We propose an axiomatic ontological framework for informational templates based on former works on slot-mereology. Templates and their parts can be filled by information content entities, but can exist even if unfilled. Thus, we represent templates as slots that can themselves have slots, where the “slot of” relation between slots can be interpreted as a parthood relation. Some slots of a slot are mandatory and need to be filled for the parent slot to be filled, whereas others are optional and do not have this restriction. The structure of a filler is also described by slots of a different nature. We discuss how parthood between slots can be connected with parthood between their fillers.
The ontology of informational entities has been investigated from a variety of perspectives and
general ontological frameworks [
Templates are of prime importance to characterize documents, especially documents that have a normative dimension. Consider a diploma. It typically encompasses a first name and a last name. However, we would want to be able to represent the fact that such a diploma could encompass a middle name. In open world systems, such facts are typically represented by closure axioms: for example, we could formally state that a diploma has as direct parts only the following: first name, middle name, last name, title of the grade, etc. However, such a formalization assumes that we know exactly all direct parts a diploma can have. By contrast, there might be cases where we know that a diploma must include a first name and a last name, could include a middle name in some cases, but might also include other unknown parts. In such situations, a closure axiom would not be adapted anymore.
An alternative, more expressive way would thus be to characterize a template for diplomas (rather than the diplomas themselves), stating that it has fields for last name and first name that must be filled in any diploma based on this template, and a field for middle names that may be filled or not. In such a representation, we can leave it open whether the template encompasses other unknown fields or not in an open world formalization. Thus, it is desirable to be able to represent not only filled documents but also templates, and the relations between both.
A former work [
To account for the fact that a document of a certain type must encompass some fields and can encompass other fields, we can introduce a distinction between slots in a template that are mandatory and others that are optional. For example, a full name slot would have as mandatory slots a first name slot and a last name slot, and as optional slot a middle name slot. It could be filled by a full name filler that would have as slots only a first name slot and a last name slot.
Therefore, we will propose an ontology of slots and fillers where the slot structure of a slot does not need to be identical to the slot structure of its filler, and where a slot can have both optional and mandatory slots. The next section will specify the methods used. A third section will propose an axiomatic representation of templates, independently of their fillers. A fourth section will axiomatize parthood between fillers. A fifth section will propose theoretical and realistic models based on clinical documents. A sixth section will discuss how this system could be extended in various directions. A conclusion will follow.
In this paper, we will work in transitive closure logic, that is, FOL enriched with the operation of
transitive closure TC, where the transitive closure of a predicate X will be written “CX” [
All the axioms reflecting the taxonomic structure in Figure 1 will be accepted, that is, if A is a subclass of B, we accept Aa → Ba. We will introduce the following classes (unary predicates), that will be explained below:
Informational slot [IS]
Slot of filler [ISF] Slot of slot [ISS]
Unattached slot [ISU]
Informational filler [IF]
We consider templates as “unattached slots” or U-slots (ISU), that are not slots of anything – that is, they do not belong to anything (we leave open here the question of whether all U-slots are templates, as well as whether all documents follow a template, at least an implicit one). Such slots can themselves have slots, which can themselves have slots, etc. This latter kind of slots is called “slot of slots” or S
We accept the binary relations summarized in Table 1, where R has domain X and range Y is translated as: Rab → Xa & Yb.
We accept an axiom of disjointness between fillers and slots: (A1) Disjointness ¬(ISa & IFa)
and an axiom of covering of all entities by fillers and slots:
(A2 )- Covering ISa ∨ IFa That is, slots and fillers form a partition of the domain of quantification considered in this paper.
