# Final Exam Spring 2018

### Directions:

• Answer the problems on the exam pages.
• There are four problems, each with multiple parts, for 120 total points. Actual scale was A = 110, C = 75.
• Some useful definitions precede the questions below.
• No books, notes, calculators, or collaboration.
• In case of a numerical answer, an arithmetic expression like "217 - 4" need not be reduced to a single integer.

```  Q1: 30 points
Q2: 20+10 points
Q3: 40 points
Q4: 30 points
Total: 120 points
```

Here are definitions of some terms, sets, predicates, and statements used on this exam.

Question 1 deals with the following scenario:

Many of the dogs in my neighborhood are fans of rap music. One day a set D of four dogs: Cardie (c), Duncan (d), Rio (r), and Scout (s) met to discuss their preferences among a set R of four rap artists: Cardi B (CB), Drake (DR), Kendrick Lamar (KL), and Kanye West (KW).

Note that variables and constants with lower case letters denote dogs, and those with capital letters denote rappers. Note also that Cardie the dog is spelled differently from Cardi B the rapper.

The predicate P(x, Y, Z) means "dog x prefers rapper Y to rapper Z". Each dog's preferences establish a total order on the rappers, so that for each dog x the predicate "(Y = Z) ∨ P(x, Y, Z)" is reflexive, antisymmetric, transitive, and total.

Question 1 also refers to the following five statements, where the variables are of type "dog" or type "rapper".

The statements are:

• Statement I: ∀Y: (Y ≠ CB) → P(c, CB, Y)

• Statement II: Every dog prefers both Drake and Kendrick Lamar to Kanye West.

• Statement III: (P(d, DR, CB) → P(d, CB, KL)) ∧ ¬(P(s, KL, CB) → P(d, CB, DR)

• Statement IV: Scout and Rio each prefer Cardi B to at least one other rapper. (Note added in test: There are two possible interpretations of this English statement. Either one may be used in Question 1, but if you see both of them you should explain in English which you intend.)

• Statement V: ∃Y: ∀x: ∀Z: (Y = Z) ∨ P(x, Z, Y)

N is the set of naturals (non-negative integers, {0, 1, 2, 3,...}.

Question 2 uses a recursive function f from N to N defined by the rules f(0) = 1, and for n ≥ 0, f(n+1) = f(n) + n2 + n - 1.

Question 2 also refers to the language L, over the alphabet {a, b, c}, which is denoted by the regular expression (∅* + b + bb)(a + ab + abb + acc)*. Note that a similar but different regular language is the subject of Question 3.

Question 2 also uses a recursive function g from N to N, defined by the rules g(0) = 1, g(1) = 2, g(2) = 4, and, for n ≥ 2, g(n+1) = g(n) + g(n-1) + 2g(n-2).

Question 3 uses the following λ-NFA N, which has state set {1, 2, 3, 4}, start state 1, final state set {1}, alphabet {a, b, c}, and transition relation Δ = {(1, a, 2), (2, λ, 1), (2, b, 3), (2, c, 4), (3, b, 1), (3, λ, 1), and (4, c, 1)}.

Here is a diagram of N, with "L" representing λ:

``````

c
----------(4)
|           ^
|           | c
|           |
V     a     |
>((1)) -----> (2)
^  <------  |
|     L     |
|           | b
|           |
|           |
|   b, L    V
--------- (3)

``````

• Question 1 (30): This question deals with the scenario described above, and with the five statements about a set of dogs D, consisting of exactly the four dogs Cardie (c), Duncan (d), Rio (r), and Scout (s), and a set of rappers R, consisting of exactly the four rappers Cardi B (CB), Drake (DR), Kendrick Lamar (KL), and Kanye West (KW). Note that variables and constants with lower case letters denote dogs, and that those with capital letters denote rappers. Note also that Cardie the dog is spelled differently from Cardi B the rapper.

The predicate P(x, Y, Z) means "dog x prefers rapper Y to rapper Z". Each dog's preferences establish a total order on the rappers, so that for each dog x the predicate "(Y = Z) ∨ P(x, Y, Z)" is reflexive, antisymmetric, transitive, and total.

• (a, 10) Translate each of these five statements as indicated.

• Statement I: (to English) ∀Y: (Y ≠ CB) → P(c, CB, Y)

• Statement II: (to symbols) Every dog prefers both Drake and Kendrick Lamar to Kanye West.

• Statement III: (to English) (P(d, DR, CB) → P(d, CB, KL)) ∧ ¬(P(s, KL, CB) → P(d, CB, DR)

• Statement IV: (to symbols) Scout and Rio each prefer Cardi B to at least one other rapper. (Note added in test: There are two possible interpretations of this English statement. Either one may be used in Question 1, but if you see both of them you should explain in English which you intend.)

