Try to formalise the following natural language sentences into FOL now or in DL later in the next chapter, or both. (optional practice). * 1\. All lions are animals 2\. Each professor teaches at least course 3\. All humans eat some fruit and cheese 4\. Animals are either herbivores or carnivores 5\. The sister of one's mother is one's aunt 6\. Something can't be both pap and pizza 7\. If a person works for a company, then that person is an employee 8\. Anything that manages something is a manager, and vice versa. 9\. All flies have exactly two eyes as part 10\. Anything has at most one life 11\. The participation relation is defined by relating physical objects to processes 12\. Having something as part is the inverse of being part of something 13\. connection is symmetric (if one thing is connected to something else, that other thing is also connected to the one thing) 14\. A vehicle can be either motorised or not, but not both 15\. Several snails are slow 16\. Each patient is registered at some hospital on a certain weekday 17 . All students are at some time in their life not a student

Step by step solution

01

- Formalize Sentence 1

Sentence: 'All lions are animals'. FOL: ∀x (Lion(x) → Animal(x))

02

- Formalize Sentence 2

Sentence: 'Each professor teaches at least one course'. FOL: ∀x (Professor(x) → ∃y (Course(y) ∧ Teaches(x, y)))

03

- Formalize Sentence 3

Sentence: 'All humans eat some fruit and cheese'. FOL: ∀x (Human(x) → ∃y ∃z (Fruit(y) ∧ Cheese(z) ∧ Eats(x, y) ∧ Eats(x, z)))

04

- Formalize Sentence 4

Sentence: 'Animals are either herbivores or carnivores'. FOL: ∀x (Animal(x) → (Herbivore(x) ∨ Carnivore(x)))

05

- Formalize Sentence 5

Sentence: 'The sister of one's mother is one's aunt'. FOL: ∀x ∀y ∀z (Sister(y, Mother(x)) → Aunt(y, x))

06

- Formalize Sentence 6

Sentence: 'Something can't be both pap and pizza'. FOL: ¬∃x (Pap(x) ∧ Pizza(x))

07

- Formalize Sentence 7

Sentence: 'If a person works for a company, then that person is an employee'. FOL: ∀x ∀y (Person(x) ∧ Company(y) ∧ WorksFor(x, y) → Employee(x))

08

- Formalize Sentence 8

Sentence: 'Anything that manages something is a manager, and vice versa'. FOL: ∀x ∀y (Manages(x, y) ↔ Manager(x))

09

- Formalize Sentence 9

Sentence: 'All flies have exactly two eyes as part'. FOL: ∀x (Fly(x) → ∃y ∃z (Eye(y) ∧ Eye(z) ∧ PartOf(y, x) ∧ PartOf(z, x) ∧ ¬∃w (Eye(w) ∧ PartOf(w, x) ∧ w ≠ y ∧ w ≠ z)))

10

- Formalize Sentence 10

Sentence: 'Anything has at most one life'. FOL: ∀x (∃y (Life(y) ∧ Has(x, y)) → ∀z (Life(z) ∧ Has(x, z) → y = z))

11

- Formalize Sentence 11

Sentence: 'The participation relation is defined by relating physical objects to processes'. FOL: ∀x ∀y (PhysicalObject(x) ∧ Process(y) → ParticipationRelation(x, y))

12

- Formalize Sentence 12

Sentence: 'Having something as part is the inverse of being part of something'. FOL: ∀x ∀y (HasPart(x, y) ↔ PartOf(y, x))

13

- Formalize Sentence 13

Sentence: 'Connection is symmetric (if one thing is connected to something else, that other thing is also connected to the one thing)'. FOL: ∀x ∀y (Connected(x, y) → Connected(y, x))

14

- Formalize Sentence 14

Sentence: 'A vehicle can be either motorized or not, but not both'. FOL: ∀x (Vehicle(x) → (Motorized(x) ∨ ¬Motorized(x)) ∧ ¬(Motorized(x) ∧ ¬Motorized(x)))

