Department of Computer Science, Faculty of Physics and Applied Informatics,University of Łódź ul. Pomorska nr 149/153, 90-236 Łódź, Poland
Author to whom correspondence should be addressed.
Received: 27 February 2014; in revised form: 12 May 2014 / Accepted: 13 May 2014 / Published: 29 May 2014
Allegories are enriched categories generalizing a category of sets and binary relations. In this paper, we extend a new, recently-introduced conceptual data model based on allegories by adding support for modal operators and developing a modal interpretation of the model in any allegory satisfying certain additional (but natural) axioms. The possibility of using different allegories allows us to transparently use alternative logical frameworks, such as fuzzy relations. Mathematically, our work demonstrates how to enrich with modal operators and to give a many world semantics to an abstract algebraic logic framework. We also give some examples of applications of the modal extension.
allegories; data modeling; modal logic
MSC 18B10; 03B45; 18D20; 03B52; 68P15; 03B70
Allegories, introduced by Freyd and Scedrov , are enriched categories modeled upon the category of sets and binary relations, , in such a way that allows for other realizations of allegorical axioms than , while still preserving many of the useful properties. This is similar to the way in which toposes generalize the category, , of sets. For a logician, allegories can be seen as the generalizations (with some axioms relaxed) of the relation algebra . Because of the deep relationship between relational databases and relational algebra and also a lively research program of using category theory for conceptual modeling (see e.g., [3,4,5,6,7,8,9,10]), it is surprising that allegories were hardly ever used for database modeling. The use of allegories for this purpose was suggested in , and recently, the new allegorical data model was introduced by the authors in ; barring that, the authors were unable to secure more references.
In the majority of works applying category theory to conceptual database modeling, the arrows in a category are interpreted as functional dependencies. While this approach was successfully used, e.g., in tackling the view update problem ([12,13]) and in the treatment of incomplete data (), the advantage of using the allegories instead of plain categories is that they furnish interpreting the arrows as arbitrary binary relationships (including non-functional ones), and makes the use of auxiliary constructs, like spans or power sets, unnecessary.
Conceptual database modeling is an important part of the database design process, assisting in creating better database schema and applications and also isolating the designer from the technical details of a particular database technology, which might be relational, object-oriented, XML, etc. One of the rationales for using category theory (and allegories, in particular) for conceptual modeling instead of plain first-order logic or relational algebra is that it allows one to abstract away also some ontological and logical commitments : the data model formulated in terms of an allegory can be interpreted both in the allegory of (sharp) relations, but also in some allegory of fuzzy relations; thus, the database designer can, to some extent, free herself from the dependence on the particular choice of logic.
A desired feature of the conceptual model is the possibility of extending the model to increase its expressivity, when such a need arises. An example of such an extension is adding the modalities and many world interpretations, useful e.g., in the context of temporal or distributed databases. A main contribution of this paper is adding support for modal operators to the allegorical conceptual data model described in . Here, we add the many world semantics to the model using only the language of allegories. This means that we are doing it independently of a concrete allegory; we can do it, e.g., for both sharp and fuzzy relations.
For both the allegorical model to be useful and the introduction of many worlds semantics to be possible requires more structure than provided by the plain allegories. A structure we work with here and which we conjecture to be minimal is that of a complete distributive allegory with all relational products (which are just products in the subcategory of maps). Our sentences are equalities between type safe terms formed using language of these categories. In particular, when our model is interpreted in , this language is sufficient to express any first-order sentence (cf. ). The specification of the model resembles the algebraic sketch (see, e.g., ), but is a lot more expressive; in fact, limits and colimits in the subcategory of maps (arrows that are like total functions) are definable within our formalism (see ).
The many worlds semantics of the allegorical conceptual model is introduced as the family of interpretations of the model in a family of allegories (each corresponding to the “world”), as well as a special allegory providing a kind of common interpretation and within which one can define the modal operators using only the allegorical operators, together with the structure connecting “worlds” with one another and the common interpretation.
