1. BFO 2.0 Modularization and Verification
John Beverley

2. Big Picture for Today Current Axiomatization of BFO 2.0
Modularization Project
Semantic Verification Project (using COLORE)
Future Work

3. Axiom Alignment
Source for the FOL implementation of BFO 2.0 found in the BFO 2.0 Reference Manual (and other works)
With respect to these sources, BFO-FOL 2.0 is current; theorems proven, e.g. s-depends-on-at irreflexivity; located-in-at transitivity
Prover9 useful in automating theorem proofs; Mace4 useful for finding models of subsets of axioms

4. Finite Model Checking
In fact, Mace4 was able to find models for many subsets of axioms; notably the mereology underlying continuant/occurrent entities
These mereologies do not require unrestricted composition making finite models with over 4 atoms computationally feasible
Compare: Stronger mereologies require more sophisticated computational tools to prove meta-theoretic properties, evidenced by Mossakawski & Kutz on DOLCE

5. Displaying Dependencies
BFO-FOL axioms organized to display logical dependencies
Example: ‘roleOfAt’ relation logically depends on class ‘Role’ and relation ‘inheresInAt’, which in turn have logical dependencies
Focusing on logical dependencies; helps with:
Spotting axioms used by all/most classes/relations in BFO 2.0
Spotting redundant axioms
Spotting useful modularizations of the BFO 2.0 axioms (as discussed in next section)

6. Modularization Project
Modules: Extracted subsets of axioms from BFO-FOL 2.0
Example: Proper subset of BFO-FOL 2.0 axioms compose the “Taxonomy Module”, i.e. structural axioms governing all entities
Example: Proper subset of BFO 2.0 axioms compose the “Temporal Region Mereology Module”, i.e. axioms governing temporal region mereology
Example: Proper subset of BFO-FOL 2.0 axioms compose the “Exists At Module”, i.e. axioms governing the Exists At relation

7. Modularization Procedure
Start with the most general subset of axioms; proceed to next most general, and so on…
Generality is a heuristic characterized in terms of logical dependence; the proper subset of axioms on which all other subsets of axioms are logically dependent is the most general
The proper subset(s) of axioms on which no others depend, are the least general

8. Modularization Procedure Continued
Hypothesis: The BFO Taxonomy Module is the most general, followed in order of (loosely) decreasing generality:
Taxonomy Module
Exists At Module
Temporal Region Mereology Module
Occurrent Mereology Module
Continuant Mereology Module…

9. Example: Taxonomy
Subset of axioms governing taxonomy for BFO; sample axioms:

10. Example: Temporal Region Mereology
Subset of axioms applying to temporal regions; sample axioms:

11. Example: Exists At
Subset of axioms governing the exists at relation; sample axioms:

12. Semantic Verification Project
To semantically verify BFO we characterize its models up to isomorphism, and check whether these models are elementarily equivalent to its intended models (i.e. satisfy the same sentences as its intended models satisfy)
Claim: Intended models of BFO 2.0 are captured with the BFO-FOL 2.0 implementation
COLORE provides a background against which verification can be conducted

13. COLO(REpository) and Hierarchies
COLORE is an open repository of ontologies represented as sets of axioms in Common Logic (framework for family of logic languages)
COLORE consists of hierarches, or partially ordered finite set of theories (modules closed under entailment) with same signature, where:
Distinct theories in the same non-logical lexicon, may be compared based on conservative/non-conservative extension properties
Distinct theories in distinct lexicons, may be compared by definable equivalence for specific theories, and reducibility for sets of theories

14. Hierarchy Properties (same Signature)
Ordering Hierarchy houses theories concerning orderings of points; includes (among others):
Theory of Linear Ordering (theory of linear order over points)
Theory of Dense Linear Ordering (theory of dense linear order over points)
Dense Linear Point is a non-conservative extension of Linear Point (Linear Point is ‘agnostic’ about density)

15. Hierarchy Properties (different Signatures)
Relationships can be examined between theories in distinct hierarchies, via definable equivalence:
A theory is (model-theoretically) interpretable into another if there is a mapping from the former to latter that preserves theorems
Faithful interpretations are maps that preserve models too
Two theories are definably equivalent if there is a faithful mapping between their non-logical lexicons

16. Hierarchy Properties (different Signatures)
Relationships can be examined between theories within distinct hierarchies, via reducibility:
A theory is reducible to a set of other theories if there is a faithful interpretation between the former theory, and the latter set
Translation definitions between two theories are (roughly) axiomatizations of interpretations, and are useful in proving definable equivalence and reducibility