15

- Formalize Sentence 15

Sentence: 'Several snails are slow'. FOL: ∃x ∃y ∃z (Snail(x) ∧ Snail(y) ∧ Snail(z) ∧ Slow(x) ∧ Slow(y) ∧ Slow(z) ∧ x ≠ y ∧ y ≠ z ∧ x ≠ z)

16

- Formalize Sentence 16

Sentence: 'Each patient is registered at some hospital on a certain weekday'. FOL: ∀x (Patient(x) → ∃y ∃z ∃w (Hospital(y) ∧ Weekday(z) ∧ RegisteredAt(x, y, w)))

17

- Formalize Sentence 17

Sentence: 'All students are at some time in their life not a student'. FOL: ∀x (Student(x) → ∃t (Time(t) ∧ ¬StudentAtTime(x, t)))

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Formalization

In computer science, formalization involves transforming natural language sentences into logically precise statements. This enables the use of computational methods to analyze and process these statements. For instance, when we say 'All lions are animals,' we can formalize this into First-Order Logic (FOL) as \( \forall x (Lion(x) \rightarrow Animal(x)) \). This precise representation helps in performing automated reasoning and proofs.
By formalizing statements, we reduce ambiguity and increase clarity. This is essential in fields like natural language processing and ontology engineering.
To formalize, follow these steps:

• Identify the subjects and predicates.
• Define relationships and quantifiers.
• Use logical connectives (e.g., ∧, ∨, →, ↔) to form complete statements.

With practice, converting natural sentences into logical statements becomes intuitive.

Natural Language Processing

Natural Language Processing (NLP) deals with the interaction between computers and human languages. It involves teaching computers to understand, interpret, and respond to natural language inputs. Formalization plays a crucial role in NLP as it helps machines convert ambiguous human language into precise logical forms.
By formalizing language, NLP systems can effectively analyze and process data. For example, consider the sentence, 'All humans eat some fruit and cheese'. To an NLP system, this becomes \(\forall x (Human(x) \rightarrow \exists y \exists z (Fruit(y) \wedge Cheese(z) \wedge Eats(x, y) \wedge Eats(x, z)))\).
Key applications of NLP include:

• Text summarization
• Sentiment analysis
• Machine translation
• Speech recognition

These applications hinge on transforming and understanding natural language through formalization.

First-Order Logic

First-Order Logic (FOL) is a formal system used in mathematics, philosophy, linguistics, and computer science. FOL allows us to express statements with variables and quantifiers and make logical deductions. For example, to express 'If a person works for a company, then that person is an employee,' we write \(\forall x \forall y (Person(x) \wedge Company(y) \wedge WorksFor(x, y) \rightarrow Employee(x))\).
Key components of FOL include:

• Variables: Symbols that can represent any element in a domain.
• Quantifiers: 'For all' ( ∀ ) and 'There exists' ( ∃ ).
• Predicates: Functions that return true or false.
• Logical connectives: And ( ∧ ), Or ( ∨ ), Not ( ¬ ), Implies ( → ), If and only if ( ↔ ).

Using FOL, we can build complex logical structures, enabling automated reasoning, verification, and inference.

Description Logic

Description Logic (DL) is a family of formal knowledge representation languages. DL is used in ontology development, semantic web technologies, and knowledge-based systems. Unlike FOL, DL emphasizes class definitions and relationships. For example, in DL, 'A vehicle can be either motorized or not, but not both' can be expressed as \(\forall x (Vehicle(x) \rightarrow (Motorized(x) \vee eg Motorized(x)) \wedge eg (Motorized(x) \wedge eg Motorized(x)))\).
DL is key in ontology-based applications due to its strong expressiveness and computational properties. It provides a structured way to define concepts, roles, and individuals. Core features of DL include:

• Concepts: Classes representing sets of individuals.
• Roles: Relationships between individuals.
• Individuals: Identifiers for objects.

DL supports constructing detailed and complex ontologies, facilitating semantic reasoning over data.