1. Introduction By definition, Raven’s Progressive Matrices provide test takers with a set number of possible answers to choose from. Therefore, given the set of possible answers is finite, I based my solution on the Generate and Test approach to solving problems, but created three different ways of generating test cases based on two different data representation models. As the main object of the Raven’s Progressive Matrices Test is to evaluate how well test takers are able to comprehend the changes in relationships objects share as shown in a set of unique figures it made sense to use data models which easily represent changes. Therefore, I chose to base my representation of the transition of a Raven’s figure from one state to the next by using Semantic Networks and Frames.

2. Reasoning At the highest level, this agent uses three different approaches to solving Raven’s Progressive Matrices Problems, each of which produces a weighted score suggesting how likely a given answer is to be correct. A higher weight has been given to those approaches which most consistently produce the correct answer for the largest number of problem, while a lower weight is given to approaches which are more applicable to a only a specific subset of problems. These three scores are then combined and the possible answer with the highest combined score is selected as the best answer to the problem. This strategy of combining the three scores was adopted in hopes that one approach might be able to compensate for the others, in the case they should it have difficulty solving a certain type of Raven’s Matrix. It was additionally thought that using a combination of approaches to solve a logical problem would more accurately mimic human problem solving strategies. 2.1 Image Processing To facilitate image processing the Agent uses the standard Java BufferedImage class to create a 2D Array of bytes to represent the image. The 2D Array is created by checking the color of each pixel in the BufferedImage and placing a ‘1’ in the array for a black pixel and a ‘0’ for a white/nonblack pixel. Once the Agent has completed the 2D Array, it begins to process the array as shown in Figures 7 & 8. This process begins by looking for a ‘1’ in a row in the 2D Array, with the Agent processing the array rows left to right, top to bottom, this is represented in Figure 7 & 8 by the number ’1’. When the Agent does find a ‘1’ in the 2D Array, it pauses its search for additional ‘1’ in the 2D Array and starts the process of trying to determine what shape it has found. Since the Agent is moving top to bottom, left to right through the 2D Array, the Agent knows that the first ‘1’ it finds should be the top left corner of the shape. With this knowledge the Agent can begin to trace around the outside of the shape in order to gain additional insight into the shape’s properties. The first step the Agent takes is to determine how far to the right in the same row the 2D Array contains consecutive 1’s, with the last consecutive ‘1’ being the top right corner of the shape. This step is shown as step ‘2’ in Figures 7 & 8. At this point the Agent knows where the top right and left corners are and therefore it can determine the width and the middle point of the top of the shape. Now that the Agent knows the middle, top coordinate it can begin the next step, which is to trace its way around the shape until it reaches the bottom middle coordinate of the shape. The Agent traces by finding the next closest ‘1’, with priority given to coordinates in the following order, as defined by their relationship to the last identified: right of, below and right of, below, below and left of and finally left of. Once the agent has found a ‘1’ which has the same xcoordinate value as the middle and a reasonable ycoordinate, the Agent considers the right half of the shape to have been traced. The tracing step is shown as step ‘3’ in Figures 7 & 8. During the tracing process the agent is also keeping track of how many times it has to change directions in order maintain tracing the outside of the shape. This information will help it determine if the line was curved or runs on a dianangle. Additionally the Agent is also changing every ‘1’ it encounters to current count of how many objects it has run into. For example if this is the third shape it has traced in the image then each ‘1’ will be replaced by a ‘3’. This helps the agent to discern between shapes which overlap. After completing it’s tracing of the right side of the shape, the Agent compares the ycoordinate of the bottom middle point to the ycoordinate of the top middle point, as shown in Figures 7 & 8 as step ‘4’ in order to calculate the height of the shape. Next the Agent determines if the shape is filled or not by tracing down from the top middle point for as long as it can before finding a ‘0’, as shown as step ‘5’ in Figures 7 & 8. If the agent can trace longer than 10 pixels the shape is considered to be filled. Finally the agent determines the width of the shape at its middle and bottom using the center line and the same process describe for step ‘2’ of the figures. This process is shown as step ’7’ in Figures 7 & 8. This entire mapping process gives the Agent the following information: shape width at top, middle, bottom, shape is/is not filled, shape is/is not curved, shape’s top right, top left, bottom right and bottom left coordinates. With this information the Agent can begin to make an educated guess as to which shape the object is. For example if the height and middle width are equal, the top and bottom widths are small and the traced line was curved it is most likely a circle. Another example would be if the top width is