BFO 2.0: Axiomatization, Modularization, and Semantic Verification

BFO 2.0: Axiomatization, Modularization, and Semantic Verification
John Beverley

Big Picture for Today The Basic Formal Ontology, and Axiomatization in First Order Logic Modularization of the Axioms Semantic Verification of the Axioms Summary and Future Work

The Basic Formal Ontology (BFO)
Everything you think you know about BFO is….

The Basic Formal Ontology
Everything you think you know about BFO is…. Probably about right, but here’s a refresher.

The BFO 2.0 Taxonomy

Some Important Distinctions
Continuant Entities vs. Occurrent Entities Dependency vs. Independency Types vs. Instances

Continuant vs. Occurrent
Continuants are enduring entities, wholly present whenever they are, and which lack temporal parts Continuant thing, quality …

Continuant vs. Occurrent
Occurrents are perduring entities, not wholly present (usually) at a single instants, and which have temporal parts Occurrent process, event Continuant thing, quality …

Dependence vs. Independence
Continuant Occurrent process, event Independent Continuant thing Dependent Continuant quality e.g. temperature depends on a bearer

Dependence vs. Independence
Continuant Occurrent process, event Independent Continuant thing Dependent Continuant quality, … e.g. an event depends on some participant

.... ..... ....... Types vs. Instances Continuant Occurrent
process, event Independent Continuant thing Dependent Continuant quality

.... ..... ....... Types vs. Instances types Continuant Occurrent
process, event Independent Continuant thing Dependent Continuant quality

.... ..... ....... Types vs. Instances types instances Continuant
Occurrent process, event Independent Continuant thing Dependent Continuant quality instances

Dependence vs. Independence Again
Continuant Occurrent process Independent Continuant thing Dependent Continuant quality temperature depends on bearer

Instance of Independent Continuant
Occurrent process Independent Continuant thing Dependent Continuant quality temperature depends on bearer John

Instance of Quality Dependent on…
Continuant Occurrent process Independent Continuant thing Dependent Continuant quality temperature depends on bearer John John’s 101 Degree Temperature

Instance of Process Dependent on…
Continuant Occurrent process Independent Continuant thing Dependent Continuant quality temperature depends on bearer John John’s 101 Degree Temperature John’s running process

Example Binary Relations
Type-Type Relations human is_a mammal human heart part_of human Instance-Type Relations John instance_of the type human Sally allergic_to the type codeine Instance-Instance Relations John’s heart part_of John John’s aorta connected_to John’s heart

More Basic Formal Ontology Details
Basic Formal Ontology History Developed by Smith/Grenon Based, in part, on Theory of Granular Partitions of Smith/Bittner/Donnelly Designed as upper level, domain neutral, ontology Computationally tractable fundamental ontology Upper level ontologies are response to data silo problems Also allow easy computational inferencing due to class/subclass structure

BFO Versioning Developing BFO is a community effort, see here:
ontology/BFO/master/releases/2.0/bfo.owl BFO has undergone significant changes since introduced Recent version, 2.0, codified in a reference document, and MIT Press published “Building Ontologies with the Basic Formal Ontology”

BFO Implementations A BFO implementation is the realization of a codified BFO technical specification (e.g. reference manual) as a program or as a software component Implementations: OWL – Used in web development/Protégé FOL – Commonly used; theorem prover friendly CLIF – Common logical language (FOL with abstract semantic language)

BFO 2.0 Implementation New versions require either new implementations, or updating old implementations Given BFO 2.0 changes, implementations are: OWL – Description Logic Base FOL – First Order Logic CLIF – Common Logic

BFO 2.0 Implementation New versions require either new implementations, or updating old implementations Given BFO 2.0 changes, implementations are: OWL – Description Logic Base (up-to-date!) FOL – First Order Logic CLIF – Common Logic

BFO 2.0 Implementation New versions require either new implementations, or updating old implementations Given BFO 2.0 changes, implementations are: OWL – Description Logic Base (up-to-date!) FOL – First Order Logic (up-to-date!) CLIF – Common Logic

