Modelling Space & Time

Question 1

Topological Space

Answer 1

A set X (of things we think of as points)
and
a set of subsets of X (called open sets)
which satisfy certain axioms.

Question 2

Definition of a Part

Answer 2

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.

Question 3

Definition of a Proper Part

Answer 3

PP(x,y) ≡ P(x, y) ∧ ¬P(y,x)

Special case: being part but not being equal.

Question 4

Definition of Overlap

Answer 4

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.

Question 5

Definition of External Connection

Answer 5

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.

Question 6

Definition of Tangential Proper Part

Answer 6

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.

Question 7

Definition of Non-tangential proper part

Answer 7

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.

Question 8

RCC axioms:
R

Answer 8

R is a set, the elements of which are the regions

Question 9

RCC Axioms:
u

Answer 9

A special region for the universe, or the whole space.

Question 10

RCC Axioms:
null

Answer 10

An element, which is not a region, for a null element.

Question 11

RCC Axioms:
compl

Answer 11

An operation compl (complement) gives for each region (except u) the outside of the region.

Question 12

RCC Axioms:
sum

Answer 12

Gives the union of a pair of regions.

Question 13

RCC Axioms:
prod

Answer 13

Gives the intersection of regions (when it exists), or the null element when the regions do not overlap.

Question 14

RCC Axioms:
C

Answer 14

The binary relation of connection.

Question 15

8 RCC Axioms

Answer 15

1. Every region is connected to itself
2. Connection is symmetric
3. All regions are connected to the universal region
4. Being NTPP of y means not being connected to the complement of y. And being part of y means not overlapping the complement of y.
5. Connected to a sum means connected to a summand.
6. Connected to an intersection means connected to a region which is a part of both the intersecting regions.
7. ∀ x, y ∈ R ( prod(x, y) ∈ R ⇔ O(x, y) )
8. Every region has a non-tangential proper part.

Axiom R7 was found to be redundant. It can be proved from other ones.

Question 16

What are the RCC8 Relations?

Answer 16

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).

Question 17

8 RCC8 Relations

Answer 17

1. Tangential Proper Part
2. Non-Tangential Proper Part
3. Non-Tangential Proper Part Inverse
4. Tangential Proper Part Inverse
5. Partial Overlap
6. Equal
7. Externally Connected
8. Disconnected

1, 2, 3, 4

Question 18

Definition of Self-connectedness

Answer 18

`SCON(X) = ∀y ∀z (( x = sum(y, z)) → C(y, z))

Question 19

Convex hull

Answer 19

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.

Question 20

3 Axioms of a linear order

Answer 20

1. ∀t₁ ∀t₂ ∀t₃ (( t₁ ≤ t₂ ≤ t₃ ) → t₁ ≤ t₃ )
2. ∀t₁ ∀t₂ ( t₁ ≤ t₂ ∨ t₂ ≤ t₁ )
3. ∀t₁ ∀t₂ (( t₁ ≤ t₂ ∧ t₂ ≤ t₁ ) → ( t₁ = t₂ )

Question 21

Define a strict ordering relation

Answer 21

`t₁ < t₂ iff t₁ ≤ t₂ ∧ ¬ ( t₁ = t₂ )

Question 22

Axiom: time will continue forever in the future

Answer 22

∀t ∃t' (t < t')

Question 23

Axiom: time is infinitely divisible

a.k.a. density

Answer 23

`∀t₁ ∀t₂ (( t₁ < t₂ ) → ∃ t₃ (( t₁ < t₃ ) ∧ ( t₃ < t₂ ))

Question 24

13 Allen relations between two intervals

Answer 24

Question 25

4 Components of propositional modal logic

Answer 25

• Propositional variables, that stand for atomic propositions just as in ordinary propositional logic
• Logical connectives: ¬ ∧ ∨ → ↔
• Two additional logical symbols:
  
  • ☐ called box
  • ♢ called diamond
• The usual parenthesis to avoid ambiguity.

Question 26

Formulas of modal logic

Answer 26

Expressions that can be true or false.

Unlike first order logic there are no terms in the language that can refer to individual things.

Question 27

Rules for construction of a formula

Answer 27

• Every atomic proposition is a formula
• If ⍺ and β are formulas then the following are formulas:
  
  • ¬⍺
  • ⍺ ∨. β
  • ⍺ ∧ β
  • ⍺ → β
  • ⍺ ↔ β
  • ☐⍺
  • ♢β

Question 28

Informal idea of modal logic

Answer 28

In propositional logic, when we give the meaning of an atomic proposition we simply say whether it is true or false.

In model lagic, we can think of propositions not simply being true or false, but being true at some places and false at others.

“Places” might be different times, or objects in the world.

Question 29

Formal definition:
frame & valuation

Answer 29

A frame is a pair (W, R) where
- a set W
- a binary relation R ⊆ W x W

a valauation V provides for each atomic proposition p, a subset of W so V(p) ⊆ W.

Question 30

Key points on ♢ and ☐

Answer 30

• ♢p is true at w when p is true somewhere accessible from w.
• ☐p is true at w when p is true everywhere accessible from w.

Question 31

Truth, satisfaction and validity
in ordinary propositional logic

Answer 31

A formula ɸ is satisfiable if it is true under some valuation and valid if it is true under all valuations.

Question 32

Truth, satisfaction and validity
in propositional modal logic

Answer 32

A formula ɸ can be:
- True at a particular world in a Frame F under valuation V
- True at all worlds in a frame F under valuation V (satisfied in F by V)
- True at all worlds in a frame F under all valuations (valid in F)
- Valid in a frames (universally valid)