thinking. The first puzzle uses the homotopy lifting property
for inductive reasoning. The invariant shape showing the change of remaining or borrowing gas has been obtained by projecting the specific patterns of E to the generalized pattern to B. The process is mathematically explained by the commutative diagram of the homotopy lifting property how the information of gas stations in X is mapped to the graph of remaining or borrowing gas in E according to the starting gas station of I using informative figures. The figures of E are different in absolute values but the common shape is projected into B, which is invariant and gives an answer to this puzzle. The second puzzle is a problem of mathematical induction. The puzzle also uses the homotopy lifting property in a bottom-up way as a tool for inductive reasoning. It is assumed that the condition is satisfied for the case of n and has to be proved for the case of n + 1. The case of n is described in X, where all situations are not represented as graphs, but a typical graph that is equivalent to the all graphs is depicted in X. I is constructed as a sequence of cases for forming one way roads between the old new cities and a new city, which constitutes the case for n + 1. The new situations in E are classified into three layers, which are mapped to B to obtain a common property of the n + 1 case. As the problem-solving process is explained structurally by the diagram and intuitively using graphs, the homotopy application to the second puzzle has succeeded in obtaining a good tool for logical thinking. The third puzzle is a divide-and-conquer problem. The problem is solved by deductive reasoning. Therefore, the homotopy extension property is used in a top-down way. The homotopy extension property is dual to the homotopy lifting property. However, as it uses a path space, which is unfamiliar concept, it is considered that the homotopy extension property is hard to be applied in cyberworlds. The third puzzle has succeeded in not only solving a divide-andconquer problem but also expanding how a path space is usefully applied to cyberworlds. Through solving these problems, we have succeeded in introducing homotopy as a tool of logical thinking, which leads to improve skills of logical thinking sophisticatedly. References [1] H. W. Dettmer, The Logical Thinking Process: A Systems Approach to Complex Problem Solving. Milwaukee, Wisconsin: ASQ Quality Press, 2007. [2] D. McInerny, Being Logical: A Guide to Good Thinking. New York: Random House, 2004. [3] P. George, How To Solve It. Princeton, New Jersey: Princeton University Press, 2004. [4] C. Dodson and P. E. Parker, A User s Guide to Algebraic Topology. Boston: Kluwer Academic Pub, 1997. [5] A. J. Sieradski, An introduction to topology and homotopy. Boston: PWS-Kent Publishing Company, 1992. [6] E. H. Spanier, Algebraic topology. New York: Springer-Verlag, 1966. [7] K. Ohmori and T. L. Kunii, A pi-calculus modeling method for cyberworlds systems using the duality between a fibration and a cofibration, Int. Conf. on Cyberworlds 2008, pp. 363 370, September 2008. [8], Mathematical foundation for designing and modeling cybeworlds, Int. Conf. on Cyberworlds 2009, pp. 80 87, September 2009. [9], Designing and modeling cyberworlds using the incrementally modular abstraction hierarchy based on homotopy theory, The Visual Computer, vol. 26, no. 5, pp. 297 309, May 2010. [10] Z. Michalewicz and M. Michalewicz, Puzzle-based Learning: Introduction to critical thinking, mathematics, and problem solving. Melbourne: Hybrid Publishers, 2008.