# Final Exam

### Directions:

• Answer the problems on the exam pages.
• There are five problems, some with multiple parts, for 120 total points plus 10 extra credit. Actual scale was A = 100, C = 65.
• 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: 10 points
Q2: 30 points
Q3: 30 points
Q3: 20+10 points
Q4: 30 points
Total: 120+10 points
```

#### Definitions:

Question 2 refers to the following five statements, where the variables are of type "dog", the set of dogs includes the dog Duncan (d) and possibly others, and we use the following predicates:

• F(x) means "dog x is in the forest"

• I(x) means "dog x is illegal", ¬I(x) means "dog x is legal"

• L(x) means "dog x is on a leash"

• V(x) means "dog x is under voice control"

• Y(x) means "dog x is in the yard"

The statements are:

• Statement I: A dog in the forest is legal if and only it is either on a leash or under voice control, or both.

• Statement II: ∀x: ¬(Y(x) ∨ F(x)) → (L(x) ⊕ I(x))

• Statement III: Duncan is on a leash unless he is in the yard.

• Statement IV: ¬∃z: I(z) ∧ Y(z)

• Statement V: No dog on a leash is illegal.

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

Question 3 uses a function f from N to N. The function arises in a popular video game where tiles are labeled by integers that are powers of two. The smallest tiles are labeled 2, and they are created without any points being scored. (We will assume for this exam that only tiles with label 2 are created, though actually the game also creates some with label 4.) When two tiles each with label 2k are merged, a single tile with label 2k+1 is created and the player gets 2k+1 points.

For any positive natural k, f(k) is defined to be the number of points scored in the process of creating a tile with label 2k. If k > 1, this includes the 2k points for the last merge creating the tile, plus all the points scored in the process of creating the tiles that were merged.

Question 4 uses the following two formal languages over the alphabet {a, b}. The language X is the set of all strings that do not have both two a's in a row and two b's in a row. The language Y is defined inductively as follows. Note that the text in green was omitted when the test was given.

1. The string λ is in Y.

2. If the string w in Y can be written as uaa for some string u, then wa = uaaa is in Y.

3. If the string w in Y can be written as ubb for some string u, then wb = ubbb is in Y.

4. If the string w in Y cannot be written as either uaa or ubb, then both wa and wb are in Y.

5. The only strings in Y are those that are forced to be by rules (1)-(4).

Question 5 uses the following λ-NFA N. The alphabet is {a, b}, the state set is {1, 2, 3, 4, 5}, the start state is 1, the final state set is {5}, and the transition relation Δ is

{(1, a, 2), (1, a, 3), (2, b, 4), (3, a, 5), (4, λ, 2), (4, λ, 3), (5, a, 4)}.

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

``````
a         a
>(1) ----> (3) ----> ((5))
|         ^         /
|         |        /
|         |       /
|         |      /
|a        |L    / a
|         |    /
|         |   /
|         |  /
|         | /
V    b    |V
(2) ----> (4)
<----
L
``````

• Question 1 (10): Identify each of the following five concepts, giving enough detail to make it clear that you are familiar with them (2 points each):

• (a, 2) a Turing recognizable language

• (b, 2) the shortest path from node x to node y in a weighted directed graph

• (c, 2) a partial order on a set X

• (d, 2) the transition function of a deterministic one-tape Turing machine

• (e, 2) a pairwise relatively prime set of naturals

• Question 2 (30): This question deals with five statements about a set of dogs that includes the named dog Duncan (d) and possibly others. The predicate F(x) means "dog x is in the forest", I(x) means "dog x is illegal", L(x) means "dog x is on a leash", V(x) means "dog x is under voice control", and Y(x) means "dog x is in the yard". Don't draw inferences from the English meanings -- for example, F(x) ∧ Y(x) might be true.

• (a, 5) Translate the five statements as indicated:

• Statement I (to symbols): A dog in the forest is legal if and only if it is either on a leash or under voice control, or both.

• Statement II (to English): ∀x:¬(Y(x) ∨ F(x)) → (L(x) ⊕ I(x))

• Statement III (to symbols): Duncan is on a leash unless he is in the yard.

• Statement IV (to English): ¬∃z: I(z) ∧ Y(z)

• Statement V (to symbols): No dog on a leash is illegal.

• (b, 5) For each of the statements I, II, and IV, give a propositional statement about Duncan that is implies by the statement. Which predicate proof rule did you use?

• (c, 10) Using Statements I-IV (but not Statement V!) prove by propositional logic that Duncan is legal. (Statement III is a propositional statement about Duncan, and you should use the three propositional statements about Duncan derived in part (b).) You may use a truth table or a propositional proof, and if you choose the latter you may use any format that makes your arguument clear.

• (d, 10) Using Statements I-IV, prove Statement V. Make clear the predicate proof rules that you are using.

• Question 3 (30): This question uses the function f from N to N, defined above.

• (a, 5) State a rule giving f(k+1) in terms of f(k), with a base case for f(1). Use this rule to compute f(1), f(2), f(3), f(4), and f(5).

• (b, 10) Prove by induction, on all naturals n such that n ≥ 3, that f(n) ≥ 2n+1.

• (c, 15) Find a formula that describes f(n) for all positive naturals n. Prove your formula correct by induction. (You should use ordinary induction starting with n = 1.)

• Question 4 (20+10): This question uses the definition of the languages X and Y above. The language X is the set of all strings that do not have both two a's in a row and two b's in a row. The language Y is defined inductively as follows (Note that the text in green was omitted when the test was given):

1. The string λ is in Y.

2. If the string w in Y can be written as uaa for some string u, then wa = uaaa is in Y.

3. If the string w in Y can be written as ubb for some string u, then wb = ubbb is in Y.

4. If the string w in Y cannot be written as either uaa or ubb, then both wa and wb are in Y.

5. The only strings in Y are those that are forced to be by rules (1)-(4).

• (a, 15) Prove Y ⊆ X. I think it is easiest to use induction on the definition of Y, but you might also use ordinary induction on the length of strings in Y.