The structure of this paper is as follows. In Section 2, we recall some basic facts about category theory and allegories, as well as the definition (slight extension) of the allegorical conceptual model from . In Section 3, we develop our modal extension and its semantics, and finally, in Section 4, we give some examples of modal specifications and queries.
We recall some basic categorical definitions in order to fix the clean, but somewhat idiosyncratic, notation we use. The bulk of this preliminaries is taken, first by introductory material on allegories, which are not widely known and for which  is the basic textbook, and, then, by modified definitions of an allegorical conceptual data model from .
The reader might be disturbed by our consequent use of the term “set” where “class” might be more appropriate. We do it consciously, in order not to clutter the presentation with the set-theoretical considerations. The reader should also remember that while we use the operator to compose arrows in diagrammatic order, we also denote by the composition in the assignment order (i.e., ), of any two actual maps (that is, procedures that assign values to values) whenever those maps are not used as arrows in a category.
2.1. Categories and Graphs
A graph, , consists of a set of vertices, , a set of arrows, , and a pair of maps called source and target, respectively. We denote by the set of arrows with source A and target B, where . A graph morphism from a graph, , to a graph, , is a pair of maps:
such that , , for all . A composition of graph morphisms is defined componentwise, i.e., .
A category, , is a graph with associative arrow composition , defined whenever (note the diagrammatic order), and identity map such that for all . We write . When using single letter arrow names, we will often omit the semicolon composition operator abbreviating . A graph morphism between categories is called a functor when it maps identity arrows to identity arrows and respects composition, i.e., and .
An allegory  is a category enriched with intersection and reciprocation operators, respectively, for all :
We require these operators to satisfy the following conditions: Intersections make each hom-set a meet semi-lattice (see, e.g., ), where we denote the associated partial order by ⊑, i.e., , for all . In addition, and are to satisfy:
for all such that the above formulas are well-defined.
Allegories generalize the allegory, , of sets (objects) and binary relations (arrows). Because of this, we may refer to arrows in any allegory as “relations”. In , we write iff . The identity in is ; intersection is the set intersection, i.e., ; reciprocation is defined by ; and the composition of relations such that is defined by:
We distinguish the following classes of arrows in allegory:
If , then R is called total.
If , then R is called functional.
If R is functional and total, it is called a map. A set of all maps in an allegory, , is denoted by: .
If , then R is called injective.
If , then R is called surjective.
A family with a common source, A, is called jointly monic if: . For any maps with target A, if for all i, then .
For any , we denote by and the top and bottom elements of , respectively, if they exist. In , we have , .
Suppose that the top element exists. An object, C, together with maps (called projections on components), where the arrow , , is called a relational product iff:
Relational products are products in , and they are unique up to an isomorphism. On the other hand, a product in is not necessarily a relational product (see, e.g., ). Let be such that , , . Then, the unique map, , such that , can be expressed explicitly as:
In an allegory, , there might be a canonical choice of a relational product of each . In such a case, we denote the canonical product object by and the projections by and . In , the canonical relational product is the usual Cartesian product.
A distributive allegory is an allegory, , in which for all hom-sets, , there exists a bottom element, , and a binary operation:
which together with makes a distributive lattice with as the infimum and as the supremum. In addition, we require that and satisfy:
for all such the above formulae are type-correct. One proves easily that:
A locally complete distributive allegory is a distributive allegory in which each of the hom-sets is a complete lattice, and moreover, the following infinite distributive laws hold:
for any arrow R and a family such that the above formulas make sense. As a consequence, each of the hom-sets in a complete distributive allegory, , is a locale (a complete lattice in which finite meets distribute over any joins) or, equivalently, a complete Heyting algebra. More precisely, the above distributivity condition implies the existence of a binary hom-set operator, →, called implication, defined uniquely by the property:
for all R, S and T in the same hom-set. The implication can be expressed explicitly by the formula:
We denote for brevity . A hom-set, , is a Boolean algebra if and only if one of the equivalent conditions below hold for all :
If this is the case, then . It is immediate that is a locally complete distributive allegory with the union defined as a set union, i.e., , and in which every hom-set is a Boolean algebra where .
