1102/1102.1889.md

OLOGS: A CATEGORICAL FRAMEWORK FOR KNOWLEDGE REPRESENTATION

DAVID I. SPIVAK AND ROBERT E. KENT

ABSTRACT. In this paper we introduce the olog, or ontology log, a category-theoretic model for knowledge representation (KR). Grounded in formal mathematics, ologs can be rigorously formulated and cross-compared in ways that other KR models (such as semantic networks) cannot. An olog is similar to a relational database schema; in fact an olog can serve as a data repository if desired. Unlike database schemas, which are generally difficult to create or modify, ologs are designed to be user-friendly enough that authoring or reconfiguring an olog is a matter of course rather than a difficult chore. It is hoped that learning to author ologs is much simpler than learning a database definition language, despite their similarity. We describe ologs carefully and illustrate with many examples. As an application we show that any primitive recursive function can be described by an olog. We also show that ologs can be aligned or connected together into a larger network using functors. The various methods of information flow and institutions can then be used to integrate local and global world-views. We finish by providing several different avenues for future research.

CONTENTS

1.	Introduction	1
2.	Types, aspects, and facts	6
3.	Instances	14
4.	Communication between ologs	17
5.	More expressive ologs I	31
6.	More expressive ologs II	40
7.	Further directions	48
	References	50

1. INTRODUCTION

Scientists have a pressing need to organize their experiments, their data, their results, and their conclusions into a framework such that this work is reusable, transferable, and comparable with the work of other scientists. In this paper, we will discuss the “ontology log” or olog as a possibility for such a framework. Ontology is the study of what something is, i.e the nature of a given subject, and ologs are designed to record the results of such a study. The structure of ologs is

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This project was supported by Office of Naval Research grant: N000141010841 and a generous contribution by Clark Barwick, Jacob Lurie, and the Massachusetts Institute of Technology Department of Mathematics.based on a branch of mathematics called category theory. An olog is roughly a category that models a given real-world situation.

The main advantages of authoring an olog rather than writing a prose description of a subject are that

• • an olog gives a precise formulation of a conceptual world-view,
• • an olog can be formulaically converted into a database schema,
• • an olog can be extended as new information is obtained,
• • an olog written by one author can be easily and precisely referenced by others,
• • an olog can be input into a computer and “meaningfully stored”, and
• • different ologs can be compared by functors, which in turn generate automatic terminology translation systems.

The main disadvantage to using ologs over prose, aside from taking more space on the page, is that writing a good olog demands a clarity of thought that ordinary writing or conversation can more easily elide. However, the contemplation required to write a good olog about a subject may have unexpected benefits as well.

A category is a mathematical structure that appears much like a directed graph: it consists of objects (often drawn as nodes or dots, but here drawn as boxes) and arrows between them. The feature of categories that distinguishes them from graphs is the ability to declare an equivalence relation on the set of paths. A functor is a mapping from one category to another that preserves the structure (i.e. the nodes, the arrows, and the equivalences). If one views a category as a kind of language (as we shall in this paper) then a functor would act as a kind of translating dictionary between languages. There are many good references on category theory, including [LS], [Sic], [Pie], [BW1], [Awo], and [Mac]; the first and second are suited for general audiences, the third and fourth are suited for computer scientists, and the fifth and sixth are suited for mathematicians (in each class the first reference is easier than the second).

A basic olog, defined in Section 2, is a category in which the objects and arrows have been labeled by English-language phrases that indicate their intended meaning. The objects represent types of things, the arrows represent functional relationships (also known as aspects, attributes, or observables), and the commutative diagrams represent facts. Here is a simple olog about an amino acid called arginine ([W]):

(1)

graph TD
    D["D  
an amino acid found in dairy"]
    A["A  
arginine"]
    E["E  
an electrically-charged side chain"]
    X["X  
an amino acid"]
    R["R  
a side chain"]
    N["N  
an amine group"]
    C["C  
a carboxylic acid"]

