Lesson 18 Analogical Reasoning

Preview

In this lesson, we will discuss Analogical Reasoning. This involves understanding new problems, in terms of family of problems.

It also involves addressing new problems, but transferring knowledge of relationships from known problems across domains. We introduced a notion of transfer previously in explanation based learning.

We also have talked about Case Based Reasoning.

Today we’ll talk about transfer in a much more general manner. We’ll start by talking about similarity then revisit case based reasoning.

Then we’ll talk through the overall process of analogical reasoning, including retrieval, and mapping, and transport. Then we’ll close by talking about a specific application of analogy, called Design by Analogy.

Exercise Similarity Ratings

To illustrate the notion of similarity, let us consider an example. Consider that a woman is climbing up a ladder. Here are seven situations. Can you please rank these seven situations by their order of similarity to the given situation?

Exercise Similarity Ratings

Interesting answer, David, note that there are several factors in David’s answers. In the two situations that he thought were most similar to a woman climbing up a ladder, in the similarity in the relationship, climbing up as well as similarity between objects, woman and ladder. In contrast, one that he did not think was really similar to a woman climbing up a ladder, woman painting a ladder. Although there is some similarity between the objects woman, and ladder, the relationship is very different here it is climbing up a ladder, here it is painting a ladder which are two very different activities. Between one and two we notice, that both of them have the same relationship, climbing up, but an object is different, In one case it is the step ladder and in another case it is a set of stairs. So one can have similarities in relationships, one can have similarities in objects, of course some of you may have different rankings with the similarities of the one they would give. Because, your background knowledge might be different or your priorities might be different, but the point here is the similarity can be measured around several dimensions. Around the dimension of relationships, around the dimension of objects, around the dimensions of features of objects, and around the dimensions of values of features of objects that are participating in relationships, we’ll talk more about this in just a few minutes.

Cases Revisited

We have come across the notion of similarity earlier in this course. When we were discussing learning cases, that particular point, we came across the matter of finding the nearest neighbor. At that point we found the nearest neighbor simply by looking at the distance between the new situation, and the familiar situations. We came across the notion of similarity when we were discussing case reasoning as well at that point, we came across at least two different methods of organizing the case library. And that, in one method, we could simply organize all the cases in array here’s an array of several cases in the domain of navigation and urban area, each case here is represented, by the x and y location of the destination. A different and smarter method, also organizes cases that are discriminatory, the leaf nodes of this discrimination tree represented the cases. The root node and the interior nodes in the discrimination tree represented discrimination, or decisions about the values of specific features for example, east of 5th Street or not east of 5th street, both of these schemes are based on measures of similarity. In the first scheme the similarity is based on the similarity between the tags, If a new problem were to come along it would be more or less similarly one of these cases depending on whether or not its tags match the tags of a particular case here. In the second scheme of this
tree, similarity is based on, traversing this particular tree, If a new problem came along, we would use the features of that new problem to traverse this tree and find the case whose features best match your new problem. Note that the new problem, and the source cases in all of these examples so far have been in the same domain. Here for example, both the new problem and the source case are in the same domain of navigating in an urban area, in the previous example, the new problem and the source case were the domain of colored blocks in the blocks world. What happens if the new problem and the SOS case are not in the same domain? So consider the example of, a woman walking up the ladder and walking up the wall. The two dimension are the same, we’re talking about woman in one case and in other case, a ladder in one case, a wall in other case yeah, there’s some similarity. Situations like this, where the new problem and the source case are from different domains, lead to cross-domain analogies. So the question now becomes, how can we find leaf similarity between the new problem, the target problem, and the source case, if they happen to be in different domains?