It is worth noting that a locally complete distributive allegory is a division allegory , that is, a distributive allegory with a partial binary operation (called a right division), defined if and only if both arguments have the same targets, and satisfying for all arrows R, S and T:
In a distributive allegory which is locally complete, the (unique) operation satisfying the above axioms is given by . In the present paper, we do not utilize this operation, but it would be easy to extend the language of the conceptual model considered here with the right division operator, in order to increase its expressivity.
The allegorical conceptual data model introduced in  requires that the allegories in which the model is interpreted contain canonical relational products of all pairs of objects and top and bottom elements in all hom-sets. Here, because the modal structure imposed on the model in this paper requires the locally complete distributive allegories anyway for the definitions to make sense, we enrich the language of specification to include unions and implications and, thus, require from the allegories in which models are interpreted to be distributive. For an easy reference, we call such allegories PTBDI (product, top, bottom distributive and containing implications) allegories. In particular, is a locally complete PTBDI allegory.
2.3. Allegorical Conceptual Data Model
Before we can recall the definition of the model, we need first the following two auxiliary notions:
An extended set of objectsof a graphis defined as a groupoid (i.e., a set with a binary operation (, Chapter V)) with operation ⊗ generated by , i.e., if , then , etc.
Letbe a graph. A set, , of allegorical terms associated to (cf. )is defined as the smallest set of expressions such that:
for all ,
for all objects ,
if , then ,
ifare such that , then ,
ifare such thatand , then , and .
We omit unnecessary parentheses when writing elements of . In the above conditions, we used the functionsdefined in an obvious way using induction on the structure of terms as the extension of source and target functions, i.e., , , , , and so on, for any terms t, , .
We are now ready to formulate the definition of our conceptual model.
An allegorical conceptual data model, , consists of a finite graph, , and a finite set, , of equations of the form , where .
We write the equations of the form as . Equations allow us to specify constraints and dependencies between data elements.
A conceptual modelloosely based on an example HR database of Oracle is depicted in Figure 1 (graph) and Figure 2 (equations). Equation (a) states that the arrows in the list are (total) maps; Equation (b) states that maps , , , andare jointly monic (that is anis uniquely determined by the values of these attributes). Equation (c) makesa linear order andtransitive (it is meant as a transitive closure of the chain of commands) and irreflexive relation. The first two equations in Equation (d) specify thatis an injective map (i.e., each department has a manager, and no manager manages two different departments). Then, the last of equations in (d) link relationsandby specifying that either an employee is a manager of the department he/she works in or he/she is under the departmental manager. Finally, Equation (e) limits the wages by enforcing that an employee doing a certain job type may not earn less than the minimal wage nor more than the maximal wage associated with the given job type.
The model is intended to be interpreted in some PTBDI allegory. Each interpretation corresponds to some actual realization and state of the database. The choice of allegory determines the kind of database; different interpretations in the same allegory may be interpreted as different states of the same database. Interpretation of the allegorical conceptual model in is the typical choice. In order to formulate the precise definition of interpretation, we first need the following simple observation:
An example of an allegorical conceptual model graph.
An example of an allegorical conceptual model graph.
Equations specifying the properties of relations from Figure 1.
Equations specifying the properties of relations from Figure 1.
Letbe a graph, and letbe a PTBDI allegory. Letbe a graph morphism. Then,can be extended to a mapusing induction on the ⊗-groupoid structure of ; that is, by definingfor all . Similarly,can be extended to a mapusing induction on the structure of terms in ; that is, by defining , , and so on.
The following remark clarifies the above lemma. Suppose that: is a subgraph contained in a graph , and that is a graph morphism from into a PTBDI allegory, , such that . Consequently, by the above definition of the extension of , we have , where . Note that even if T is intended to be interpreted as a composition of R and S, this intention cannot be communicated through the structure of the graph, , and hence, it does not introduce any inconsistency in the extension of . On the other hand, the intended identification of T with the composition of R and S can be declared through equations. In particular, if , then F cannot be an interpretation of , as defined below.
