Modelling Space & Time Flashcards
Topological Space
A set X (of things we think of as points)
and
a set of subsets of X (called open sets)
which satisfy certain axioms.
Definition of a Part
Relationship of being a part:
P(x, y) ≡ ∀ z ( C(z,x) → C(z,y) )
Where C(x, y)
means that x
is connected to y
.
The idea of one region being a part of another is that the part is somewhere inside the whole.
Definition of a Proper Part
PP(x,y) ≡ P(x, y) ∧ ¬P(y,x)
Special case: being part but not being equal.
Definition of Overlap
Relationship of regions overlapping
O(x, y) ≡ ∃z ( P(z, x) ∧ ¬P(z, y) )
Overlapping is defined as sharing a region.
Regions that only just touch are not overlapping.
Definition of External Connection
Relationship of regions being externally connected.
EC(x,y) ≡ C(x, y) ∧ ¬O(x, y)
This is the idea of just touching at the boundary and not overlapping.
Definition of Tangential Proper Part
TPP(x,y) ≡ PP(x, y) ∧ ∃z ( EC(z, x) ∧ EC(z, y) )
Special kind of proper part relationship.
Idea of the part being inside the whole but the part does touch the boundary of the whole from the inside.
Definition of Non-tangential proper part
NTPP(x,y) ≡ PP(x, y) ∧ ∃z ( EC(z, x) ∧ EC(z, y) )
Idea of the part being so clearly inside the whole that the part does not touch the boundary of the whole.
RCC axioms: R
R
is a set, the elements of which are the regions
RCC Axioms:u
A special region for the universe, or the whole space.
RCC Axioms:null
An element, which is not a region, for a null element.
RCC Axioms:compl
An operation compl
(complement) gives for each region (except u
) the outside of the region.
RCC Axioms:sum
Gives the union of a pair of regions.
RCC Axioms: prod
Gives the intersection of regions (when it exists), or the null element when the regions do not overlap.
RCC Axioms:C
The binary relation of connection.
8 RCC Axioms
- Every region is connected to itself
- Connection is symmetric
- All regions are connected to the universal region
- Being NTPP of
y
means not being connected to the complement ofy
. And being part ofy
means not overlapping the complement ofy
. - Connected to a sum means connected to a summand.
- Connected to an intersection means connected to a region which is a part of both the intersecting regions.
∀ x, y ∈ R ( prod(x, y) ∈ R ⇔ O(x, y) )
- Every region has a non-tangential proper part.
Axiom R7 was found to be redundant. It can be proved from other ones.
What are the RCC8 Relations?
8 Relationships having the special property that every pair of regions will be in exactly one of these relationships.
They form a Jointly Exhaustive and Pairwise Disjoint set (JEPD).
8 RCC8 Relations
- Tangential Proper Part
- Non-Tangential Proper Part
- Non-Tangential Proper Part Inverse
- Tangential Proper Part Inverse
- Partial Overlap
- Equal
- Externally Connected
- Disconnected
Definition of Self-connectedness
`SCON(X) = ∀y ∀z (( x = sum(y, z)) → C(y, z))
Convex hull
In geometry, the Convex Hull of x
is the smallest region y
including x
and such that any line between points in y
lies entirely in y
.
I.e. the smallest convex region including x.
3 Axioms of a linear order
∀t₁ ∀t₂ ∀t₃ (( t₁ ≤ t₂ ≤ t₃ ) → t₁ ≤ t₃ )
∀t₁ ∀t₂ ( t₁ ≤ t₂ ∨ t₂ ≤ t₁ )
∀t₁ ∀t₂ (( t₁ ≤ t₂ ∧ t₂ ≤ t₁ ) → ( t₁ = t₂ )
Define a strict ordering relation
`t₁ < t₂ iff t₁ ≤ t₂ ∧ ¬ ( t₁ = t₂ )
Axiom: time will continue forever in the future
∀t ∃t' (t < t')
Axiom: time is infinitely divisible
a.k.a. density
`∀t₁ ∀t₂ (( t₁ < t₂ ) → ∃ t₃ (( t₁ < t₃ ) ∧ ( t₃ < t₂ ))
13 Allen relations between two intervals