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- [20 points] Convert each of the following English sentences to predicate calculus using the following predicates which should have an obvious meaning: Student(x), Major(x,y), CScourse(y), Take(x,y), Room(x,y),CampusBuilding(y), Visited (x,y). Please treat CS and Chemistry as constants (Objects).
- All students major either in CS or in Chemistry but not in both.
- Every student takes at least one CS course.
- There is a student who has taken at most one CS course.
- There is a student who has visited every room of at least one building on campus.
- Every student has visited at least one room of every building on campus.
- [20 points] You are given the following sentence:
”Any person who owns a cat does not own a bird” and the predicate calculus expressions listed in items 1 – 5 below.
Convert each logical expression to English. Indicate if any sentences are the same and explain briefly the differences between the expressions when there are differences. Are any of the sentences below the correct representation of the sentence given in English above?
∀x ∀y [[Cat(y) ∧ Owns(x,y)] → [∃z Bird(z)∧ ¬ Owns(x,z)]]∀x ∀y ∃z [[Cat(y) ∧ Bird(z) ∧Owns(x,y)] → ¬ Owns(x,z)]
∀x ∀y ∀z [[Cat(y) ∧ Bird(z) ∧Owns(x,y)] → ¬ Owns(x,z)]
∀x ∀y ∀z [[Cat(y) ∧ Owns(x,y) ∧ Owns(x,z)] → ¬ Bird(z)]
∃x ∃y ∃z Cat(y) ∧ Bird(z) ∧ Owns(x,y) ∧ ¬ Owns(x,z)
- [20 points] Represent the following knowledge in predicate calculus, using the following predicates: Pet(x), Rabbit(y), Dog(z), Own(x,y), Love(x,y), Eat(x,y):
- Anyone who owns a pet loves it.
- Dogs and rabbits are pets.
- James does not love pets who eat rabbits.
- A dog that James owns ate a rabbit.
Convert the sentences to CNF, and using resolution show that there is a contradiction, i.e. you will reach the empty clause (Note: in this case you are not given a sentence and asked to show that it is entailed, you just have to show there is a contradiction.
Show each step in the proof by contradiction using resolution until you reach the empty clause and show the unifications done at each step.
- [20 points] Using resolution, prove by contradiction (refutation) that”¬Dog(Bro)” is entailed by the knowledge base below (so, alpha = “¬Dog(Bro)” ).
Capital letters used as arguments indicate constants (objects), lower case letters are variables. First, convert the expressions to CNF, then show the steps in the proof by resolution – showing the unifications done at each step. Make sure you reach the empty clause.
- ∀x Dog(x) ⇒ Furry(x)
- ∀x Furry(x) ⇒ Lovable(x)
- ∀x ∀p [Furry(x) ∧ Own(p,x)] ⇒ Love(x,p)
- ∀x ∀y Lovable(x) ⇒ ¬ Love(x,y)
- Own(Binh,Bro)5. [20 points] This question is from material in Chapter 11. The problem given below is very similar to the problem in the textbook in 11.1.1. Carefully review the problem in the textbook before you start answering this question.
Suppose you have a tray with a dinner plate on it. The tray is on the dining table in the dining room. You want to take the tray with the dinner plate on it to the kitchen, put the dinner plate on the kitchen table, and bring the tray back to the dining room. You need to write the problem description and the actions using predicate calculus and action schemas.
Use the binary predicates below:
At(object, location) and On(object, support), and the unary predicates Object(x),Tray(y), Room(z):
- Define one action schema for taking an object from a location and putting it on a tray, one for moving a tray and its contents from one location to another location,and one for taking the object off the tray and placing it at the location. When writing the actions schemas use the predicates given above.
- Specify the initial state and the goal state.
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