Letbe an allegorical conceptual data model, and letbe a PTBDI allegory. A graph morphismis called an interpretation ofiniffor all equations . A set of all interpretations ofis denoted by , and the set of all interpretations in an allegoryis denoted by .
Reasoning about data is furnished by the following semantic definition of a derived equality:
Letbe an allegorical conceptual model. We write , where F is an interpretation ofif . We write (the equalityholds in ) iffor any interpretation F of .
3. Modal Terms and Modal Interpretations
Recall that a modal similarity type consists of a set, , of symbols and a map called an arity. A frame with modal similarity type is a set, , together with a family of relations.
Let be a graph, and let be a modal similarity type. We extend the set, , of allegorical terms associated with to a set, , of modal allegorical terms of modal similarity type by introducing new term constructors called modal operators: for each , such that , and each term such that and also terms . We also declare:
(i.e., modal operators associate arrows with arrows from the same hom-set) so that the terms with modal operators can be used as subterms. Arity zero modal operators (that is, modal constants) require special treatment. For all such that , and all pairs of objects , we declare to be a term in with and .
Letbe a frame of similarity type ; letbe a locally complete PTBDI allegory, and letbe an allegorical conceptual data model. A modal interpretationofinwith frameconsists of a family of interpretations , a mapand a map , where we denote , satisfying, for alland:
We denote bythe set of all modal interpretations ofinwith frame .
The last of the equalities (10) specifies to be a unique map which makes a diagram in Figure 3 commute (cf. Equation (4)). In fact, Formula (10c) could be taken as a definition of , because it guarantees by itself that is an injective map. Indeed, as defined by Equation (10c) is a map because and are maps and , are projections for a relational product. It remains to prove that :
Similarly, exchanging A and B, we prove the inequality
Observe that a modal interpretation defines a family of maps given explicitly by:
Alternatively, is determined by the commutativity of the following family of diagrams (for all ):
The construction of injection for a relational product (note that and ).
The construction of injection for a relational product (note that and ).
satisfies the following properties:
For all , such thatand , if , then .
For all , such thatand , we haveand .
For allsuch that , we have .
For all , the following properties hold:
ad. 1: Suppose that . Then:
ad. 2: follows immediately from the distributivity of composition over union. The composition does not distribute over the intersection. However, it is well known (see, e.g., ) that in any allegory and if R is functional, that is, if . Hence, the result follows from the injectivity of and , which implies the functionality of and .
ad. 4: , as is functional. The first of Equations (5) implies that . Inequality holds trivially. Then, (cf.Figure 3):
Note that does not in general preserve identities, tops, bottoms and relational product projections, and thus, it is not an interpretation. The objects , for should be considered a representation of “total domains”—the collections of all possible values (whatever that means in a general allegory). On the other hand, is an interpretation of the conceptual model of the database in the concrete world (say a moment of time), and in particular objects represent actual values stored in the database in the world m.
Note also that does not preserve implications in general. This is obvious, as, in particular, in , the complement in the arrow-set is much larger than the embedding in of the complement of in . However, Lemma 10 alleviates that a bit. First, we need the following general result, interesting on its own.
Letandbe injective maps in some allegory, , and suppose that the meet semilattice of arrowscontains the top element, . Let . Consider the following statements (see the diagram below for illustration):
for some .
Statement (1) implies Statements (2) and (3), and Statement (3) implies Statements (1) and (2).
If , then, because and the composition is monotonic, we must also have . Furthermore, as and, similarly, . This finishes the proof of the first set of implications.
Assume now that . We will prove first that . Because is functional, i.e., , and composition is monotonic, we have . Similarly, . In order to prove the other inequalities, we first define for any arrow, T, the domain of T:
It follows from Formula (13c) that if and only if . By assumption, however, , and so, by the transitivity of the order relation and the monotonicity of (Equation (13b)), it is enough to prove that . However, by Equation (13a), we have:
In order to prove that , first note that because reciprocation is monotonic and , if , then , and hence, by the just proven result, . Taking the reciprocation of both sides of this inequality yields as desired.
Now, we are ready to prove the last implication. Assuming , itfollows from the just proven result that . Hence, defining yields , as desired. ☐
For allsuch thatand , we have:
Using explicit Formula (8) for an implication in a locally complete distributive allegory, and the distributivity law (6), we have:
In the second equality, in order to change the ⨆ subscript, we made use of the fact that . Note that by Lemma 9 for some . It follows immediately, because extending the set of which the supremum is taken increases the supremum value that:
Then, because the injectivity of and implies that if and only if , we have that:
It remains to prove that:
The above inequality is equivalent to the conjunction of (which holds by the monotonicity of ) and . Using explicit Formula (8) for implication and the fact that decreasing the set for which the supremum is taken decreases the supremum value, we have:
The last expression is identical to (**), which is equal (as shown above) to . ☐
The following immediate corollary to Lemma 10 shows that for certain safe expressions, preserves implications:
For allwith the same source and target and all , we have:
We are now ready to define the interpretation of the modal formulas in the allegory using the interpretation .
Letbe a frame of similarity type ; letbe a locally complete PTBDI allegory, and letbe an allegorical conceptual data model. A modal interpretationofinwith framedefines for alla mapdefined by induction on the structure of terms inas follows (we omit the subscript wheneveris clear from the context):
, , , , .
Ifis such thatand , then:
Ifis such thatandare such thatand , then:
Note thatpreserves neither top elements, relational projections and identities nor implications (becausedoes not preserve them either). Note, however, that by definition,andfor all .
Here, we construct an allegorical counterpart of the basic modal logic. Letbe a frame of similarity type . Letbe an allegorical conceptual data model and consider a modal interpretation . We denote customarilyandfor all . Let , . Then:
Note that the above formulas are the same as in the definition of satisfaction of modal operators in the (first-order) basic modal logic (see, e.g., [19,20]).
In this example, we will consider modal interpretations in the allegorywhere L is a locale, of locale-valued relations (see, e.g., ). In , objects are sets, and for any , the hom-setconsists of maps . Locale-valued relations are usually interpreted as fuzzy relations, where the value, , gives the degree to which the relation, R, between a and b actually holds. Composition forsuch that , is given by:
For all , the identity morphisms are given explicitly byifandif . The intersection, union and reciprocation operators are defined inby , , .
Let us now consider the formulas defining modal interpretation in the same setting as in Example 2, but withinstead of . Then:
Sometimes, the modal operators associated directly to the frame relations are not enough. For instance (see Example 4), the counterpart of until and since operators from temporal logic cannot be expressed using the formalism described so far. Therefore, we allow one to extend the set of terms, , to the set, , using an arbitrary set of constructors corresponding to the set of operators, Ω, instead of using only the constructors tied in a canonical way to frame relations and attaching to those operators arbitrary (though defined using frame relations) semantics.
Letbe a frame of similarity type , such that ≤ is a linear order. Letbe an allegorical conceptual data model. We extendtoby adding two new term constructors: For all termssuch thatand , also(until) and(since).
Let , and let . Then, the semantics ofandis given by:
where we denote . In particular, if , then for alland , those definitions boil down to the classical definition of the since and until temporal modalities:
Formal equalities between terms from can be used to specify the dependencies between data from different “worlds”, where the “world” can mean, depending on the application, a time moment or perhaps a database server instance. Here is the precise definition.
Let ; letbe a frame of similarity type , and let the modal interpretation . We say that a formal equalityis satisfied in modal interpretationatif , which we denote by . We say that a formal equalityis satisfied in modal interpretation , which we denote byiffor all .
4. Applications to Database Modeling
In this section, we consider how our formalism can be used to express business rules and queries in the context of temporal and distributed databases. In all the examples, we use the allegorical conceptual model depicted in Figure 1 and Figure 2.
4.1. Temporal Databases
Here, we work with frame of similarity type , where we denote , . We abbreviate and similarly for the other symbols. We assume that ≤ is a linear order on and that <, > and ≥ are dependent on ≤ in the usual way. We interpret the set of worlds, , as the time axis, with ≤ being the linear time ordering.
