The Traveling Salesman Problem $($TSP$)$ is defined as follows: the input is a set of cities and a distance

between each pair of cities. The problem is to find a tour of all the cities so that the salesman ends up at

the same place he started so that he visits each city exactly once and travels the shortest possible

distance. In the decision warsion of the problem, the input also includes a threshold $k_{{?}_{?}}$ and the quastion is

whether there is a tour that visits every city exactly once and whoes total distance is at most $k.$

Here is an encoding of the Traveling Salesman Problem. The input alphabet is $\{0,1,2,3,4,5,6,7,8,9\}$

along with a comma, semicolon, left and right parentheses. A valid encoding of a Traveling Salesman

Problem will start with two numbers, each followed by a semicolon. The first number is the number of

cities, and the second number is the threshold for the tour. $/$Note that since we don't have the character $"-"$

or $".",$ all numbers are non$-$negative integers$).$ Each city will hawe a unique name which is a number in the

range from $1$ through $n,$ where $n$ is the number of cities. The encoding then lists triplets of the form $($i$,j,d)$

indicating that the distance between cities $j$ and $j$ is $d.$ If the distance between two cities is given more than

once or not at all, the encoding is not valid. The diagram below shows a picture of $4$ cities and their

distances along with a valid encoding for that particylac. inpht to the Traveling Salesman Problem:

Number

of cities

Tour length

threshold

The Traveling Salesman language is the set of all strings that correspond to valid encodings and are "yes"

inputs for the Traveling Salesman decision problem. Which of the following strings are in the language and

why? If not valid, also say why.

$4$;$10$;$(1,2,1)(1,3,5)(2,3,4)(1,4,5)(2,4,2)(3,4,3)(2,4,2)$

$4$;$11$;$(1,2,1)(1,3,5)(2,3,4)(1,4,5)(2,4,2)(3,4,3)$