BFO 2.0 Implementation New versions require either new implementations, or updating old implementations Given BFO 2.0 changes, implementations are: OWL – Description Logic Base (up-to-date!) FOL – First Order Logic (up-to-date!) CLIF – Common Logic (given FOL implementation, easily generated)

BFO 2.0 Implementation New versions require either new implementations, or updating old implementations Given BFO 2.0 changes, implementations are: OWL – Description Logic Base (up-to-date!) FOL – First Order Logic (up-to-date!) CLIF – Common Logic (given FOL implementation, easily generated) The work I’ll discuss today concerns the FOL implementation…

Axiom Alignment The FOL implementation is generated by perusing BFO source material, and representing philosophical claims in first order logic To my knowledge, all claims in the literature concerning BFO have been axiomatized, i.e. represented as axioms in FOL But let’s look at an example of what I mean…

Example: Continuant Axiom
Perusing the reference manual, you’ll find claims like the following: “Every continuant is an entity.” Which can be represented in FOL as:

Theorem Alignment Additionally, theorems are often stated (in the source material) to follow from given philosophical claims, and these must be proven To my knowledge, all claimed theorems have been proven…by me and some relatively new, and very powerful, software:

Theorem Alignment Additionally, theorems are often stated (in the source material) to follow from given philosophical claims, and these must be proven To my knowledge, all claimed theorems have been proven…by me and some relatively new, and very powerful, software Prover9 is useful in automating theorem proofs, and it comes bundled with Mace4 which is useful for finding models of subsets of axioms

Prover9: What? FOL axioms can be used to prove theorems about BFO 2.0 within Prover9 Prover9’s user interface

Toy Example: Sentential Logic
Sentential logic input illustrated with MP: Set Goal(s):

Toy Example: Sentential Logic
Sentential logic MP ‘resolution’ proof:

Resolution Proof? Common automated prover technique.
Claims ‘clausified’ in conjunctive normal form Goal is negated (for reductio ad absurdam) Contradictions sought and resolved A formula is said to be in conjunctive normal form when it is expressed as a series of disjunctions connected by conjunctions. Automated theorem provers (such as Prover9) use a narrowed definition where each formula is represented as clauses. A clause is a disjunction or atomic letter.

Toy Example: FOL First Order Logic input illustrating DS: Set Goal(s):

Toy Example: FOL Disjunctive Syllogism resolution proof:
Quantifiers are implicit in resolution methods; Universals are understood while Existentials are replaced by Skolem functions, i.e. a function that maps, via a functional rule, a stand-invariable for the variable being replaced when evaluated for satisfaction. The function replacement (Skolematization) does not produce an equivalent formula of course, but the produced formula is equally satisfiable, i.e. it is satisfiable iff the original formula is.

Time to Put Away the Toys: BFO 2.0
BFO 2.0 modularized txt files can be uploaded to prover9:

Generating axioms (and associated Reference Manual citations + comments) This is the Continuant_Axiom file.

Look! Here’s our example from earlier, all continuants are entities… This is the Continuant_Axiom file.

BFO 2.0-FOL Theorem: S-DependsOnAt
But let’s not waste time with nostalgia; let’s get provin’ ‘Specifically Depends On At’ is a BFO 2.0 relation Holds between a,b,t such that b and a share no parts; if b exists c must exist; and b is neither a boundary nor Site of c Similar to “existential dependence”, e.g. pain on organism; gait on walking, etc.

Intuitively, this relation is irreflexive, i.e. it is not the case for any x, x s-depends-on x This is because BFO accepts parthood is reflexive; everything is a (maximal) part of itself Hence, every continuant shares a part with itself; but s-depends-on requires relata share no parts; we should be able to prove irreflexivity then for this relation (in the presence of parthood axioms)

S-depends-on-at irreflexive proof:

BFO 2.0-FOL Theorem: Material Entity
If an Entity has a continuant part that is a material entity, then the Entity in question is a Continuant Entities are…everything…but not everything is a continuant…(e.g. there are occurrents) Import the source file Material Entity Axioms and Continuant Axioms…

Voila, the theorem is proved: Note, this is similar to the axiom in the draft [ ] though instead of Continuant, the draft claims x is a Material Entity.

