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 , each column represents an attribute of , and each row represents an example of . 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 in an olog represents a type of thing and every arrow emanating from represents an attribute of (whose results are of type ). 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 FDM.

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 FDM).

1.3.2 Ologs and RDF/OWL.

In FDM, 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 is the mother of .” 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 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 ). 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 [11].

Moreover, commutative diagrams and other expressive abilities held by ologs are not generally part of the semantic network concept (see [12]). 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 demonstates 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.

2.1.1 Types with compound structures.

Many types have compound structures; i.e., they are composed of smaller units. Examples include (5)

It is good practice to declare the variables in a “compound type”, as we did in the last two cases of (5). In other words, it is preferable to replace the first box above with something like (6)so that the variables are clear.

2.1.2 Rules of good practice.

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;

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 “nicknamed” various boxes. So may stand as a nickname for and as a nickname for.

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*)(8*)

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 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 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) can be read by first reading the label on its source box (domain of definition) , then the label on the arrow , and finally the label on its target box (set of result values) . For example, the arrow(9)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 a common example from the way we write ologs.

(10)

Neither arrow is readable by the protocol given above (e.g. “a pair where and are integers an integer” is not an English sentence), and yet it is obvious what each map means. For example, given the pair which belongs in box , application of arrow would yield in box . The label can be thought of as a nickname for the full name “yields, via the value of ,” and similarly for . 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 where is a woman and is a man, which yields, via the value of , 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 , and sometimes it is not, as in . (Obviously, a father may have more than one child.) A quick fix would be to replace the latter by . This is ok, but the relationship between and set of then becomes an issue to deal with later. There is another way to indicate such “non-functional” relationships.

In mathematics, a relation between sets , and so on through is defined to be a subset of the Cartesian product(12)

The set represents those sequences that are so-related. In an olog, we represent this as follows(13)

For example,

(14)Whereas includes all possible triples where is a person, is a paper, and is a journal, it is obvious that not all such triples are found in . Thus represents a proper subset of .

Rules of good practice 2.1.2. 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;

2.3.2 Example.

How would one record a fact like “a truck weighs more than a car”? We suggest something like this:(17)

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:(18)

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 (18) 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 , rather than as a type in and of itself, and similarly see 150 as an instance of . This idea of an olog having a “data level” is the subject of the Section 3.

2.3.4 Rules of good practice.

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.

3.1.1 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(19)

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 , 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 (19) 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 between finite sets and can always be written as a 2-column table: the first column is filled with the instances of type and the second column is filled with their -values, which are instances of type .

For example, consider the aspect (20)

We can document some instances of this relationship using the following table(21)

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 the aspect in (20) and Table (21) 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:(15)and consider the following instances of the three aspects in it(22)

When we declare that the diagram in (15) commutes (using the check-mark), we are saying that for every instance of (of which we have three: Cain, Abel, and Chelsey), the two paths to give the same answers. Indeed, for Cain the two paths are:

i. Cain (Eve, Adam) Eve;

ii. Cain Eve;

and these answers agree. If one changed any instance of the word “Eve” to the word “Steve” in one of the tables in (22), some pair of paths would fail to agree. Thus the “fact” that the diagram in (15) 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.

3.2.1 Example.

We can convert Olog (23) 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.(24)

Note that every arrow of Olog (23) is represented in Database (24) as a column of table , and that every cell in that column can be found in the Id column of table . 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 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 , such that the declared facts hold, is called an assignment of instance data for . More precisely, instance data on is a functor , as in Definition 3.2.3.

3.2.3 Definition.

Let be a category (olog) with underlying graph , and let denote the category of sets. An instance of (or an assignment of instance data for ) is a functor . That is, it consists of

• a set for each object (type) in ,

• a function for each arrow (aspect) in , and

• for each fact (path-equivalence or equation)

declared in , an equality of functions

The symbol ‘’ in paths denotes concatenation or formal composition. If we let and denote two paths, then we often write to denote the fact that these paths are equivalent.