very small, the middle width is greater than the top width and the bottom width is greater than the middle width than the shape is most likely a triangle which is pointing up. 2.2 Mapping The agent’s approach to solving Raven’s Progressive Matrices can be broken down into two paths which both start with the process of mapping objects between figures in the matrix. In order to draw any conclusions about the transitions a matrix undergoes, the agent has to be able to determine which objects correspond to one another between the figures within the matrix. The agent uses a highlevel Analogical Reasoning approach to compare the deep similarities between the objects. The agent then uses a weighted scale to determine which objects are most likely to be the same object between frames. This process takes into account that scenarios can exist where the best match for one object could be an even better match for a different object. Therefore an object’s second, third or even fourth best match might be the same object in a different figure. Additionally, the agent accounts for the possibility that the object could have been deleted from one frame to the next and there might not be a match for it in the subsequent frame. 2.3 Semantic Network The first of the agent’s three approaches is to use a Semantic Network as a data representation structure for the differences between to Raven’s Figures. First, the agent establishes a baseline set of Semantic Networks by calculating the Semantic Network for the differences which exist between each of the figures within the question prompt. The agent accomplishes this task by comparing the attributes of each object to the attributes of the same object in the second figure and capturing any differences it finds between the two. The collection of these differences for all the objects within a pair of frames makes up the Semantic Network for their differences (See Figure 1.1 & 1.2). For both a 3x3 and a 2x2 matrix a single collection will represent all of the differences found for an entire row or column which is complete. In other words the agent will only calculate the differences which exist in columns/rows which do not include the possible answer. In order to facilitate the testing step of the process later on, each Semantic Network is calculated in relationship to a column or row in the 3x3 or 2x2 matrix. When looking at a 2x2 matrix, the agent will calculate the columns and rows in relation to Figure A. For a 3x3 matrix this is not an option as every column and row which does not contain the possible answer does not also include Figure A. Therefore, the agent just follows the rule of calculating differences for any complete column or row. This means the agent will complete this process for the following pairs of Figures: A&B and A&C for a 2x2 matrix and for ((A&B) & (B&C)), ((D&E) & (E&F)), ((A&D) & (D&G)) and ((B&E) & (E&H)) for a 3x3 matrix. 2.4 Generate & Test Semantic Networks Once the agent has established a baseline of Semantic Networks for the figures in the question prompt is can begin testing to see if any of the possible answers generate similar Semantic Networks when compared to the question prompt figures. The agent generates each test case by selecting the next possible answer and creating each of the Semantic Networks it created in 2.2, except this time the agent calculates the networks in relation to the possible answer for a 2x2 Matrix and in comparison to the other columns and rows for a 3x3 matrix. For example in a 3x3 Matrix, the agent compares the Semantic Network for the column (C F I) containing the possible answer to the Semantic Network for the other columns (A D H) & (B E G). The agent also compares the Semantic Network for the row (G H I) containing the possible answer to the Semantic Networks for the other rows (A B C) & ( D E F). For the testing phase the agent compares the baseline set of Semantic Networks, which differences between the Figures within the first two columns and the first two rows in a 3x3 Matrix to the two generated set of Semantic Networks, which represent the differences between the possible answer and the rest of the Figures in the last row and the last column respectively (See Figure 6). During this comparison, the agent awards a weighted score to each of the similarities it finds between the Semantic Networks of the two sets (see Figure 1.3). The sum of the awarded scores represents confidence the agent has that the answer is going to be correct. Each score is then combined with the score generated from the process detailed in sections 2.4 2.6; after which, the highest scoring answer is submitted by the agent as it’s best answer. 2.5 Frames The second approach the agent uses to solve Raven’s Progressive Matrices relies on the Frames data representation. To start, the agent creates a frame for each Raven’s Figure by extracting everything the agent can find out about the figure and organizing it. As shown in Figure 2 for Figure A, the agent can create a collection of frames in order to represent the relationships between objects within a Raven’s Figure. Each of these frames are able to preserve the relationships between a Figure’s Frames and the objects within it. 2.6 MeansEnd Analysis Once the agent has created a collection of frames for each Figure it begins to compare the frames with the goal of discovering the transition objects take from one frame to a next. This transition from one frame to the next is represented by yet another Frame as shown in Figure 3. By comparing each of the figures in the question to one another and creating a Frame to represent the difference between each of them, as a collection, the agent captures all of the transformations that exist within all of the columns and rows of the matrix. This information will not always happen to be completely helpful as the changes don’t always exist throughout the entire matrix. Therefore, this approach is given less weight than the Propositional Logic approach in Section 2.7. 