Chapter 3

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Chapter 3 A Little Bit of Model Theory

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Section 3.1: Bisimulation

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Dresden © Franz Baader

56 A Little Bit of Model Theory

e5

c F F

f₂ f₃

f₆ F f₅ c

e2 e3

c d2

c e₁

c

c c

c c c c

e4 f₄

f₁ d₁

d3 f₇

M

M M M

M

M M M

M M

M F

M

d4

c

Fig. 3.1. Three interpretations I¹,I²,I³ represented as graphs

of a bisimulation needs to be parametrized w.r.t. the employed set of concept names C and role names R. In the following, we assume that these two sets are fixed, and thus do not mention them explicitly. It should also be noted that the interpretations I¹ and I² in Definition 3.1 are not required to be distinct. In addition, the empty relation is always a bisimulation, though not a very interesting one.

Given the three interpretations depicted in Figure 3.1 (where c is supposed to represent the role child, M the concept Male and F the concept Female), it is easy to see that (d₁,I¹) and (f₁,I³) are bisimilar, whereas (d₁,I¹) and (e₁,I²) are not.

The followimg theorem states that ALC cannot distinguish between bisimilar elements.

Theorem 3.1 If (I1, d₁) ⇠ (I2, d₂), then the following holds for all ALC concepts C:

d₁ 2 C^I¹ i↵ d₂ 2 C^I².

Proof: Since (I¹, d₁) ⇠ (I², d₂), there is a bisimulation ⇢ between I¹ and I² such that d₁⇢d₂. We prove the theorem by induction on the structure of C. Since, up to equivalence, any ALC concept can be con- structed using only the constructors conjunction, negation, and existen- tial restriction (see Lemma 2.7), we consider only these constructors in the induction step. The base case is the one where C is a concept name.

• Assume that C = A 2 C. Then

d1 2 A^I¹ i↵ d2 2 A^I²

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Proof: blackboard

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Section 3.2: Expressive power

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Expressive power

Proof: blackboard

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Expressive power

Proof: blackboard

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Expressive power

Proof: blackboard

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Section 3.3: Closure under disjoint union

Blackboard

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Section 3.3: Closure under disjoint union

Proof: blackboard

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Proof: blackboard

Section 3.3: Closure under disjoint union

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Proof: blackboard

Section 3.3: Closure under disjoint union

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Proof first requires some definitions and auxiliary results.

Section 3.4: Finite model property

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Size

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Subconcepts

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Type

Proof: obvious

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Filtration

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Filtration

Proof: blackboard

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Proof: blackboard

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Proof: blackboard

No finite model property

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Section 3.5: Tree model property

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Unraveling

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Unraveling

Blackboard

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Proof: blackboard

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