Limitations of Theorem Proving
Proving theorems is useful for uncovering relationships among axioms and axiom commitments But capturing the philosophical underpinning of BFO requires a large number of axioms; proving theorems here and there just isn’t going to help us understand BFO very well… We need to prove meta-theoretic results about BFO; we accomplish this by working with the logical structure of BFO directly, i.e. the relations each class bears to other classes

BFO 2.0 Logical Structure

Displaying Logical Dependencies
BFO-FOL axioms have been organized to display logical dependencies

BFO-FOL axioms have been organized to display logical dependencies Example: Object class logically depends on Continuant class

The BFO 2.0 Hierarchy

Why Logical Dependencies?
Focusing on logical dependencies helps with:

Focusing on logical dependencies helps with: Spotting axioms used by all/most classes/relations in BFO 2.0

Focusing on logical dependencies helps with: Spotting axioms used by all/most classes/relations in BFO 2.0 Spotting redundant axioms

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

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 What is, and why should I care about, modularization, I hear you cry…

Modularization: What? Modules are 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 (the examples I just gave!)

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

Modularization: What? 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

Modularization: What? 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 (a parthood relation governing regions of time)

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

Modularization: What? 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

Modularization: What? 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 (an existence at a time relation ranging over all entities in BFO)

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

Modularization: How? 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: How? 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… As mentioned, modularization is a “divide and conquer” approach to proving meta-theoretic properties about BFO

Modularization: Why? Proving meta-theoretic properties concerning the BFO axioms is easier if we use a ‘divide-and-conquer’ approach when dealing with the axioms; ultimately we are trying to answer: What are BFO’s ontological and relational commitments? This is too tough to answer directly, since BFO is so big, but if we divide the axioms up, we can approach this question in steps… Still, a natural follow-up: Why do I want to find such commitments?

Modularization: Why? Given two theories, if they have different commitments which satisfy them, then they cannot be computationally integrated Because one theory might say “X” is true under one interpretation, while the second says “~X” is true under the same interpretation; if we attempt to integrate these theories together in a computing environment, disaster will result We want to be able to predict disaster, so we can avoid it…finding commitments allows us to make predictions; we find them by a method of semantic verification, which uses the modules we construct…

Semantic Verification: What?
Logical theories are satisfied or not; if they are, the axioms representing the theory has a true interpretation, we say the interpretation is a model Theories have intended models, or standard interpretations E.g. The intended models of Peano Arithmetic include just the natural numbers as the domain and arithmetic operations as relations Semantically verification consists in (1) finding, for a given theory, all the models which satisfy that theory, and (2) making sure the intended models of the theory line up with the models that satisfy it

Semantic Verification: What?
But we accomplish this in pieces, by using proper modules of BFO… Claim: Intended models of BFO 2.0 are captured with the BFO-FOL 2.0 implementation To verify this claim for BFO-FOL 2.0, we must characterize the models of the axioms (i.e. find the ontological and relational commitments) Then check to see if they line up with BFO 2.0 (i.e. the reference manual, Barry’s intuitions, etc.)

Semantic Verification: How?
This is no easy task… It requires tedious sorting through well-understood mathematical theories to uncover axiom sets that constrain specific models

This is no easy task… It requires tedious sorting through well-understood mathematical theories to uncover axiom sets that constrain specific models There are…so many well-understood mathematical theories…and trust me, staring at, and thinking hard about, how axioms relate to one another is not as productive as you might first think.

