# TU Wien:Knowledge-based Systems VU (Egly, Eiter, Tompits)/Übungstest 13.06.2018

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## First-order logic & Rules[edit]

### Example 1[edit]

For the following sentences, where c is a constant, R a binary and P a unary predicate:

- a) Give an interpretation structure under which sentence (1)-(4) become true.
- b) Give a formal definition of Entailment in first-order logic.
- (1)
- (2)
- (3)
- (4)

### Example 2[edit]

- a) Give a formale definition of a rule. When is a rule applicable?
- b) For the following problem write some rules in CLIPS to model the problem.
- A person has either a dog or a cat, but not both. If a person is allergic, then he can't have a cat. If a person has a dog, then he needs at least 1 hour of sparetime to take the dog for a walk. A person who doesn't have enough money to pay a vet, can't afford a pet.

## Description Logics and Truth Maintenance Systems[edit]

### Example 3[edit]

- Construct a knowledge base (we do not need an RBox here) expressing the following knowledge concerning employees and their hierarchy.
- Every person who is an employee is either a supervisor or a worker.
- A supervisor is a person who manages at least 1 or more workers.
- No worker manages himself.
- Alice and Bob have jointsupervisor.

Use the class name , the role names and the individual names

- What does it mean in that a knowledge base is satisfiable.

### Example 4[edit]

- Using the tableau algorith for concept satisfiability, show that the GCI (didn't remember it :/) is not satisfied by each interpretation.
- What is an interpretation?

## Answer Set Programming[edit]

### Example 5[edit]

- a) What is grounding?
- b) The combinatorial graph problem vertex cover is defined as follow:

**INSTANCE:** Given a graph G = (V,E) and a positive integer k ≤ |V|.

**QUESTION:** Does there a subset D ⊆ V such that |D| ≤ k and such that for each edge(x,y) ∈ E is either x ∈ D or y ∈ D.

Define all required predicates, rules, and constraints to represent the vertex cover problem as an anser-set program such that

- each vertex v ∈ V of G is denoted by a fact v(v),
- each edge(u,v) ∈ E of G is denoted by a fact edge(u,v),
- the integer k is given by a single fact size(k), and
- answer sets of c(X) correspond to solutions D.

### Example 6[edit]

Consider the following normal logic program P, where p,q,r and s are atoms.

- a) Given the program.

- Reason if the sets , are stable models of P and if so explain why.

- b) Which kind of negation in ASP do you know and what are their differences?