Five of the above-mentioned unary and binary predicates are primitive: IS (“is a slot”), S (“slot of”), MS (“mandatory slot of”), OS (“optional slot of”) and F (“fills”). All the others will be derived from those five predicates. In particular, the four derived classes in the above taxonomy depicted in Figure 1 can be defined as follows: - Fillers are entities filling a slot: (D1 )- Filler definition IFx:=def ∃t Fxt - U-slots are slots belonging to nothing: (D2) U-slot definition ISUs:=def ISs & ∀a ¬Ssa - S-slots are slots belonging to a slot: (D3 )- S-slot definition ISSs:=def ∃t Sst & ISt - F-slots are slots belonging to a filler: (D4) F-slot definition ISFs:=def ∃x Ssx & IFx
ISs ↔ [ISSs ∨ ISFs ∨ ISUs]
Note that fillers can only exist in slots, but slots can exist independently of fillers, as argued for in
former work [
Second, F-slots belong only to fillers: (A4) F-slot owner ISFs & Ssx → IFx
We can then deduce that slots are exactly S-slots, F-slots and U-slots: (T1) Covering of IS Proof: The right-to-left inference is trivial. For the left-to-right inference, suppose that ISs. If there is no a such that Ssa, then ISUs. If there is some a such that Ssa, then a is either a slot or a filler by A2. If a is a filler, then ISFs. If a is a slot, then ISSs. QED
We can also show that the three kinds of slots are disjoint: (T2) S-slots, F-slots and U-slots are disjoint
¬(ISSs & ISFs) & ¬(ISSs & ISUs) & ¬(ISFs & ISUs) Proof: Consider s such that ISFs, thus there is some filler x such that Ssx. By definition D2 of ISU, ¬ISUs. Thus, ¬(ISFs & ISUs). If ISSs, there is some slot t such that Sst. Since ISFs, by A4, t is a filler. Thus t is both a slot and a filler: absurd by A1. Thus ¬(ISSs & ISFs).
Consider now s such that ISSs. By definition D2 of ISU, we have ¬ISUs, thus ¬(ISSs & ISUs). QED
Thus, S-slots, F-slots and U-slots form a partition of slots. We will now turn in the next part to the analysis of mere slot structures, independently of their fillers.
As mentioned above, slots can themselves have slots, as represented by the relation S. Consider for example a template “complete name” that is an unattached slot ‘cn0[]’, which has as slots ‘fn0[]’ (for “first name”) and ‘ln0[]’ (for “last name”), and can exist even if it is not filled.
We define the transitive closure of the S relation: CS. (D5) Slot_of transitive closure CS :=def TC(S)
We will say that s is a “C-slot” of t when CSst. As we will see, CS will be interpreted as a relation of proper parthood between slots, and thus we want S to be irreflexive (nothing is a slot of itself). Indeed, we do not want to have cycles by the S relation such as represented on Figure 2.a. (A5) Slot_of closure irreflexivity ¬CStt
That is, slots are structured in directed acyclic graphs by the relation S. This implies trivially that S is irreflexive. It also implies that S is asymmetric (two slots s and t cannot own each other): (T3) Slot_of asymmetry Stu → ¬Sut Proof: If Stu and Sut, then CStt by definition D5: absurd by A5. QED
We can now turn to relational properties of slots of slots, namely being mandatory or optional relatively to their owner. First, slots of slots (and only those) are either mandatory or optional relatively to their owners: (A6) Slots of slots are mandatory or optional (ISt & Sst) ↔ (MSst ∨ OSst) (in the models graphically represented in this paper, either S or MS/OS relations will be pictured, keeping in mind that if S holds between two slots, then either MS or OS holds, and vice versa; also, CS relations will not be pictured but can easily be inferred from S relations)
Second, no slot is both optional and mandatory relatively to the same owner: (A7) Optional and mandatory slots are disjoint ¬(OSst & MSst)
Note however that a slot can be optional relatively to a slot, and mandatory relatively to another slot: the mandatory or optional characters of a slot are relational properties. On Figure 2.b, for example, u is a mandatory slot for t, but an optional slot for s. A realist model would consist of t being a slot for a name that must encompass a last name slot (v) and can – but must not – encompass a first name slot (u); and s being a slot for a name that must encompass both a first name slot and a last name slot. 3.3.
