3. How are air exchange and water movement in soils related to the amount of pore space? Include $t$ role of pore size distribution in your answer. 4. How would total pore space and pore size distribution change with depth in a soil that has a sandy loam surface texture and a silty clay loam subsoil?

A graph G has an independent set of size k if there is a set V of k nodes in G so that no two nodes in V share an edge. Let INDEPENDENT-SET = { <G, k> | G is a graph with an independent set of size k }. a. (5) Show that INDEPENDENT-SET exists within NP by writing either a verifier or an NDTM. b. (15) Show that INDEPENDENT-SET is NP-complete by reduction from VERTEX-COVER.

Part VI: Compare algorithms, and try to speed things up! In this last part of the project, you will write code to facilitate the comparison of the different state-space search algorithms, and you will also attempt to speed things up so that your code can more easily solve puzzles that require many moves. Your tasks In eight_puzzle.py, write a function named process_file(filename, algorithm, param). It should take the following three inputs: a string filename specifying the name of a text file in which each line is a digit string for an eight puzzle. For example, here is a sample file containing 10 digit strings, each of which represents an eight puzzle that can be solved in 10 moves: 10_moves.txt a string algorithm that specifies which state-space search algorithm should be used to solve the puzzles ('random', 'BFS', 'DFS', 'Greedy', or 'A*') a third input param that allows you to specify a parameter for the searcher – either a depth limit (for the uninformed search algorithms) or a choice of heuristic function (for the informed search algorithms). Your function should open the file with the specified filename for reading, and it should use a loop to process the file one line at a time. We discussed line-by-line file processing earlier in the semester. For each line of the file, the function should: obtain the digit string on that line (stripping off the newline at the end of the line) take the steps needed to solve the eight puzzle for that digit string using the algorithm and parameter specified by the second and third inputs to the function report the number of moves in the solution, and the number of states tested during the search for a solution In addition, the function should perform the cumulative computations needed to report the following summary statistics after processing the entire file: number of puzzles solved average number of moves in the solutions average number of states tested For example: >>> process_file('10_moves.txt', 'BFS', -1) 125637084: 10 moves, 661 states tested 432601785: 10 moves, 1172 states tested 025318467: 10 moves, 534 states tested 134602785: 10 moves, 728 states tested 341762085: 10 moves, 578 states tested 125608437: 10 moves, 827 states tested 540132678: 10 moves, 822 states tested 174382650: 10 moves, 692 states tested 154328067: 10 moves, 801 states tested 245108367: 10 moves, 659 states tested solved 10 puzzles averages: 10.0 moves, 747.4 states tested (In this case, because BFS finds optimal solutions, every solution has the same number of moves, but for other algorithms this won’t necessarily be the case.) Notes/hints: You can model your code for solving a given puzzle on the code that we’ve given you in the eight_puzzle driver function. In particular, you can emulate the way that this function: creates Board and State objects for the initial state calls the create_searcher helper function to create the necessary type of searcher object, and handles the possible return value of None from that function Important: Make sure to create a new searcher object for each puzzle (i.e., for each line of the file). Otherwise, the searcher will still have information inside it from previous searches when it starts to solve a new puzzle. When calling the searcher object’s find_solution method, you should do so as follows: soln = None try: soln = searcher.find_solution(s) except KeyboardInterrupt: print('search terminated, ', end='') Making the call to find_solution() in this way will allow you to terminate a search that goes on too long by using Ctrl-C. In such cases, soln will end up with a value of None (meaning that no solution was found), and you should make sure to properly handle such cases. You should not use a timer in this function. This function should not return a value. Testing your function: If you haven’t already done so, download the 10_moves.txt file, putting it in the same folder as the rest of your files for the project. Try to reproduce the results for BFS shown above. Try applying Greedy Search to the same file. You may find that it takes Greedy a very long time to solve some of the puzzles, at least when using h1 (the num-misplaced-tiles heuristic). If this happens, use Ctrl-C as needed to terminate the problematic searches. When we processed 10_moves.txt using our implementation of Greedy, we ended up using Ctrl-C to terminate two of the searches: >>> process_file('10_moves.txt', 'Greedy', h1) 125637084: 204 moves, 658 states tested 432601785: 12 moves, 13 states tested 025318467: search terminated, no solution 134602785: 78 moves, 221 states tested 341762085: 26 moves, 339 states tested 125608437: 162 moves, 560 states tested 540132678: 68 moves, 749 states tested 174382650: search terminated, no solution 154328067: 10 moves, 16 states tested 245108367: 48 moves, 49 states tested solved 8 puzzles averages: 76.0 moves, 325.625 states tested It’s also possible for none of the puzzles to have a solution, and you should handle that situation appropriately. For example, this can happen if you impose a depth limit that is too small: >>> process_file('10_moves.txt', 'DFS', 5) # depth limit of 5 125637084: no solution 432601785: no solution 025318467: no solution 134602785: no solution 341762085: no solution 125608437: no solution 540132678: no solution 174382650: no solution 154328067: no solution 245108367: no solution solved 0 puzzles Note that you can’t compute any averages in this case, so you shouldn’t print the averages line in the results. Create a plain-text file named results.txt, and put your name and email (and those of your partner, if any) at the top of the file. Conduct an initial set of experiments