    D -- is --> X
    A -- is --> X
    A -- has --> E
    E -- is --> R
    X -- has --> R
    X -- has --> N
    X -- has --> C
  
```The idea of representing information in a graph is not new. For example the Resource Descriptive Framework (RDF) is a system for doing just that [CM]. The key difference between a category and a graph is the consideration of paths, and that two paths from  $A$  to  $B$  may be declared identical in a category (see [Spi3]). For example, we can further declare that in Diagram (1), the diagram

(2)



*commutes*, i.e. that the two paths  $A \rightsquigarrow R$  are equivalent, which can be translated as follows. Let  $A$  be a molecule of arginine. On the one hand  $A$ , being an amino acid, has a side chain; on the other hand  $A$  has an electrically-charged side-chain, which is of course a side chain. We seem to have associated *two* side-chains to  $A$ , but in fact they both refer to the same physical thing, the same side-chain. Thus, the two paths  $A \rightarrow R$  are deemed equivalent. The fact that this equivalence may seem trivial is not an indictment of the category idea but instead reinforces its importance — we must be able to indicate obvious facts within a given situation because what is obvious is the most essential.

While many situations can be modeled using basic ologs (categories), we often need to encode more structure. For this we will need so-called sketches. An olog will be defined as a finite limit, finite colimit sketch (see [BW2]), meaning we have the ability to encode objects (“types”), arrows (“aspects”), commutative diagrams (“facts”), as well as finite limits (“layouts”) and finite colimits (“groupings”).

Throughout this paper, whenever we refer to “the author” of an olog we are referring to the fictitious person who created it. We will refer to ourselves, David Spivak and Robert Kent, as “we” so as not to confuse things.

*Warning 1.0.1.* The author of an olog has a world-view, some fragment of which is captured in the olog. When person A examines the olog of person B, person A may or may not “agree with it.” For example, person B may have the following olog

which associates to each marriage a man and a woman. Person A may take the position that some marriages involve two men or two women, and thus see B’s olog as “wrong.” Such disputes are not “problems” with either A’s olog or B’s olog, they are discrepancies between world-views. Hence, throughout this paper, a reader R may see a displayed olog and notice a discrepancy between R’s world-view and our own, but R should not worry that this is a problem. This is not to say that ologs need not follow rules, but instead that the rules are enforced to ensure that an olog is structurally sound, rather than that it “correctly reflects reality,” whatever that may mean.

**1.1. Plan of this paper.** In this paper, we will define ologs and give several examples. We will state some rules of “good practice” which help one to author ologsthat are meaningful to others and easily extendable. We will begin in Section 2 by laying out the basics: types as objects, aspects as arrows, and facts as commutative diagrams. In Section 3, we will explain how to attach “instance” data to an olog and hence realize ologs as database schemas. In Section 4, we will discuss meaningful constraints between ologs that allow us to develop a higher-dimensional web of information called an information system, and we will discuss how the various parts of such a system interact via information channels. In Sections 5 and 6, we will extend the olog definition language to include “layouts” and “groupings”, which make for more expressive ologs; we will also describe two applications, one which explicates the computation of the factorial function, and the other which defines a notion from pure mathematics (that of pseudo-metric spaces). Finally, in Section 7, we will discuss some possible directions for future research.

For the remainder of the present section, we will explain how ologs relate to existing ideas in the field of knowledge representation.

**1.2. The semantic advantage of ologs: modularity.** The difference between ologs and prose is modularity: small conceptual pieces can form large ideas, and these pieces work best when they are reusable. The same phenomenon is true throughout computer science and mathematics. In programming languages, modularity brings not only vast efficiency to the writing of programs but enables an “abstraction barrier” that keeps the ideas clean. In mathematics, the most powerful results are often simple lemmas that are reusable in a wide variety of circumstances.

Web pages that consist of prose writing are often referred to as *information silos*. The idea is that a silo is a “big tube of stuff” which is not organized in any real way. Links between web pages provide some structure, but such a link does not carry with it a precise method to correlate the information within the two pages. Similarly in science, one author may reference another paper, but such a reference carries very little structure — it just points to a silo.

Ologs can be connected with links which are much richer than the link between two silos could possibly be. Individual concepts and connections within one olog can be “functorially aligned” with concepts and connections in another. A functor creates a precise connection between the work of one author and the work of another so that the precise nature of the comparison is not left to the reader’s imagination but explicitly specified. The ability to incorporate mathematical precision into the sharing of ideas is a central feature of ologs.

**1.3. Relation to other models.** There are many languages for knowledge representation (KR). For example, there are database languages such as SQL, ontology languages such as RDF and OWL, the language of Semantic Nets, and others (see [Bor]). One may ask what makes the olog concept different or better than the others.

The first response is that ologs are closely related to the above ideas. Indeed, all of these KR models can be “categorified” (i.e. phrased in the language of category theory) and related by functors, so that many of the ideas align and can be transferred between the different systems. In fact, as we will make clear in Section 3, ologs are almost identical to the categorical model of databases presented in [Spi2].

However, ologs have advantages over many existing KR models. The first advantage arises from the notion of commutative diagrams (which allow us to equate different paths through the domain, see Section 2.3) and of limits and colimits(which allow us to lay out and group things, see Sections 5 and 6). The additional expressivity of ologs give them a certain semantic clarity and interoperability that cannot be achieved with graphs and networks in the usual sense. The second advantage arises from the notion of olog morphisms, which allow the definition of meaningful constraints between ologs. With this in hand, we can integrate a set of similar ologs into a single information system, and go on to define information fusion. This will be discussed further Section 4.

In the remainder of this section we will provide a few more details on the relationship between ologs and each of the above KR models: databases, RDF/OWL, and semantic nets. The reader who does not know or care much about other systems of knowledge representation can skip to Section 1.4.

**1.3.1. *Ologs and Databases.*** A database is a system of tables, each table of which consists of a header of columns and a set of rows. A table represents a type of thing  $T$ , each column represents an attribute of  $T$ , and each row represents an example of  $T$ . An attribute is itself a “type of thing”, so each column of a table points to another table.

The relationship between ologs and databases is that every box  $B$  in an olog represents a type of thing and every arrow  $B \rightarrow X$  emanating from  $B$  represents an attribute of  $B$  (whose results are of type  $X$ ). Thus the boxes and arrows in an olog correspond to tables and their columns in a database. The rows of each table in a database will correspond to “instances” of each type in an olog. Again, this will be made more clear in Section 3 or one can see [Spi2] or [Ken5].

The point is that every olog can serve as a database schema, and the schemas represented by ologs range from simple (just objects and arrows) to complex (including commutative diagrams, products, sums, etc.). However, whereas database schemas are often prescriptive (“you must put your data into this format!”), ologs are usually descriptive (“this is how I see things”). One can think of an olog as an interface between people and databases: an olog is human readable, but it is also easily converted to a database schema upon which powerful applications can be put to work. Of course, if one is to use an olog as a database schema, it will become prescriptive. However, since the intention of each object and arrow is well-documented (as its label), schema evolution would be straightforward. Moreover, the categorical structure of ologs allows for *functorial data migration* by which one can transfer the instance data from an older schema to the current one (see [Spi2]).

**1.3.2. *Ologs and RDF / OWL.*** In [Spi2], the first author explained how a categorical database can be converted into an RDF triple store using the Grothendieck construction. The main difference between a categorical database schema (or an olog) and an RDF schema is that one cannot specify commutativity in an RDF schema. Thus one cannot express things like “the woman parent of a person  $x$  is the mother of  $x$ .” Without this expressivity, it is hard to enforce much rigor, and thus RDF data tends to be too loose for many applications.

OWL schemas, on the other hand, can express many more constraints on classes and properties. We have not yet explored the connection, nor compared the expressive power, of ologs and OWL. However, they are significantly different systems, most obviously in that OWL relies on logic where ologs rely on category theory.

**1.3.3. *Semantic Nets.*** On the surface, ologs look the most like semantic networks, or concept webs, but there are important differences between the two notions. First,arrows in a semantic network need not indicate functions; they can be relations. So there could be an arrow  $\ulcorner \text{a father} \urcorner \xrightarrow{\text{has}} \ulcorner \text{a child} \urcorner$  in a semantic network, but not in an olog (see Section 2.2.3 for how the same idea is expressible in an olog). There is a nice category of sets and relations, often denoted **Rel**, but this category is harder to reason about than is the ordinary category of sets and functions (often denoted **Set**). Thus, as mentioned above, semantic networks are categorifiable (using **Rel**), but this underlying formalism does not appear to play a part in the study or use of semantic networks. However, some attempt to integrate category theory and neural nets has been made, see [HC].

Moreover, commutative diagrams and other expressive abilities held by ologs are not generally part of the semantic network concept (see [Sow1]). For these reasons, semantic networks tend to be brittle: minor changes can have devastating effects. For example, if two semantic networks are somehow synced up and then one is changed, the linkage must be revised or may be altogether broken. Such a disaster is often avoided if one uses categories: because different paths can be equivalent, one can simply add new ideas (types and aspects) without changing the semantic meaning of what was already there. As section 4.4 demonstrates with an extended example, conceptual graphs, which are a popular formalism for semantics nets, can be linearized to ologs, thereby gaining in precision and expressibility.

#### 1.4. Acknowledgements.

1.4.1. *David Spivak's acknowledgments.* I would like to thank Mathieu Anel and Henrik Forssell for many pleasant and quite useful conversations. I would also like to thank Micha Breakstone for his help on understanding the relationship between ologs and linguistics. Finally I would like to thank Dave Balaban for helpful suggestions on this document itself.

1.4.2. *Robert Kent's acknowledgments.* I would like to thank the participants in the Standard Upper Ontology working group for many interesting, spirited, rewarding and enlightening discussions about knowledge representation in general and ontologies in particular; I especially want to thank Leo Obrst, Marco Schorlemmer and John Sowa from that group. I want to thank Jon Barwise for leading the development of the theory of information flow. I want to thank Joseph Goguen for leading the development of the theory of institutions, and for pointing out the common approach to knowledge representation used by both the Information Flow Framework and the theory of institutions.

## 2. TYPES, ASPECTS, AND FACTS

In this section we will explain basic ologs, which involve types, aspects, and facts. A basic olog is a category in which each object and arrow has been labeled by text; throughout this paper we will assume that text to be written in English.

The purpose of this section is to show how one can convert a real-world situation into an olog. It is probably impossible to explain this process precisely in words. Instead, we will explain mainly by example. We will give “rules of good practice” that lead to good ologs. While these rules are not strictly necessary, they help to ensure that the olog is properly formulated. As the Dalai Lama says, “Learn the rules so you know how to break them properly.”2.1. **Types.** A type is an abstract concept, a distinction the author has made. We represent each type as a box containing a *singular indefinite noun phrase*. Each of the following four boxes is a type:

(3)

<table style="width: 100%; text-align: center;">
<tr>
<td style="border: 1px solid black; padding: 5px;">a man</td>
<td style="border: 1px solid black; padding: 5px;">an automobile</td>
</tr>
<tr>
<td style="border: 1px solid black; padding: 5px;">a pair <math>(a, w)</math>, where <math>w</math> is a woman and <math>a</math> is an automobile</td>
<td style="border: 1px solid black; padding: 5px;">a pair <math>(a, w)</math> where <math>w</math> is a woman and <math>a</math> is a blue automobile owned by <math>w</math></td>
</tr>
</table>

Each of the four boxes in (3) represents a type of thing, a whole class of things, and the label on that box is what one should call *each example* of that class. Thus ‘a man’ does not represent a single man, but the set of men, each example of which is called “a man”<sup>1</sup>. Similarly, the bottom right-hand box in (3) represents an abstract type of thing, which probably has more than a million examples, but the label on the box indicates a common name for each such example.

Typographical problems emerge when writing a text-box in a line of text, e.g. the text-box a man seems out of place here, and the more in-line text-boxes one has in a given paragraph, the worse it gets. To remedy this, we will denote types which occur in a line of text with corner-symbols, e.g. we will write ‘a man’ instead of a man.

2.1.1. *Types with compound structures.* Many types have compound structures; i.e. they are composed of smaller units. Examples include

(4)

<table style="width: 100%; text-align: center;">
<tr>
<td style="border: 1px solid black; padding: 5px;">a man and a woman</td>
<td style="border: 1px solid black; padding: 5px;">a food <math>f</math> and a child <math>c</math> such that <math>c</math> ate all of <math>f</math></td>
<td style="border: 1px solid black; padding: 5px;">a triple <math>(p, a, j)</math> where <math>p</math> is a paper, <math>a</math> is an author of <math>p</math>, and <math>j</math> is a journal in which <math>p</math> was published</td>
</tr>
</table>

It is good practice to declare the variables in a “compound type”, as we did in the last two cases of (4). In other words, it is preferable to replace the first box above with something like

<table style="width: 100%; text-align: center;">
<tr>
<td style="border: 1px solid black; padding: 5px;">a man <math>m</math> and a woman <math>w</math></td>
<td style="padding: 0 10px;">or</td>
<td style="border: 1px solid black; padding: 5px;">a pair <math>(m, w)</math> where <math>m</math> is a man and <math>w</math> is a woman</td>
</tr>
</table>

so that the variables  $(m, w)$  are clear.

*Rules of good practice 2.1.1.* A type is presented as a text box. The text in that box should

- (i) begin with the word “a” or “an”;
- (ii) refer to a distinction made and recognizable by the author;
- (iii) refer to a distinction for which instances can be documented;
- (iv) not end in a punctuation mark;

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<sup>1</sup>In other words, types in ologs are intentional, rather than extensional — the label on a type describes its intention. The extension of a type will be captured by *instance data*; see Section 3.(v) declare all variables in a compound structure.

The first, second, and third rules ensure that the class of things represented by each box appears to the author as a well-defined set; see Section 3 for more details. The fourth and fifth rules encourage good “readability” of arrows, as will be discussed next in Section 2.2.

We will not always follow the rules of good practice throughout this document. We think of these rules being followed “in the background” but that we have “nick-named” various boxes. So ‘Steve’ may stand as a nickname for ‘a thing classified as Steve’ and ‘arginine’ as a nickname for ‘a molecule of arginine’.

**2.2. Aspects.** An aspect of a thing  $x$  is a way of viewing it, a particular way in which  $x$  can be regarded or measured. For example, a woman can be regarded as a person; hence “being a person” is an aspect of a woman. A man has a height (say, taken in inches), so “having a height (in inches)” is an aspect of a man. In an olog, an aspect of  $A$  is represented by an arrow  $A \rightarrow B$ , where  $B$  is the set of possible “answers” or results of the measurement. For example when observing the height of a man, the set of possible results is the set of integers, or perhaps the set of integers between 20 and 120.

(5)

(6)

We will formalize the notion of aspect by saying that aspects are functional relationships.<sup>2</sup> Suppose we wish to say that a thing classified as  $X$  has an aspect  $f$  whose result set is  $Y$ . This means there is a functional relationship called  $f$  between  $X$  and  $Y$ , which can be denoted  $f: X \rightarrow Y$ . We call  $X$  the *domain of definition* for the aspect  $f$ , and we call  $Y$  the *set of result values* for  $f$ . For example, a man has a height in inches whose result is an integer, and we could denote this by  $h: M \rightarrow \mathbf{Int}$ . Here,  $M$  is the domain of definition for height and  $\mathbf{Int}$  is the set of result values.

A set may always be drawn as a blob with dots in it. If  $X$  and  $Y$  are two sets, then a *a function from  $X$  to  $Y$* , denoted  $f: X \rightarrow Y$  can be presented by drawing arrows from dots in blob  $X$  to dots in blob  $Y$ . There are two rules:

- (i) each arrow must emanate *from* a dot in  $X$  and point *to* a dot in  $Y$ ;
- (ii) each dot in  $X$  must have precisely *one* arrow emanating from it.

Given an element  $x \in X$ , the arrow emanating from it points to some element  $y \in Y$ , which we call *the image of  $x$  under  $f$*  and denote  $f(x) = y$ .

Again, in an olog, an aspect of a thing  $X$  is drawn as a labeled arrow pointing from  $X$  to a “set of result values.” Let us concentrate briefly on the arrow in (5). The domain of definition is the set of women (a set with perhaps 3 billion elements); the set of result values is the set of persons (a set with perhaps 6 billion elements). We can imagine drawing an arrow from each dot in the “woman” set to a unique dot in the “person” set. No woman points to two different people, nor to zero people — each woman is exactly one person — so the rules for a functional relationship are satisfied. Let us now concentrate briefly on the arrow in (6). The

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<sup>2</sup>In type theory, what we here call aspects are called *functions*. Since our types are not fixed sets (see Section 3), we preferred a term that was less formal.domain of definition is the set of men, the set of result values is the set of integers  $\{20, 21, 22, \dots, 119, 120\}$ . We can imagine drawing an arrow from each dot in the “man” set to a single dot in the “integer” set. No man points to two different heights, nor can a man have no height: each man has exactly one height. Note however that two different men can point to the same height.

2.2.1. *Invalid aspects.* We tried above to clarify what it is that makes an aspect “valid”, namely that it must be a “functional relationship.” In this subsection we will present two arrows which on their face may appear to be aspects, but which on closer inspection are not functional (and hence are not valid as aspects).

Consider the following two arrows:

(7\*) 
$$\boxed{{\displaystyle \text{a person}}}\stackrel{\text{has}}{\to }\boxed{{\displaystyle \text{a child}}}\backslash boxed\{\backslash text\{a\; person\}\}\; \backslash xrightarrow\{\backslash text\{has\}\}\; \backslash boxed\{\backslash text\{a\; child\}\}$$

(8\*) 
$$\boxed{{\displaystyle \text{a mechanical pencil}}}\stackrel{\text{uses}}{\to }\boxed{{\displaystyle \text{a piece of lead}}}\backslash boxed\{\backslash text\{a\; mechanical\; pencil\}\}\; \backslash xrightarrow\{\backslash text\{uses\}\}\; \backslash boxed\{\backslash text\{a\; piece\; of\; lead\}\}$$

A person may have no children or may have more than one child, so the first arrow is invalid: it is not functional because it does not satisfy rule (2) above. Similarly, if we drew an arrow from each mechanical pencil to each piece of lead it uses, it would not satisfy rule (2) above. Thus neither of these is a valid aspect.

Of course, in keeping with Warning 1.0.1, the above arrows may not be wrong but simply reflect that the author has a strange world-view or a strange vocabulary. Maybe the author believes that every mechanical pencil uses exactly one piece of lead. If this is so, then  $\ulcorner \text{a mechanical pencil} \urcorner \xrightarrow{\text{uses}} \ulcorner \text{a piece of lead} \urcorner$  is indeed a valid aspect! Similarly, suppose the author meant to say that each person *was once* a child, or that a person has an inner child. Since every person has one and only one inner child (according to the author), the map  $\ulcorner \text{a person} \urcorner \xrightarrow{\text{has as inner child}} \ulcorner \text{a child} \urcorner$  is a valid aspect. We cannot fault the author for such a view, but note that we have changed the name of the label to make its intention more explicit.

2.2.2. *Reading aspects and paths as English phrases.* Each arrow (aspect)  $X \xrightarrow{f} Y$  can be read by first reading the label on its source box (domain of definition)  $X$ , then the label on the arrow  $f$ , and finally the label on its target box (set of result values)  $Y$ . For example, the arrow

(9) 
$$\boxed{{\displaystyle \text{a book}}}\stackrel{\text{has as first author}}{\to }\boxed{{\displaystyle \text{a person}}}\backslash boxed\{\backslash text\{a\; book\}\}\; \backslash xrightarrow\{\backslash text\{has\; as\; first\; author\}\}\; \backslash boxed\{\backslash text\{a\; person\}\}$$

is read “a book has as first author a person”, a valid English sentence.

Sometimes the label on an arrow can be shortened or dropped altogether if it is obvious from context. We will discuss this more in Section 2.3 but here is acommon example from the way we write ologs.

(10)

Neither arrow is readable by the protocol given above (e.g. “a pair  $(x, y)$  where  $x$  and  $y$  are integers  $x$  an integer” is not an English sentence), and yet it is obvious what each map means. For example, given the pair  $(8, 11)$  which belongs in box  $A$ , application of arrow  $x$  would yield 8 in box  $B$ . The label  $x$  can be thought of as a nickname for the full name “yields, via the value of  $x$ ,” and similarly for  $y$ . We do not generally use the full name for fear that the olog would become cluttered with text.

One can also read paths through an olog by inserting the word “which” after each intermediate box. For example the following olog has two paths of length 3 (counting arrows in a chain):

(11)

The top path is read “a child is a person, which has as parents a pair  $(w, m)$  where  $w$  is a woman and  $m$  is a man, which yields, via the value of  $w$ , a woman.” The reader should read and understand the content of the bottom path.

**2.2.3. Converting non-functional relationships to aspects.** There are many relationships that are not functional, and these cannot be considered aspects. Often the word “has” indicates a relationship — sometimes it is functional as in “a person”  $\xrightarrow{\text{has}}$  “a stomach”, and sometimes it is not, as in “a father”  $\xrightarrow{\text{has}}$  “a child”. (Obviously, a father may have more than one child.) A quick fix would be to replace the latter by “a father”  $\xrightarrow{\text{has}}$  “a set of children”. This is ok, but the relationship between “a child” and “a set of children” then becomes an issue to deal with later. There is another way to indicate such “non-functional” relationships.

In mathematics, a relation between sets  $A_1, A_2$ , and so on through  $A_n$  is defined to be a subset of the Cartesian product

$$R\subseteq A_1\times A_2\times \cdots \times A_n\mathrm{.}R\; \backslash subseteq\; A\_1\; \backslash times\; A\_2\; \backslash times\; \backslash cdots\; \backslash times\; A\_n.$$The set  $R$  represents those sequences  $(a_1, a_2, \dots, a_n)$  that are so-related. In an olog, we represent this as follows

graph TD
    R[R] --> A1[A1]
    R --> A2[A2]
    R --> dots[...]
    R --> An[An]

For example,

graph TD
    subgraph R
        R["a sequence (p, a, j) where p is a paper, a is an author of p, and j is a journal in which p was published"]
    end
    R -- p --> A1["A1<br/>a paper"]
    R -- a --> A2["A2<br/>an author"]
    R -- j --> A3["A3<br/>a journal"]

Whereas  $A_1 \times A_2 \times A_3$  includes all possible triples  $(p, a, j)$  where  $a$  is a person,  $p$  is a paper, and  $j$  is a journal, it is obvious that not all such triples are found in  $R$ . Thus  $R$  represents a proper subset of  $A_1 \times A_2 \times A_3$ .

*Rules of good practice 2.2.1.* An aspect is presented as a labeled arrow, pointing from a source box to a target box. The arrow text should

- (i) begin with a verb;
- (ii) yield an English sentence, when the source-box text followed by the arrow text followed by the target-box text is read;
- (iii) refer to a functional dependence: each instance of the source type should give rise to a specific instance of the target type;

**2.3. Facts.** In this section we will discuss facts and their relationship to “path equivalences.” It is such path equivalences, which exist in categories but do not exist in graphs, that make category theory so powerful. See [Spi3] for details.Given an olog, the author may want to declare that two paths are equivalent. For example consider the two paths from  $A$  to  $C$  in the olog

(12)

graph LR
  A["A  

a person"] -- "has as parents" --> B["B
a pair (w, m)
where w is a woman
and m is a man"] A -- "has as mother" --> C["C
a woman"] B -- "w" --> C subgraph CommutativeRegion A B C end

We know as English speakers that a woman parent is called a mother, so these two paths  $A \rightarrow C$  should be equivalent. A more mathematical way to say this is that the triangle in Olog (12) *commutes*.

A *commutative diagram* is a graph with some declared path equivalences. In the example above we concisely say “a woman parent is equivalent to a mother.” We declare this by defining the diagonal map in (12) to be *the composition* of the horizontal map and the vertical map.

We generally prefer to indicate a commutative diagram by drawing a check-mark,  $\checkmark$ , in the region bounded by the two paths, as in Olog (12). Sometimes, however, one cannot do this unambiguously on the 2-dimensional page. In such a case we will indicate the commutative diagrams (fact) by writing an equation. For example to say that the diagram



commutes, we could either draw a checkmark inside the square or write the equation  $f; g = h; i$  above it. Either way, it means that “ $f$  then  $g$ ” is equivalent to “ $h$  then  $i$ ”.

2.3.1. *More complex facts.* Recording real-world facts in an olog can require some creativity. Whereas a fact like “the brother of ones father is ones uncle” is recorded as a simple commutative diagram, others are not so simple. We will try to show the range of expressivity of commutative diagrams in the following two examples.*Example 2.3.2.* How would one record a fact like “a truck weighs more than a car”? We suggest something like this:

where both top and bottom commute. This olog exemplifies the fact that simple sentences sometimes contain large amounts of information. While the long map may seem to suffice to convey the idea “a truck weighs more than a car,” the path equivalences (declared by check-marks) serve to ground the idea in more basic types. These other types tend to be useful for other purposes, both within the olog and when connecting it to others.

2.3.3. *Specific facts at the olog level.* Another fact one might wish to record is that “John Doe’s weight is 150 lbs.” This is established by declaring that the following diagram commutes:

(13)

If one only had the top line, it would be less obvious how to connect its information with that of other ologs. (See Section 4 for more on connecting different ologs).

Note that the top line in Diagram (13) might also be considered as existing at the “data level” rather than at the “olog level.” In other words, one could see John Doe as an “instance” of “a person”, rather than as a type in and of itself, and similarly see 150 as an instance of “a real number”. This idea of an olog having a “data level” is the subject of the Section 3.

*Rules of good practice 2.3.4.* A fact is the declaration that two paths (having the same source and target) in an olog are equivalent. Such a fact is either presented as a checkmark between the two paths (if such a check-mark is unambiguous) or by an equation. Every such equivalence should be declared; i.e. no fact should be considered too obvious to declare.### 3. INSTANCES

The reader at this point hopefully sees an olog as a kind of “concept map,” and it is one, albeit a concept map with a formal structure (implicitly coming from category theory) and specific rules of good practice. In this section we will show that one can also load an olog with data. Each type can be assigned a set of instances, each aspect will map the instances of one type to instances of the other, and each fact will equate two such mappings. We give examples of these ideas in Section 3.1.

In Section 3.2, we will show that in fact every olog can also serve as the layout for a database. In other words, given an olog one can immediately generate a *database schema*, i.e. a system of tables, in any reasonable data definition language such as that of SQL. The tables in this database will be in one-to-one correspondence with the types in the olog. The columns of a given table will be the aspects of the corresponding type, i.e. the arrows whose source is that type. Commutative diagrams in the olog will give constraints on the data.

In fact, this idea is the basic thesis in [Spi2], even though the word olog does not appear in that paper. There it was explained that a category  $\mathcal{C}$  naturally can be viewed as a database schema and that a functor  $I: \mathcal{C} \rightarrow \mathbf{Set}$ , where  $\mathbf{Set}$  is the category of sets, is a database state. Since an olog is a drawing of a category, it is also a drawing of a database schema. The current section is about the “states” of an olog, i.e. the kinds of data that can be captured by it.

**3.1. Instances of types, aspects, and facts.** Recall from Section 2 that basic ologs consist of types, displayed as boxes; aspects, displayed as arrows; and facts, displayed as equations or check-marks. In this section we discuss the instances of these three basic constructions. The rules of good practice (2.1.1, 2.2.1, and 2.3.4) were specifically designed to simplify the process of finding instances.

**3.1.1. Instances of types.** According to Rules 2.1.1, each box in an olog contains text which should refer to a **distinction made and recognizable by the author for which instances can be documented**. For example if my olog contains a box

$$(14)\boxed{{\displaystyle \begin{array}{l}\text{a pair }(p,c)\text{ where }p\\ \text{is a person, }c\text{ is a cat,}\\ \text{and }p\text{ has petted }c\end{array}}}(14)\; \backslash quad\; \backslash boxed\{\backslash begin\{array\}\{l\}\; \backslash text\{a\; pair\; \}\; (p,\; c)\; \backslash text\{\; where\; \}\; p\; \backslash \backslash \; \backslash text\{is\; a\; person,\; \}\; c\; \backslash text\{\; is\; a\; cat,\}\; \backslash \backslash \; \backslash text\{and\; \}\; p\; \backslash text\{\; has\; petted\; \}\; c\; \backslash end\{array\}\}$$

then I must have some concept of when this situation occurs. Every time I witness a new person-cat petting, I document it. Whether this is done in my mind, in a ledger notebook, or on a computer does not matter; however using a computer would probably be the most self-explanatory. Imagine a computer program in which one can create ologs. Clicking a text box in an olog results in it “opening up” to show a list of documented instances of that type. If one is reading the CBS news olog and clicks on the box “an episode of 60 Minutes”, he or she should see a list of all episodes of the TV show “60 Minutes.” If we wish to document a new person-cat petting incident we click on the box in (14) and add this new instance.

**3.1.2. Instances of aspects.** According to Rules 2.2.1, each arrow in an olog should be labeled with text that refers to a functional relationship between the source box and the target box. A functional relationship  $f: A \rightarrow B$  between finite sets  $A$  and  $B$  can always be written as a 2-column table: the first column is filled with theinstances of type  $A$  and the second column is filled with their  $f$ -values, which are instances of type  $B$ .

For example, consider the aspect

(15) 
 A diagram showing a relationship between two types. On the left is a box labeled 'a moon'. An arrow labeled 'orbits' points from this box to a box on the right labeled 'a planet'.

We can document some instances of this relationship using the following table:

(16)

<table border="1">
<thead>
<tr>
<th colspan="2">orbits</th>
</tr>
<tr>
<th>a moon</th>
<th>a planet</th>
</tr>
</thead>
<tbody>
<tr>
<td>The Moon</td>
<td>Earth</td>
</tr>
<tr>
<td>Phobos</td>
<td>Mars</td>
</tr>
<tr>
<td>Deimos</td>
<td>Mars</td>
</tr>
<tr>
<td>Ganymede</td>
<td>Jupiter</td>
</tr>
<tr>
<td>Titan</td>
<td>Saturn</td>
</tr>
</tbody>
</table>

Clearly, this table of instances can be updated as more moons are discovered by the author (be it by telescope, conversation, or research).

The correspondence between aspect (15) and Table (16) makes it clear that ologs can serve to hold data which exemplifies the author's world-view. In Section 3.2, we will show that ologs (which have many aspects and facts) can serve as bona fide database schemas.

3.1.3. *Instances of facts.* Recall the following olog:

(12)

 A diagram showing three types and their relationships. 
 - Type A: 'a person' (in a box).
 - Type B: 'a pair (w, m) where w is a woman and m is a man' (in a box).
 - Type C: 'a woman' (in a box).
 - An arrow labeled 'has as parents' points from 'a person' to 'a pair (w, m)'.
 - An arrow labeled 'has as mother' points from 'a person' to 'a woman'.
 - A checkmark is placed near the 'has as mother' arrow.
 - A vertical arrow labeled 'w' points from 'a pair (w, m)' down to 'a woman'.

and consider the following instances of the three aspects in it:

(17)

<table border="1">
<thead>
<tr>
<th colspan="2">has as parents</th>
</tr>
<tr>
<th>a person</th>
<th>a pair <math>(w, m)</math> ...</th>
</tr>
</thead>
<tbody>
<tr>
<td>Cain</td>
<td>(Eve, Adam)</td>
</tr>
<tr>
<td>Abel</td>
<td>(Eve, Adam)</td>
</tr>
<tr>
<td>Chelsea</td>
<td>(Hillary, Bill)</td>
</tr>
</tbody>
</table>

<table border="1">
<thead>
<tr>
<th colspan="2"><math>w</math></th>
</tr>
<tr>
<th>a pair <math>(w, m)</math> ...</th>
<th>a woman</th>
</tr>
</thead>
<tbody>
<tr>
<td>(Eve, Adam)</td>
<td>Eve</td>
</tr>
<tr>
<td>(Hillary, Bill)</td>
<td>Hillary</td>
</tr>
<tr>
<td>(Margaret, Samuel)</td>
<td>Margaret</td>
</tr>
<tr>
<td>(Emily, Kris)</td>
<td>Emily</td>
</tr>
</tbody>
</table>

<table border="1">
<thead>
<tr>
<th colspan="2">has as mother</th>
</tr>
<tr>
<th>a person</th>
<th>a woman</th>
</tr>
</thead>
<tbody>
<tr>
<td>Cain</td>
<td>Eve</td>
</tr>
<tr>
<td>Abel</td>
<td>Eve</td>
</tr>
<tr>
<td>Chelsea</td>
<td>Hillary</td>
</tr>
</tbody>
</table>

When we declare that the diagram in (12) commutes (using the check-mark), we are saying that for every instance of 'a person' (of which we have three: Cain,Abel, and Chelsey), the two paths to  $\ulcorner \text{a woman} \urcorner$  give the same answers. Indeed, for Cain the two paths are:

- (i)  $\text{Cain} \mapsto (\text{Eve, Adam}) \mapsto \text{Eve}$ ;
- (ii)  $\text{Cain} \mapsto \text{Eve}$ ;

and these answers agree. If one changed any instance of the word “Eve” to the word “Steve” in one of the tables in (17), some pair of paths would fail to agree. Thus the “fact” that the diagram in (12) commutes ensures that there is some internal consistency between the meaning of parents and the meaning of mother, and this consistency must be born out at the instance level.

All of this will be formalized in Section 3.2.2.

**3.2. The relationship between ologs and databases.** Recall from Section 3.1.1 that we can imagine creating an olog on a computer. The user creates boxes, arrows, and compositions, hence creating a category  $\mathcal{C}$ . Each text-box  $x$  in the olog can be “clicked” by the computer mouse, an action which allows the user to “view the contents” of  $x$ . The result will be a set of things, which we might call  $I(x) \in \mathbf{Set}$ , whose elements are things of type  $x$ . So clicking on the box  $\ulcorner \text{a man} \urcorner$  one sees  $I(\ulcorner \text{a man} \urcorner)$ , the set of everything the author has documented as being a man. For each aspect  $f: x \rightarrow y$  of  $x$ , the user can see a function from the set  $I(x)$  to  $I(y)$ , perhaps as a 2-column table as in (17).

The type  $x$  may have many aspects, which we can put together into a single multi-column table. Its columns are the aspects of  $x$ , and its rows are the elements of  $I(x)$ . Consider the following olog, taken from [Spi2] where it was presented as a database schema.

(18)

The type  $\ulcorner \text{Employee} \urcorner$  has four aspects, namely **manager** (valued in  $\ulcorner \text{Employee} \urcorner$ ), **works in** (valued in  $\ulcorner \text{department} \urcorner$ ), and **first name** and **last name** (valued in  $\ulcorner \text{string} \urcorner$ ). As a database, each type together with its aspects form a multi-column table, as in the following example.

*Example 3.2.1.* We can convert Olog (18) into a database schema. Each box represents a table, each arrow out of a box represents a column of that table. Here is an example state of that database.<table border="1">
<thead>
<tr>
<th colspan="5">employee</th>
<th>string</th>
</tr>
<tr>
<th>Id</th>
<th>first name</th>
<th>last name</th>
<th>manager</th>
<th>works in</th>
<th>Id</th>
</tr>
</thead>
<tbody>
<tr>
<td>101</td>
<td>David</td>
<td>Hilbert</td>
<td>103</td>
<td>q10</td>
<td>a</td>
</tr>
<tr>
<td>102</td>
<td>Bertrand</td>
<td>Russell</td>
<td>102</td>
<td>x02</td>
<td>b</td>
</tr>
<tr>
<td>103</td>
<td>Alan</td>
<td>Turing</td>
<td>103</td>
<td>q10</td>
<td>⋮</td>
</tr>
</tbody>
</table>

  

<table border="1">
<thead>
<tr>
<th colspan="3">department</th>
</tr>
<tr>
<th>Id</th>
<th>name</th>
<th>secretary</th>
</tr>
</thead>
<tbody>
<tr>
<td>q10</td>
<td>Sales</td>
<td>101</td>
</tr>
<tr>
<td>x02</td>
<td>Production</td>
<td>102</td>
</tr>
</tbody>
</table>

  

<table border="1">
<tbody>
<tr>
<td>z</td>
</tr>
<tr>
<td>aa</td>
</tr>
<tr>
<td>ab</td>
</tr>
<tr>
<td>⋮</td>
</tr>
</tbody>
</table>

Note that every arrow  $f: x \rightarrow y$  of Olog (18) is represented in Database (19) as a column of table  $x$ , and that every cell in that column can be found in the Id column of table  $y$ . For example, every cell in the “works in” column of table **employee** can be found in the Id column of table **department**.

The point is that ologs can be drawn to represent a world-view (as in Section 2), but they can also store data. Rules 1,2, and 3 in 2.1.1 align the construction of an olog with the ability to document instances for each of its types.

**3.2.2. Instance data as a set-valued functor.** Let  $\mathcal{C}$  be an olog. Section 3 so far has described instances of types, aspects, and facts and how all of these come together into a set of interconnected tables. The assignment of a set of instances to each type and a function to each aspect in  $\mathcal{C}$ , such that the declared facts hold, is called an assignment of *instance data* for  $\mathcal{C}$ . More precisely, instance data on  $\mathcal{C}$  is a functor  $\mathcal{C} \rightarrow \mathbf{Set}$ , as in Definition 3.2.3.

**Definition 3.2.3.** Let  $\mathcal{C}$  be a category (olog) with underlying graph  $|\mathcal{C}|$ , and let  $\mathbf{Set}$  denote the category of sets. An *instance* of  $\mathcal{C}$  (or an *assignment of instance data for  $\mathcal{C}$* ) is a functor  $I: \mathcal{C} \rightarrow \mathbf{Set}$ . That is, it consists of

- • a set  $I(x)$  for each object (type)  $x$  in  $\mathcal{C}$ ,
- • a function  $I(f): I(x) \rightarrow I(y)$  for each arrow (aspect)  $f: x \rightarrow y$  in  $\mathcal{C}$ , and
- • for each fact (path-equivalence or equation) <sup>3</sup>

$$f_1;f_2;\dots ;f_n=f_1^{\mathrm{\prime }};f_2^{\mathrm{\prime }};\dots ;f_m^{\mathrm{\prime }}f\_1;\; f\_2;\; \backslash dots;\; f\_n\; =\; f\text{'}\_1;\; f\text{'}\_2;\; \backslash dots;\; f\text{'}\_m$$

declared in  $\mathcal{C}$ , an equality of functions

$$I(f_1);I(f_2);\dots ;I(f_n)=I(f_1^{\mathrm{\prime }});I(f_2^{\mathrm{\prime }});\dots ;I(f_m^{\mathrm{\prime }})\mathrm{.}I(f\_1);\; I(f\_2);\; \backslash dots;\; I(f\_n)\; =\; I(f\text{'}\_1);\; I(f\text{'}\_2);\; \backslash dots;\; I(f\text{'}\_m).$$

For more on this viewpoint of categories and functors, the reader can consult [Spi3].

#### 4. COMMUNICATION BETWEEN OLOGS

The world is inherently heterogeneous. Different individuals <sup>4</sup> in the world naturally have different world-views — each individual has its own perspective on the

<sup>3</sup>If we let  $f = f_1; f_2; \dots; f_n$  and  $f' = f'_1; f'_2; \dots; f'_m$ , then we often write  $(f = f'): i \rightarrow j$  to denote the fact that these paths are equivalent.

<sup>4</sup>By an individual we mean either an individual person acting on their own, a community acting as a single entity, a software agent, etc. Later in this section we will use the notion of a community acting as a distributed collection of linked, yet independent, individuals.world. The conceptual knowledge (information resources) of an individual represents its world-view, and is encoded in an ontology log, or olog, containing the concepts, relations, and observations that are important to that individual. An olog is a formal specification of an individual's world-view in a language representing the concepts and relationships used by that individual. In addition to the formulation of an expressive language, a specification needs to contain axioms (facts) that constrain the possible interpretations of that language.

Since the ologs of different individuals are encoded in different languages, the important need to merge disparate ologs into a more general representation is difficult, time-consuming and expensive. The solution is to develop appropriate communication between individuals to allow interoperability of their ologs. Communication can occur between individuals when there is some commonality between their world-views. It is this commonality that allows one individual to benefit from the knowledge and experience of another. In this section we will discuss how to formulate these channels of communication, thereby describing a generalized and practical technique for merging ologs.

The mathematical concept that makes it all work is that of a functor. A functor is a mapping from one category to another that preserves all the declared structure. Whereas in Definition 3.2.3 we defined a functor from an olog to **Set**, here we will be discussing functors from one olog to another.

Suppose we have two ologs,  $\mathcal{C}$  and  $\mathcal{D}$ , that represent the world-views of two individuals. A functor  $F: \mathcal{C} \rightarrow \mathcal{D}$  is basically a way of matching each type (box) of  $\mathcal{C}$  to a type of  $\mathcal{D}$ , and each aspect (arrow) in  $\mathcal{C}$  to an aspect (or path of aspects) in  $\mathcal{D}$ . Once ologs are aligned in this way, communication can occur: the two individuals know what each other is talking about. In fact, mathematically we can show that instance data held in  $\mathcal{C}$  can be transformed (in coherent ways) to instance data held in  $\mathcal{D}$ , and vice versa (see [Spi2]). In simple terms, once individuals understand each other in a certain domain (be it social, mathematical, etc.), they can communicate their views about it.

While the basic idea is not hard, the details can be a bit technical. This section is written in a more formal and logical style, and is decidedly more difficult than the others. For this section only, we assume the reader is familiar with the notion of fibered categories, colimits in the category **Cat** of categories, etc. We return to our more informal style in Section 5, where we discuss how an individual can author a more expressive olog.

**4.1. Categories and their presentations.** We never defined categories in this paper, but we defined ologs and said that the two notions amounted to the same thing. Thus, we implied that a category consists of the following: a set of objects, a set of arrows (each pointing from one object to another), and a congruence relation on paths.<sup>5</sup> This differs from the standard definition of categories (see [Mac]), which replaces our congruence relation with a composition rule and associativity law (obtained by taking the categorical quotient). One could say that an olog is a presentation of a category by generators (objects and arrows) and relations (path congruences). Any category can be resolved and presented in such a way, which we

---

<sup>5</sup>A congruence relation on paths is an equivalence relation on paths that respects endpoints and is closed under composition from left and right (see the axioms in 20).will call a specification. Likewise any functor can be resolved and presented as a morphism between specifications.<sup>6</sup>

In fact, this presentation form for categories (and the analogous one for functors) is preferable for our work on communication between ologs, because it separates the strictly graphical part of an olog (its types and aspects, regarded as the olog language) from the propositional part (its facts, regarded as the olog formalism). This presentation form is standard in the institutions [GB] and information flow [BS] communities, since it separates the mechanism of flow from the content of flow; in this case the formal content. Our work here applies the general theories of institutions and information flow to the specific logical system that underlies categories and functors,<sup>7</sup> demonstrating how this logical system can be used for knowledge representation. Using the presentation forms for categories and functors, we show how communication between individuals is effected by the flow of information along channels.

**4.2. The architecture underlying information systems.** We think of a community of people, businesses, etc. in terms of the ologs of each individual participant together with the information channels that connect them. These channels are functors between ologs, which allow communication to occur. The heterogeneity of multiple differing world-views connected through such links can lead to a flexibility and robustness of interaction. For example, heterogeneity allows for multiple schemas to be employed in the design of database systems in particular, and multiple languages to be employed in the design of knowledge representation systems in general.

For any olog, consider the underlying graph of types and aspects. We regard this graph as being the language of the olog,<sup>8</sup> with the facts of the olog being a subset of all the possible assertions that one can make within this language. Any two ologs with the same underlying graph of types and aspects have the same language, and since the facts of each olog are expressed in the same language, they can be “understood” by each other without translation. As such, we think of the collection of all ologs with the same language (underlying graph) as forming a homogeneous *context*, with the ologs ordered in a specialization-generalization hierarchy.

Whereas an olog represents (the world-view of) a single individual, an information system (of ologs) represents a community of separate, independent and distributed individuals. Here we consider an information system to be a diagram of ologs of some shape  $\mathbf{I}$ ; that is, a collection of ologs and constraints indexed by a base category  $\mathbf{I}$ . The parts of the system represent either the ologs of the various individuals in the system or common grounds needed for communication between the individuals. Each part of the system specifies its world-view as facts expressed in terms of its language. The system is heterogeneous, since each part has a separate language for the expression of its world-view. The morphisms between the parts are the alignment (constraint) links defining the common grounds.

---

<sup>6</sup>We take an agnostic approach to foundations here. With the presentation form, we show how categories and functors are definable in terms of sets and functions, indicating how category theoretic concepts could be defined in terms of set theory. However, we fully understand that **Set**, the category of sets and functions, is but one example of a topos, indicating how set theoretic concepts could be defined in terms of category theory.

<sup>7</sup>For the expert, this refers to the sketch logical system **Sk**, in its various manifestations.

<sup>8</sup>Section 4.4 indicates how natural languages can be encoded into ologs.As will be made clear in a moment, there is an underlying distributed system consisting of the language (underlying graph) for each component part of the information system and a translation (graph morphism) for each alignment link. We can think of this distributed system as an underlying system of languages linked by translating dictionaries. This distributed system determines an information channel with core language (graph) and component translation links (graph morphisms) along which the specifications of each component part can flow to the core. We can think of this core as a universal language for the whole system and the channel as a translation mechanism from parts to whole. At the core, the direct flow of the component specifications are joined together (unioned) and allowed to interact through entailment. The result of this interaction can then be distributed back to the component parts, thereby allowing the separate parts of an information system to interoperate.

In this section, we will make all this clear and rigorous. As mentioned above, we will work with category presentations (here called *specifications*) rather than categories. We will discuss the homogeneous contexts called fibers in detail and give the axioms of satisfaction. We will then discuss how morphisms between graphs (the translating dictionaries between the ologs) allow for direct and inverse information flow between these homogeneous fiber contexts. Finally, we discuss specifications (also known as *theories*) and the lattice of theories construction for ontologies.

In Section 4.3 we will discuss how the information in ologs can be aligned by the use of common grounds. This alignment will result in the creation of *information systems*, which are systems of ologs connected together along functors. We will discuss how to take the information contained in each olog of a heterogeneous system and integrate it all into a single whole, called the fusion olog. Finally we will discuss how the consequence of bringing all this information together, and allowing it to interact, can be transferred back to each part of the system (individual olog) as a set of local facts entailed by remote ologs, allowing for a kind of interoperability between ologs. In Section 4.4 we will discuss conceptual graphs and their relationship to ologs.

**4.2.1. Fibers.** A graph  $G$  contains types as nodes and aspects as edges. The graphs underlying an olog is considered its *language*. Any category  $\mathcal{C}$  has an underlying graph  $|\mathcal{C}|$ . In particular,  $|\mathbf{Set}|$  is the graph underlying the category of sets and functions. Olog (12) has an underlying graph containing the three types  $\ulcorner \text{person} \urcorner$ ,  $\ulcorner \text{person-pair} \urcorner$  and  $\ulcorner \text{woman} \urcorner$  and the three aspects ‘has a parent’, ‘woman’ and ‘has as mother’. Olog (17) has an underlying graph containing the three types  $\ulcorner \text{employee} \urcorner$ ,  $\ulcorner \text{department} \urcorner$ , and  $\ulcorner \text{string} \urcorner$  and the six aspects ‘manager’, ‘works in’, ‘secretary’, ‘name’, ‘first name’ and ‘last name’. Let  $\mathbf{eqn}(G)$  denote the set of all facts (equations) that are possible to express using the types and aspects of  $G$ . A  $G$ -specification is a set  $E \subseteq \mathbf{eqn}(G)$  consisting of some of the facts expressible in  $G$ . The singleton set with the one fact that “the female parent of a person is his/her mother” is a specification for the graph of Olog (12). The set with the two facts that “the manager has the same department as any employee” and “the secretary of a department is an employee in that department” is a specification for the graph of Olog (17). Let  $\mathbf{spec}(G)$  denote the collection of all  $G$ -specifications ordered by inclusion  $E_1 \subseteq E_2$ .4.2.2. *Satisfaction.* It will be useful here to define an instance of a graph  $G$ , instead of an instance of a category  $\mathcal{C}$ . An instance of a graph populates the graph by assigning instance data to it. An instance of a graph  $G$  is a graph morphism  $D: G \rightarrow |\mathbf{Set}|$  mapping each type  $x$  in  $G$  to a set  $D(x)$  of instances and mapping each aspect  $e: x \rightarrow y$  in  $G$  to an instance function  $D(e): D(x) \rightarrow D(y)$ . Using database terminology, we also call  $D$  a key diagram, since it gives the set of row identifiers (primary keys) of tables and the cell contents defined by key maps.

A key diagram  $D: G \rightarrow |\mathbf{Set}|$  satisfies (is a model of) a  $G$ -fact  $\epsilon \in \mathbf{eqn}(G)$  (see Definition 3.2.3), symbolized  $D \models_G \epsilon$ , when we have an equality of functions  $D^*(\epsilon_0) = D^*(\epsilon_1)$ . We also say that  $\epsilon$  (holds in) is true when interpreted in  $D$ . An identity  $(f =_G f): i \rightarrow j$  holds in all key diagrams (hence, is a tautology), and vice-versa for any set  $A \in |\mathbf{Set}|$  a constant key diagram  $\Delta(A): G \rightarrow |\mathbf{Set}|$  satisfies any fact  $\epsilon \in \mathbf{eqn}(G)$ . A key diagram  $D: G \rightarrow |\mathbf{Set}|$  satisfies (is a model of) a  $G$ -specification  $E$ , symbolized  $D \models_G E$ , when it satisfies every fact in the specification. For any graph  $G$ , a  $G$ -specification  $E$  entails a  $G$ -fact  $\epsilon$ , denoted by  $E \vdash_G \epsilon$ , when any model of the specification satisfies the fact. The consequence  $E^\bullet$  of a  $G$ -specification  $E$  is the set of all entailed equations. The consequence operator  $(-)^\bullet$  is a closure operator, and the consequence of a specification is a congruence. For any  $G$ -specification  $E$ , entailment satisfies the following axioms.



These are converted to inference rules in Table 1. To construct  $E^\bullet$ , we first take the reflexive, symmetric, and transitive closure  $E^*$  of  $E$  (so that  $E^*$  is a  $G$ -specification and also the smallest equivalence relation containing  $E$ ), and then we get  $E^\bullet$  by closing up under composition on left and right. We extend specification inclusion with the entailment order, where  $E_1 \leq_G E_2$  when  $E_1$  entails each equation in  $E_2$ ; that is, when  $E_1^\bullet \supseteq E_2$  or equivalently when  $E_1^\bullet \supseteq E_2^\bullet$ . The statement “ $E_1 \leq_G E_2$ ” asserts that  $E_1$  is at least as specialized as  $E_2$ . The entailment order  $\langle \mathbf{spec}(G), \leq_G \rangle$ , which is a specialization-generalization order, represents a local version of the “lattice of theories” construction of Sowa [Sow2] (see Section 4.2.5). The opposite entailment order  $\mathbf{fbr}(G) = \langle \mathbf{spec}(G), \geq_G \rangle$  is called the fiber order.<sup>9</sup> In the lattice<sup>10</sup>  $\mathbf{spec}(G)$ , the meet is union  $\wedge = \cup$  and the join is intersection  $\vee = \cap$ ; whereas in the lattice  $\mathbf{fbr}(G)$ , the join is union  $\vee = \cup$  and the meet is intersection  $\wedge = \cap$ . Any specification  $E$  is entailment equivalent to its consequence  $E \cong E^\bullet$ . A specification  $E$  is closed when it is equal to its consequence  $E = E^\bullet$ . There is a one-one correspondence between closed  $G$ -specifications and categories over graph  $G$ . The

<sup>9</sup>For consistency in discussion, we follow the terminology of formal concept analysis [GW], information flow [BS] and the theory of institutions [GB]. This includes the polarity induced by concept lattices and the directionality of infomorphisms.

<sup>10</sup>This is a complete preorder, loosely called a “lattice”.conceptual intent of a key diagram  $D$ , implicit in satisfaction, is the closed specification  $\mathbf{int}(D)$  consisting of all facts satisfied by the key diagram. Hence,  $D \models_G E$  iff  $E \subseteq \mathbf{int}(D)$  iff  $\mathbf{int}(D) \leq_G E$ .<sup>11</sup>

**4.2.3. Elementary flow.** A graph morphism  $H: G_1 \rightarrow G_2$  maps the types and aspects of  $G_1$  to the types and aspects of  $G_2$ . Graph morphisms are the translations between ologs. A functor  $\mathcal{F}: \mathcal{C}_1 \rightarrow \mathcal{C}_2$  has an underlying graph morphism  $|\mathcal{F}|: |\mathcal{C}_1| \rightarrow |\mathcal{C}_2|$ . For any graph morphism  $H: G_1 \rightarrow G_2$ , there is a fact function  $\mathbf{eqn}(H): \mathbf{eqn}(G_1) \rightarrow \mathbf{eqn}(G_2)$  that maps a  $G_1$ -equation  $(f_1 =_{G_1} f'_1): i_1 \rightarrow j_1$  to the  $G_2$ -equation  $(H^*(f_1) =_{G_2} H^*(f'_1)): H(i_1) \rightarrow H(j_1)$ , and a key diagram functor  $\mathbf{dgm}(H): \mathbf{dgm}(G_2) \rightarrow \mathbf{dgm}(G_1)$  that maps a key diagram  $D_2: G_2 \rightarrow |\mathbf{Set}|$  to the key diagram  $H \circ D_2: G_1 \rightarrow |\mathbf{Set}|$ .<sup>12</sup> The fact function is the fundamental unit of information (formal) flow for ologs, and the key diagram functor is the fundamental unit of semantic flow for ologs.<sup>13</sup> Formal flow is adjoint to semantic flow — satisfaction is invariant under flow:  $\mathbf{dgm}(H)(D_2) \models_{G_1} \epsilon_1$  iff  $D_2 \models_{G_2} \mathbf{eqn}(H)(\epsilon_1)$  for any graph morphism  $H: G_1 \rightarrow G_2$ , source fact  $\epsilon_1$  and target diagram  $D_2$ . Specifications can be moved along graph morphisms by extending the fact (equation) function. For any graph morphism  $H: G_1 \rightarrow G_2$ , define the *direct flow* operator  $\mathbf{dir}(H) = \wp \mathbf{eqn}(H): \mathbf{spec}(G_1) \rightarrow \mathbf{spec}(G_2)$ <sup>14</sup> and the *inverse flow* operator  $\mathbf{inv}(H) = \mathbf{eqn}(H)^{-1}((-)^\bullet): \mathbf{spec}(G_2) \rightarrow \mathbf{spec}(G_1)$ . Direct and inverse flow are adjoint monotonic functions  $(\mathbf{dir}(H) \dashv \mathbf{inv}(H)): \mathbf{fbr}(G_1) \rightarrow \mathbf{fbr}(G_2)$  w.r.t. fiber order:  $\mathbf{dir}(H)(E_1) \geq_{G_2} E_2$  iff  $E_1 \geq_{G_1} \mathbf{inv}(H)(E_2)$ . For any graph morphism  $H: G_1 \rightarrow G_2$ , any  $G_1$ -specification  $E_1$ , and any  $G_2$ -specification  $E_2$ , entailment satisfies the following axioms.

- (direct flow) If  $E_1$  entails the equation  $(f =_{G_1} f'): i \rightarrow j$ , then  $\mathbf{dir}(H)(E_1)$  entails the equation  $(H^*(f) =_{G_2} H^*(f')): H(i) \rightarrow H(j)$ .
- (inverse flow) If  $E_2$  entails the equation  $(H^*(f) =_{G_2} H^*(f')): H(i) \rightarrow H(j)$ , then  $\mathbf{inv}(H)(E_2)$  entails the equation  $(f =_{G_1} f'): i \rightarrow j$ .

These are converted to inference rules in Table 1. A graph morphism  $H: G_1 \rightarrow G_2$  defines a consequence operator  $(-)^{\diamond H} = \mathbf{dir}(H) \circ \mathbf{inv}(H)$  on the fiber preorder  $\mathbf{fbr}(G_1)$ , where  $E_1 \geq_{G_1} E_1^\bullet \geq_{G_1} E_1^{\diamond H}$ .

**4.2.4. Specifications.** A specification  $\mathcal{S} = \langle G, E \rangle$  is an indexed notion consisting of a graph  $G$  and a  $G$ -specification  $E \in \mathbf{spec}(G)$ . It is sometimes convenient to use the symbol ‘ $\mathcal{S}$ ’ in place of ‘ $E$ ’; for example, to say that “ $\mathcal{S} \in \mathbf{spec}(G)$ ”. A category  $\mathcal{C}$  can be resolved and presented as a specification  $\mathbf{spec}(\mathcal{C}) = \langle G, E \rangle$  consisting of the underlying graph  $G = |\mathcal{C}|$  containing the types and aspects of  $\mathcal{C}$  and the collection  $E$  of all facts that hold in  $\mathcal{C}$ . In the other direction, any specification  $\mathcal{S}$  induces a (quotient) category  $\mathbf{cat}(\mathcal{S})$ . Olog (12) and Olog (17) are described as specifications in Section 4.2.1. A specification morphism  $H: \langle G_1, E_1 \rangle \rightarrow \langle G_2, E_2 \rangle$  is a graph morphism  $H: G_1 \rightarrow G_2$  that preserves entailment:  $E_1 \vdash_{G_1} \epsilon_1$  implies  $E_2 \vdash_{G_2} \mathbf{eqn}(H)(\epsilon_1)$  for any  $\epsilon_1 \in \mathbf{eqn}(G_1)$ ; or equivalently that satisfies the adjointness conditions,  $\mathbf{dir}(H)(E_1) \geq_{G_2} E_2$  iff  $E_1 \geq_{G_1} \mathbf{inv}(H)(E_2)$ . Being a graph morphism, it maps types to types and aspects to aspects. Moreover, it also maps facts in  $E_1$  to facts in  $E_2$ ; that is, it preserves all the declared structure. A functor  $\mathcal{F}: \mathcal{C}_1 \rightarrow \mathcal{C}_2$  can be resolved and presented as a specification morphism

<sup>11</sup>This is the first step in the algebraization of Tarski’s “semantic definition of truth” [Ken4].

<sup>12</sup>The composition of graph morphisms is written in diagrammatic order.

<sup>13</sup>This is so, at the abstraction of institutions [Ken3].

<sup>14</sup>The symbol  $\wp$  denotes the power-set operator.$\mathcal{F}: \mathbf{spec}(\mathcal{C}_1) \rightarrow \mathbf{spec}(\mathcal{C}_2)$ . Hence, the presentation form for a functor does exactly what the functor does. The fibered category of specifications  $\mathbf{Spec}$  has specifications as objects and specification morphisms as morphisms. Thus, it is defined in terms of information flow. There is an underlying graph functor  $\mathbf{gph}: \mathbf{Spec} \rightarrow \mathbf{Gph}$  from specifications to graphs  $\langle G, E \rangle \mapsto G$ . The subcategory over any fixed graph  $G$  is the fiber  $\mathbf{fbr}(G)$ ; because of the opposite orientation, we say that “the category of specifications points downward in the concept lattice”. Throughout this section we identify ologs with specifications and olog morphisms with specification morphisms.

4.2.5. *The lattice of theories construction.* Sowa’s “lattice of theories” construction (LOT) describes a modular framework for ontologies [Sow2]. The Olog formalism follows the approach to LOT described in [IFF2].<sup>15</sup> In the Olog formalism, LOT is locally represented by the entailment preorders  $\mathbf{spec}(G)$ , and globally represented by the category of specifications  $\mathbf{Spec}$ . We follow the discussion in section 6.5 “Theories, Models and the World” of Sowa [Sow2]. From each olog (specification) in the “lattice of theories”, the entailment ordering defines paths to the more generalized ologs above and the more specialized ologs below. Sowa defines four ways for moving along paths from one olog to another: contraction, expansion, revision and analogy.

**Contraction:** Any olog can be contracted or reduced to a smaller, simpler olog, moving upward in the preorder  $\mathbf{spec}(G)$ , by deleting one or more facts.

**Expansion:** Any olog can be expanded, moving downward in the preorder  $\mathbf{spec}(G)$ , by adding one or more facts.

**Revision:** A revision step is composite, moving crosswise in the preorder  $\mathbf{spec}(G)$ ; it uses a contraction step to discard irrelevant details, followed by an expansion step to added new facts.

**Analogy:** Unlike contraction and expansion, which move to nearby ologs in an entailment preorder  $\mathbf{spec}(G)$ , analogy moves to an olog in a remote entailment preorder in the category  $\mathbf{Spec}$  via the flow along an underlying graph morphism  $H: G_1 \rightarrow G_2$  by systematically renaming the types and aspects that appear in the facts: any olog  $E_1$  in  $\mathbf{spec}(G_1)$  is moved (by systematic renaming) to the olog  $\mathbf{dir}(H)(E_1)$  in  $\mathbf{spec}(G_2)$ .

According to Sowa, the various methods used in nonmonotonic logic and the operators for belief revision correspond to movement through the lattice of theories.

### 4.3. Alignment and integration of information systems.

4.3.1. *Common ground.* Given the world-views of two individuals, as represented by ologs  $\mathcal{S}_1 = \langle G_1, E_1 \rangle$  and  $\mathcal{S}_2 = \langle G_2, E_2 \rangle$ , there is little hope that one of them completely contains the other (even after allowing for renaming of types and aspects), and there is correspondingly little chance of finding a meaningful olog morphism between the two. Instead, in order to communicate the two individuals could attempt to find a common ground, a third olog  $\mathcal{S} = \langle G, E \rangle$  and meaningful morphisms<sup>16</sup>

<sup>15</sup>The IFF term ‘theory’ is replaced by the Olog term ‘specification’ or ‘olog’.

<sup>16</sup>Roughly speaking, an olog morphism  $F: \mathcal{C} \rightarrow \mathcal{D}$  is *meaningful* when for each type  $X$  in  $\mathcal{C}$ , every intended instance of  $X$  in  $\mathcal{C}$  would be considered an instance of  $F(X)$  by the author of  $\mathcal{D}$  (in which case we say the intention for types is respected), and in a similar way the intention for aspects is respected. Precisely speaking, if  $I: \mathcal{C} \rightarrow \mathbf{Set}$  and  $J: \mathcal{D} \rightarrow \mathbf{Set}$  are instance data for$H_1: \mathcal{S} \rightarrow \mathcal{S}_1$  and  $H_2: \mathcal{S} \rightarrow \mathcal{S}_2$ .<sup>17</sup> This connection is a 1-dimensional knowledge network  $\mathcal{S}_1 \xleftarrow{H_1} \mathcal{S} \xrightarrow{H_2} \mathcal{S}_2$  of shape  $\bullet \leftarrow \bullet \rightarrow \bullet$  called a span (in **Spec**), where each node is an olog and each edge is a morphism between ologs. The requirements of this span are that  $\mathbf{dir}(H_1)(E) \geq_{G_1} E_1$  and  $\mathbf{dir}(H_2)(E) \geq_{G_2} E_2$ , two requirements involving local flow. Equivalently, that  $E \geq_G \mathbf{inv}(H_1)(E_1) \vee_G \mathbf{inv}(H_2)(E_2)$ . The latter precise expression can be rendered in natural language as “the world-view of the common ground is contained in the combined world-views of the two individuals”. The various local direct/inverse flows allow world-views to be compared. Such a common ground can be expanded and improved over time. The basic idea is that one individual can attempt to explain a new idea (type, aspect or fact) to another in terms of the common ground. Then the other individual can either interpret this idea as they already have, learn from it (i.e. freely add it to their olog), or reject it. We view an olog morphism  $H_1: \mathcal{S}_1 \rightarrow \mathcal{S}_2$  as an atomic constraint (alignment) link between  $\mathcal{S}_1$  and  $\mathcal{S}_2$ .<sup>18</sup> We view a common ground span  $\mathcal{S}_1 \xleftarrow{H_1} \mathcal{S} \xrightarrow{H_2} \mathcal{S}_2$  as a molecular constraint between  $\mathcal{S}_1$  and  $\mathcal{S}_2$ , which is weakest when  $\mathcal{S} = \emptyset$  and strongest when  $\mathcal{S}_1 = \mathcal{S} = \mathcal{S}_2$ .

4.3.2. *Systems of ologs.* In the general case, more than two individuals will share a common ground. For example, companies that do business together may have a common-ground olog as part of a legal contract; or, the various participants at a conference will have some common understanding of the topic of that conference. In fact, for any finite set of ologs  $\mathbb{X} = \{\mathcal{S}_1, \mathcal{S}_2, \dots, \mathcal{S}_n\}$ , there should be a common ground world-view (even if empty), say  $\mathcal{S}_{\mathbb{X}}$ . If  $\mathbb{Y} \subseteq \mathbb{X}$  is a subset, then there should be a map  $\mathcal{S}_{\mathbb{X}} \rightarrow \mathcal{S}_{\mathbb{Y}}$  because any common understanding held by the individuals in  $\mathbb{X}$  is held by the individuals in  $\mathbb{Y}$ . For example, the triangular-shaped diagram

(21)

represents three individuals  $\{1, 2, 3\}$ , their ologs  $\{\mathcal{S}_1, \mathcal{S}_2, \mathcal{S}_3\}$ , their pair-wise commonality ologs  $\{\mathcal{S}_{12}, \mathcal{S}_{13}, \mathcal{S}_{23}\}$ , and their three-way commonality olog  $\mathcal{S}_{123}$ . This diagram, which stands for the interaction between individuals  $\{1, 2, 3\}$ , does not stand alone, but is part of an intricate web of other ologs and alignment constraints. In particular, individuals 1 and 3 may be part of some different interacting group, say of individuals  $\{1, 3, 6, 7\}$ , and hence the right edge of the diagram would be part of some tetrahedron-shaped diagram with vertices  $\{1, 3, 6, 7\}$ . If we take the point-of-view that “a collection of ologs representing the world-views of various individuals” is a system, then we can think of the ologs as being the types of that

$\mathcal{C}$  and  $\mathcal{D}$ , then  $F$  is meaningful relative to  $I$  and  $J$  if one can exhibit a natural transformation  $\mu: I \Rightarrow F \circ J$  as in [Spi2].

<sup>17</sup>A common ground olog is also called a reference ontology in knowledge representation.

<sup>18</sup>This is so, at the abstraction of institutions [Ken3].system, the morphisms connecting the ologs as being the aspects of that system, with the shape of a system being its underlying graph. In essence, we can apply ologs to themselves. In the system represented by diagram (21), there are seven types  $\{\mathcal{S}_1, \mathcal{S}_2, \mathcal{S}_3, \mathcal{S}_{12}, \mathcal{S}_{13}, \mathcal{S}_{23}, \mathcal{S}_{123}\}$  and nine aspects  $\{\dots, \mathcal{S}_{123} \rightarrow \mathcal{S}_{13}, \dots\}$ , and the shape looks like this

In addition, we can introduce certain facts to represent the meaning of that system and then enforce those facts.

A *distributed system* is a diagram (functor)  $\mathcal{G}: \mathbf{I} \rightarrow \mathbf{Gph}$  of shape  $\mathbf{I}$  within the ambient category  $\mathbf{Gph}$ . As such, it consists of an indexed family  $\{G_n \mid n \in \mathbf{I}\}$  of graphs together with an indexed family  $\{G_e: G_n \rightarrow G_m \mid (e: n \rightarrow m) \in \mathbf{I}\}$  of graph morphisms. Let  $\mathbf{Dist}(\mathbf{I})$  denote the collection of distributed systems of shape  $\mathbf{I}$ . An *information system* is a diagram  $\mathcal{S}: \mathbf{I} \rightarrow \mathbf{Spec}$  of shape  $\mathbf{I}$  within the ambient category  $\mathbf{Spec}$ . As such, it consists of an indexed family  $\{\mathcal{S}_n = \langle G_n, E_n \rangle \mid n \in \mathbf{I}\}$  of ologs together with an indexed family  $\{\mathcal{S}_e: \mathcal{S}_n \rightarrow \mathcal{S}_m \mid (e: n \rightarrow m) \in \mathbf{I}\}$  of olog morphisms. Some of these ologs might represent the world-views of various individuals, whereas others could be common grounds; also included might be portals between individual ologs and common grounds, as in the CG example of Section 4.4. Let  $\mathbf{Info}(\mathbf{I})$  denote the collection of information systems of shape  $\mathbf{I}$ . An information system  $\mathcal{S}$  with component ologs  $\mathcal{S}_n = \langle G_n, E_n \rangle$  has an underlying distributed system  $\mathcal{G}$  of the same shape with component graphs  $G_n$  for  $n \in \mathbf{I}$ . For any distributed system  $\mathcal{G}$ , let  $\mathbf{info}_{\mathbf{I}}(\mathcal{G})$  denote the collection of information systems over  $\mathcal{G}$  of shape  $\mathbf{I}$ . There is a pointwise entailment order  $\mathcal{S} \leq_{\mathcal{G}}^{\mathbf{I}} \mathcal{S}'$  on  $\mathbf{info}_{\mathbf{I}}(\mathcal{G})$  when component ologs satisfy the same entailment ordering  $E_n \leq_{G_n} E'_n$  for  $n \in \mathbf{I}$ , and by taking the coproduct there is a pointwise entailment order on  $\mathbf{Info}(\mathbf{I}) = \coprod_{\mathcal{G} \in \mathbf{Dist}(\mathbf{I})} \mathbf{info}_{\mathbf{I}}(\mathcal{G})$ . A constant distributed system  $\Delta(G) \in \mathbf{Dist}(\mathbf{I})$  is a distributed system  $\Delta(G): \mathbf{I} \rightarrow \mathbf{Gph}$  with the same language  $G$  for any index  $n \in \mathbf{I}$ . Any constant distributed system defines join and meet monotonic functions  $\bigvee_G^{\mathbf{I}}, \bigwedge_G^{\mathbf{I}}: \mathbf{info}_{\mathbf{I}}(\Delta(G)) \rightarrow \mathbf{fbr}(G)$  mapping an information system  $\mathcal{S} \in \mathbf{info}_{\mathbf{I}}(\Delta(G))$  to the join and meet ologs  $\bigvee \mathcal{S} = \bigcup_{n \in \mathbf{I}} E_n$  and  $\bigwedge \mathcal{S} = \bigcap_{n \in \mathbf{I}} E_n$  in  $\mathbf{fbr}(G)$ . The join monotonic function is adjoint to the constant monotonic function  $\Delta_G^{\mathbf{I}}: \mathbf{fbr}(G) \rightarrow \mathbf{info}_{\mathbf{I}}(\Delta(G))$  that distributes an olog  $\mathcal{S}' \in \mathbf{fbr}(G)$  to the various locations  $n \in \mathbf{I}$  forming a constant information system  $\Delta(\mathcal{S}') \in \mathbf{info}_{\mathbf{I}}(\Delta(G))$ , since  $\bigvee \mathcal{S} \geq_G \mathcal{S}'$  iff  $\mathcal{S} \geq_{\Delta(G)}^{\mathbf{I}} \Delta(\mathcal{S}')$  for any system  $\mathcal{S} \in \mathbf{info}_{\mathbf{I}}(\Delta(G))$  and any olog  $\mathcal{S}' \in \mathbf{fbr}(G)$ .

**4.3.3. System morphisms.** Just as ologs are linked by morphisms, information systems are also linked by morphisms. For these there is the new complication of shape. In this paper we define fixed-shape system morphisms, but a more general definition would allow the shape to vary. A distributed system morphism  $\theta: \mathcal{G} \Rightarrow \mathcal{G}'$  in  $\mathbf{Dist}(\mathbf{I})$  consists of a collection  $\{\theta_n: G_n \rightarrow G'_n \mid n \in \mathbf{I}\}$  of component graph morphisms, which are systematically coordinated in the sense that they satisfy the naturality conditions  $G_e \circ \theta_m = \theta_n \circ G'_e$  for any indexing link  $e: n \rightarrow m$  in  $\mathbf{I}$ . A direct flow operator  $\mathbf{dir}_{\mathbf{I}}(\theta): \mathbf{info}_{\mathbf{I}}(\mathcal{G}) \rightarrow \mathbf{info}_{\mathbf{I}}(\mathcal{G}')$  along  $\theta$  can be defined, which maps an information system  $\mathcal{S} \in \mathbf{info}_{\mathbf{I}}(\mathcal{G})$  to an information system  $\mathbf{dir}_{\mathbf{I}}(\theta)(\mathcal{S}) \in \mathbf{info}_{\mathbf{I}}(\mathcal{G}')$defined by  $\mathbf{dir}_{\mathbf{I}}(\theta)(\mathcal{S})_n = \mathbf{dir}(\theta_n)(E_n)$  for  $n \in \mathbf{I}$ .<sup>19</sup> An inverse flow operator  $\mathbf{inv}_{\mathbf{I}}(\theta) : \mathbf{info}_{\mathbf{I}}(\mathcal{G}') \rightarrow \mathbf{info}_{\mathbf{I}}(\mathcal{G})$  can similarly be defined. Direct and inverse flow are adjoint monotonic functions  $\langle \mathbf{dir}_{\mathbf{I}}(\theta) \dashv \mathbf{inv}_{\mathbf{I}}(\theta) \rangle : \mathbf{info}_{\mathbf{I}}(\mathcal{G}) \rightarrow \mathbf{info}_{\mathbf{I}}(\mathcal{G}')$ , since  $\mathbf{dir}_{\mathbf{I}}(\theta)(\mathcal{S}) \geq_{\mathcal{G}'}^{\mathbf{I}} \mathcal{S}'$  iff  $\mathcal{S} \geq_{\mathcal{G}}^{\mathbf{I}} \mathbf{inv}_{\mathbf{I}}(\theta)(\mathcal{S}')$ . An information system morphism  $\theta : \mathcal{S} \Rightarrow \mathcal{S}'$  in  $\mathbf{Info}(\mathbf{I})$  consists of a collection  $\{\theta_n : \mathcal{S}_n \rightarrow \mathcal{S}'_n \mid n \in \mathbf{I}\}$  of component olog morphisms, which are systematically coordinated and preserve alignment in the sense that they satisfy the naturality conditions  $\mathcal{S}_e \circ \theta_m = \theta_n \circ \mathcal{S}'_e$  for any indexing link  $e : n \rightarrow m$  in  $\mathbf{I}$ ; equivalently,  $\theta$  is a morphism between the underlying distributed systems  $\theta : \mathcal{G} \Rightarrow \mathcal{G}'$  and the direct flow of  $\mathcal{S}$  is at least as general as  $\mathcal{S}'$ :  $\mathbf{dir}_{\mathbf{I}}(\theta)(\mathcal{S}) \geq_{\mathcal{G}'}^{\mathbf{I}} \mathcal{S}'$ . The ordering  $\mathcal{S} \geq_{\mathcal{G}}^{\mathbf{I}} \mathcal{S}'$  is an information system morphism  $\theta : \mathcal{S} \Rightarrow \mathcal{S}'$  with identity component translations  $\theta_n = id_{G_n}$  for each index  $n \in \mathbf{I}$ .

**4.3.4. Channels.** We continue with our systems point-of-view. Since we have represented the whole system as a diagram  $\mathcal{S}$  of parts (ologs)  $\mathcal{S}_n$  with part-part relations (alignment constraints)  $\mathcal{S}_n \rightarrow \mathcal{S}_m$ , we also want to represent the whole system as an olog  $\mathcal{C}$  with part-whole relations  $\mathcal{S}_n \rightarrow \mathcal{C}$ .<sup>20</sup> An *information channel*  $\langle \gamma : \mathcal{M} \Rightarrow \Delta(C), C \rangle$  consists of an indexed family  $\{\gamma_n : G_n \rightarrow C \mid n \in \mathbf{I}\}$  of graph morphisms called flow links with a common target graph  $C$  called the core of the channel. A channel  $\langle \gamma, C \rangle$  covers a distributed system  $\mathcal{G}$  of shape  $\mathbf{I}$  when the part-whole relationships respect the alignment constraints (are consistent with the part-part relationships):  $\gamma_n = G_e \circ \gamma_m$  for each indexing morphism  $e : n \rightarrow m$  in  $\mathbf{I}$ . A covering channel is a distributed system morphism  $\gamma : \mathcal{G} \Rightarrow \Delta(C)$  in  $\mathbf{Dist}(\mathbf{I})$  from distributed system  $\mathcal{G}$  to constant distributed system  $\Delta(C) : \mathbf{I} \rightarrow \mathbf{Gph}$ . Such a channel defines a direct flow operator  $\mathbf{dir}_{\mathbf{I}}(\gamma) : \mathbf{info}_{\mathbf{I}}(\mathcal{G}) \rightarrow \mathbf{info}_{\mathbf{I}}(\Delta(C))$  and an inverse flow operator  $\mathbf{inv}_{\mathbf{I}}(\gamma) : \mathbf{info}_{\mathbf{I}}(\Delta(C)) \rightarrow \mathbf{info}_{\mathbf{I}}(\mathcal{G})$ . For any two covering channels  $\langle \gamma', C' \rangle$  and  $\langle \gamma, C \rangle$  over the same distributed system  $\mathcal{G}$ , a refinement  $H : \langle \gamma', C' \rangle \rightarrow \langle \gamma, C \rangle$  is a graph morphism between cores  $H : C' \rightarrow C$  that respects the part-whole relationships of the two channels:  $\gamma'_n \circ H = \gamma_n$  for  $n \in \mathbf{I}$ . In such a situation, we say the channel  $\langle \gamma', C' \rangle$  is a refinement of the channel  $\langle \gamma, C \rangle$ . A channel  $\langle \iota, \coprod \mathcal{G} \rangle$  is a minimal cover<sup>21</sup> or optimal(ly refined covering) channel of a distributed system  $\mathcal{G}$  when it covers  $\mathcal{G}$  and for any other covering channel  $\langle \gamma, C \rangle$  there is a unique refinement  $[\gamma, C] : \coprod \mathcal{G} \rightarrow C$  from  $\langle \iota, \coprod \mathcal{G} \rangle$  to  $\langle \gamma, C \rangle$ .

**4.3.5. System flow.** In order to represent an information system  $\mathcal{S} = \{\mathcal{S}_n \xrightarrow{\mathcal{S}_e} \mathcal{S}_m\}$  as a single olog  $\coprod \mathcal{S}$ , called the fusion of  $\mathcal{S}$ , with part-whole relations  $\mathcal{S}_n \rightarrow \coprod \mathcal{S}$ , we follow the colimit theorem of [TBG] by recognizing the following three properties.

- • Optimal channels exist for any distributed system  $\mathcal{G}$ .
- •  $\mathbf{fbr}(G)$  is a complete preorder for any graph  $G$ , loosely called a “lattice”.
- • For any graph morphism  $H : G_1 \rightarrow G_2$ , direct and inverse flow are adjoint monotonic functions  $\langle \mathbf{dir}(H), \mathbf{inv}(H) \rangle : \mathbf{fbr}(G_1) \rightarrow \mathbf{fbr}(G_2)$ .

Let  $\mathcal{G} \in \mathbf{Dist}(\mathbf{I})$  be a distributed system of shape  $\mathbf{I}$  with optimal channel  $\langle \iota, \coprod \mathcal{G} \rangle$ . The optimal core  $\widehat{\mathcal{G}} = \coprod \mathcal{G}$  is called the sum of the distributed system  $\mathcal{G}$ , and the optimal channel components (graph morphisms)  $\{\iota_n : G_n \rightarrow \coprod \mathcal{G} \mid n \in \mathbf{I}\}$  are called flow links. There is a *direct system flow* monotonic function (see Figure 1)

<sup>19</sup>Well-defined, since  $\mathbf{dir}(G'_e)(\mathbf{dir}(\theta_n)(E_n)) = \mathbf{dir}(\theta_m)(\mathbf{dir}(G_e)(E_n)) \geq_m \mathbf{dir}(\theta_m)(E_m)$ .

<sup>20</sup>The theory of part-whole relations is called mereology. It studies how parts are related to wholes, and how parts are related to other parts within a whole.

<sup>21</sup>Information flow terminology [BS].FIGURE 1. System Flow

$\mathbf{dir}_{\langle \mathbf{I}, \mathcal{G} \rangle} = \mathbf{dir}_{\mathbf{I}}(\iota) \cdot \mathbf{V}_{\widehat{\mathcal{G}}}^{\mathbf{I}}: \mathbf{info}_{\mathbf{I}}(\mathcal{G}) \rightarrow \mathbf{fbr}(\widehat{\mathcal{G}})$ . Direct system flow has two steps: (i) direct (fixed shape) system flow of an information system along the optimal channel ( $\mathbf{Dist}(\mathbf{I})$ -morphism)  $\iota: \mathcal{G} \Rightarrow \Delta(\widehat{\mathcal{G}})$  and (ii) lattice join combining the contributions of the parts into a whole. In the opposite direction, there is an *inverse system flow* monotonic function (see Figure 1)  $\mathbf{inv}_{\langle \mathbf{I}, \mathcal{G} \rangle} = \Delta_{\widehat{\mathcal{G}}}^{\mathbf{I}} \cdot \mathbf{inv}_{\mathbf{I}}(\iota): \mathbf{fbr}(\widehat{\mathcal{G}}) \rightarrow \mathbf{info}_{\mathbf{I}}(\mathcal{G})$ . Inverse system flow has two steps: (i) mapping an olog with core language  $\widehat{\mathcal{G}}$  to a constant information system over  $\Delta(\widehat{\mathcal{G}})$  with shape  $\mathbf{I}$  by distributing the olog to the locations  $n \in \mathbf{I}$ , and (ii) inverse (fixed shape) system flow of this constant information system back along the optimal channel  $\iota: \mathcal{G} \Rightarrow \Delta(\widehat{\mathcal{G}})$ . Direct system flow is adjoint to inverse system flow  $\langle \mathbf{dir}_{\langle \mathbf{I}, \mathcal{G} \rangle} \dashv \mathbf{inv}_{\langle \mathbf{I}, \mathcal{G} \rangle} \rangle: \mathbf{info}_{\mathbf{I}}(\mathcal{G}) \rightarrow \mathbf{fbr}(\widehat{\mathcal{G}})$ , since the composition components are adjoint. For any distributed system  $\mathcal{G} \in \mathbf{Dist}(\mathbf{I})$  with optimal core  $\widehat{\mathcal{G}} = \coprod \mathcal{G}$ , any information system  $\mathcal{S} \in \mathbf{info}_{\mathbf{I}}(\mathcal{G})$ , and any olog  $\widehat{\mathcal{S}} \in \mathbf{fbr}(\widehat{\mathcal{G}})$ , entailment satisfies the following axioms.

- (direct flow) If  $E_n$  entails the equation  $(f =_{G_n} f'): i \rightarrow j$ , then  $\mathbf{dir}_{\langle \mathbf{I}, \mathcal{G} \rangle}(\mathcal{S})$  entails the equation  $(\iota_n^*(f) =_{\mathcal{G}} \iota_n^*(f')): \iota_n(i) \rightarrow \iota_n(j)$  for any  $n \in \mathbf{I}$ .
- (inverse flow) If  $\widehat{\mathcal{S}}$  entails the equation  $(\iota_n^*(f) =_{\widehat{\mathcal{G}}} \iota_n^*(f')): \iota_n(i) \rightarrow \iota_n(j)$ , then  $\mathbf{inv}_{\langle \mathbf{I}, \mathcal{G} \rangle}(\widehat{\mathcal{S}})_n$  entails the equation  $(f =_{G_n} f'): i \rightarrow j$  for any  $n \in \mathbf{I}$ .

These are converted to inference rules in Table 1. Information flow can be used to compute the fusion olog for an information system and to define the consequence of an information system. Fusion is direct system flow, and consequence is the composition of direct and inverse system flow. Let  $\mathcal{S} \in \mathbf{info}_{\mathbf{I}}(\mathcal{G})$  be any information system. The fusion  $\coprod \mathcal{S} = \mathbf{dir}_{\langle \mathbf{I}, \mathcal{G} \rangle}(\mathcal{S}) = \langle \coprod \mathcal{G}, \bigvee_{n \in \mathbf{I}} \mathbf{dir}_{\langle \mathbf{I}, \mathcal{G} \rangle}(E_n) \rangle \in \mathbf{fbr}(\widehat{\mathcal{G}})$  is an olog that represents the whole system in a centralized fashion [Ken2],[Ken3]. The consequence  $\mathcal{S}_{\langle \mathbf{I}, \mathcal{G} \rangle}^\diamond = \mathbf{inv}_{\langle \mathbf{I}, \mathcal{G} \rangle}(\mathbf{dir}_{\langle \mathbf{I}, \mathcal{G} \rangle}(\mathcal{S})) = \mathbf{inv}_{\langle \mathbf{I}, \mathcal{G} \rangle}(\coprod \mathcal{S}) = \{\mathbf{inv}_{\langle \mathbf{I}, \mathcal{G} \rangle}(\mathcal{S}) \mid n \in \mathbf{I}\} \in \mathbf{info}_{\mathbf{I}}(\mathcal{G})$  is an information system that represents the whole system in a distributed fashion [Ken3]. It is inverse flow of the fusion olog along the optimal channel, transferring the entailed facts of the whole system to the component parts.

The consequence operator  $(-)^{\diamond}$ , which is defined on information systems, is a closure operator on the complete preorder  $\mathbf{info}_{\mathbf{I}}(\mathcal{G})$ , and by taking the coproduct it is a closure operator on the complete preorder  $\mathbf{Info}(\mathbf{I}) = \coprod_{\mathcal{G} \in \mathbf{Dist}(\mathbf{I})} \mathbf{info}_{\mathbf{I}}(\mathcal{G})$ : (increasing)  $\mathcal{S} \geq \mathcal{S}^\diamond$ , (monotonic)  $\mathcal{S} \geq \mathcal{S}'$  implies  $\mathcal{S}^\diamond \geq \mathcal{S}'^\diamond$  and (idempotent)  $\mathcal{S}^{\diamond\diamond} = \mathcal{S}^\diamond$ . <sup>22</sup> Pointwise entailment order  $\leq$  on  $\mathbf{Info}(\mathbf{I})$  is only a preliminary order, since it does not incorporate interactions between system component parts. System

<sup>22</sup>By allowing system shape to vary, channels can be generalized to morphisms of distributed systems. Then a notion of relative fusion (direct system flow) can be defined in terms of left Kan<table border="0">
<tbody>
<tr>
<td rowspan="3"><b>equivalence:</b></td>
<td>(reflexive)</td>
<td><math>\frac{}{(f =_G f) : i \rightarrow j}</math></td>
</tr>
<tr>
<td>(symmetric)</td>
<td><math>\frac{(f_1 =_G f_2) : i \rightarrow j}{(f_2 =_G f_1) : i \rightarrow j}</math></td>
</tr>
<tr>
<td>(transitive)</td>
<td><math>\frac{(f_1 =_G f_2) : i \rightarrow j, (f_2 =_G f_3) : i \rightarrow j}{(f_1 =_G f_3) : i \rightarrow j}</math></td>
</tr>
<tr>
<td rowspan="2"><b>algebra:</b></td>
<td>(compositional)</td>
<td><math>\frac{(f_1 =_G f_2) : i \rightarrow j, (g_1 =_G g_2) : j \rightarrow k}{(f_1 ; g_1 =_G f_2 ; g_2) : i \rightarrow k}</math></td>
</tr>
<tr>
<td>(bi-closed)</td>
<td><math>\frac{(g_1 =_G g_2) : j \rightarrow k}{(f ; g_1 =_G f ; g_2) : i \rightarrow k, (g_1 ; h =_G g_2 ; h) : j \rightarrow l}</math></td>
</tr>
<tr>
<td rowspan="2"><b>morphic flow:</b></td>
<td>(direct)</td>
<td><math>\frac{(f_1 =_{G_1} f'_1) : i_1 \rightarrow j_1}{(H^*(f_1) =_{G_2} H^*(f'_1)) : H(i_1) \rightarrow H(j_1)}</math></td>
</tr>
<tr>
<td>(inverse)</td>
<td><math>\frac{(H^*(f_1) =_{G_2} H^*(f'_1)) : H(i_1) \rightarrow H(j_1)}{(f_1 =_{G_1} f'_1) : i_1 \rightarrow j_1}</math></td>
</tr>
<tr>
<td rowspan="2"><b>system flow:</b></td>
<td>(direct)</td>
<td><math>\frac{(f =_{G_n} f') : i \rightarrow j}{(\iota_n^*(f) =_{\mathcal{G}} \iota_n^*(f')) : \iota_n(i) \rightarrow \iota_n(j)}</math></td>
</tr>
<tr>
<td>(inverse)</td>
<td><math>\frac{(\iota_n^*(f) =_{\mathcal{G}} \iota_n^*(f')) : \iota_n(i) \rightarrow \iota_n(j)}{(f =_{G_n} f') : i \rightarrow j}</math></td>
</tr>
</tbody>
</table>

TABLE 1. Inference Rules

entailment order  $\preceq$  on  $\mathbf{Info}(\mathbf{I})$  is defined by  $\mathcal{S}_1 \preceq \mathcal{S}_2$  when  $\mathcal{S}_1^\diamond \leq \mathcal{S}_2^\diamond$ ; equivalently,  $\mathcal{S}_1^\diamond \leq \mathcal{S}_2^\diamond$ . Pointwise order is stronger than system entailment order:  $\mathcal{S}_1 \leq \mathcal{S}_2$  implies  $\mathcal{S}_1 \preceq \mathcal{S}_2$ . This is a specialization-generalization order. Any information system  $\mathcal{S}$  is entailment equivalent to its consequence  $\mathcal{S} \cong \mathcal{S}^\diamond$ . An information system  $\mathcal{S}$  is closed when it is equal to its consequence  $\mathcal{S} = \mathcal{S}^\diamond$ .

The whole effect of taking the system consequence may be greater than the sum of its parts, in the sense that  $\mathcal{S}_n \geq_n \mathcal{S}_n^{\diamond_{\iota_n}} \geq_n \bigvee_m \mathbf{inv}(\iota_n)(\mathbf{dir}(\iota_m)(\mathcal{S}_m)) \geq_n \mathcal{S}_n^\diamond$  for any  $n \in \mathbf{I}$ , since separate parts may have a productive interaction at the channel core. A final part of an information system is a part with no non-trivial constraint links from it. (The graphical subsystem beneath) nonfinal parts are necessary for the alignment of information systems, resulting in the equivalencing of types and aspects through quotienting. However, because of the covering condition  $\iota_n = G_e \circ \iota_m$  and the entailment order  $\mathbf{dir}(G_e)(E_n) \geq_m E_m$  for constraint links  $\mathcal{S}_e : \mathcal{S}_n \rightarrow \mathcal{S}_m$ , only the fact(ual) content of final parts of information systems are necessary to compute the system fusion and consequence.

**4.3.6. General examples.** Here are some examples of system fusion/consequence.

---

extension, and a notion of relative system consequence can be defined as the composition of direct followed by inverse system flow.- • An information system  $\mathcal{S}$  with a constant underlying distributed system,  $G_i = G$  for all  $n \in \mathbf{I}$ , gathers together all the component parts of the information system and forms their consequence. It has identity flow links  $\{\iota_n = id_G: G \rightarrow G = \coprod \mathcal{G} \mid n \in \mathbf{I}\}$ , component join fusion  $\coprod \mathcal{S} = \bigvee_{n \in \mathbf{I}} \mathcal{S}_n = \langle G, \bigcup_{n \in \mathbf{I}} E_n \rangle$ , and constant system consequence  $\mathcal{S}_n^\bullet = (\bigvee_{n' \in \mathbf{I}} \mathcal{S}_{n'})^\bullet$  for all  $n \in \mathbf{I}$ .
- • A discrete information system  $\mathcal{S} = \{\mathcal{S}_n = \langle G_n, E_n \rangle \mid n \in \mathbf{I}\}$  with no constraint links  $G_e: \mathcal{S}_n \rightarrow \mathcal{S}_m$  for  $n \neq m$ , has coproduct injection flow links  $\iota_n: G_n \rightarrow \bigoplus_{n \in \mathbf{I}} G_n$ , non-restricting fusion, and inverse flow projecting back to individual component consequence  $\mathcal{S}_n^\bullet = \mathcal{S}_n^\bullet$  for all  $n \in \mathbf{I}$ . No alignment (constraint) links means no interaction.
- • An information system  $\mathcal{S} = \{\mathcal{S}_1 \xleftarrow{H_1} \mathcal{S} \xrightarrow{H_2} \mathcal{S}_2\}$  consisting of a single common ground  $\mathcal{S} = \langle G, E \rangle$  between two component ologs  $\mathcal{S}_1 = \langle G_1, E_1 \rangle$  and  $\mathcal{S}_2 = \langle G_2, E_2 \rangle$ , with underlying distributed system (span)  $\mathcal{G} = \{G_1 \xleftarrow{H_1} G \xrightarrow{H_2} G_2\}$ , has pushout injection flow links  $G_1 \xrightarrow{\iota_1} \coprod \mathcal{G} \xleftarrow{\iota_2} G_2$ , direct image union fusion  $\coprod \mathcal{S} = \langle \coprod \mathcal{G}, \mathbf{dir}(\iota_1)(E_1) \cup \mathbf{dir}(\iota_2)(E_2) \rangle$ , and system consequence components  $\mathcal{S}_n^\bullet = \langle G_n, \mathbf{inv}(\iota_n)(\mathbf{dir}(\iota_1)(E_1) \cup \mathbf{dir}(\iota_2)(E_2)) \rangle$  for  $n = 1, 2$ . The flow links will quotient any types and aspects that are connected through the common ground allowing for the appropriate interaction in the fusion consequence  $(\mathbf{dir}(\iota_1)(E_1) \cup \mathbf{dir}(\iota_2)(E_2))^\bullet$ , then the inverse flow will reconnect this with the component types and aspects.

**4.4. Conceptual graphs.** The conceptual graph formalism (CG) for knowledge representation [Sow2], was initially formulated to represent database systems (DBS), but is now used in natural language processing (NLP) and first-order logic (FOL). Verbs in NLP can often be represented relationally by star(-shaped conceptual) graphs. For example, the sentence “John is going to Boston by bus” might be represented by the conceptual graph

(22)

graph TD
    go((go))
    John[Person: John]
    Boston[City: Boston]
    Bus[Bus]
    go -- 1 --> John
    go -- 2 --> Boston
    go -- 3 --> Bus

In a sentence of natural language, thematic roles are semantic descriptions of the way (the entities described by) a noun phrase functions with respect to (the action of) the verb. These entities are the participants in the occurrent expressed by the verb. For the action of ‘going’ in the above sentence there are three participants and hence three thematic roles. ‘John’ plays the role of the agent of the action, a ‘Bus’ is the instrument used in the action and ‘Boston’ is the destination of the action. Translations using thematic roles can be used to align two ontologies with respect to a common ground. A CG-style translation of conceptual graph (22) would replace the verb relation ‘going’ with a concept ‘Go’ and replace the edges that form the signature of the ‘going’ relation with binary relations for the threeroles ‘agent’, ‘instrument’ and ‘destination’.

(23)

graph LR Go[Go] -- agent --> Person[Person: John] Go -- dest --> City[City: Boston] Go -- inst --> Bus[Bus]

However, the case relations that semantically describe the thematic roles should be viewed as functional in nature; that is, for any instance of the action of a sentence’s verb there is a unique entity described by a noun phrase of the sentence. When this semantics is respected, the translation to thematic roles becomes a process of “linearization”, which is best described abstractly as: (1) the identification of relation types with entity types, (2) the translation of a sorted multiarity relation to a span of functions, one function for each role, and (3) the functional interpretation of thematic roles.

The Olog formalism, which also represents DBS and NLP, is a version of equational logic. Both the Olog and CG formalisms were designed as graphical representations. However, the CG formalism is binary and relational, whereas the Olog formalism is unary and functional. The CG formalism is binary since it has two kinds of type, concepts and relations; it is relational in the way it interprets edges. The Olog formalism is unary since it has only one kind of type, the abstract concept; it is functional in the way it interprets aspects (edges). However, much of the semantics of the CG formalism can be transformed to the Olog formalism by the process of linearization<sup>23</sup>, thereby gaining in efficiency and conciseness. For example, conceptual graph (22) can be linearized to the olog graph<sup>24</sup>

(24)

graph LR 1_John[1 John] -- agent --> Person[Person] Go[Go] -- agent --> Person Go -- dest --> City[City] 1_Boston[1 Boston] -- dest --> City Go -- inst --> Bus[Bus]

Since olog aspects are interpreted functionally, the functional nature of thematic roles is respected. In this manner, the olog formalism could be used to replace the CG representation of ontologies. For example, a community (acting as an individual) could build its ontology  $\mathcal{C}$  from ground up by aligning it with some top-level reference ontology  $\mathcal{T}$  (such as in the appendix of [Sow2]), thereby importing some formal semantics from  $\mathcal{T}$ . The following fragment demonstrates how this works.

Assume that ontology  $\mathcal{T}$  contains the concept of “spatial process” as represented by the general concept type  $\llbracket \text{Spatial-Process} \rrbracket$  with aspects  $\llbracket \text{Spatial-Process} \rrbracket \xrightarrow{\text{agent}} \llbracket \text{Agent} \rrbracket$ ,  $\llbracket \text{Spatial-Process} \rrbracket \xrightarrow{\text{inst}} \llbracket \text{Vehicle} \rrbracket$  and  $\llbracket \text{Spatial-Process} \rrbracket \xrightarrow{\text{dest}} \llbracket \text{Location} \rrbracket$ . At some stage assume that the community ontology  $\mathcal{C}$  has specified the concept type orderings  $\llbracket \text{Person} \rrbracket \leq \llbracket \text{Agent} \rrbracket$ ,  $\llbracket \text{Bus} \rrbracket \leq \llbracket \text{Vehicle} \rrbracket$  and  $\llbracket \text{City} \rrbracket \leq \llbracket \text{Location} \rrbracket$  with corresponding injective aspects  $\llbracket \text{Person} \rrbracket \xrightarrow{\text{is}} \llbracket \text{Agent} \rrbracket$ ,  $\llbracket \text{Bus} \rrbracket \xrightarrow{\text{is}} \llbracket \text{Vehicle} \rrbracket$  and

<sup>23</sup>The linearization process works for any binary/relational knowledge representation, such as CGs, entity-relationship data modelling [JRW], relational database systems [Ken5] or the Information Flow Framework [IFF1]. In the entity-relationship data modelling,  $n$ -ary relationship links are replaced by  $n$ -ary spans of aspects and attributes are included as types.

<sup>24</sup> $\llbracket 1 \rrbracket$  is the universal type to which all types have a unique aspect.