Binary Relations in OWL: Generic and Specific

A General Challenge

OWL is a semantic web ontology language based on a guarded binary fragment of first-order logic (FOL). Restricting FOL to this guarded fragment provides computationally valuable properties, such as decidability. As a consequence of this restriction, however, OWL is less expressive than FOL. In particular, FOL sentences involving ternary relations cannot be straightforwardly expressed in OWL. Ternary relations, however, are a natural way to capture certain relationships among entities of a given domain during and over time. To illustrate, consider three examples from the biological domain. First, any brain is always part of the same host. That is, specific brains are permanently related to their host. Call this type of relatedness Permanent Specific Relatedness. Second, organic tissue is always composed of cells, but not necessarily the same collection of cells. That is, generic cells are permanently related to a given tissue. Call this Permanent Generic Relatedness. Lastly, certain organisms have wings at some time, but not necessarily at all times, of their development. That is, wings are temporarily related to a given organism. Call this Temporary Relatedness. In FOL, these varieties of relatedness are easily distinguishable with ternary relations (letting ‘R’ stand for a relevant ternary relation):

(1)   (x)(y)(t) [R(x,y,t)]           Permanent Specific Relatedness
(2)  (x)(t)(y) [R(x,y,t)]           Permanent Generic Relatedness
(3)  (x)(t)(y) [R(x,y,t)]            Temporary Relatedness

Not so in OWL. A well-known general challenge for any ontology represented in OWL modeling a domain in which these varieties of relatedness are considered distinct, is to differentiate (1), (2) and (3) with OWL’s limited resources.
Attempts to address the general challenge range from reification to extending the expressiveness of the logic underwriting OWL. We will not rehearse the advantages and disadvantages of these much-discussed proposals. Rather, we examine a recent attempt to address the general challenge in the context of the widely-used Basic Formal Ontology (BFO).

A Particular Challenge

BFO is a top-level, domain-neutral ontology used by biologists, among others, to provide a common starting point for the creation of domain ontologies in various areas of science. Given its wide purview, BFO is motivated to distinguish the preceding varieties of relatedness. BFO currently has two distinct formal language implementations, one employing FOL, BFO-FOL, and the other OWL, BFO-OWL. Where needed, BFO-FOL distinguishes among the varieties of relatedness with ternary relations along the lines of (1), (2), and (3). Clearly BFO-OWL must adopt a different strategy, making the general challenge salient for this implementation. The general challenge is made more difficult in this context, however, since developers adopt as a design principle that characterizations of BFO concepts in BFO-OWL be translatable into BFO-FOL. Hence, the general challenge in this context is to distinguish motivated varieties of relatedness represented in BFO-FOL with the limited resources of BFO-OWL, while ensuring there is a translation from the latter to the former. It is this particular version of the general challenge we address in what follows.
Only certain relationships among concepts in BFO are in purview of our particular challenge. Our motivating examples each plausibly concern parthood, and BFO adopts distinct primitive mereological relations among entities, one between continuant entities, and the other between occurrent entities. Roughly, continuant entities do not have temporal parts, and are disjoint from occurrent entities which do. Given the intended reading of the occurrent mereology in BFO, e.g. occurrent entities never gain or lose parts, distinguishing the preceding varieties of relatedness for the relation is unmotivated. This is reflected in BFO-FOL, where the occurrent parthood relation, named occurrentPartOf and restricted to entities of the class Occurrent, is binary and so has a straightforward translation from BFO-FOL into BFO-OWL. On the other hand, given the intended reading of the continuant mereology in BFO and observing our motivating examples are plausibly characterized as parthood among continuant entities at times, distinguishing among varieties of relatedness for this parthood relationship is desirable. In BFO-FOL, the continuant parthood relation, named continuantPartOfAt, restricts the first two entities to instances of the class Continuant and the third to instances of the class Temporal Region, a subclass of Occurrent. Since ternary, continuantPartOfAt does not have a straightforward translation from BFO-FOL to BFO-OWL. Hence, providing a translation of BFO-FOL’s characterizations of (1), (2), and (3) in BFO-OWL is within our particular challenge.

Temporally Qualified Continuant Strategy

There have been attempts to address the particular challenge. The “Graz Release” of BFO, for example, proposed an underdeveloped first pass solution - replacing ternary relations in BFO-FOL with tensed binary relations in BFO-OWL - which has since become a starting point for addressing the particular challenge. A recent proposal found here builds on the Graz recommendations introduces what advocates call temporally qualified continuants to BFO-OWL, an ontologically neutral class of computational artifacts. In more detail, advocates make the following recommendations for BFO-OWL:

(i) Universally tensed binary relation corresponding to the ternary continuantPartOfAt relation of BFO-FOL
(ii) Temporally qualified continuant individuals time-stamped with the temporal regions over which a corresponding continuant exists
(iii) Classes and Relations linking temporally qualified continuants to other BFO-OWL entities, such as temporal regions