Need for Cross-Domain Analogy

To dig into this issue of similarity between a target problem and a source case in different domains, let us look at another example. Let us suppose it is a patient who has a tumor in his stomach. There is a physician who has a laser gun. She knows that if the laser light were to shine on this tumor, the tumor will be killed and the patient will be cured. But the physician has a problem. The laser light is so strong that it will also kill all the healthy tissue on its way to the tumor, and thereby killing the patient. What should the physician do? This is actually a very famous problem in cognitive science. It was first used by a psychologist called Don Curry around 1926. What do you think the physician should do in this situation? Take a moment and think about it.
We’ll return to the physician and the patient example in just a minute. First let me tell you another story. Once there was a kingdom ruled by a ruthless king, there was a rebel army approaching the fortress in which the king lived. Well there was a problem. The kings’ men had mined, all the roads approaching the fort. As a result, if an army was to walk over the roads, the mines would go off, and the soldiers would be killed. So what did the army decide to do? The army decided to decompose itself into smaller groups, so that each group could come from a different road, and reach the fort at the same time. Because each group was small enough, the mines on the roads did not go off. The soldiers were able to attack the fort at the same time and overthrow the bad king. Now let’s go back to the problem of the physician and the patient. What do you think now? Has the answer to the problem changed? Some of you indeed may have changed your answer because of this story I told you about the king and the rebel army. One solution to this problem is that a physician would divide a very intense laser beam into several smaller, less intense beams. As these beams come from different directions, they do not harm the healthy tissue. However, they reach the tumor at the same time and manage to kill the tumor. You will note that this is an example of cross-domain analogy. Here the target problem had to do with the physician and the patient. The source case had to do with the king and the rebel army. The objects in these two situations were clearly very different. In one case we had a physician and the patient, the laser beam and the tumor. And in the other case we had the king and the rebel army, the fort and the mines. Some of the relationships were very similar. In capturing the fort case, we had a resource, the army, which was decomposed into several small armies, which was sent to the goal location at the same time. We took this battle, we took this strategy, abstracted it out, and then applied it to the patient and physician example. A physician used the same strategy. Resource decomposes into several smaller resources, and sent to the goal at the same time. Now you can also see why the ant climbing a wall is similar to a woman climbing a ladder. The objects are different, ant and wall, woman, and ladder, but the relationship is similar. Climbing up. The cross-domain analogy is then, the objects and the features and the values of the objects can be different. The similarity is based on the relationship. It is the relationship that is important. It is the relationship that gets transferred from the source case to the target problem.

Spectrum of Similarity

We can think of a spectrum of similarity. At one end of the spectrum, the target problem and the source case are identical. At the other extreme end of the similarity spectrum, the target problem and the source case have nothing in common. We can evaluate the similarity between the target problem and source case in similar dimensions. In terms of the relationships occurring in the source case and the target problem. In terms of the objects occurring between the two. In terms of the features of the objects and in terms of the values that the features of the objects take. At the end of the spectrum, where the target problem and the source case are very similar, with relationships, objects, features, and values are all similar. At the other end the values, features and objects may be different, but the relationships are similar. If the relationships too are different, then there’ll be nothing in common with the target problem in the source case. When the relationships, objects, features and values are all similar, then that is an example of recording cases, and we have come across it. An example of that was from the colored blocks in the blocks world. When the similarity between the target problem and the source case is along the dimension to relationships and objects, but not along the dimensions of values or values and features, then that’s an example of case-based reasoning. We discuss this method in the domain of navigation and urban areas. The objects of the concept between the target problem and the source case being the same means that the domains are the same. So cased-based reasoning is within domain analogy. An analogical reasoning in general, objects in the target column and the source case too, might be different. We saw an example of analogical reasoning in the Dunker radiation problem, when we were talking about cross-domain analogical transfer. Actually recording cases in case based reasoning are also examples of analogical reasoning, except that they occur in the same domain. The target firm and the source cases in the same domain, which is why we consider them earlier. By analogical reasoning here, we mean cross domain analogical transfer. As in the Dunker radiation problem.

