Ontology Flashcards

Question

# Building concepts Universal value restriction

Answer 1

`∀r. C` where `C` is a concept and `r` is a role. The things in the class where everything to which they stand in relation `r` belongs to the class `C`. | E.g. `∀ eats. Vegetable` is the class of things that only eat vegetables

Answer 2

`∃r. C` where `C` is a concept and `r` is a role. The things in the class where something to which they stand in relation `r` belongs to the class `C`. ## Footnote E.g. `∃ eats. Vegetable.` is the class of things that eat at least one thing which is a Vegetable.

Answer 3

`C ≣ D` Asserting that means concepts have the same instances.

Answer 4

`C ⊑ D` Asserting that means `C` is a sub-concept of `D`.

Answer 5

A typical Description logic consists of: - **TBox**: A set of terminological axioms - **ABox**: A set of assertions about individuals

Answer 6

- functional - inverse functional - transitive - symmetric - asymmetric - reflexive - irreflexive

Answer 7

If `A` is any individual then it **never happens** that: - there are individuals `B` and `C` which are different from each other and `R(A, B)` and `R(A, C)`

Answer 8

If `A` is any individual, then it **never happens** that: - there are individuals `B` and `C` which are different from each other, and - `R(B, A)` and `R(C, A)`

Answer 9

If `A`, `B`, `C` are any individuals (including possibly some of them being equal to others) then it **always happens that**: - if `R(A, B)` and `R(B, C)`, then `R(A, C)`

Answer 10

If `A` and `B` are any individuals (including possibly both being the same) then it **always happens** that: - if `R(A, B)`, then `R(B, A)`.

Answer 11

If `A` and `B` are any individuals (including possibly both being the same) then it **never happens** that: - if `R(A, B)`, then `R(B, A)`.

Answer 12

If `A` is any individual, then it **always happens** that - `R(A, A)` holds

Answer 13

If `A` is any individual then it **never happens** that - `R(A, A)` holds

Answer 14

Given properties `R` and `S` `R ○ S` is the property where `x` is linked to `z` if: 1. `x` is linked to something `y` by `R`, and 2. `y` is linked to `z` by `S`

Answer 15

If `R` and `S` are roles then we can construct these roles: 1. `R ⨆ S` (union) 2. `R ⨅ S` (intersection) 3. `¬ R` (complement)

Answer 16

Two things are related by this role if they are related by `R` or by `S` or by both.

Answer 17

Two things are related by this role if they are related by both `R` and `S`.

Answer 18

Two things are rleated by this role if they are not related `R`.

Answer 19

`R ᐩ` Two things `x` and `y` are related by this role if there is a sequence of `n+1` things `x₀, x₁, ..., xₙ` where `x = x₀`, `xₙ = y`, **`n ≥ 1`** and `R( xᵢ , xᵢ₊₁ )` for all `0 ≤ i < n`.

Answer 20

`R*` Two things `x` and `y` are related by this role if there is a sequence of `n+1` things `x₀, x₁, ..., xₙ` where `x = x₀`, `xₙ = y`, **`n ≥ 0`** and `R( xᵢ , xᵢ₊₁ )` for all `0 ≤ i < n`.

Answer 21

`R*(x, x)` always holds `Rᐩ(x, x)` only holds in the case that `R(x, x)`.

Answer 22

`R ⁻ ¹` or `R ⁻` (inverse / converse) This is the role where `R⁻ (x, y)` iff `R (y, x)`.

Answer 23

`id(C)` (identity role on concept `C`) This is the role where the domain and range are `C` and `x` is related to `y` iff `x = y`.

Answer 24

This is a set (domain) `Δᴵ` and for each concept `C`, a set `Cᴵ ⊆ Δᴵ`, and for each role a binary relation on `Δᴵ`, and elements of `Δᴵ` for each individual, subject to the TBox and ABox assertions. ## Footnote It is possible that `Cᴵ` is empty for some concepts.

Answer 25

For concepts `A` and `B` (with T-Box axioms `T`) we say that `A` is **subsumed by** `B`, or that `B` **subsumes** `A`, and write `A ⊑ B` if for every model `I` of `T` we have `Aᴵ ⊆ Bᴵ`. If `A ⊑ B` then `B` is a more general concept than `A`. ## Footnote E.g. `Mother` subsumes `Grandmother` because every grandmother is a mother.

Answer 26

A concept `A` is said to be **satisfiable** if there is a model `I` in which `Aᴵ ` is non-empty. The concept is **unsatisfiable** when `Aᴵ ` is empty in every model. `A ⊑ B` if and only if `A ⨅ ¬B` is unsatisfiable.

Answer 27

* Replace `¬ ( ⍺ ⨆ β )` by `¬⍺ ⨅ ¬β`, and replace `¬ ( ⍺ ⨅ β )` by `¬⍺ ⨆ ¬β`. * Replace `¬ ∃ r. ɑ` by `∀ r. ¬ɑ` and replace `¬∀ r. ɑ` by `∃ r. ¬ɑ` * Replace `¬¬⍺` by `⍺` | The `⍺` and `β` can be any concept descriptions, not just concept names.

Answer 28

Construct a tree in stages and continue until no further development is possible. Paths may become closed and if all paths close then the concept you started with is unsatisfiable. Each path constructed by building the tree is a part of a possible model. A path becomes closed when it contains both `A(x)` and `¬A(x)` for some concept name `A`. This means that particular attempt to build a model has to be abandoned. To show that a concept description `⍺` is satisfiable, we start with a note `⍺(x)` and develop a tree using the rules. | `x` in `⍺(x)` is not a variable. It is a name for an individual.

Answer 29

- `⍺(x)` where `⍺` is a concept description and `x` is an individual name. - `r(x, y)` where `r` is a role name and `x` and `y` are individual names. ## Footnote The assertion `⍺(x)` means that `x` satisfies `⍺` and `r(x, y)` means that `x` stands in relation `r` to `y`