Call this the Temporally Qualified Continuant proposal, or TQC. Each commitment deserves discussion.
Commitment to (i) stems directly from the Graz recommendation. However, where the Graz proposal advocated introducing two tensed binary relations to BFO-OWL for each ternary relation in BFO-FOL, reflecting universal and existential quantification over temporal indices, TQC adopts only the universally quantified versions. Proponents of TQC claim only universally tensed relations are needed to distinguish among varieties of relatedness in the presence of (ii) and (iii). To illustrate (i), assume for some pair of continuants, John and John’s Hand, it is the case that John’s Hand is continuant part of John for as long as John’s Hand exists. According to (i), this fact may be characterized in BFO-OWL with a universally tensed binary relation named continuantPartOf, satisfied (roughly) when the first and second entities stand in the continuant parthood relation at any temporal region at which the first entity exists.
Commitment (ii) is best introduced by example. Assume continuant John lives over 85 years. Then according to (ii), John has a corresponding temporally qualified continuant, John2000-2085, with accompanying 2000-2085 time-stamp. Additionally, John2000-2085 may have temporally qualified sub-continuants, corresponding to smaller time-stamps over which John exists. For example, John2000-2085 may have a temporally qualified continuant corresponding to JohnTuesdayFebruary22-2017, with a time-stamp smaller than and intuitively contained within John2000-2085.  Prima facie, commitment to (ii) appears to blur the BFO distinction between continuant and occurrent entities, but advocates of (ii) deny any ontological commitment. Rather, temporally qualified continuants are mere ad hoc entities designed to solve the particular challenge.
Concerning (iii), observe that formally, introducing temporally qualified continuant individuals hides complexity in a mere name, e.g. “John2000-2085” is indistinguishable from “John2000-2285”. Recovering complexity requires appealing to formal machinery, such as predicates and relations. Proponents of TQC reuse native BFO concepts to that end where possible. As above, we may introduce without much loss BFO concepts as they are characterized in BFO-FOL. For example, TQC reflects the BFO-FOL class Material Entity, restricted to certain continuant entities which have matter as parts, such as John. Instances of Material Entity bear the useful BFO-FOL hasHistory binary relation to unique instances of the BFO-FOL class History, occurrent sums of processes transpiring in the spatiotemporal region a Material Entity occupies. TQC extends BFO-FOL, however, by adopting the temporally qualified continuant class Tqc, instances of which bear the binary tqcOf relation to a corresponding instance of Material Entity. Each Tqc may be divided into instances of the novel class Phase which bear the novel phaseOf relation to the Tqc (an inverse hasPhase is defined as one would expect). Relevant histories of material entities are then bridged to phases through the binary hasOccurrentPart of BFO-FOL. Diagrammatically (borrowed from the cited paper):

                                         Figure 1. Overview of TQC

                                         Figure 1. Overview of TQC

Altogether, TQC represents material entity John who lives between 2000 and 2085 as having a unique history, the sum of all processes occurring in the spatiotemporal region John occupies, which, we may say without loss of generality, has phase occurrent parts, such as John2000, John2001, John2002, etc., that are the phases of a Tqc John2000-2085, itself the tqcOf John. 

TQC Strategy Applied

Adding these commitments to BFO-OWL permits the following characterizations of varieties of relatedness involving continuant parthood (we include characterizations in BFO-FOL for comparison):

(FOL-1)     (x)(y)(t)(continuantPartOfAt(x,y,t))           
(TQC-1)  (x)(y)(continuantPartOf(x,y))    
(FOL-2)    (x)(t)(y)(hasContinuantPartAt(x,y,t))          
(TQC-2)  (x)(y)(z)(w)(u)(hasHistory(x,y) & hasOccurrentPart(y,z) & phaseOf(z,w) & hasContinuantPart(w,u))
(FOL-3)    (x)(t)(y)(hasContinuantPartAt(x,y,t))           
(TQC-3)  (x)(y)(z)(w)(u)(hasPhase(x,y) & hasOccurrentPart(y,z) & phaseOf(z,w) & continuantPartOf(w,u))

Recall, our motivating example for Permanent Specific Relatedness, reflected in BFO-FOL as (FOL-1), is every brain is always continuant part of the same host. Observe, (TQC-1) captures this relationship by appealing only to the binary tensed universal continuant parthood relation. Turning to Permanent Generic Relatedness, reflected in BFO-FOL with the inverse of continuantPartOfAt, named hasContinuantPartOfAt, as (FOL-2), our motivating example was every tissue has some cell as part at all times. Observe, with (TQC-2) this becomes every tissue has a history which has a phase as occurrent part, which is the phase of some temporally qualified continuant that has some continuant cell part at all times. Turning finally to Temporary Relatedness, reflected in BFO-FOL as (FOL-3), our motivating example was organisms having wing parts at some, but not all, portions of their development. Observe with (TQC-3) this becomes every organism has a history with a phase as occurrent part that is the phase of some temporally qualified continuant which is itself continuant part of some continuant wing.
In short, proponents of TQC claim the binary universal tensed relation is adequate for (1), and distinguish (2) and (3) in terms of whether a given temporally qualified continuant has a continuant part or is a continuant part of some relevant continuant. The TQC strategy thus appears to address one aspect of our particular challenge.

