Module 4 Worksheet
In Class Questions
1) (S4) Decision sorting:
Input: list of integers
Yes/ No Question: is the list in nondecreasing order?
For the decision sorting problem, specify the following sets
Universe of legal input instances:
Set of yes input instances:
Set of no input instances:
2) (S5) Define the set of no input instances for the following decision problem
Universe of legal input instances: Set of all positive integers
Universe of yes input instances: Set of positive numbers that are divisible by 3
Universe of no input instances:
2) (S7) Specify the language recognition problem for L = the set of even length strings
over {a,b} using both formats (traditional, 3 sets).
Input:
Yes/No question:
Universe of legal input instances:
Set of yes input instances:
Set of no input instances:
4) (S9) Define an encoding scheme E for converting the following problem into a language recognition problem over alphabet {a}.
Input: Non-negative integer n
Yes/No question: Is n a prime number?
What would E(4) be under your scheme?
What is the resulting language L?
5) (S11) Suppose a program P is designed to solve some decision problem

. What does
P’s declaration look like?
6) (S11) What should P do on a yes input instance/no input instance?
7) (S23) Which is true? a) A problem is solvable implies the problem is half-solvable. b) A problem is half-solvable implies the problem is solvable.

Take home review questions
1) What is the type of an input instance in a language recognition problem?
2) Give both formulations (Input & Y/N question, 3 sets) of the LRP corresponding to the language L = the set of strings beginning with aaa over {a,b}.
3) What are Y(P), N(P), C(P), and I(P) for the following program?
bool main(string x) {
if ((x.length( ) % 2) = = 0) return (yes);
else return(no);
}
Y(P) =
N(P) =
C(P) =
I(P) =
4) For any program P that takes as input a string, can P ever solve a language other than
Y(P)?