The Fuzzification of an Information Architecture for Information Integration
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We present a new information architecture based on one recently introduced to structure categorized but otherwise unstructured information. The new architecture is based on fuzzy set theory subset operations that define graph theory nodes. Two types of graph edges are defined such that a user interface based on this architecture can logically minimize the number of visible navigable edges and atoms of information. This minimization is understood to be one of the primary advantages of the architecture for human-computer interaction due to its mitigation of information overload. The advantages of hierarchical, organic, and sequential information architectures are fused by the new architecture and the dialectical method is also integrated—all of which are intended to enhance human-computer interaction. The new architecture can easily incorporate quantitative information, which can be converted into a fuzzy set theory representation with fuzzy clustering and other techniques. Moreover, traditionally qualitative information such as narrative, audio, and video, although naturally represented with crisp sets, can be represented with fuzzy sets. Therefore, the new architecture can fuse traditionally disparate types of information.
KeywordsMembership Function Organic Hierarchy Information Architecture Fuzzy Intersection Dialectical Structure
Memory plays a central role in intelligence; furthermore, the computer’s relative stability of memory is one of its most promising features for the enhancement of intelligence, be it artificial or human. For these reasons, computers have long used forms of categorization to store and present information: the two most striking architectures are the venerable hierarchy and the organic tag-based systems. The authors have previously presented an information architecture—the dialectical architecture—with a structure explicitly designed to incorporate the advantages of each of these . In the present work, we fuzzify this architecture in order to include information best categorized in each category to a certain degree—that is, fuzzily categorized information. This type of information is especially important in applications such as robotics in which sensor information is quantitative. Well-established techniques such as fuzzy clustering [6, 13] and direct assignment of membership functions can assign each atom of information fuzzy categories (sets). The architecture is developed as a method for human-computer interaction to enhance human intelligence through integrating disparate types of information—narrative, audio, video, and now quantitative data—into a single representation fundamentally based on categorization. We believe beginning with human intelligence enhancement (a worthy application) may provide insight into artificial intelligence development as well.
We can recognize paintings of Picasso as well as adept imitations of Picasso. When we recognize a painting we have never seen as a Picasso or as an imitation, we are doing more than recalling earlier impressions. We are categorizing: Picasso and fakes. Our recognition of paintings or of people is the recognition of a category, not of a specific item. People are never exactly what they were moments before, and objects are never seen in exactly the same way.1 
The fuzzy dialectical architecture, like the “crisp” dialectical architecture, unites three information “planes.” The first is the structure plane, which is built from fuzzy set operations that define relations among nodes defined by category (set) intersections. The second is the flow plane, which allows information to be distributed through the structure in a sequential fashion. Flows can represent many types of information: narrative, audio, video, and data streams. The third and final plane is the dialectic plane, which provides a mechanism for flows to evolve within the framework of thesis–antithesis–synthesis. Together, these three planes comprise the fuzzy dialectical architecture, as will be described in detail in Sect. 2, which especially focuses on the fuzzification of the dialectical architecture. In Sect. 3, algorithmic considerations are explored. The human interface and a specific instantiation are described in Sect. 4.
2 Fuzzifying the Dialectical Architecture
The dialectical information architecture was introduced as a way of enhancing human intelligence by a synthesis of structure, flow, and dialectic. These teleological foundations remain intact and the methodology has been developed to include an additional type of information, that which is quantitative. The primary tool for this is fuzzy set theory, with which quantitative information can be interpreted qualitatively in the form of categories. The original or “crisp” architecture defined its structure from crisp set theoretic relations from unstructured categorized information; in this section, the structure of the fuzzy architecture will be defined from fuzzy set theoretic relations from unstructured (fuzzily) categorized information.
Consider a collection of data, each member of which we call an atom. Each atom is associated to a certain degree with a collection of categories which are represented as fuzzy sets. Crisp set operations union \(\cup \) and intersection \(\cap \) are analogous the fuzzy set operations union and intersection [17, 25]. We exploit this analog to define the fuzzy dialectical architecture in a way similar to the definition of the crisp dialectical architecture, which made much use of the crisp set operations.