4.2.1 Fibers.

A graph contains types as nodes and aspects as edges. The graphs underlying an olog is considered its language. Any category has an underlying graph . In particular, is the graph underlying the category of sets and functions. Olog (12) has an underlying graph containing the three types , and and the three aspects ‘has a parent’, ‘woman’ and ‘has as mother’. Olog (17) has an underlying graph containing the three types ,, and and the six aspects ‘manager’, ‘works in’, ‘secretary’, ‘name’, ‘first name’ and ‘last name’. Let denote the set of all facts (equations) that are possible to express using the types and aspects of . A -specification is a set consisting of some of the facts expressible in . 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 denote the collection of all -specifications ordered by inclusion .

4.2.2 Satisfaction.

It will be useful here to define an instance of a graph , instead of an instance of a category . An instance of a graph populates the graph by assigning instance data to it. An instance of a graph is a graph morphism mapping each type in to a set of instances and mapping each aspect in to an instance function . Using database terminology, we also call 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 satisfies (is a model of) a -fact (see Definition 3.2.3), symbolized , when we have an equality of functions . We also say that (holds in) is true when interpreted in . An identity holds in all key diagrams (hence, is a tautology), and vice-versa for any set a constant key diagram satisfies any fact . A key diagram satisfies (is a model of) a -specification , symbolized , when it satisfies every fact in the specification. For any graph , a -specification entails a -fact , denoted by , when any model of the specification satisfies the fact. The consequence of a -specification is the set of all entailed equations. The consequence operator is a closure operator, and the consequence of a specification is a congruence. For any -specification , entailment satisfies the follow axioms.(25)

These are converted to inference rules in Table 1. To construct , we first take the reflexive, symmetric, and transitive closure of (so that is a -specification and also the smallest equivalence relation containing ), and then we get by closing up under composition on left and right. We extend specification inclusion with the entailment order, where when entails each equation in ; that is, when or equivalently when . The statement “” asserts that is at least as specialized as . The entailment order , which is a specialization-generalization order, represents a local version of the “lattice of theories” construction of Sowa [15] (see Section 4.2.5). The opposite entailment order is called the fiber order. For consistency in discussion, we follow the terminology of formal concept analysis [16], information flow [14] and the theory of institutions [13]. This includes the polarity induced by concept lattices and the directionality of infomorphisms. In the lattice (this is a complete preorder, loosely called a “lattice”), the meet is union and the join is intersection ; whereas in the lattice , the join is union and the meet is intersection . Any specification is entailment equivalent to its consequence . A specification is closed when it is equal to its consequence . There is a one-one correspondence between closed -specifications and categories over graph . The conceptual intent of a key diagram , implicit in satisfaction, is the closed specification consisting of all facts satisfied by the key diagram. Hence, iff iff . This equivalence between satisfaction and entailment order is the first step in the algebraization of Tarski’s “semantic definition of truth”.

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Table 1. Inference Rules.

https://doi.org/10.1371/journal.pone.0024274.t006

4.2.3 Elementary flow.

A graph morphism maps the types and aspects of to the types and aspects of . Graph morphisms are the translations between ologs. A functor has an underlying graph morphism . For any graph morphism , there is a fact function that maps a -equation to the -equation , and a key diagram functor that maps a key diagram to the key diagram (the composition of graph morphisms is written in diagrammatic order). At the abstraction of institutions [17], 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. Formal flow is adjoint to semantic flow – satisfaction is invariant under flow: iff for any graph morphism , source fact and target diagram . Specifications can be moved along graph morphisms by extending the fact (equation) function. For any graph morphism , define the direct flow operator to be the direct image function, and the inverse flow operator to be the composition of the specification consequence operator followed by the inverse image function. Direct and inverse flow are adjoint monotonic functions w.r.t. fiber order: . For any graph morphism , any -specification , and any -specification , entailment satisfies the following axioms.

These are converted to inference rules in Table 1. A graph morphism defines a consequence operator on the fiber preorder , where .