Process of Analogical Reasoning

Analogical reasoning allows us to look at new problems in terms of familiar problems. It also allows us to transfer knowledge from familiar problems to new problems. A hierarchy process of analogical reasoning is shown here. It consists of five major phases, retrieval, mapping, transfer, evaluation, and storage. We’ll discuss all five stages, in detail. Let us compare for a moment, the process for an illogical reasoning in general, with a process for, case based reasoning, within domain and illogical reasoning that we discussed earlier. Notice that retrieval, evaluation, and storage, are common between the two processes. In case based reasoning the target problem and this first case, were from the same domain. They had the same kind of relationships, and the same kind of objects. We simply had to adapt the source case to address the target properly. An analogical reasoning in general, the target form and the source case need not be from the same domain. When they are not from the same domain, we can’t just take the source codes and adapt it. We first have to map the target problem with the source case that is, we need to address the correspondence problem. What in the target problem, corresponds to what in this source case as an example? The laser beam and the target Duncker’s radiation problem corresponds to the rebel army in the source case. Once we have mapped the conceptual relationship in the target problem to the conceptual relationships in the source case, then, we can try to transfer some of the relationships in the source case to the target problem. We can first abstract those relationships and then transfer them to the target problem. As an example, the Duncker’s radiation problem we first did the alignment, that is, we just did the correspondence problem, what in the target [UNKNOWN] corresponds to what in the source case? Then we took the relationship, and abstructed it. The relationship in that case was, take the resource and decompose it into several smaller resources, and send them to the goal at the same location. That particular relationship, that particular pattern, is what we abstracted and transferred to the target problem. Note that this is just one theory of analogical reasoning. In other theories, some of these boxes are configured differently. For example, in another theory, mapping is a part of retrieval. We do mapping in order to do retrieval.

Analogical Retrieval

Let us look at analogical retrieval more closely. Once again, we have come across this idea earlier. We did analogical retrieval in recording cases by the k nearest neighbor method. We did analogical retrieval on a case based reasoning using two different methods, the array method and the discrimination tree method. Here the criteria for evaluating similarity were very clear, as we discussed earlier. [UNKNOWN] distance, same tags, as well as placement in this discrimination tree. The question now becomes, what criteria should be used to decide on the similarity between the target problem and the source case, given that they come from different domains? On surface, there seems to be little similar between the two. None of the objects are similar. None of the values of features are similar. Yet, there is a deep similarity there. We can distinguish now between superficial similarity between two situations and deep similarity between two situations. Superficial similarity deals with features of objects or counts of objects or objects themselves. Deep similarity deals with relationships between objects, or sometimes relationships between relationships. Examples of this arise from the variables test with which you are already familiar. Features here refer to the size of the square, the size of the circle, or perhaps where there is a hollow square, or a solid dot. The count refers to the number of squares, or number of circles, in a particular image. Objects here refer to circles and squares and dots. Let us look at relationship between objects. Two situations are said to be deeply similar, if the relationship between the objects is similar. As an example, a and b are similar, in that, that the dot is outside the circle here and the square is outside the circle here. A and b are also similar in that the dot is above the circle here and the square is above the circle here. What about relationships between relationships? Let us compare a and b. In going from A to B, the dot has disappeared, and a square has come outside the circle, and become bigger. Now we can compare this relationship between a and b, with some of the relationship between a c and a b, in which too, some object might be disappearing, and another object which may be in the center of the circle comes out of the circle. I’m sure you’ve come across problems like that on the variable test. This is an example of a binary relationship, a relationship between two objects. This is an example of a higher order relationship, a tertiary relationship if you wish. This is a relationship between the relationship between objects. You might even say that these are examples of unary relationships. These are just examples of objects and their features and counts. In general, as we go from unary relationships to binary relationships to tertiary relationships to even higher order relationships, the similarity becomes deeper and deeper. This means that mind decides two situations to be more similar if the similarity is at the level of relationship between objects rather simply at the level of objects or features or counts and objects.

Three Types of Similarity

Semantic similarity used with conceptual similarity between the target problem and the source case. If we recall the original exercise that David had answered, in that exercise, a woman climbing up a ladder is conceptually similar, semantically similar to a woman climbing up a step ladder. The same kind of concepts occur in both situations. Woman, and step ladder or ladder. Pragmatic similarity concerns with external factors. Factors external to the presentation, such as goals. As an example, in the Dunker radiation problem, the physician had a goal of killing the tumor, which was similar to the goal of capturing the fort in case of the rebel army and the king. Pragmatic similarity refers to similarity of external factors, factors external to the representation, such as similarity of goals. The Dunker radiation problem for example, the physician had the goal of killing the tumor, which was similar to the goal of capturing the fort in case of the rebel army in the king example. The third measure of similarity is structural similarity. Structure here refers to the structure of presentations, not to physical structure. Now structural similarity of the first two, similarity between the representational structures of the target problem and the source case, and we’ll look at an example of this in just a few minutes. Know that one can assign different kinds of weights to these three measures of similarity. So some queries of analogy focus on structural similarity. Other theories of analogy focus on semantic and pragmatic similarity. That is also why you may have given slightly different answers to the questions in the first exercise than David did.

Exercise Analogical Retrieval I

Let us do another exercise together now that we know about deep similarity and superficial similarity. Consider the situation again, a woman is climbing up a ladder. Give this set of situations, mark whether each of the situations is deeply similar or superficially similar to this given situation. Know that some might be both and others might be neither

Exercise Analogical Retrieval I

This is good, David. Once again, different people may give different answers to this exercise. Why do we do so? Well, let’s examine it next.

Exercise Analogical Retrieval II

Many science textbooks in middle school or high school explain the atomic structure in terms of the solar system. Here’s a representation for the solar system, here’s a representation for the atomic structure. Let us see how this model of the solar system helps us make sense of the atomic structure. We’ll use this example often going forward. And this representation of the solar system is arrows are denoting causality. So the sun’s mass is greater than the planet’s mass, which causes the planet to revolve around the sun. Similarly for the atomic structure, there is a force within the nucleus and the electron, and that causes the nucleus to attract the electron and the electron to attract the nucleus. Given these two models, what are the deep similarities between them?

Exercise Analogical Retrieval II

Now we can see why these textbooks write about the solar system, and the atomic structure in such a way that these relationships become salient. They help us make sense of the atomic structure, by pointing to the deep similarities between the relationship that occur in the atomic structure, and the relationship that are occurring in the solar system.

Analogical Mapping

Now let us consider analogical mapping. The problem here is called the correspondent’s problem. There are a number of obvious relationships in this target problem. There are a number of obvious relationships in this source case. What in the target problem corresponds to what in the source case? If we can address the correspondence problem. If we can say, for example, that the laser beam corresponds to the rebel army, then we can start aligning the target problem and the source case so it makes the deep similarities between relationships salient. Note there are several parts of a target problem and several in the source case. In principle, any of these objects of the target problem could correspond to any of the objects in the source case. In which case we would have an m to n mapping, and that becomes computationally inefficient. If you and I, often do not have much of a problem deciding, if the laser beam must correspond to the devil army. How do we do it? And how can we help AI agents make similar kind of correspondences? Our answer is, we’ll make use of relationships. In fact, we’ll make use of higher order relationships, whenever possible. We’ll give precedence to higher order relationships, over other relationships. As a unary relationship, we might say that a patient is a person here, and king is a person there. The binary relationship we might say, that physician has a resource, the laser beam. And that the rebel army has a resource, the army itself. It’s a higher ordered relationship, a tertiary relationship between say, that between the goal and the resource is an obstacle. They held a tissue in this case. Similarly between the goal and the resource is an obstacle, the minds in this case. We focus on the higher ordered relationship there, that’s where the deepest similarity between the two situations lies. This is how we know to mark between the king and the tumor and not between the king and the patient. Although the king and the patient are superficially similar, a deeper similarity lies in viewing the king and the tumor in terms of goals which need to be cured or captured using a resource when there is an obstacle in between them.

Exercise Analogical Mapping

Let us do an exercise on deep relationships. Let’s get back to for example the solar system and the atomic structure. Let us suppose that you’ve given this representation of the solar system and this representation of the atomic structure. How would you map the solar system to the atomic structure?

Exercise Analogical Mapping

This is right, David. Another thing to take away from here is note the depth of understanding it requires in order to be able to make your right kind of correspondences. If one didn’t have the right kind of model for the solar system and atomic structure that captures the deep relationships, then the mapping may not be done. The alignment wouldn’t work, and we would not be able to understand the atomic structure in terms of the solar system. Thus, models, deep and rich models of the two systems, the target problem and the source case, are essential to deciding how to align them, how to map them, and as will see in a moment, what to transfer and how to transfer it.

Analogical Transfer

Now let us consider analogical transfer. So, given this target problem, analogical retrieval has led to this source case. Given a model of the target problem and a model of the source case, analogical mapping has also occurred, correspondence has been established. This we now know that a king corresponds to the tumor, not to the patient. And that the rebel army corresponds to laser beam. For the source case we also know the solution. The rebel army divided itself into smaller groups, and the smaller groups all arrived at the fort at the same time. Now the question becomes how can we transfer this solution to our original target problem. >From the source case now, we’re inducing a pattern of relationships, a strategy. In this case the pattern is that if there is a goal, capturing the king, and a resource, the rebel army and an obstacle between the resource and the goal. They march on the road, then decompose the resource into several smaller resources. And send them to the goal from different directions at the same time. This abstract pattern is now transferred to the target problem and instantiated. Because we know that the goal is the tumor, the resource is the laser gun, and the obstacle is the healthy tissue, we know what to do. We must decompose the resource, the laser gun, into smaller pieces, smaller, less intense laser beams, and send them to the tumor, the goal, at the same time from different directions. This is how we can transfer the problem-solving strategy, from the source case to the target problem. Note that this transfer depended upon the correct mapping, the correct alignment between the target problem and the source case, which in turn depended upon the retrieval of the source case corresponding to this target problem. Note the important rule or goal here. The goal was to capture the king. So this is an example of pragmatic similarity. With a lot of similarities at the level of the goal, capturing the king, curing the tumor.

Exercise Analogical Transfer

Let us do an exercise on analogical transfer together. Back to our example of this sort of system in the atomic structure. Given this representation of the solar system and this representation of the atomic structure, what would be transferred from the solar system into the atomic structure model?

Exercise Analogical Transfer

That’s a smart answer David. Recall that originally I had said that I’ll explain structural similarity, and I have not done it so far. I’m going to use the spherical example and David’s answer to explain it now. Given the solar system as the source case, and the atomic structure as target problem. We can see that there is little semantic similarity between them. The kinds of objects that occur in the solar system are not at all like the kinds of objects that occur in the atomic structure. We can also see that pragmatic similarity is not a major issue here. We’re not talking about the goal of the solar system or the goal of the atomic structure. Although we might have the goal of understanding atomic structure in the solar system, there is nothing in the solar system, or in the atomic structure, which has a goal. Yet, David was able to answer this question correctly. This is because of structural similarity. Let us consider the top part of the model of the solar system. You can think of this top part like a graph. The vertices in this graph, correspond to objects and their properties. The edges in this graph correspond to relationships, such as force between Sun and Planet. Or, Sun attracts Planet, and Planet attracts Sun. Once again, this graphical representation of the model atomic structure. The words are representing the objects and the features. And the edges are representing the relationships between the objects. Although there’s little semantic similarity, or pragmatic similarity between the two situations, we can see a structural similarity. A similarity in the structure of the graphs. Because this part of the graph of the representative of the solar system. The similar this part of the graph of the representative of the atomic structure. We can differ infer that we can transfer this part of the graph of the solar system to infer this part of the graph of the atomic structure. Structural similarity then captures relational similarity. What is common between these two situations is neither the objects or the goals. Where as common here as the relational similarity, and that is what structure similarity captures.

Evaluation and Storage in Analogical Reasoning

Let us briefly talk about evaluation in storage. These evaluation and storage steps in analogical reasoning are very similar to the evaluation and storage steps in case based reasoning. Analogical reasoning by itself does not provide guarantees of correctness. So the solution that it proposes must be evaluated by some manner. For the down correlation problem, for example, we may evaluate the proposed solution by doing a simulation. Once the evaluation has been done, then the new problem and a solution can be encapsulated as a new case and stored back in memory for later potential reuse. To return to the down correlation problem, as an example. Once we have the solution of decomposing the laser beam into several smaller beams and sending them to a tumor at the same time from different directions, we can do a simulation of this solution and see whether they are successful. If it is, then we can encapsulate the target problem and the proposed solution as a case, and store it in memory. It might be useful later. It could potentially become a source case for a new target problem to come later. Once again, in this way, the AI agent learns incrementally. Each time it solves a problem, the new problem and its solution becomes a case for later reuse.

Design by Analogy

It is often useful to look at specific problem domains, both to see how we can apply current theories of chronological reasoning to them and also to see how we can use those problem domains to build new theories of chronological reasoning. So let us turn now to the domain of design. In design, there is a new movement that is sometimes called biologically inspired design, or biomimicry. This movement is pulled by the need for environmental sustainability and is pushed by the desire for creativity and innovation and design. On the top left here is a picture of the Shinkansen 500 train in Japan. This is a bullet train. It’s called a bullet train because of the shape of its nose. This particular shape is inspired by the shape of the beak of the Kingfisher. The story goes something like this. Japanese railway engineers were interested in building faster trains. However, they had a problem, these trains had to go through tunnels. And as they went through tunnels, they created shock waves, which created a lot of noise, bothering the neighbors. The shock wave was created because outside the tunnel and inside the tunnel were two different mediums. By serendipity, the railway engineers looked at how the Kingfisher goes from the medium of air into a medium of water, dips its beak and catches its prey. The shape of its nose allows us to create a smaller shock wave. We use the same principle to create the design of the nose of the bullet train. Shinkansen 500 travels faster than previous trains and also makes less noise than previous trains, mostly because of the nose shape. Another example often cited in biomimicry is the example of a Mercedes Benz box car, designed by inspiration to the Boxfish. Notice as biological inspired design entails analogical reasoning. There’s a target problem. There’s a source case. There is cross-domain analogical transfer.

Design by Analogy Retrieval

To illustrate analogical reasoning and design, or analogical design, we’ll talk about a specific problem, let us suppose we design a robot that can walk on water. Nature already offers several examples of organisms that can walk on water, this is a picture of the basilisk lizard, which can walk very well on water and catch it’s prey. Recall that we said earlier that for analogical mapping and crossword worker row, requires a deep understanding of the source case and the target problem. That is true here as well in case of analogical design, we require a deep understanding of the locomotion of the basilisk lizard, in order to be able to design a robot that can walk on water, inspired by the design of the basilisk lizard. Here is a model of the basilisk lizard, this model is sometimes called structure behavior function model. This particular picture doesn’t show the structure, it just shows the function and the behavior. The function is shown at the top here, It is shown by it’s initial state and it’s goal state, and it’s function is achieve by behavior shown here. The behavior is represented as a series of states, and transitions between those states. We will not talk about the representations in more detail here, readings given in the class notes give this sort of representations a lot more detail if you are so interested.

Design by Analogy Mapping Transfer

Recall that we started with a problem of designing a robot, that can walk on water. Let us suppose that, that particular target problem resolves in the retrieval of a source case, of a robot design that we already encountered. One that can walk on ground. Now the question becomes, how can we adapt this particular design of the robot that can walk on the ground, into a robot design that can walk on water? Let us now suppose, if we reuse this particular problem of designing a robot to walk on water. As a probe into the case memory. And now the case returns to us, the design of the basilisk lizard. That might happen, because the design of the basilisk lizard, is indexed by it’s functional model, walk on water. So there is a pragmatic similarity between the two. We now have the design of a robot who can walk on ground, and we have the design of a biological organism, the Basilisk Lizard, that can walk on water. For the Basilisk lizard, we also have a complete model, a complete explanation of how its structure achieves its function. Now that we have a partial design for the robot, this is a design of the robot that can walk on ground. And we have a design of an organism that can walk on water. We can try to do an alignment between these two. This alignment will be based on the similarity between relationships. Clearly, the objects here, and objects there are very different. Once we have aligned these structural models, or the robot that can walk on ground, and the basilisk lizard that can walk on water. Then, we can start doing transfer. We can transfer specific features, of the structure of the basilisk lizard. For example, the shape of its feet, to this model, of the robot that can walk on ground. In order to convert it into a robot, it can walk on water. Having constructed a structural model, for this robot that can walk on water then we can try to transport the behavioral model, and then the functional model. And then this way we have a complete model of a robot that can walk on water. Along with an explanation of how it will achieve it’s function. This is sometimes called compositional analogy. We’ll first do mapping at the level of structure, and that mapping at a level of structure helps us transfer some information. That in turn allows us to transfer information at the behavioral level. Once we have transferred information at the behavioral level, we can climb up this abstraction hierarchy, and transfer information at a functional level. We can now revisit our computational process, and our logical reasoning. Initially we had presented this particular process like, a linear chain, Retrieval, Mapping, Transfer, Evaluation and Storage. In general, however, there can be many loops here. We may do some initial mapping, for example, that may result in some transfer of information. But that transfer then, may lead to additional mapping, and then to additional transfer and so on. Here is another brief example, from biological inspired design, in this case we want to design a robot that can swim under water in a very slowly manner. This particular function of swimming underwater in a stealthy manner, reminds a design team of a copepod. A copepod is a biological organism, that has a large number of appendages. It moves underwater, in such a way that in generates minimum wake, especially when it moves very slowly. On the other hand, when it moves rapidly, then the wake becomes large, when the wake is small then its motion is very steady, when the wake is large, its motion is no longer steady. An analogically transfer of knowledge about this particular copepod, gives a design for the microbot for slow velocity. This analogy, decomposes our original design problem. We had the original design problem, as moving underwater in a stealthy manner. Now that we have a design of an organism, for moving underwater at low velocities, we are still left with the sub goal of moving underwater at high velocities. The goal of designing a microbot, that can move underwater in a stealthy manner, at fast velocities, may remind the design team of the squid. The squid uses a special mechanism, like the jet propulsion mechanism to move underwater in a stealthy manner at pretty high velocities. Now we have created a designed for microbot. Where part of the design comes from the design of the copepod, and the other part comes from the design of the squid. Instead of borrowing the design from one source case, we are borrowing parts of the design of multiple source cases. This is a compound analogy. Notice that there’s a problem evolution going on, which started with one problem. We arrived at a partial solution to that. Which then leads us to a problem evolution. And the problem transformation. We then have a new understanding of the problem. So, this example we saw, how we first did analogical retrieval of the coco powder organism. Then Mapping, then Transfer. That then lead to addition retrieval, in this case with a squid. Once again this process is not linear. Just like we can iterate between Mapping and Transfer, similarly we can iterate between Transfer and Retrieval.

Design by Analogy Evaluation Storage

Evaluation too can play a very important role in the iterative loops in this analogical reasoning process. One can use several different methods for doing evaluation. In the robot that can walk on water, for example, we can do a simulation, or we can build a physical prototype. If the evolution succeeds, then well and good, we can encapsulate the target polymer solution as a case and store it in case memory. If the evaluation fails, we may need to revisit transfer and see whether we want to transfer some of the knowledge or revisit mapping, and perhaps align things differently or revisit retrieval and perhaps try to retrieve a different source case. As an example, supposing that the evaluation shows, then the robot that we designed for walking on water is a little too heavy. In that particular case, we may change the problem specification and retrieve a different kind of organism that perhaps is a little lighter. Let us suppose that we evaluate the design of the robot that can walk on water and find that the design is a little too heavy. In that case, we can go back to the transfer stage and see whether we can transfer some of the relationship that might make the robot a little lighter. Or we can go back to the mapping stage and see whether we can align the source case and the target problem slightly differently so that we can transfer a different relationship. Or alternatively, we can go back to the retrieval state and try to retrieve a source case, a different kind of biological organism altogether. Thus, we see that the process of analogical reasoning is not linear at all and see it can have many different kinds of iterations. Analogical reasoning continues to be an important topic in our research and biological-inspired design is becoming one. We provide several readings with both topics in the class notes.

Advanced Open Issues in Analogy

There’re a number of advanced and open issues in analogical reasoning, that are the subject for current research. First, because analogical reasoning entails cross-domain transfer, does it mean that we necessarily need a common vocabulary across all the domains? Consider the example of the atomic structure and the solar system once again. Suppose I were to use this term revolve, to say the electron revolves around the nucleus. But use the term rotate to say the planet rotates in an orbit around the sun. I have used two different terms. How then can I do alignment between these two situations? Should I use the same vocabulary? If I don’t use the same vocabulary, what alternative is there? Second, analogical reasoning entails problem abstraction and transformation. So far we have talked as if the problem remain fixed, it’s source case is retrieved and transferred across. But often, the agent needs to abstract and transfer the problem, in order to be able to retrieve the source case. A third issue in analogical reasoning concerns compound and compositional analogies. So far we have talked that given a problem, we can retrieve a case and transfer some knowledge from that case to the problem. But often we retrieve not one case, and we transfer knowledge from not one case, but from several cases. If you’re designing a car, you might design the engine binology to one vehicle and the chassis binology to some other vehicle. This is an example of compound analogy. But how can we make compound analogy work? In compositional analogy, analogy works at several levels of abstraction. Supposing we were to make an analogy between your business organisation and some other business organisation. We might make this compositional analogy, first at the level of people. Next to the level of processes. Third of level of the organisation as a whole. This is another example of compositional analogy, where mapping at one level supports transfer to the next level. How do we do compositional analogy in AI agents? Fourth, visuospatial analogies. So far we have talked about analogies in which it transferred necessarily engages causal knowledge. But a large number of analogies in which causality is at most implicit. We’ll consider these visuospatial analogies later in the class. Fifth, conceptual combination. A powerful learning mechanism is learning a new concept by combining parts of familiar concepts. Analogical reasoning is one mechanism for conceptual combination. I have a one concept, [UNKNOWN] concept, that of the atomic structure, another concept, the solution concept. The concept of the solar system. I take some part of the solar system knowledge, combine it with my concept of the atom to get a new concept of the atom. If you’re interested in any of these issues, I invite you to join the PhD program in Computer Science.

Assignment Analogical Reasoning

So how would you use analogical reasoning to design an agent to answer Raven’s progressive matrices? This might be a tough question at first, because the agents we’re designing only operate in one domain, taking the Raven’s test. They don’t look at other areas. So, we’re going to get the knowledge necessary to do cross domain analogical transfer. In this instance instead of the agent doing the analogical reasoning, maybe it’s you doing the analogical reasoning. Can you take inspiration from other activities to inspire how you design your agent? Or can you take knowledge from other activities and put them in your agent, so that it can do the analogical reasoning?

Wrap Up

So today, we’ve been talking about analogical reasoning. We started by talking about similarity. As we saw in our opening exercise, similarity is something that we evaluate very easily without even really thinking about it. How can we design agents that can do the same kind of similarity evaluation? We then talked about analogical retrieval, which can be difficult, because we’re trying to retrieve examples across other domains. How can we structure our knowledge to facilitate this kind of retrieval? How can a system know the given a model of the atom, it should retrieve a model of the solar system? Then we talked about mapping, which is figuring out which parts of different systems correspond. For example, how can figure out that the troops in the four example correspond to the lasers in the tumor example? We then talk about transfer, which is moving knowledge from the concept we know to the concept we don’t. For example, we used what we knew about the solar system to fill in our knowledge of the atom. Then next, we talked about evaluation and storage. How do we evaluate our analogies? In the tumor example, we might actually try that medical procedure. But for other analogies, how do we evaluate them? And then how do we store them for future use? Last, we talked about a special kind of analogy, design by analogy, where we use something that we know a lot about to inform our design of something new. We’ll talk a lot more about this, especially design by analogy, when we come to the design unit later in our course.

The Cognitive Connection

Analogy is often considered to be a core process of cognition. A common example of analogy we encounter everyday is that of metaphors. For example, you can imagine someone saying, I had to break up with her. We had grown very far apart. Far apart here is a spatial metaphor. One of the famous examples of metaphors comes from Shakespeare. All the world’s a stage, all the men and women merely players. The theater here is a metaphor for the world. A third connection is the Rubin’s test of intelligence. The Rubin’s test is considered to be one of the most common and reliable test of intelligence, and as you well know by now, it is based entirely on analogies. An analogy is that central to cognition.