in which you try all of the algorithms on the puzzles in the following files: 5_moves.txt - puzzles that can be solved in 5 moves 10_moves.txt - puzzles that can be solved in 10 moves 15_moves.txt - puzzles that can be solved in 15 moves More specifically, you should run the following algorithms on each file: random BFS DFS with a depth limit of 20 DFS with a depth limit of 50 Greedy (using h1 – the num-misplaced-tiles heuristic) A* (using the same heuristic) Note that it may be necessary to use Ctrl-C to terminate searches that take too much time. You might want to pick a given amount of time (e.g., 30 seconds or 1 minute), and use Ctrl-C to terminate any search that doesn’t complete in that time. In your results.txt file, create three tables that summarize the results of these initial experiments. Use a separate table for each file’s results. For example, the results for 5_moves.txt should be put into a table that looks like this: puzzles with 5-move optimal solutions ------------------------------------- algorithm num. solved avg. moves avg. states tested ------------------------------------------------------------------------ random BFS DFS (depth limit 20) DFS (depth limit 50) Greedy Search (using h1) A* (using h1) Below these tables, write a short reflection (one or two paragraphs) in which you summarize the key points that are illustrated by the results. For example, you might discuss whether the algorithms produce reliably optimal results, and how much work they do in finding a solution. There’s no one correct reflection; we just want to see that you have reflected intelligently on the results and drawn some conclusions. Even informed search algorithms like Greedy Search and A* can be slow on problems that require a large number of moves. This is especially true if the heuristic function used by the algorithm doesn’t do a good enough job of estimating the remaining cost to the goal. Our h1 heuristic function–which uses the number of misplaced tiles as the estimate of the remaining cost–is one example of a less than ideal heuristic. For example, consider the following two puzzles: Both of them have 4 misplaced tiles (the ones displayed in red), but the puzzle on the left can be solved in 4 moves, whereas the puzzle on the right requires 24 moves! Clearly, it would be better if our heuristic could do a better job of distinguishing between these two puzzles. Come up with at least one alternate heuristic, and implement it as part of your classes for informed searchers (GreedySearcher and AStarSearcher). To do so, you should take the following steps: As needed, add one or more methods to the Board class that will be used by your new heuristic function. (Adding a new method to the Board class is not required, but it can be helpful to add one so that the heuristic function can obtain the information needed for its estimate.) Add your new heuristic function(s) to searcher.py, and follow these guidelines: Continue the numbering scheme that we established for the earlier heuristic functions. Call your first alternate heuristic function h2, your next heuristic function (if any) h3, etc. Make sure that each heuristic function is a regular function, not a method. In addition, make sure that it takes a single State object and returns an integer. When conducting tests using a new heuristic function, use its name in the same ways that you would use h0 or h1. For example: >>> g = GreedySearcher(h2) >>> eight_puzzle('142358607', 'Greedy', h2) >>> process_file('15_moves.txt', 'A*', h2) You are welcome to design more than one new heuristic function, although only one is required. When testing and refining your heuristic(s), you can use the files that we provided above, as well as the following files: 18_moves.txt - puzzles that can be solved in 18 moves 21_moves.txt - puzzles that can be solved in 21 moves 24_moves.txt - puzzles that can be solved in 24 moves 27_moves.txt - puzzles that can be solved in 27 moves. Compare the performance of Greedy and A* using the h1 heuristic to their performance using your new heuristic(s). Keep revising your heuristic(s) as needed until you are satisfied. Ideally, you should see the following when using your new heuristic(s): Both Greedy and A* are able to solve puzzles more quickly – testing fewer states on average and requiring fewer searches to be terminated. Greedy Search is able to find solutions requiring fewer moves. A* continues to find optimal solutions. (If it starts finding solutions with more than the optimal number of moves, that probably means that your heuristic is overestimating the remaining cost for at least some states.) Although you are welcome to keep more than one new heuristic function, we will ultimately test only one of them. Please adjust your code as needed so that the heuristic function that you want us to test is named h2. Also, please make sure that we will still be able to test the num-misplaced-tiles heuristic if we specify h1 for the heuristic. In your results.txt file, briefly describe how your best new heuristic works: heuristic h2 ------------ This heuristic ... If your code includes other alternate heuristics, you are welcome to describe them as well, although doing so is not required. Conduct a second set of experiments in which you try both your new heuristic function(s) and the h1heuristic function on the puzzles in the four new files provided above (the ones that require 18, 21, 24, and 27 moves). In your results.txt file, create four tables that summarize the results of these experiments. Use a separate table for each file’s results. For example, the results for 18_moves.txt should be put into a table that looks like this: puzzles with 18-move optimal solutions -------------------------------------- algorithm num. solved avg. moves avg. states tested ---------------------------------------------------------------------- Greedy (heuristic h1) Greedy (heuristic h2) # Greedy with any other heuristics A* (heuristic h1) A* (heuristic h2) # Greedy with any other heuristics Below these tables, write a short reflection (one or two paragraphs) in which you summarize the key points that are illustrated by the results. Here again, there is no one correct reflection; we just want to see that you have reflected intelligently on the results and drawn some conclusions.

