CS 181留学生代做、代写algorithms、R程序语言调试、代做R设计

日期：2020-03-18 11:00

CS 181 FINAL EXAM

WINTER 2020

You have 3 hours to complete this exam. You may state without proof any fact

taught in class or assigned as homework. Your algorithms must terminate on all

inputs.

1 Construct one-tape, deterministic Turing machines that decide the languages below. You

must provide both a state-transition diagram and a detailed verbal description:

a. binary strings of the form (0n1)k

(3 pts) , where n ≥ 0 and k ≥ 0;

(3 pts) b. binary strings (including ε) that represent properly nested parentheses, where 0

denotes an open parenthesis and 1 denotes a close parenthesis.

(3 pts) 2 Give an algorithm that takes as input a DFA D and determines whether D accepts at

least two palindromes.

(3 pts) 3 A language L is called downward-closed if for every string w that L contains, L also

contains every prefix of w. Give an algorithm that takes as input a regular expression R

and decides whether the language that R generates is downward-closed.

(3 pts) 4 A variable T in a context-free grammar G is called essential if every derivation of every

string by G uses the variable T. Give an algorithm that takes as input a context-free

grammar G and a variable T, and decides whether T is essential in G.

(3 pts) 5 A string w is said to cover a given PDA P if there is a computation by P on w such

that every state of P is visited. Give an algorithm that takes as input a PDA P and

decides whether there is a string that covers P.

(2 pts) 6 Prove that every infinite decidable language can be partitioned into two disjoint infinite

decidable languages.

7 For each problem below, determine whether it is decidable and prove your answer:

(2.5 pts) a. on input a Turing machine M and a symbol σ, decide whether M ever writes σ on

the tape in any computation;

(2.5 pts) b. on input a Turing machine M, decide if M recognizes a decidable language.