Contrarily to what was assumed in former work [
To take a clinical example inspired by an ontology of drug prescriptions [
Note that we do not want to impose the functionality of S here, that is, that slots would have only one owner. Indeed, it would exclude slot-overlap as presented on Figure 3.a, which might be acceptable. A realistic model of Figure 3.a would be e.g. having u being a slot for a day number (between 1 and 31, e.g. 24), v a slot for a month (e.g. October), and w a slot for a year (e.g. 1981) (t being a slot for a day in a month without specifying the year, e.g. October 24, t’ a slot for a month of a specific year, e.g. October 1981, and s a slot for a specific day of a specific month of a specific year, e.g. October 24 1981). Another example of slot-overlap, this time in the clinical domain, would be a template for a drug prescription of several drugs (suppose here there are only two) made by the same prescriber: such a document is composed by two slots t and t’ that should each be filled by the information about the prescriber and the prescription of a single drug, and where t and t’ overlap on the prescriber slot v (u being the slot for the prescription of the first drug, and w the slot for the prescription of the second drug).
CS can be seen as a proper parthood relation on the domain ISU ∨ ISS. It already satisfies axioms of
strict partial order: it is irreflexive, asymmetric and transitive. Arguably, for it to be an authentic proper
parthood relation, it must also satisfy some supplementation axiom [
C-SOst:=def (ISs & ISt) & (s=t ∨ CSst ∨ CSts ∨ ∃u (CSus & CSut))
An axiom of C-slot weak supplementation would be the following: if t is a C-slot of s, there is a Cslot of s that does not C-slot-overlap t: (A8) C-slot weak supplementation CSts → ∃u (CSus & ¬CSOut)
Note that we do not want a stronger axiom of slot weak supplementation on S as follows (where SO would be defined like CSO, replacing CS by S): Sts → ∃u (Sus & ¬SOut). Indeed, the model presented above in Figure 3.a is intended, but does not satisfy the slot weak supplementation. As a matter of fact, in this model, Sts holds, but there is no slot a such that Sas and ¬SOta (since the two only slots of s do S-overlap). However, the model does satisfy the C-slot weak supplementation.
Do we also want an axiom of strong supplementation by CS as follows: ¬CSst → ∃u (CSus & ¬CSOut)? The answer is negative, as this would imply extensionality, and thus prohibit models such as the one presented on Figure 2.b. Note that we could even accept a model like on Figure 2.b but where u and v would be mandatory slots of both s and t. Indeed, templates might have the exact same slots but with different additional relations holding between those slots. For example, we might want to accept two templates for complete names ‘cn[]1’ and ‘cn[]2’ which would have the same two mandatory slots ‘first_name[]’ and ‘last_name[]’, but in different orders (where such order relations would be ternary: ‘first_name[]’ is before ‘last_name[]’ in ‘cn[]1’, and ‘last_name[]’ is before ‘first_name[]’ in ‘cn[]2’).
The slot structure of fillers is typically different from the slot structure of slots. Indeed, since a slot
of a filler cannot itself have slots (by A3 and T2), on fillers, CS collapses to S. On F-slots, S and CS
never apply and CSO collapses to identity. This implies that on fillers, C-slot weak supplementation
collapses to the following theorem of “company” (reusing the vocabulary of Varzi [
However, a slot of a filler can be filled by another filler that has itself slots: this is how a filler can have a mereological structure spanning several levels, and how the slot structure of a template can be mirrored in a filler of this template (as will be discussed in Section 6). Let us now turn to more substantial statements on fillers. 4.1.
We adapt a first axiom former works [
This implies that when we informally say that the “same” template is used for several documents constituted by different fillers, according to this theory, we are actually referring to different U-slots that are similar by being instances of a same class of templates. Indeed, a slot can have at most one filler. On the other hand, a filler can occupy several slots: an entity can “have a part twice over”, as Bennett claims.
Like in traditional slot mereology [
Pxy:=def PPxy ∨ (IFx & x=y) 4.3.