Relationship to BFO-FOL

However, our particular challenge requires an adequate characterization of varieties of relatedness in BFO-OWL be translatable into BFO-FOL. With respect to BFO-FOL, the path to (ii) can be understood as a series of relation parametrizations resulting in additions to the domain.  Roughly, to parametrize a relation is to define a lower arity relation with satisfaction conditions dependent on the higher arity relation. To illustrate, consider the sentence “John’s hand is part of John on Tuesday” characterized with the BFO-FOL continuantPartOfAt relation:

(I) continuantPartOfAt(John’s Hand, John, Tuesday)

Which, assuming standard first-order semantics, is satisfied (roughly) iff the ordered triple <John’s Hand, John, Tuesday> is a member of a subset of DxDxD, where “D” denotes the domain. This ternary expression can be parametrized by introducing a binary relation with satisfaction conditions tied to the ternary relation. Parametrizing with respect to time (and replacing ‘at’ in the name for readability) we have:

(II) continuantPartOfTuesday(John’s Hand, John)

Satisfied iff the ordered pair <John’s Hand, John> is a member of a subset of DxD. We might continue parametrizing, this time with respect to John, resulting in:

(III) continuantPartOfTuesdayJohn(John’s Hand)

Satisfied iff <John’s Hand> is a member of a subset of D. Finally, we might complete the parametrization by introducing an individual:

(IV) continuantPartOfTuesdayJohnJohn’sHand

Satisfied iff the individual is a member of D. Parametrization resulting in an individual is called total.  Otherwise, the parametrization is called partial. (II) and (III) are thus partial parametrizations, and (IV) a total parametrization.
Our choice of example was only illustrative, of course, since rather than parametrizing ternary BFO-FOL relations, TQC replaces them. Nevertheless, temporally qualified continuants may be understood as the result of parametrizing the binary existsAt relation of BFO-FOL, with the domain understood as restricted to continuants (rather than applying to all entities), and the range restricted to temporal regions (and so unchanged). For example, in BFO-FOL the sentence “John exists during 2000 and 2085” might be characterized as (where “2000-2085” names a temporal region):

(V) existsAt(John,2000-2085)

Satisfied iff <John,2000-2085> is a member of CxT, where C and T are, respectively, continuant and temporal sorts of D. Total parametrization results in the individual:

(VI) existsAtJohn2000-2085

Satisfied iff the individual is a member of D. If we assume the existsAt relation is implicit for any individual, then we can, without loss, drop this portion of the name, resulting in the familiar:

(VII) John2000-2085

More generally we might represent individuals such as “John2000-2085” with the variable notation reflecting continuants existing at temporal regions as: xt, yt, zt, with “t” subscripts reflecting temporal indices. These observations suggest some reason to think individuals postulated by the TQC strategy may be translated into BFO-FOL. 
Additionally, commitment to (i) has a clear relationship with BFO-FOL, as straightforward translations for tensed binary relations in BFO-OWL follow the pattern:

R(x,y) =def (t)(existsAt(x,t) & existsAt(y,t) & R(x,y,t))

R(x,y) =def (t)(existsAt(x,t) & existsAt(y,t) & R(x,y,t))

In light of our observations concerning (ii), we might then characterize TQC’s commitment to (i) as accepting only the first definitional pattern, replacing the existsAt conjuncts with our temporally qualified continuant notation:

            R(x,y) =def (t)(R(xt,yt,t))

Again, these observations provide some reason to think were BFO-OWL to accept (i) and (ii), the resulting implementation would be translatable into BFO-FOL.
The same cannot be said for commitment to (iii). To be sure, some proposed TQC classes and relations have apparently straightforward translations into BFO-FOL, e.g. the class Phase appears only terminologically distinct from a certain BFO-FOL class discussed below. Others, however, have neither obvious parallels in nor translations to BFO-FOL, e.g. the class Tqc, the binary relation tqcOf, etc. This is perhaps to be expected for computational artifacts, but it is a noteworthy cost that TQC commitment to (iii) appears to conflict with a major design principle adopted by BFO developers. Proponents of TQC will likely reply these computational artifacts are needed to address the particular challenge, and that the native machinery of BFO-FOL is inadequate for the job. Hence, they might continue, the cost is worth paying.
But such a response exaggerates the need for these computational artifacts. We demonstrate why in a follow-up post, where we introduce an alternative strategy with a clear translation into BFO-FOL.