4.1 Specify the independent variable (IV) and dependent variable (DV) for the proposed study. (4 marks) 4.2 Formulate the null and alternative hypotheses for the proposed study. (4 marks) 4.3 Based on the outcome of the analysis of the data gathered from the twenty employees in the preliminary investigation, discuss the strength, direction and significance of the correlation between employee quiet quitting score and employee performance rating. (6 marks) 4.4 Provide a comprehensive interpretation of the output of the simple linear regression analysis (quote the relevant statistical figures in the interpretation). (8 marks) 4.5 Proffer a practical recommendation to the management of Mayfair Enterprises on how best to mitigate the recent decline in employee and organisational outcomes. (3 marks)

A vehicle moves along a straight road. The vehicle's position is given by $f(t)$, where $t$ is measured in seconds since the vehicle starts moving. During the first 10 seconds of the motion, the vehicle's acceleration is proportional to the cube root of the time since the start. Which of the following differential equations describes this relationship, where $k$ is a positive constant? A $\frac{d f}{d t}=k \sqrt[3]{t}$ (B) $\frac{d f}{d t}=k \sqrt[3]{f}$ C $\frac{d^{2} f}{d t^{2}}=k \sqrt[3]{t}$ (D) $\frac{d^{2} f}{d t^{2}}=k \sqrt[3]{f}$

Using LC-3 Programming Exercise 1 Use .BLKW to set up a remote array of 10 values, starting at memory location x4000, as in lab 3. Now programmatically populate the array with the numbers 0 through 9 (note - NOT the characters '0' through '9'!) – i.e. hard-code just the first value (0), then calculate the rest one at a time, storing them in the array as you go. Remember the proper way of setting a register to 0! After you've stored them all, grab the seventh value (i.e. 6) and store it in R2. Clearly, you can't access this location via a label, so you'll need its actual address. How will you obtain that? And how will you use that address to get the value stored there? As always, step through your program and examine the values as they are stored in the array, and examine the final value stored in R2, to make sure your program works as expected. Exercise 2 You'll notice that Exercise 1 didn't ask you to output to console the array you built. Why not? Because as you know by now, numbers are not characters! So now copy your exercise 1 code into lab4_ex2.asm, and add an output loop, making it output the characters corresponding to the numbers stored in your array. Exercise 3 Let's try another modification of our well-used array program from Exercise 1: This time, instead of calculating and storing the numbers from 0 to 9 in the array, calculate and store the first ten powers of 2, starting with 20 = 1 in array index 0. Finally, grab the seventh value (26) from the array, and store it in R2. In order to do this, you will have to figure out how to calculate powers of 2. Some hints: Mathematically speaking: How do I obtain 2n+1 if I know 2n? What LC-3 operation could I use to multiply a number by 2? As always, place a breakpoint in your program, and step through it, examining the values as they are being stored in the array, and examine the final value stored in R2,  to make sure your program works as expected. Pay attention to both the decimal and hex representations of the values being stored. You already understand that you can't simply output the values in the array to the console “as is”, so we have to manipulate them somehow to turn them into characters. But this time all but the first four are multi-digit numbers when represented as decimal values, so our trick from the last exercise won't work – it can only convert the numbers from 0 to 9 into the single-digit ascii characters '0' through '9'.  Exercise 4 For now, we will instead observe what happens if we attempt to do what we just said can't be done - output the contents of the array "as is", i.e. as if they were ascii codes (which they are not!). Copy your exercise 3 code into your lab4_ex4.asm file, and add a second loop that loads the content of each element of the array into R0 and outputs it to the console using OUT. Now place a breakpoint at the start of your display loop and step through it: You will need to keep an eye on the Console to understand what is going on!! Please add comments!!!

7. The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with mean of 0.04 flaws per square foot of plastic panel. Assume automobile interior contains 10 square feet of plastic panel.(a) What is the probability that there are no surface flaws in an auto's interior?(b) If 10 cars are sold to a rental company, what is the probability that none of the 10 cars has any surface flaws?(c) If 10 cars are sold to a rental company, what is the probability that at most two cars have any surface flaws?