Former work [
Since we defined proper parthood as the transitive closure of direct proper parthood, proper parthood is trivially transitive: (T5) Proper parthood transitivity PPxy & PPyz → PPxz
From this, we derive trivially parthood transitivity: (T6) Parthood transitivity Pxy & Pyz → Pxz
Former works [
Finally, parthood reflexivity is a trivial consequence of the definition of parthood: (T7) Parthood reflexivity IFx → Pxx
This implies proper parthood asymmetry: (T8) Proper parthood asymmetry PPxy → ¬PPyx Proof: If PPxy and PPyx, then PPxx by transitivity of PP: absurd by A10. QED
From this, we derive parthood anti-symmetry trivially as usual: (T9) Parthood anti-symmetry Pxy & Pyx → x=y Proof: Suppose that Pxy and Pyx. By T8, either ¬PPxy or ¬PPyx. Thus, we have x=y. QED 4.4.
(A11) Slot of a filler is filled IFx & Stx → ∃y Fyt
We can deduce trivially from this axiom that a slot of a filler is always filled by a proper part of the
filler:
(T10) Slot of a filler is filled by a proper part IFx & Stx → ∃y (Fyt & PPyx)
For example, the filler ‘John Doe’ has a slot for the first name ‘fnJD[]’ filled by ‘John’. As in former
work [
ISt & (∃s Sst) & [∀u (Sut → ∃x Fxu)] → ∃y Fyt Note that we need the condition “(∃s Sst)” in A12, otherwise the condition ∀u (Sut → ∃xFxu) would be trivially true for any slot that does not have any slot, and thus such slots would always be filled. This axiom excludes models such as represented on Figure 3.b, where t is not filled, despite all its sub-slots u and v being filled.
However, the reciprocal is not true: a filled slot does not need to have all its slots filled. Indeed, a slot can be filled while some of its optional slots are not filled – this is, as a matter of fact, the motivation for introducing optional slots. Moreover, a slot that has all its mandatory slots filled is not always filled. For example, if someone named ‘John D. Brown’ fills the mandatory slots for the first name and for the last name of a full name slot, but does not fill the optional slot for the middle name – that is, if he writes only ‘John Brown’, then he has not filled the full name slot.
However, we claim that if a slot is filled, then all its mandatory slots are also filled2: (A13) Mandatory slots of a filled slot are also filled
∃y Fyt → ∀u (MSut → ∃x Fxu)
We also impose that if a slot that has slots is filled, then at least one of its slots is filled. (A14) Some slot of a filled slot with slots is filled
(∃t Sts) & Fxs → ∃u,y (Sus & Fyu) (note the importance of the proviso ∃s Sts, as otherwise a slot that has no slot could never be filled)
This excludes models in which a slot that would have only optional slots could be filled without any of its slots being filled, such as pictured on Figure 3.c. For example, in an informal registry of a small classroom, there could be a slot “name” with two optional slots “first name” and “last name” (that is, people could be referred either by their first name or by their last name). However, it is not possible in such a registry that a name would be indicated without indicating either the first name or the last name. A14 ensures that this constraint is satisfied.
It enables also to represent a situation where either a first name and last name (or both) must be specified, and a middle name can (but must not) be specified: this can be represented with the templates having a first mandatory slot that has two optional slots, one for the first name and one for the last name, and a second optional slot for the middle name.
The theory was implemented in Alloy4 [
2 An alternative formalization would be to introduce a neutral element ∅ such that all slots of a filled slot are filled, but its optional slot can be filled by this neutral element. However, the ontological status of such a neutral element would raise complex issues. For example, we might not want to endorse that optional slots are pre-filled with the neutral element, since we want slots to be able to exist even if unfilled. But then, does it imply that when someone fills a template, she fills each optional slot with a filler (either the neutral element or something else)? Then, how do we account for situations where the author of the filler did not realize the existence of such slots?
Such a system might be applied to clinical documents. For example, every drug prescription
specifies a drug product; however, some drug prescription specify a route of administration, but other
do not [
Another example would concern stays at a healthcare institution. A (simplistic) template t for such documents could have three slots: s1 for the unit of care, s2 for the starting date and s3 for the end date. One could have t owning s1 and s2 as mandatory slots and s3 as optional slot, such that s3 would be filled for terminated stays and would be left unfilled for ongoing stays. An alternative template structure would be to introduce two intermediate optional slots of t, t1 for terminated stays and t2 for ongoing stays, such that t1 has as mandatory slots s1, s2 and s3 (as described above) and t2 has as mandatory slots s1’ (also for the unit of care) and s2’ (also for the starting date). The second template might be more desirable to make sure that no user would leave an end date slot unfilled when entering the information for a terminated stay.
