NOTALL3SAT = {p: $ is a 3CNF formula over n variables and has a satisf ying assignment in which every clause has at least one FALSE literal.}
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- Problem 5 (#2.1.36).Find the truth set of each of these predicates where the domain is the set of integers. a) P(x) : “x3≥1” b) Q(x) : “x2= 2” c) R(x) : “x < x2”Problem 1: Let P(x, y) denote the statement “Student x has taken class y”, where the domain for x consists of all students in your class, and for y consists of all computer science courses at your school. Express each of the following quantifications in English.1. x y P(x, y)2. x y P(x, y)3. x y P(x, y)4. y x P(x, y)5. y x P(x, y)6. x y P(x, y)Show that each of the following restrictions of SAT are NP-complete: a) Each clause contains three or fewer literals and each variable appears in three or fewer clauses. b) Each clause contains exactly three literals and each variable appears in four or fewer clauses.
- 6. Consider a modification to the rod-cutting problem in which, in addition to a value pi for each rod, there is handling cost ci that is one and a half times the length of the rod cut plus a flat fee of 3 (i.e., the handling cost to cut a rod of length 5 is 7.5+3 =10). The revenue generated is the sum of the value of the pieces cut minus the sum of handling costs of the cuts. Provide a dynamic programming approach to solve this problem given an initial rod of length n with potential cut lengths of 1, 2, 5, and 7 [Adapted from ITA, pg. 370, 15.1-3] length of cut (i) 1 2 5 7 value (pi) 5 7 30 45 handling cost (ci) 4.5 6 10.5 13.5Using c++ Apply both breadth-first search and best-first search to a modified version of MC problem. In the modified MC, a state can contain any number of M’s and any number of C’s on either side of the river. Assume the goal is always to move all the persons on the left side to the right side. The Initial state should be a parameter given to the program at beginning of execution. As in the original problem, boat capacity =2, the boat cannot move by itself, and on either side C’s should not outnumber M’s. For best-first search, you need to come up with an appropriate heuristic. In addition to solving the problem, your grade will also be based on th effectiveness of the heuristic. As an example, the program should execute as follows. Initial state… Enter number of M’s on left side of the river: 3 Enter number of C’s on left side of the river: 1 Enter number of M’s on right side of the river: 0 Enter number of C’s on right side of the river: 0 Enter location of the boat: L The output…Consider the version of the dining-philosophers problem in which the chopsticks are placed at the center of the table and any four of them can be used by a philosopher. In other words, a philosopher needs four chopsticks to eat. Assume that requests for chopsticks are made one at a time. Assuming that there are m=4k chopsticks and n=6k philosophers around the table, (i) How many maximum philosophers can eat simultaneously? (ii) Describe a simple rule for determining whether a particular request can be satisfied without causing deadlock given the current allocation of chopsticks to philosophers. (Hint: Use rules similar to the Banker’s algorithm.)
- To demonstrate that a problem L2 is NP-hard, it is sufficient to demonstrate that L1 L2, where L1 is an NP-hard issue. Justify.Problem 1. Consider the regular expression (0|1)*1 over the alphabet {0, 1} What language does the regular expression describe? Use the Thompson's construction method to derive a non-deterministic finite automaton (NFA) that recognizes L. It is enough to draw the final NFA Use the powerset construction method to derive an equivalent deterministic finite automaton (DFA). You must show the computation of e-closures and draw the DFA Use the partitioning method to derive an equivalent, minimal DFA. You must show the partitioning steps and draw the minimal DFAThe missionaries and cannibals problem is usually stated as follows. Three missionariesand three cannibals are on one side of a river, along with a boat that can hold one ortwo people. Find a way to get everyone to the other side without ever leaving a group of missionariesin one place outnumbered by the cannibals in that place. This problem is famous inAI because it was the subject of the first paper that approached problem formulation from ananalytical viewpoint (Amarel, 1968).b. Implement and solve the problem optimally using an appropriate search algorithm. (Mention the name of the search algorithm, and write the complete answers, Draw the answer using the searching algorithms with complete and all paths and branches.)
- This characteristic is met by a problem if it is feasible to develop an optimum solution for it by first creating optimal solutions for its subproblems. a) Subproblems that overlap; b) substructure that's optimum; c) memorization; d) greedy) Show that ∀xP(x) ∧ ∃xQ(x) is logically equivalent to ∀x∃y(P(x) ∧ P(y)) The quantifiers have the same non empty domain. I know that to prove a proposition is logically equivalent to another one, I have to show that ∀xP(x) ∧ ∃xQ(x) ↔ ∀x∃y(P(x) ∧ P(y)) Which means I have to prove that (∀xP(x) ∧ ∃xQ(x)) → ∀x∃y(P(x) ∧ P(y)) ∧ ∀x∃y(P(x) ∧ P(y)) → (∀xP(x) ∧ ∃xQ(x)) I don't know the answer, so I saw the textbook answer. It says (1) Suppose that ∀xP(x) ∧ ∃xQ(x) is true. Then P(x) is true for all x and there is an element y for which Q(y) is true. I get this part. Because P(x) ∧ Q(x) is true for all x and there is a y for which Q(y) is true, ∀x∃y(P(x) ∧ P(y)) is true. Emm... I think ∀x∃y(P(x) ∧ P(y)) is true because ∀x only affects P(x) and ∃y only affects P(y) since their alphabets are different. So, it has the exact same meaning as ∀xP(x) ∧ ∃yQ(y). And since the domains are the same, ∀xP(x) ∧ ∃yQ(y) is actually equal to ∀xP(x) ∧ ∃xQ(x). But the textbook states that "P(x) ∧ Q(x) is…2. A safe has 5 locks v, w, y and z; all of which must be unlocked for the safe to open. The keys to the locks are distributed among five executives in the following manner.Mr. A has keys for locks v and x. Mr. B has keys for locks v and y. Mr. C has keys for locks w and y. Mr. D has keys for locks x and z. Mr. E has keys for locks v and z.a) Determine the minimal number of executives required to open the safe.b) Find all the combinations of executives that can open the safe; write and expression f(A, B, C, D, E) which specifies when the safe can be opened as a function of what executives are present.c) Who is the essential executive?