CSCE 625 – Homework #4due: Tuesday, 10/20/091. Prove F from {A, C, AB, CD, BDEF, E} using a resolution-refutation proof. 2. Derive a proof of the proposition A from the following set of sentences by natural deduction (using rules of inference). Recall that a proof contains a sequence of derivations labeled with the indexes of the sentences they were derived from, along with an indication of the inference rule used.{D ^ E  A, F ^ C  D, ¬(F ^ C  B),C v B  E}3. Determine whether or not the following pairs of predicates are are unifiable. If they are, give the most-general unifier and show the result of applying the substitution to each predicate. If they are not unifiable, indicate why. Variables are in capital letters; constants and function names are lowercase.a) P(a, X, f(g(Y ))) P(Z, f(Z), f(U))b) Q(f(a), g(X)) Q(Y, Y )c) R(f(Y ), Y,X) R(X, f(a), f(V ))d) P(a, Y, f(X)) P(X, f(b), f(b))e) Q(g(f(a)), g(X),Z) Q(Y, Y, f(X))f) Q(f(a, a), Y,Z) Q(X, f(Z,Z), Y )4. Translate the following sentences into first-order logic: A poodle is a small, fluffy dog. Someone from the post-office is at the front door of my house. Tomatos are either a fruit or vegetable. Beef is a type of meat, rather than fruit or vegetable. All Romans were either loyal to Ceasar or hated him. People only try to assassinate rulers they are not loyal to. If it is night, then drivers should stop at yellow lights. When a horse gallops, no more than 2 legs touch the ground at any time. American football rules: For a receiver to make a catch, he must have possession of the ball, the ball must not be touching the ground, and both feet must touch in-bounds unless the ball crosses the endzone plane or contact by an opponent pushes him out of bounds. (note: be specific about what it means for both feet to touch in-bounds)5. Using first-order rules of inference, prove that “there exists a vegetarian” from the following description: anyone who does not eat meat is a vegetarian, tomatoes are not meat, carrots are not meat, and there is someone who eats only tomatoes and carrots. Be sure to explicitly label each new sentence with the one(s) it was derived from, along with the inference rule and any substitution used.6. Determine whether the following set of sentences is satisfiable using DPLL. Label each step with detailed information on why each decision is made (i.e. choice, pure literal, unit literal, backtrack; include line number). Show a satisfying assignment. When there is a choice, choose among the remaining variables in lexicographic order.1. A v B 2. C vA3. B v C4. D v E5. G v D6. B v C7. H v G8. A v B9. A v D10. E v