This work can be completed in several directions that will be described now. 6.1.
This theory would need to be further developed to avoid all unwanted models. In particular, this work would need to be completed with an analysis of how the intrinsic structure of a slot constrains the structure of its filler. For example, if a template has as mandatory slots a first name slot and a last name slot, then any filler of this template must encompass two parts that respectively fill those slots. On the other hand, a filler can have some slot structure that is not reflected in the template. For example, in a document template, there could be a slot for names that has no slot itself; it could be completed by a filler “John Brown” which has itself one F-slot for the first name “John” and one F-slot for the last name “Brown”, even if there is no equivalent in the template.
We could for this introduce a relation of mirroring between the slots of a slot and the slots of a filler.
For example, if s is a slot filled by f, and s has two mandatory slots, then f would have two mirrored
slot that are filled. A slot and its mirror slot would be always filled by the same filler. Thus, the
mandatory structure of the slot would be found in its filler. This is one motivation for focusing the
formal system on a non-transitive “slot of” relation rather than on its transitive closure (as was done in
[
In this framework, we can propose a solution to the counting problem different from the one
proposed by Tarbouriech et al. [
Thus, if x is part of y, we cannot count the number of times that x is part of y by counting the number of slots of y that x fills. Instead, we propose the following: the mereological structure of a filler can be represented by a directed acyclic graph (whose edges are constituted by the relations S and F), and to know how many times y is part of x, we count the number of paths that link y to x.
For example, in Figure 5, y is part of x twice as it fills two slots (s and t) of x. y has as parts z and
w, each of which is part of x twice, as there are two chains by S and F that link z to x and w to x. In
Bennett’s mereology, y would also be part of x twice as it fills two slots of x; however, z
(respectively w) would be (incorrectly) part of x only once, as the slot u (respectively v) would be
inherited only once by x (as explained by Tarbouriech et al. [
The topic at hand can be related with the work on documents and their structures in the digital document traditions, among which the most well-known is the Text-Encoding Initiative standards. The Text-Encoding Initiative offers a variety of slot-like tags such as <speaker> (to tag the mention of the speaker in a theater play), <quote> or <note>. However, the TEI is geared mostly at the analysis of texts, whereas our formalism aims first and foremost at representing templates that might be filled or not. Also, our focus here was on the mereological axiomatization of such a system, something which is, as far as we know, absent from the TEI; and indeed, inside a tag, not all parts of a texts need to be further tagged (whereas all parts of a text need to fill a slot in our system). There is thus a potential to strengthen the foundations of TEI thanks to the theory proposed above. 6.4.
Slot could also enable to add a semantic layer to fillers. For example, the same filler “diabetes” could be used in several clinical documents: a first time as a diagnostic for a given patient, another time as a diagnosis of a family member of the patient, and a third time as a possible diagnosis that should be ruled out by a test. The filler “diabetes” refers to the disease in all the situations, but the slots in which it can be found can add this semantic layer to each occurrence of the term.
We have proposed a mereological system for informational templates. The relation S (slot of) defines
a mereology on templates. It also applies on fillers and enables to define direct proper parthood between
fillers as filling a slot. This system eschews the limit of a former system [
[14] C. Masolo, L. Vieu, Graph-Based Approaches to Structural Universals and Complex States of Affairs, in: S. Borgo, P. Hitzler, O. Kutz (Eds.), Formal Ontology in Information Systems: Proceedings of the 10th International Conference (FOIS 2018), IOS Press, 2018, pp. 69–82. [15] K. Koslicki, The structure of objects, Oxford University Press, 2008. [16] M. Brochhausen, M.B. Almeida, L. Slaughter, Towards a formal representation of document acts and resulting legal entities, in: Johanssonian Investigations: Essays in Honour of Ingvar Johansson on His Seventieth Birthday, Walter de Gruyter, 2013.