| i, j, k ≥ 0, i ≥ j + k} is context-free.

浙江大学 2005-2006 学年 秋冬 学期期末考试

《计算理论》课程试卷

考试时间：120 分钟 开课学院： 计算机学院 专业：

任课教师： 姓名： 学号：

题序 1 2 3 4 5 6 7 总分

得分 评阅人

Zhejiang University

Theory of Computation, Fall-Winter 2005 Final Exam

1. (30%) Determine whether the following statements are true or false. If it is true write a X otherwise a × in the bracket before the statement.

(a) ( ) Language {a^m(bc)ⁿ : m, n ∈ N} is not regular.

(b) ( ) Language {aⁱ

b

c

| i, j, k ≥ 0, i ≥ j + k} is context-free.

(c) ( ) Let F = {f : f be a primitive recursive function from N to N}, then 2^F (Power set of F ) is uncountable.

(d) ( ) Let L₁

, L

, · · · , L

, · · · be all regular languages, so is ∪

L

(e) ( ) Suppose language L is context-free and L⁰ is a regular, then L^∗

L

(f) ( ) Every computable function is primitive recursive.

(g) ( ) The complement of every recursive enumerable language is recursive enu- merable.

(h) ( ) a^∗

b

∩ c

d

(i) ( ) Every regular language is recursively enumerable.

(j) ( ) Let L be a language and there is a Turing machine M halts on x for every

x ∈ L, then L is decidable.

2. (14%) Decide whether the following languages are regular or not and provide a formal proof for your answer.

(a) L₁ = {aⁿ

b

Theory of Computation Final Exam(Page 2 of 4) Jan. 11, 2006 (b) L2 = {w ∈ {a, b}^∗ : w 6= w^R

}

3. (18%)

(a) Give a Context-Free Grammar that generates the language

L

, |x| = |y| and x and y

For example, abbbbaba, abbbbbbb ∈ L₃, but aababb 6∈ L₃.

(b) Design a PDA M = (K, Σ, Γ, ∆, s, F ) accepting the language L₃. Solution: (a)

(b) The PDA M = (K, Σ, Γ, ∆, s, F ) is defined below:

K =

Γ =

s = F =

(q, σ, β) (p, γ)

Theory of Computation Final Exam(Page 3 of 4) Jan. 11, 2006 4. (16%) Let the following Turing machine M computes f (x, y), the alphabet is

{0, 1, ; }. The head of M begins from the most left blank; t is the symbol of blank;

x and y are presented by binary strings respectively and separated with ;.

(a) Describe the key configurations when M started from the configuration .t10111; 111.

(b) Try to give the function f (x, y) that M can compute.

Theory of Computation Final Exam(Page 4 of 4) Jan. 11, 2006 5. (12%) Let P (x, y) be primitive recursive predicate. Prove the following predicate

∃y

P (x, y), ∀u ∈ N

6. (10%) Show that the following language

H = {“M” | M is a Turing Machine and halts on empty string}

is recursively enumerable. An informal description suffices.