4.2.4 Specifications.

A specification is an indexed notion consisting of a graph and a -specification . It is sometimes convenient to use the symbol ‘’ in place of ‘’; for example, to say that “”. A category can be resolved and presented as a specification consisting of the underlying graph containing the types and aspects of and the collection of all facts that hold in . In the other direction, any specification induces a (quotient) category . Olog (12) and Olog (17) are described as specifications in Section 4.2.1. A specification morphism is a graph morphism that preserves entailment: implies for any ; or equivalently that satisfies the adjointness conditions, . Being a graph morphism, it maps types to types and aspects to aspects. Moreover, it also maps facts in to facts in ; that is, it preserves all the declared structure. A functor can be resolved and presented as a specification morphism . Hence, the presentation form for a functor does exactly what the functor does. The fibered category of specifications has specifications as objects and specification morphisms as morphisms. Thus, it is defined in terms of information flow. There is an underlying graph functor from specifications to graphs . The subcategory over any fixed graph is the fiber ; 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 [15]. The Olog formalism follows the approach to LOT described in [18], where the IFF term ‘theory’ is replaced by the Olog term ‘specification’ or ‘olog’. In the Olog formalism, LOT is locally represented by the entailment preorders , and globally represented by the category of specifications . We follow the discussion in Section 6.5 “Theories, Models and the World” of Sowa [15]. 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 , by deleting one or more facts.

Expansion: Any olog can be expanded, moving downward in the preorder , by adding one or more facts.

Revision: A revision step is composite, moving crosswise in the preorder ; 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 , analogy moves to an olog in a remote entailment preorder in the category via the flow along an underlying graph morphism by systematically renaming the types and aspects that appear in the facts: any olog in is moved (by systematic renaming) to the olog in .

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

4.3.1 Common ground.

Roughly speaking, an olog morphism is meaningful when for each type in , every intended instance of in would be considered an instance of by the author of (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 and are instance data for and , then is meaningful relative to and if one can exhibit a natural transformation as in FDM.

Given the world-views of two individuals, as represented by ologs and , 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 and meaningful morphisms and (a common ground olog is also called a reference ontology in knowledge representation). This connection is a 1-dimensional knowledge network of shape called a span (in ), where each node is an olog and each edge is a morphism between ologs. The requirements of this span are that and , two requirements involving local flow. Equivalently, that . 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. At the abstraction of institutions [17], an olog morphism is an atomic constraint (alignment) link between and . Following this, we view a common ground span as a molecular constraint between and , which is weakest when and strongest when .

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 , there should be a common ground world-view (even if empty), say . If is a subset, then there should be a map because any common understanding held by the individuals in is held by the individuals in . For example, the triangular-shaped diagram(26)

represents three individuals , their ologs , their pair-wise common ground ologs , and their three-way commonality olog . This diagram, which stands for the interaction between individuals , 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 , and hence the right edge of the diagram would be part of some tetrahedron-shaped diagram with vertices . 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 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 (26), there are seven types and nine aspects , and the shape is the graph in the diagram (27).(27)

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) of shape within the ambient category . As such, it consists of an indexed family of graphs together with an indexed family of graph morphisms. Let denote the collection of distributed systems of shape . An information system is a diagram of shape within the ambient category . As such, it consists of an indexed family of ologs together with an indexed family 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 denote the collection of information systems of shape . An information system with component ologs has an underlying distributed system of the same shape with component graphs for . For any distributed system , let denote the collection of information systems over of shape . There is a pointwise entailment order on when component ologs satisfy the same entailment ordering for , and by taking the coproduct there is a pointwise entailment order on . A constant distributed system is a distributed system with the same language for any index . Any constant distributed system defines join and meet monotonic functions mapping an information system to the join and meet ologs and in . The join monotonic function is adjoint to the constant monotonic function that distributes an olog to the various locations forming a constant information system , since iff for any system and any olog .