2.7 Generate & Test Frames & MeansEnd Analysis Next, the agent can apply each of the transition Frames calculated in 2.5 to Figure A in the question set with the goal of calculating the most likely answer, as seen in Figure 4 for a 2x2 matrix. Since Frames are not a problem solving solution the agent needs to use some sort of approach to determine which answer is most similar to the one generated in Figure 4. Given that the agent has already generated a possible answer and only has a set number of possible answers to compare it to, it is logical to use the Generate and Test method to determine which answer is most likely to be correct. The agent then follows a process similar to the Test phase of the Semantic Network approach to determine which answer is most likely to be correct by using a weighted comparison. Each of the generated scores from this process are combined with the Semantic Networks score and the Propositional Logic score for the same answer. The highest combined score is then selected as the best answer to the question. 2.8 Propositional Logic The Frame representations of the Raven’s Figures is used in a second problem solving approach using Propositional Logic when a 3x3 Matrix is encountered. 3x3 Matrices are large enough to introduce the possibility that patterns between the Figures which make up a column or row can exist. This possibility presents an interesting challenge and opportunity to gather additional insight as to what the correct answer could be if the the agent is able to recognize the pattern. The agent uses Propositional Logic to compare the filler values of the Frame used to represent each Figure in a particular column or row. The basic process that the agent uses to recognize a pattern is to look at the value of the same filler across each Frame in the two complete columns of the matrix and the two complete rows in the matrix. Next the agent checks to see if that filler’s value changes in every Frame in a single row or column and if it changes in a similar way in the rest of the columns and rows. If it does, the agent assumes that it has found a pattern and stores the value of the filler it found for each Frame. With this information the agent can look at the column or row which the potential answer belongs to and determine which filler value in the pattern that column or row is missing. Through this determination the agent has learned additional information about the potential answer it is looking for, it has to have the missing filler value in the pattern it recognized. The agent will complete this process for every filler value the Frame recognized as the first Figure in the column or row contains. For each pattern it finds it awards additional points to that answer which are combined with the points awarded from the other two methods. The process of looking at Raven’s Figures as Propositional logic statements and determining how two Figures compare to one another is shown in Figure 5.

3. Observed Weaknesses The agent faces the most difficulty when it is tasked with solving a problem which requires the solver to make a logical leap to come up with the answer. For example, the agent cannot solve 2x2 Question #12 because it is impossible to tell if a circle has rotated or not. This is an assumption that a human would make after considering all possible answers and seeing that none of them allow for an unchanged circle to exist. The second type the agent has a weakness for when solving is problems where objects are added. The agent does not include any logic to map an object which did not exist before in the possible solution set to the existing objects the sample problem, which again limits the agent’s ability to determine how objects have changed within the sample question. As a result, the agent has less information to use in order to determine which answer follows the same transitions and is in turn, the correct answer. The agent additionally has difficulty with Raven’s Figures which contain a large number of objects in each partition of the Matrix, such as the 3x3 Matrices in questions 17, 18 & 20. The reasoning behind this is most likely two fold. The first challenge for the agent is determining which objects in a partition within the Matrix map to objects in a different partition when there are many very similar objects, such as the situation seen in question 18. If the agent cannot create an accurate mapping then it will be unsuccessful when it attempts to calculate how an object changes throughout the matrix, as it will not always be comparing like objects. The second challenge is recognizing patterns that a bit untraditional, such as those which exist on a dianagle, as seen in question 18 or those which use an attribute which is hard to quantify, such as “rightof” or “inside”, as seen in question 20. The agent was not programed to that level of specificity when determining a pattern. With the addition of the requirement to process Raven’s Figure images in order to determine object attributes another important weakness has begun to affect the Agent’s accuracy. The Agent only has the capability to determine the size, shape and fill of an object. This limitation on the quantity of attributes diminishes the Agent’s ability to recognize important differences in other attributes between figures which are required in order to successfully calculate the correct answer.