This is no easy task… It requires tedious sorting through well-understood mathematical theories to uncover axiom sets that constrain specific models There are…so many well-understood mathematical theories…and trust me, staring at, and thinking hard about, how axioms relate to one another is not as productive as you might first think. So, I turned to a recent development in Toronto, an axiom repository

COLO(REpository) and Hierarchies
COLORE is an open repository of ontologies represented as sets of axioms in Common Logic COLORE consists of hierarches, or partially ordered finite set of theories (modules closed under entailment) in same language:

COLORE is an open repository of ontologies represented as sets of axioms in Common Logic COLORE consists of hierarches, or partially ordered finite set of theories (modules closed under entailment) in same language: Distinct theories in the same language, may be compared based on conservative/non-conservative extension properties

COLORE is an open repository of ontologies represented as sets of axioms in Common Logic COLORE consists of hierarches, or partially ordered finite set of theories (modules closed under entailment) in same language: Distinct theories in the same language, may be compared based on conservative/non-conservative extension properties Distinct theories in distinct languages, may be compared by definable equivalence for specific theories, and reducibility for sets of theories

Definitions A Conservative extension to an axiom set, is an extension that ‘doesn’t get you anything new’, but makes stuff easier to prove E.g. defining the binary relation ‘is a member of’ in naïve set theory results in a conservative extension A non-conservative extension ‘gets you new stuff’ E.g. adding a new non-equivalent axiom to an existing axiom set, trivially gets you something new, and is a non-conservative extension

Hierarchy Properties (same Language)
Ordering Hierarchy houses theories concerning orderings of points.

Ordering Hierarchy

Ordering Hierarchy houses theories concerning orderings of points.

Ordering Hierarchy houses theories concerning orderings of points; includes (among others): Theory of Linear Ordering (theory of linear order over points)

Ordering Hierarchy

Ordering Hierarchy houses theories concerning orderings of points; includes (among others): Theory of Linear Ordering (theory of linear order over points)

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)

Ordering Hierarchy

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)

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; Dense Linear requires it)

Hierarchy Properties (different Languages)
Relationships can be examined between theories in distinct hierarchies, via definable equivalence: We start with one theory in one language, make a ‘translation manual’ that allows this theory to speak the language of other theories, then see how much of what the theories ‘say’ is the same Toy example: When I use the word ‘parthood’ how much of what you mean by ‘parthood’ matches up with what I mean?

I’ve used the COLORE repository’s extensive catalogue of well- understood mathematical theories (such as Dense Linear Point), to semantically verify BFO Mathematical theories in the repository are linked together based on language, conservativity, definable equivalence, etc. My task: Construct a BFO hierarchy in COLORE, built out of modules, and determine how each module relates to other theories in COLORE

BFO/COLORE Observations
BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others):

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”

BFO:Taxonomy and COLORE:Theory of Lines

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”

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”

Minimal Extensional Mereology (MEM)
Concerns parthood relation, minimal in that the theory only requires: Reflexive (every x is part of itself) Antisymmetric (if x part of y & y part of x then x=y) Transitive (if x part of y & y part of z then x part of z)… Applied here, the result is every temporal region is part of itself, putatively symmetrical parts of temporal regions are identical, and, of course, transitivity…

COLORE: Mereology Hierarchy

Concerns parthood relation, minimal in that the theory only requires: Reflexive (every x is part of itself) Antisymmetric (if x part of y & y part of x then x=y) Transitive (if x part of y & y part of z then x part of z)… Applied here, the result is every temporal region is part of itself, putatively symmetrical parts of temporal regions are identical, and, of course, transitivity…

Concerns parthood relation, minimal in that the theory only requires: Reflexive (every x is part of itself) Antisymmetric (if x part of y & y part of x then x=y) Transitive (if x part of y & y part of z then x part of z)… Applied here, the result is every temporal region is part of itself, putatively symmetrical parts of temporal regions are identical, and, of course, transitivity… MEM contrasts with stronger mereologies, which may require the existence of unrestricted composite objects (BFO doesn’t!)

COLORE: Mereology Hierarchy

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”

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 of “Lower MEM Weak Mereology”

Taxonomy Models: Setup
We show 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

BFO 2.0: Axiomatization, Modularization, and Semantic Verification

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