17. BFO Verification Preliminaries
COLORE includes various mathematical theories which will prove useful characterizing the models of subsets of BFO axioms
Axioms of BFO-FOL are not in COLORE; however, we can add BFO axioms, construct a BFO hierarchy, then ultimately use definable equivalence and reducibility to show relationships to theories in COLORE
I’ll focus on definable equivalencies between theories today

18. BFO/COLORE Observations
BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others):
The BFO Taxonomy constrains the same models as a COLORE “Theory of Lines”
The Temporal Region Mereology constrains the same models as the COLORE “Minimal Extensional Mereology”
Exists At constrains the same models as the “Lower MEM Weak Mereology”

19. Taxonomy Models: Setup
We show, following comparable work with DOLCE, the BFO Taxonomy semantically equivalent to the COLORE: Theory of Lines
We introduce translation defs between Taxonomy and Lines and vice versa (sample):
Entity(x) <-> (x=x)
Continuant(x) <-> L1…
Restricts the domain to entities, and translates ‘Continuant’ in BFO 2.0 to ‘Line 1’ in the COLORE Theory of Lines, and so on…

20. Taxonomy Models: Characterized
Given the appropriate two-way translation definitions, derivation of target axioms can be conducted:
In one direction, with the translation definitions from Taxonomy to Lines, we prove each axiom of the COLORE Theory of Lines
In the other direction, with the translation definitions that from Lines to Taxonomy, we prove each axiom of the BFO Taxonomy
Prover9 semi-automates both tasks, showing definable equivalence between the theories

21. Temporal Region Models: Setup
We next show the Temporal Region Mereology (TRM) semantically equivalent to the COLORE: Minimal Extensional Mereology (MEM)
We introduce translation defs between TRM and MEM and vice versa (sample):
TemporalRegion(x) <-> (x=x)
TemporalRegionPartOf(x,y) <-> part(x,y)…
Restricts the domain to Temporal Region, and translates ‘TemporalRegionPartOf’ in BFO 2.0 to ‘part’ in the COLORE mereology hierarchy

22. Temporal Region Models: Characterized
Given the appropriate two-way translation definitions, derivation of target axioms can be conducted:
In one direction, with the translation definitions from TRM to MEM, we prove each axiom of COLORE: MEM
In the other direction, with the translation definitions that from MEM to TRM, we prove each axiom of the BFO Temporal Region Mereology
Again, we use Prover9 to show definable equivalence between TRM and the COLORE theory MEM

23. Exists At Models: Setup
A more complicated example, we want to show Exists At semantically equivalent to a formal theory in COLORE…but which?
Following Gruninger & Chui work on DOLCE’s ‘present’ relation, what might be called the Lower MEM Weak Mereological Geometry seems a good place to start (but let me explain…)
Motivating Intuition: Entity existing at a time can be represented by a point incident to a line

24. Exists At Models: Setup
Lower MEM Weak Mereological Geometry is composed of theories from various hierarchies in COLORE (continued):
Mereological Geometry combines theories from the ordering, mereology, and incidence geometry hierarchies
Incidence relation concerns lines/points and is reflexive and symmetric
MEM is the underlying mereology (partial order, unique product, etc.)
‘Lower’ indicates parthood preserves incidence (e.g. if Entity x Exists At at Temporal Region t’, and Temporal Region t’’ is part of t’, then x Exists At t’’)

25. Exists At Models: Characterized
We show Exists At semantically equivalent to Lower MEM Weak Mereological Geometry (LWMG)
We introduce translation defs from LWMG to Exists at, and vice versa (sample):
Point(x) <-> (Continuant(x) v Occurrent(x))
Line(x) <-> Temporal Region(x)…
Definable equivalence shown as before with Prover9 and Mace4

26. Verification So Far…
The following definable equivalencies have been carried out (BFO modules on left side; COLORE on right):
BFO Taxonomy – Theory of Lines
Temporal Region – Minimal Extensional Mereology (MEM)
Exists At – Lower MEM Weak Mereology*
Occurrent Mereology – MEM
Continuant Mereology – Lower MEM Foliation*
*Indicates proposed theory extensions to be introduced to COLORE

27. Future Work
The remainder of BFO-FOL 2.0 needs to be modularized and verified…in works are Theory of Time and Theory of Specific Dependence
BFO theories should also be reduced, if possible, to existing theories in COLORE
Modularization and Verification of BFO-FOL 2.0 will allow for direct comparisons of semantic properties with other ontologies in COLORE (such as DOLCE and SUMO, which have been partially modularized and verified)

28. Future Work
COLORE + Hierarchies approach fleshes out the logical space of axioms, and allows information gleaned from well-known subsets of axioms to clarify other subsets of axioms
Hence, Modularization and Verification will also, likely, lead to clarification of classes of models for various subsets of axioms in general
And in particular, full verification will likely lead to a better understanding of BFO-FOL 2.0’s overall models