Two types of s-relation are defined: the suggestively named (1) has visible a priori subcategory or vs-relation and (2) has hidden a priori subcategory or hs-relation. The definition of the hs-relation first requires the concept of a metacategory. A “minimal” metacategory for a given node is a collection of subcategories that contain as a subset all atoms associated with the node. A node’s vs-relations are those that have tails connected to the node and heads connected to subcategory nodes contained in a minimal metacategory. By minimal, we mean containing the minimum number of subcategories to fully contain all atoms. An hs-relation is defined as any s-relation that is not a vs-relation.2
Finally, atoms themselves can be either “visible” or “hidden,” names suggestive of how the user interface in later sections will be defined. An atom is visible at a given node if and only if it has nonzero membership in all categories intersected to define the node and zero membership in all others. This definition requires that an atom be visible in one and only one node in the structure.
2.1 Visibility and Hiddenness
The names given to the two types of atoms and s-relations—“visible” and “hidden”—are a crucial aspect of the structure’s advantage for intelligence amplification in human-computer interaction. In a user interface (one instantiation to be discussed in Sect. 4), these signifiers will be taken literally: at a given node, hidden atoms and hidden s-relations (edges) will not be presented to the user. The definition of each guarantees that a hidden atom will be visible if one navigates via visible s-relations to a lower level. The primary advantage of this from a usability standpoint is that the user is not inundated with as much information, one of the key aspects of a hierarchy, while remaining in a logically categorized structure—the other key aspect of a hierarchy.
2.2 Structure as Estimation
Let us consider what type of structure this graph has. It is constructed from a collection of fuzzily categorized atoms (in the case of quantitative information, these atoms are data points with membership values in each category). For a given variable, say temperature, the subset relationships are pre-defined by the membership function of the data; e.g. “luke-warm” will be a subset of “warm.” However, the inter-variable relationships are typically not so; for instance, “cold” might be a subset of “high-pressure.” The structure defined here can be understood as an estimation process for these relationships, one of several applications to be discussed in later sections.
2.3 Organic Hierarchy
The term organic hierarchy was introduced when defining the crisp dialectical architecture , and it still applies to the fuzzy architecture. It is “organic” in the sense that it evolves with each new atom’s introduction to the structure. Unlike a traditional static hierarchy that requires insertion into the structure at a specific node, an organic hierarchy evolves with the information, and a user need not explicitly define the hierarchy, which is implicit in the user’s categorization of each atom.
2.4 Invariance of Path
Another aspect of the crisp architecture that ports almost directly to the fuzzy architecture is that of the invariance of path—that is, the fact that navigation of the structure is invariant to the order in which one navigates. Let us represent each navigation along a vs-relation as the “selection” of the additional category for the intersection that defines the edge’s head node. Let each selection add that category to the path, similar to a traditional file system path (e.g. /A/B/C). For the dialectical architecture, the order of the selection is inconsequential; for instance, /A/B/C, /B/A/C, and /C/A/B all point to the same node, due to the invariance of the fuzzy intersection operation.
2.5 Fuzzy Flows
The concept of a flow was introduced in the context of the crisp dialectical architecture . It’s definition—a flow is a series of atoms—applies directly to the fuzzy dialectical architecture, but unique implications emerge. Previously, flows have been used to represent the sequential aspect of several types of information, such as narrative, audio, and video. In a fuzzy dialectical architecture representing quantitative information, each data point is an atom and a data stream is a flow. Thus each atom should not be presented to a user as an isolated data point at each node, but should be displayed in a plot (more on plotting in Sect. 4) with a trace representative of a flow. This yields an additional method of navigation, as well. A flow may intersect a node and continue on another node; the user should be able to “follow the flow” to the other node in addition to navigating the categorical structure directly, via edges.
2.6 Fuzzy Dialectic
The Fichtean dialectic is the evolution of understanding. It is often represented as a position taken, a thesis; an alternative position taken, an antithesis (not necessarily in conflict with the thesis); and a sublation of the two to form a synthesis .3 Fichte goes so far as to claim that every act of thinking is a synthesis , and so it is natural for an information architecture designed to enhance human thinking to express this model.
The crisp dialectical architecture includes a special type of flow to express the dialectic called the thesis flow, which also applies to the fuzzy dialectical architecture (and it is this aspect that is its namesake). A thesis flow is defined for each node and can be considered to be a user’s description of the intersection of the categories defining the node. When another flow intersects a thesis flow, it is considered an antithesis flow to the thesis. A user would then be prompted to resolve these to form a newly informed thesis. But flow intersections are in fact not limited to thesis flows, so each intersecting flow is an antithesis to a given flow. This dialectical manner can have many instantiations; for instance, consider a thesis flow for the node \(A\cap B\) (the relationship between A and B). Perhaps a user has written a document comprising this thesis flow, and then brings in a new quantitative data set such that the flow it defines intersects \(A\cap B\). The thesis flow would then require the sublation of the thesis and the antithesis (data). In this way, when newly connected information is introduced to the information system, those flows that are affected can be immediately identified.
3 Algorithmic Instantiation of the Structure
A naïve approach to writing an algorithm to instantiate the fuzzy dialectical architecture would yield exponential computation time. In this section, we discuss some salient ideas to consider when instantiating the architecture. A highly efficient algorithm for the structure remains an open problem, but progress has been made.
A key insight is that the entire structure need not be recomputed when a new atom is inserted or removed. This allows us to incrementally build a structure, which should, of course, be invariant to the order in which atoms are inserted. This is especially important for real-time applications such as robotics.
What requires recomputation when an atom is inserted? Only the relations originating at those nodes that are constructed by categories in associated with the new node need be recomputed. That is, (typically) most of the structure is untouched by the insertion of a new atom. Furthermore, the visibility or hiddenness of an atom never needs to be computed because an atom is visible in only one node, that which is defined by the intersection of all categories associated with it.
Moreover, any node that is new to the structure requires no structural computation, since all its relations must be vs-relations because no relation can possibly contain more than the others, since only one atom (the new one) is at the “bottom” of those paths. This allows extremely quick insertions for new categories and combinations of categories.
The unavoidably most computationally intensive aspect of the computation is the re-computation of metacategories for those nodes affected by the insertion of a new node. It is important to note that once a minimal metacategory has been found at a given level, no more levels are required.
It is also of note that memory resources can become an issue if the structure is maintained in memory (especially if metacategories are stored). It is advisable to use a graph database to persist and access the structure.
4 Human Interfacing for the Fuzzy Architecture
As with any information architecture, the fuzzy dialectical architecture may have innumerable instantiations. In this section, we describe general guidelines for these instantiations and present a specific example in Sect. 4.1.
The user should be able to browse nodes like a traditional hierarchy.
The nodes represent the intersection of categories, as they typically do in a hierarchy or in tag-based browsing. The hierarchy has a long and illustrious history of value to human thinking . Although the structure is, in fact, a graph, it will be natural to most users to experience it as a hierarchy. The “hierarchy” the user interacts with will be organic in the sense that it may change when new information is added to the system. All the spatial metaphors so valuable to hierarchies will be applicable, like “up” and “down,” “in” and “out.” At each node, the visible edges should be represented as single categories—the category that would be intersected with the current node to yield the lower node.
The user should be presented only visible edges.
“Information overload” has been identified as a significant challenge to our information age [19, 24]. One of the primary advantages of the dialectical architecture is that it minimizes the amount of information a user is presented at each node, much like a traditional hierarchy, which “tucks” the information that is further-categorized into lower levels. This means “hidden” atoms and edges should not be presented, explicitly (although exceptions can be made, of course). In some instances, hidden atoms, as defined above, might also be hidden from the user’s view; however, caution is advisable here, since in some instances, the interface might call for their visibility.
The user should be able to browse “up” to any parent node.
The property of the architecture that the path order is invariant can be exploited to allow browsing the structure in a manner analogous to the hierarchical “up-one-level,” but with multiple possibilities. The user can traverse “up” to any parent node, of which there may be several, unlike in the hierarchy, which allows each node to have only a single parent. This can be visualized by allowing the user to de-select any selected category along the path, and not merely the last-selected.
The user should be able to browse by following edges or flows.
Following edges is the structural method of navigating and is isomorphic to browsing traditional hierarchies. The dialectical architecture adds the ability to browse along flows as well. A flow can intersect a node for one or more consecutive atoms, then move to another node. For instance, an article may be discussing the intersection of several topics, then drill deeper into it with an additional categorization, which would lead it to a child node. This could be navigated by “going with the flow,” such that the user continues to see the series of atoms that comprise the flow.
The user should be able to synthesize newly intersecting flows.
The dialectical aspect of the architecture requires the thesis–antithesis–synthesis structure of information development. An information attempting to enhance human thinking should certainly capture the development of that thinking, which this feature accomplishes. A flow can be “intersected” when another flow is coincident with a node the flow traverses, and this intersection may provide a new perspective to the original flow (antithesis). A user should be able to synthesize the two perspectives such that their information system remains well-curated.
The user should be able to view quantitative data in graphs.
With the inclusion of quantitative information, the fuzzy dialectical architecture should have a user interface that presents quantitative information in a concomitant manner, typically a graph. A data point (atom) that is visible at a given node may belong to a multivariate data set and belongs to the node with some membership value in the range [0, 1]. A two-dimensional graph of given data set intersecting a node is often the best option; the user’s ability to change which variables are plotted on the abscissa and ordinate axes is important. Data series should be connected and multiple series on the same graph should appear with different line properties or colors.4
The user should be presented the membership of an atom in a node.
The fuzziness of the architecture yields an interesting aspect of the information: the degree to which each atom belongs to a given node. For quantitative information, the membership value of each point in the node should be presented; we suggest opacity of the data point. For other types of information, several techniques are possible, including sorting, iconic differentiation, color, and opacity.
We now turn to an example instantiation for demonstration purposes.
4.1 A Demonstration
An exemplar set of data was generated for demonstrative purposes as if from sensors on a balloon deployed to measure atmospheric data at various altitudes. Each data point consists of four quantities: altitude, air temperature, air pressure, and air density. The Committee on Extension to the Standard Atmosphere (COESA)  has defined a mathematical model used here to synthesize sensor data. The data points were categorized using fuzzy set theory, parsed with an algorithm that computes the fuzzy dialectical structure, and presented to the user. This data set was chosen for demonstration purposes because changes in air properties with altitude are well-understood.
The fuzzy dialectical algorithm was written in Ruby [23, v 2.4.0] and powers the user interface, which was built using the Pakyow [12, v 0.11] web application framework, which features graphs generated by c3.js [11, v 0.4.11]. Future implementations will include flow visualization and traversal disparate data types (this instantiation shows only quantitative information, but this is incidental and not an inherent limitation).
5 Conclusion and Prospects
What emerges from the information architecture presented herein is a framework for fusing information structure, category, sequence, and dialectic; fusing the representation of complex information with simple interface; and fusing information that is quantitative with that which is qualitative. It is an architecture designed for information integration and intelligence amplification. Applications include many situations for which traditional architectures have been effective, yet restrictive. The integration of quantitative information has many application, but the authors are developing a human-robot interface based especially on this aspect of the architecture.
We have presented the methods of fuzzification of the dialectical information architecture, algorithmic considerations, human interfacing, and a specific demonstration of the architecture. The methods employed were motivated by concepts from neuroscience, psychology, artificial intelligence, and philosophy. This presentation has been primarily synthetic and lays the groundwork for analytical investigation.
Quoted from Pfeifer [14, pp. 311–312].
These definitions have strong parallels in , where more mathematically oriented definitions are presented. We favor a narrative approach here. The interested reader may find the explicit mathematical definitions of the previous work elucidating.
See Ref.  for a discussion of the similarities and differences between the Fichtean and Hegelian dialectics.
We suggest a designer to make liberal use of the advice given by Tufte  for the visual display of quantitative information.
While these categories are adequate for this demonstration, the Scikit Fuzzy package offers the flexibility to tune membership functions to more accurately align with user selected categories. A user would have the ability to quickly and intuitively define categories based on data type and origin.
In future instantiations, the authors envision allowing the user to select which variable is plotted on each axis.
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