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 moorphisms, but a more general definition would allow the shape to vary. A distributed system morphism in consists of a collection of component graph morphisms, which are systematically coordinated in the sense that they satisfy the naturality conditions for any indexing link in . A direct flow operator along can be define, which maps an information system to an information system defined by for . This is well-defined, since . An inverse flow operator can similarly be defined. Direct and inverse flow are adjoint monotonic functions , since iff . An information system morphism in consists of a collection of component olog morphisms, which are systematically coordinated and preserve alignment in the sense that they satisfy the naturality conditions for any indexing link in ; equivalently, is a morphism between the underlying distributed systems and the direct flow of is at least as general as : . The ordering is an information system morphism with identity component translations for each index .

4.3.4 Channels.

We continue with our systems point-of-view. Since we have represented the whole system as a diagram of parts (ologs) with part-part relations (alignment constraints) , we also want to represent the whole system as an olog with part-whole relations . 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. An information channel consists of an indexed family of graph morphisms called flow links with a common target graph called the core of the channel. A channel covers a distributed system of shape when the part-whole relationships respect the alignment constraints (are consistent with the part-part relationships): for each indexing morphism in . A covering channel is a distributed system morphism in from distributed system to constant distributed system . Such a channel defines a direct flow operator and an inverse flow operator . For any two covering channels and over the same distributed system , a refinement is a graph morphism between cores that respects the part-whole relationships of the two channels: for . In such a situation, we say the channel is a refinement of the channel . A channel is called a minimal cover (using information flow terminology [14]) or an optimal(ly refined covering) channel of a distributed system when it covers and for any other covering channel there is a unique refinement from to .

4.3.5 System flow.

In order to represent an information system as a single olog , called the fusion of , with part-whole relations , we follow the colimit theorem of [19] by recognizing the following three properties.

• Optimal channels exist for any distributed system .

• is a complete preorder for any graph , loosely called a “lattice”.

• For any graph morphism , direct and inverse flow are adjoint monotonic functions .

Let be a distributed system of shape with optimal channel . The optimal core is called the sum of the distributed system , and the optimal channel components (graph morphisms) are called flow links. There is a direct system flow monotonic function (see Diagram 28) . Direct system flow has two steps: (i) direct (fixed shape) system flow of an information system along the optimal channel (-morphism) 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 Diagram 28) . Inverse system flow has two steps: (i) mapping an olog with core language to a constant information system over with shape by distributing the olog to the locations , and (ii) inverse (fixed shape) system flow of this constant information system back along the optimal channel . Direct system flow is adjoint to inverse system flow , since the composition components are adjoint. For any distributed system with optimal core , any information system , and any olog , entailment satisfies the following axioms.

These are converted to inference rules in Table 1.(28)

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 be any information system. The fusion is an olog that represents the whole system in a centralized fashion [20],[17]. The consequence is an information system that represents the whole system in a distributed fashion [17]. It is inverse flow of the fusion olog along the optimal channel, transfering the entailed facts of the whole system to the component parts. 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 extension, and a notion of relative system consequence can be defined as the composition of direct followed by inverse system flow.

The consequence operator , which is defined on information systems, is a closure operator on the complete preorder , and by taking the coproduct it is a closure operator on the complete preorder : (increasing) , (monotonic) implies and (idempotent) . Pointwise entailment order on is only a preliminary order, since it does not incorporate interactions between system component parts. System entailment order on is defined by when ; equivalently, . Pointwise order is stronger than system entailment order: implies . This is a specialization-generalization order. Any information system is entailment equivalent to its consequence . An information system is closed when it is equal to its consequence .

The whole effect of taking the system consequence may be greater than the sum of its parts, in the sense that for any , 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 and the entailment order for constraint links , 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.

• An information system with a constant underlying distributed system, for all , gathers together all the component parts of the information system and forms their consequence. It has identity flow links , compo nent join fusion , and constant system consequence for all .

• A discrete information system with no constraint links for , has coproduct injection flow links , non-restricting fusion, and inverse flow projecting back to individual component consequence for all . No alignment (constraint) links means no interaction.

