- Answer the problems on the exam pages.
- There are four problems, each with multiple parts, for 120 total points. Actual scale A = 105, C = 70.
- The textbook, course website, and notes are allowed. No other books, calculators, or collaboration.
- In case of a numerical answer, an arithmetic expression like
"2
^{17}- 4" need not be reduced to a single integer.

Q1: 30 points Q2: 30 points Q3: 30 points Q4: 30 points Total: 120 points

- (a, 10) Translate the following statements as indicated:
- Statement I (to symbols): Every dog who sat nicely got a treat.
- Statement II (to English): (RO(m) ↔ ∃z:(z≠m) ∧ RO(z)) ∧ &neg;SN(m)
- Statement III (to symbols): Neither Becky nor Cindy sat nicely, and one of them ran off while the other got a treat.
- Statement IV (to symbols): At least two different dogs sat nicely.
- Statement V (to English): ∀y:((&neg;RO(y)) → GT(y))

- (b, 5) Consider the six propositions GT(b), GT(c), RO(b),
RO(c), SN(b), and SN(c).
There are 2
^{6}= 64 possible assignments of truth values to these propositions. How many of these are consistent with Statement III? Justify your answer. (Hint: If you make a truth table, you can simplify your life by considering only four of the six variables.) - (c, 15) Using Statements I-IV, prove Statement V. You may use English, symbols, or a combination, but make your use of quantifier proof rules clear.

- (a, 5) Prove that for all naturals n, except for n = 1, there exists a string of length n in S. (Hint: Show the n = 0 case separately and then use induction on all n with n ≥ 2.)
- (b, 5) Let f(n) be the number of strings in S of length n. Explain why, for all n with n ≥ 3, f(n) = 3f(n-2) + 2f(n-3).
- (c, 10) Prove, for all naturals n, that f(n) =
[2
^{n+2}+ (-1)^{n}(3n+5)]/9. (Hint: Use strong induction on the recurrence from part (b), with base cases for n = 0, n = 1, and n = 2. - (d, 10) The Myhill-Nerode Theorem tells us that
Σ
^{*}is divided into a finite number of S-equivalence classes. How many are there? Justify your answer. You may quote the Myhill-Nerode Theorem to justify your answer if you do so correctly.

- (a, 10)
Construct an ordinary NFA N’ from N using the construction given in
lecture. You should
get eleven a-moves and six b-moves.
- (b, 5) Construct a DFA D from N’ using the Subset Construction.
- (c, 5) Construct a minimal DFA D’ equivalent to D. If you
do not use the minimization construction from the textbook
and supplemental lecture, prove that your D’ is in fact minimal.
- (d, 5) Construct a regular expression R with L(R) = L(D). You may start from either D or D’.
- (e, 5) Construct a lambda-NFA with the same language as R, using the construction from lecture and the textbook.

- (a) If T is any rooted tree with n nodes and e edges, then it is possible that e = n-2.
- (b) Let A and B be any two languages (sets of
strings). If A
^{*}⊆ B^{*}, then A ⊆ B. - (c) If the contrapositive of a statement is true, then the converse of the statement is true as well.
- (d) A basket contains a certain number of apples. If I take them out 5 at a time, 3 apples remain. If I take them out 6 at a time, 1 apple remains. Then the smallest number of apples this basket might contain is 13.
- (e) The congruence 5x ≡ 3 (mod 10) has no solution.
- (f) For an undirected graph G and a node v in that graph, the DFS and BFS trees of G rooted at v always contain the same number of edges.
- (g) Let h
_{1}(s) and h_{2}(s) be two non-negative potential heuristics for the same search problem. If either h_{1}(s) is admissible and h_{2}(s) is inadmissible, then h_{3}(s) = min(h_{1}(s), h_{2}(s)) will be admissible. - (h) The maximum number of edges in a bipartite graph on 14 vertices is 91.
- (i) The less-than relation, <, on a set of real numbers is not a partial ordering because while it is antisymmetric, it is irreflexive.
- (j) Given any multi-tape Turing machine, there is a single-tape Turing machine that has the same language.
- (k) The regular expressions
(1
^{*}0^{*})^{*}and (0 + 1)^{*}denote the same language. - (l) Every subset of a regular language is regular.
- (m) If L and M are two languages such that M, LM and ML are all regular, then L must be regular.
- (n) There exists a finite language (i.e., a language with a finite number of words) that is not the language of any DFA.
- (o) Every language is Turing recognizable.

Last modified 19 May 2020