BFO 2.0 Modularization and Verification John Beverley

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

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

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

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)

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

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

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…

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

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

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

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

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

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)

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

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

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

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”

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…

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

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

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

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

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’’)

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

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

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)

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