4. Suggested Improvements A substantial improvement which could be made to the agent would be to use additional MetaReasoning to determine which of the problem solving methods is most likely going to result in the most accurate answer. For example, if the agent could retain knowledge of what makes a problem unique, the process it used to solve it and if that process resulted in a successful answer then it could use that knowledge to apply the same process to similar problems. This retention could result the agent being able to construct a map of which processes are most effective at answering certain types of problems after it has been able to a set of “learning” problems. Since, our agents are allowed to ask if the correct answer was submitted and the agent is not penalized for incorrect 2x1 or 2x2 problem answers, it is feasible this approach could be successfully implemented even in this project.

5. Agent Efficiency On a standard laptop the agent completes its analysis of over 25 problems in less than a couple of minutes. However, there are several inefficiencies within the agent design which could be improved to increase performance. The most prevalent violations occur during the process of mapping the objects in one Raven’s Figure to those in another. This process not only repeats the steps required to find the best match for an object, but it also repeats the search for any other objects in the figure that could be a match. This is done to insure that each object is only mapped to only one other object and that no competing object had a better similarity than the object which was selected. Additionally, each approach completes the matching process based on it’s form of data representation. While this can be a benefit, as one data representation approach might work better than the other in different cases, it is a identical task which would not have to be repeated. Finally this bottleneck is dependent on the number of objects within a figure. Therefore, as the number of objects in a figure rises, so does the runtime of the agent. A second improvement which could be made to improve the agent’s performance would be to incorporate more MetaReasoning into the process the agent takes to solve a problem. as suggested in the Suggested Improvements section, MetaReasoning could be used to determine which method of solving a problem is most likely to produce the most accurate result. If best approach is known before solution of the problem begins, then there isn’t has much of a benefit to solving the problem using multiple approaches. Therefore, this would limit the number of times that the agent has to solve the problem to one and greatly decrease processing time per problem.

6. Relationship to Human Cognition The design of this agent proves that sometimes using multiple approaches to solving a problem can prove to be beneficial as one approach will perform better in certain situations where it others might be produce as accurate of results. This approach is similar to what humans do when solving complex logic problems. It is common that we will apply different strategies that we have learned to a problem and then use the collection of information we have gathered from the application of each strategy to derive our conclusion. We often refer to this approach as “exploring” the different options we have to solve a problem. More typically, humans will only consider each approach which they are aware of to solve a problem, determine the best approach and solve the problem that way. This is more similar to the approach which was suggested in the Suggested Improvements section that would rely more heavily on MetaReasoning to determine the best strategy for solving a given problem. The agent further mimics human cognition by using Propositional Logic for pattern recognition. The easiest way to solve a Raven’s Matrix is to discover the pattern(s) that is follows and then find the answer which completes the pattern. The agent uses Propositional Logic to follow this same approach by representing each object and it’s attributes as logical statements and comparing the statements of figures in the same column or row to find any patterns in the differences between logic statements. The agent then finds the answer which completes the pattern. While this process is much more formal and specific than we as humans would typically think, it does follow the same process.

7. Referenced Figures Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Each comparison generates an integer similarity score, which combined give the overall similarity the possible answer’s differences within it’s own column and rows have to the question prompt figures within their own columns/rows. KEY: Figure 7: Figure 8: