Diagrammatic Reasoning in AI

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A Classification of Diagrammatic Reasoning

Diagrammatic Reasoning in AI illustrates several examples of diagramming.  Some of the diagrams are static, or not intended to be modified by a user or updated by a system.   Other diagrams are intended to be used in more dynamic and interesting ways.  In the table below, a diagrammatic representations is classified by (1) whether or not the structure of the diagram can be modified and (2) whether or not the information on the diagram is updated.  Whereas Quadrant I diagrams are those in which both structure and information are static, Quadrant II, III, and IV diagrams are more dynamic, enabling us to create diagrammatic user interfaces, and in some cases, infer solutions to difficult problems.   

diagrammatic reasoning table


The great bulk of diagrams in use today are Static Representations (Quadrant I).  The diagrams in Chapter 3, for the most part, are static, never really intended to be manipulated in any way.  Diagrams such organization charts, basic flowcharts, semantic networks, and activity diagrams are primarily informational and help us understand systems, but are (usually) not intended to be modified by the user.  Some of the diagrams presented in Chapter 6, such as the rule trace diagrams that show the interrelationships between the conditions and actions in a rule trace (see Figure 6.8 in Chapter 6), as well as the diagrams that model strategic knowledge (see, for example, Figure 6.14 in Chapter 6) are nothing more than static representations:  they depict how an expert system works, and help us acquire a deeper system understanding, but are not really dynamic.  Because these diagrams are static, they can easily be inserted in a static medium, such as in the pages of a book, on a read-only computer screen, or on a static Web page.

On the other hand, Propagation Structures (Quadrant II ), are diagrams that are intended to be the user interface of an intelligent system.  As such, the information on the diagram can be modified and updated by a user.   Examples of propagation networks presented in this book include TransMode Hierarchy described in Chapter 6 and the Bayesian Networks described in Chapter 8.  TransMode Hierarchy is a diagram that serves as the user interface to the expert system that makes transportation mode recommendations.  When input information is entered on the leaf nodes of the hierarchy, the hierarchy transmits and propagates information upwards through the diagrammatic structure.  Thus, a user can visualize and trace a line of reasoning through the knowledge base.  For more information about TransMode Hierarchy, click here.  Bayesian networks, in a similar manner, employ causal diagrams that can illustrate a network of cause and effect associations.  Probabilities are assigned to each node on the causal diagram, and are updated based on the introduction of new evidence and data.  While propagation structures are static (the structure itself cannot be modified), the information is dynamic because new information can be propagated on the diagram based on the addition of new evidence or data.

Constructive Diagrams (Quadrants III and IV) are dynamic in the sense that the structure itself is meant to be modified by the user.  A good example is the dynamic Venn diagrams described in Chapter 4.  These diagrams can be manipulated through rules of transformation, which refer to rules for combining and manipulating objects on the Venn diagram.  Like the rules of algebra, the rules of transformation tell us what a permissible transformation is—how one can transform a Venn diagram, step by step, into another equivalent Venn diagram.  Why do we do this?  We perform these transformations in order to test the validity of a syllogism or to prove logic theorems.  It turns out that constructive diagramming can be a very powerful technique for solving difficult problems, such as proving logic theorems.

Constructive Diagrams with Information Propagation (see Quadrant IV) are dynamic in a dual sense: both the structure itself can be modified, and information can be updated and propagated through the structure.  Chapter 7 describes a system called LogNet, which supports a network design task:  how to create a business logistics network design—that is, how to configure a network of factories, warehouses, and customer zones interconnected by transportation links.  The diagram below is an example of a logistics network created in LogNet.

Logistics Network Model

At the heart of LogNet is the network model. One way to view and model a logistics environment is as a network of nodes interconnected by transportation links.  The problem of specifying the model would be one of specifying the network structure through which manufactured goods flow.  To model this environment, three types of nodes are considered:  first, the factories, where the products are manufactured; second, the warehouses, which receive the finished products from the factories for storage and possibly for further processing; and third, the customer zones (or markets), which place orders and receive the desired products from the assigned warehouse(s). Product moves through the logistics network via different transportation options (e.g., rail, trucking, shipping, air), which are represented by the connections or links between the nodes.  There are two types of transportation links:  inbound links move products from factories to warehouses, and outbound links move products from warehouses to customers.  In this diagram above, squares represent factories, circles represent warehouses, and triangles represent customer zones. 

The network design task involves a tradeoff between consolidation (merging two or more warehouses into one thereby reducing costs) and decentralization (splitting warehouses into two or more separate locations so that they better serve customers but the downside is that the additional warehouse results in increased costs).  The diagrams are constructive because they are meant to be actively modified by the user—different network configurations can be drawn (nodes can be added/deleted to the network, and transportation links can be added and deleted to connect and disconnect nodes).  By request, the resulting benchmarks (e.g., cost and customer service level) of the network design can be calculated.  Furthermore, information propagates in the network, based on how the networks are configured.  For example, the inventory carrying cost of a warehouse is updated based on the extent to which inventory is consolidated—the more consolidated the warehouse the less the (per unit) inventory carrying cost, whereas the more decentralized the more the (per unit) inventory carrying cost.


All in all, diagrammatic reasoning is really about the use of dynamic techniques, whether through information propagation (Quadrant II diagrams) or through constructive diagramming (Quadrant III and Quadrant IV diagrams).  This book, in essence, is about illustrating some of these more interesting uses, and going  beyond the more traditional and static uses of diagramming.  The future of diagrammatic reasoning, and how it intersects with the AI discipline, is about how these techniques might be used to solve difficult, thorny, and seemingly intractable problems.  This book has showcased a number of applications, but hopefully this is only the beginning, and only a hint and a glimpse at what else may be possible.