• An information system consisting of a single common ground between two component ologs and , with underlying distributed system (span) , has pushout injection flow links , direct image union fusion , and system consequence components for . The flow links will quotient any types and aspects that are connected through the common ground allowing for the approprate interaction in the fusion conse quence , then the inverse flow will reconnect this with the component types and aspects.

5.2.1 Example.

We will now give four examples to motivate the definition of pullback. In the first example, (34), both and will be subtypes of , and in such cases the pullback will be their intersection. In the next two examples (35 and 36), only will be a subtype of , and in such cases the pullback will be the “corresponding subtype of ” (as should make sense upon inspection). In the last example (37), neither nor will be a subtype of . In each line below, the pullback of the diagram to the left is the diagram to the right. The reader should think of the left-hand olog as a kind of problem to which the new box in the right-hand olog is a solution.(34)(35)(36)(37)

See Example 5.2.3 for a justification of these, in light of Definition 5.2.2.

The following is the definition of pullbacks in the category of sets. For an olog, the instance data are given by sets (at least in this paper, see Section 3), so this definition suffices for now. See [5] for more details on pullbacks.

5.2.2 Definition.

Let and be sets, and let and be functions. The pullback of , denoted , is defined to be the set

together with the obvious maps and , which send an element to and to , respectively. In other words, the pullback of is a commutative square(38)

5.2.3 Example.

In Example 5.2.1 we gave four examples of pullbacks. For each, we will consider to be sets and functions as in Definition 5.2.2 and explain how the set follows that definition, i.e., how its label fits with the set .

In the case of (34), the set should consist of pairs where is a wealthy customer, is a loyal customer, and is equal to (as customers). But if and are the same customer then is just a customer that is both wealthy and loyal, not two different customers. In other words, an instance of the pullback is a customer that is both loyal and wealthy, so the label of fits.

In the case of (35), the set should consist of pairs where is a person, is the color blue, and the favorite color of is equal to (as colors). In other words, it is a person whose favorite color is blue, so the label of fits. If desired, one could instead label with .

In the case of (36), the set should consist of pairs where is a dog, is a woman, and the owner of is equal to (as people). In other words, it is a dog whose owner is a woman, so the label of fits. If desired, one could instead label with .

In the case of (37), the set should consist of pairs where is a piece of furniture, is a space in our house, and the width of is equal to the width of . This is fits perfectly with the label of .

5.2.4 Using pullbacks to classify.

To distinguish between two things, one must find a common aspect of the two things for which they have differing results. For example, a pen is different from a pencil in that they both use some material to write (a common aspect), but the two materials they use are different. Thus the material which a writing implement uses is an aspect of writing implements, and this aspect serves to segregate or classify them. We can think of three such writing-materials: graphite, ink, and pigment-wax. For each, we will make a layout in the olog below:(39)

One could also replace the label of box with “a pencil”, the label of box with “a pen”, and the label of box with “a crayon”; in so doing, the layouts at the top would define a pencil, a pen, and a crayon to be a writing implement that uses respectively graphite, ink, and pigment-wax.

5.2.5 Building pullbacks on pullbacks.

There is a theorem in category theory which states the following. Suppose given two commutative squares(40)

such that the right-hand square (3,4,5,6) is a pullback. It follows that if the left-hand square (1,2,3,4) is a pullback then so is the big rectangle (1,2,5,6). It also follows that if the big rectangle (1,2,5,6) is a pullback then so is the left-hand square (1,2,3,4). This fact can be useful in authoring ologs.

For example, the type is vague, but we can lay out precisely what it means using pullbacks(41)

The category-theoretic fact described above says that since and , it follows that . That is, we can deduce the definition “a cellphone that has a bad battery is defined as a cellphone that has a battery which remains charged for less than one hour.” In other words, .

5.3.1 Definition.

Given sets , their product, denoted , is the set(42)

There are two obvious projection maps and , sending the pair to and to respectively.