Consider the following business rule concerning the allegorical conceptual modeldepicted in Figure 1 and Figure 2: “A manager of a department must have worked in the department in the past and not as its manager”. For simplicity let us ignore the fact that this rule is impossible to realize for the new departments. We claim that this rule can be expressed as the following formal equality between terms in:
where we denoted for brevity . Indeed, let the modal interpretation , and let . Then:
is equivalent to the statement, that for all elementsand:
which is a direct first-order rendering of the desired business rule, where we read , as a is in relation b at the time m.
Note that in Equation (15), the negation was used in a “safe” way in accordance with Corollary 11.
Suppose that we want to create a temporal query (that is, a term in ), the evaluation of which at some momentwill return those employees, whose pay will not decrease in the future. One may use such a term as a part of a formal equality stating some business rule or as a query in a temporal database. We claim that the query can be expressed as follows:
Note the unfortunate clash of symbols here: in the above formula, we have the arrowfrom the graph, , and the relation symbol, . Any modal interpretation is bound to represent Term (16) as a coreflexive (that is contained in the identity) arrow. Coreflexive arrows are natural allegorical representations of subobjects (subsets) (see, e.g., ). In , if R is coreflexive, then by definitionand for all , we have . One can associate with R the subset . Conversely, with a subset , one can associate a coreflexive relationdefined by .
The correctness of the solution (16) of our problem depends on the following reasonable assumption: the objectshould be understood not as a collection of salaries really earned by someone, but rather as a (generalized) numerical domain (e.g.,NUMBER(6,2)). Accordingly, its interpretation (as well as the interpretation of the linear order,) should be constant, that is, we consider only those modal interpretationsofin which for all , we have:
As in the previous example, we consider the query to satisfy the desired semantics if the modal interpretation inturns out to be the translation into the first-order logic of the problem statement. Therefore, let us consider anysatisfying Conditions (17). Let T denote Term (16). In this setting,can be equivalently written using Definition 12 as:
4.2. Replicated Databases
Here, we consider applications of modal interpretations of allegorical conceptual data models to specifying properties of replicated databases. One of the frames a replicated database can be associated with is the frame of similarity type . Here, is the set of database servers, and means that d pushes data into . To avoid problems with circular pushes and consistency, we assume that the graph of ⇝ is acyclic and that that there exists a unique master, that is a database, , such that there exists no , such that . Applications can read any database, but only the master is directly modified. What remains is the specification of details of change propagation.
We want to specify the following rule: deletions from the relationare propagated within the updating transaction. However insertions are propagated after the end of the transaction. The solution can be easily seen to be as follows:
Consider now the frame of similarity type , where and . Here, , and ρ are like in SubSection 4.1, and ⇝ and are like above, except that both and ⇝ are extended in a natural way as relations on Cartesian product , e.g., and .
Here, we continue Example 7. Note that one cannot express the requirement that the inserts will be eventually propagated without temporal modalities, that is using the frame that was used in the previous example. One can do it, however, using the frame described above, combining the replicated database frame with the temporal frame from the previous subsection. The reader will easily verify that the appropriate equation is as follows:
We have presented a modal extension of a new allegorical conceptual data model, which interprets modal operators within an appropriate allegory. Our formalism can be seen as a generalization via the relational algebra of a limited form of a first-order modal logic with variable domains (see, e.g., ). We have given a few examples of applications of the modal extensions of our allegorical conceptual data model to specifying properties of temporal or distributed databases. We hope that those examples justify our conviction about the usefulness and expressiveness of our approach. More examples are needed, however, where the data should be naturally interpreted in allegories other than or for some locale, L.
It will be also interesting to compare our approach with that of , where it is shown how to enrich an arbitrary institution  with modalities and many worlds semantics. To make this comparison possible, we are currently working on the institutional formulation of our allegorical data model.
We would like to thank the reviewers for their helpful suggestions.
Bartosz Zieliński provided the main idea. Paweł Maślanka and and Ścibor Sobieski helped to develop it. All contributed to writing and editing.
Conflicts of Interest
The authors declare no conflict of interest.
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