• Statement V: (to English) ∃Y: ∀x: ∀Z: (Y = Z) ∨ P(x, Z, Y)

• (b, 10) Using Statement III only, along with the given properties of the relation P, determine the truth value of the three propositions P(d, CB, DR), P(d, CB, KL), and P(s, KL, CB).

• (c, 10) Using Statements I, II, III, and IV, along with the given properties of the relation P, prove Statement V. You may use English, symbols, or a combination, but make your use of quantifier proof rules clear.

• Question 2 (20+10): This question uses the functions f and g, and the language L, defined above. The rules defining f are f(0) = 1, and for any n with n ≥ 0, f(n+1) = f(n) + n2 + n - 1. The rules defining g are g(0) = 1, g(1) = 2, g(2) = 4, and for any n with n ≥ 2, g(n+1) = g(n) + g(n-1) + 2g(n-2). The language L, over the alphabet {a, b, c}, is denoted by the regular expression (∅* + b + bb)(a + ab + abb + acc)*.

• (a, 10) Prove by ordinary induction on all naturals n that f(n) = (n3 - 4n + 3)/3.

• (b, 10) Prove by strong induction on all naturals n that g(n) = 2n. Make sure that you have the correct base cases.

• (c, 10XC) Let h(n) be the number of strings in L of length n. Prove by strong induction for all naturals n that h(n) = g(n). Make sure that you have the correct base cases.

• Question 3 (40): This question is the usual one about Kleene's Theorem constructions, using the λ-NFA N given above.

• (a, 5) Use the construction from lecture and the text to find a λ-NFA whose language is denoted by the regular expression (a + ab + abb + acc)*. For full credit, use the construction exactly, without any simplifications.

• (b, 10) Using the construction from the text on the λ-NFA N given above (not on your answer to part (a)), build an ordinary NFA N' such that L(N') = L(N).

• (c, 10) Using the Subset Construction on N', find a DFA D such that L(D) = L(N) = L(N').

• (d, 5) Find a minimal DFA D' with L(D') = L(D). You may use the minimization construction, or prove directly that your D is already minimal.

• (e, 10) By using the State Elimination construction on either D or D', find a regular expression for L(N). (In fact, L(N) is the language of the regular expression from part (a) of this question, so you have a correct regular expression already, but I want the one from the construction, which will probably be more complicated.)

• Question 4 (30): Identify each of the following fifteen statements as true or false. There are two points for each correct answer, with no explanation needed or wanted. Note that there is no penalty for guessing. Some of these statements refer to terms defined at the beginning of the exam. sheet.

• (a) In the scenario of Question 1, define the function h from D to R so that h(x) is dog x's favorite rapper. Then, given Statements I-V, the function h is neither onto nor one-to-one.

• (b) Let x be a natural number that is congruent to 3 modulo 4 and is congruent to 5 modulo 6. Then x must be congruent to 23 modulo 24.

• (c) Let p be any prime number and let x be any positive natural. Then there exists a natural e such that pe divides x and pe+1 does not divide x.

• (d) Let G be the directed graph made from the λ-NFA N above by ignoring the edge labels and merging parallel edges. Then G is not strongly connected.

• (e) Let H be an undirected graph with node set {1, 2, 3, 4, 5, 6} and an edge between nodes i and j if and only if i + j is odd. Then H is a bipartite graph.

• (f) There exists a labeled undirected graph G, with positive integer edge weights, a start node s, a goal node g, and an admissibile and consistent heuristic function h for g, such that the path found by an A* search from s with goal node g is shorter (has smaller total cost) than the path found by a uniform-cost search from s with goal node g.

• (g) Let G be any undirected graph with n nodes, where n > 0. Then G is a tree if and only if for every node x of G, the DFS tree of any depth-first search of G from x contains exactly n-1 tree edges and no back edges.

• (h) Consider a game with a finite game tree, where each leaf is labeled as a win for White or Black and each internal node is labeled as to whether White or Black moves next from that node. Then one of the two players has a winning strategy, and there is an algorithm to determine which, given the entire tree as input.

• (i) Let X and Y be two languages over the alphabet {0, 1}. Then is it possible that Y is regular, X ⊆ Y, and X is not regular.

• (j) The language of the regular expression ab + ba, where the alphabet is {a, b}, has exactly five Myhill-Nerode equivalence classes.

• (k) For every two-way DFA M, there exists a regular expression R such that L(M) = L(R).

• (l) Let X and Y be any languages with X ⊆ Y. If Y is Turing recognizable, then X must be Turing recognizable but may not be Turing decidable.

• (m) A language X is Turing recognizable if and only if both X and its complement are Turing decidable.

• (n) An ordinary one-tape Turing machine, as originally defined in the text, may move more than one space on its tape in a single step.

• (o) The set of descriptions of DFA's that have non-empty languages is Turing decidable.