5.3.2 Example.

In Example 5.2.1, (37) we presented the idea of a piece of furniture that was the same width as a space in the house. What if we say that is any space that is between 1 and 8 inches bigger than a piece of furniture? We can use a combination of products and pullbacks to create the appropriate type(43)

Here and are products and is a pullback. This olog lays out what it means to be “a nice furniture placement” using products. The bottom horizontal aspect is an example of a map obtained by the “universal property of products”; see Section 5.6.

5.3.3 Products of more (or fewer) types.

The product of two sets and was defined in 5.3.1. One may also take the product of three sets in a similar way, so the elements are triples where and . In fact this idea holds for any number of sets. It even makes sense to take the product of one set (just ) or no sets! The product of one set is itself, and the product of no sets is the singleton set . For more on this, see Section 5.5 or [6].

6.1.1 Definition.

Given sets and , their coproduct, denoted , is the set(50)

There are two obvious inclusion maps and , sending to and to , respectively.

If and have no elements in common, then the one can drop the “” and “” labels without changing the set in a substantial way. Here are two examples that should make the coproduct idea clear.

6.1.2 Example.

In the following olog the types and are disjoint, so the coproduct is just the union.(51)

6.1.3 Example.

In the following olog, and are not disjoint, so care must be taken to differentiate common elements.(52)

Since ducks can both swim and fly, each duck is found twice in , once labeled as a flyer and once labeled as a swimmer. The types and are kept disjoint in , which justifies the name “disjoint union.”

6.2.1 Example.

In each example below, the diagram to the right is the pushout of the diagram to the left. The new object, , is the union of and , but instances of are equated to their and aspects. This will be discussed after the two diagrams.(54)(55)

In the olog (52), the shoulder is seen as part of the arm and part of the torso. When taking the union of these two parts, we do not want to “double-count” the shoulder (as would be done in the coproduct , see Example 6.1.3). Thus we create a new type for cells in the shoulder, which are considered the same whether viewed as cells in the arm or cells in the body. In general, if one wishes to take two things and glue them together, the glue serves as and the two things serve as and , and the union (or grouping) is the pushout .

In the olog (53), if every mathematics course is simply “too hard,” then when reading off a list of courses, each math course will not be read aloud but simply read as “too hard.” To form we begin by taking the union of and , and then we consider everything in to be the same whether one looks at it as a course or as the phrase “too hard.” The math courses are all blurred together as one thing. Thus we see that the power to equate different things can be exercised with pushouts.

6.2.2 Definition.

Let and be sets and let and be functions. The pushout of , denoted , is the quotient of (see Definition 6.1.1) by the equivalence relation generated by declaring (i.e., is equivalent to ) if: , and there exists with and .

6.7.1 Definition.

Let denote the set of non-negative real numbers. A pseudo-metric space is a pair where is a set and is a function with the following properties for all elements :

1. ;

2. and

3. .(63)

As long as the instances for the right-hand side of the olog are mathematically correct (i.e., we assign the set of non-negative real numbers), this olog has the same content as Definition 6.7.1. One can use ologs to define usual metric spaces (in which Property (1) in Definition 6.7.1 is strengthened), but it would have taken too much space here.

It should be clear that ologs provide a more precise and explicit description of any concept, relying less on the grammar of English and more on the mathematical “grammar” of sets and functions. Assumptions are exposed as all the working parts of an object need to be explicitly documented. Thus an olog is likely to be instantly readable by a theorem prover such as Coq ([28]), at least if one creates the olog within an appropriate Olog-Coq interface API. Moreover, various parts of this olog may be reusable in other contexts, and hence connect pseudo-metric spaces into a web of neighboring definitions and theorems.

In fact, once a corpus of mathematics has been written in olog form, evidence of conjectures not yet proven could be written down as instance data. For example, one could record every known prime as instances of a type and a machine could automatically check that Goldbach’s conjecture (written as an olog containing as a type) holds for all example “so far.” With definitions, theorems, and examples all written in the same computer-readable language of ologs, one may hope for much more advanced searching and knowledge retrieval by humans. For example, one could formulate very precise questions as database queries and use SQL on the database corresponding to a given olog (see Section 3.2).