Among the many applications of the constraint satisfaction problem (CSP) paradigm, one of the first was to perform a process called line-labeling to attempt to use a 2-dimensional image of a “blocks-world” environment to infer its 3-dimensional structure so that a robot can reconfigure the blocks to achieve a goal. In this task, you will utilize code that we provide to you, and extend it with code that you write, to develop a deeper understanding of the CSP paradigm and its application to line labeling. As an introduction to the line-labeling problem, see the separate document provided that summarizes it. Part of the line labeling code has been written for you (Python 2.7). Parts a to i will provide more instruction on finishing the code. Many of the utility functions written in will be helpful. The class Vertex is used extensively in and is defined in You will use to run the code. Example commands: python examples/cube.txt propagation -p ac_3 # only run AC3 on cube.txt python examples/cube.txt backtracking -p ac_3 -o mrv # find all solutions of AC3 and MRV python examples/cube.txt backtracking -s -p ac_3 -o mrv # find ONE solution using backtracking with AC3 and MRV python examples/cube.txt backtracking -p forward_checking -o first_unassigned # find ONE solution using backtracking with forward checking and the first unassigned heuristic Run python -h for detailed usage instructions. row# #solns propagation ordering # assignments TOWER # assignments POIUYT 1 all No-prop First-unassigned 2 all Forward-checking First-unassigned 3 all Arc-consistency First-unassigned 4 all No-prop MRV 5 all Forward-checking MRV 6 all Arc-consistency MRV 7 one No-prop First-unassigned 8 one Forward-checking First-unassigned 9 one Arc-consistency First-unassigned 10 one No-prop MRV 11 one Forward-checking MRV 12 one Arc-consistency MRV 13 all Arc-consistency** MRV a. The method for backtracking, search_solutions in was left unfinished. Finish the code to fill in the first 2 rows of the table. b. Implement arc consistency and fill in the 3rd row of the table. Fill in the ac_3 method of the D iagram class in For each of the diagrams, briefly explain the extent to which it did (or did not) affect the number of variable assignments compared to forward-checking, and why. c. In class, we will have talked about how the order in which variables are assigned can make a difference to the search effort. In the current code the first_unassigned method chooses based on alphabetical order. As a first step at considering this, modify first_unassigned to return a random unassigned vertex instead. Test out the new random heuristic and record the number of variable assignments for 10 trials for both diagrams and each of the 3 propagation methods. What do your results tell you about the impact of variable ordering on the search effort for each of these diagrams? d. One way of ordering vertices that heuristically can decrease the number of (tentative) assignments needed to find solutions, as discussed in class, is to assign the variable next that has the fewest number of possible values to assign – the Minimum Remaining Values (MRV) heuristic. Implement this heuristic and add it to your code in the method mrv. Break ties based alphabetical ordering (as was done in first_unassigned). With your new code, fill in rows 4-6 of the Table. e. Use observation(s) from the answer to part c to explain the impact of the MRV heuristic in rows 4-6 you just filled in. Particularly address the fourth row performance on the TOWER diagram. What do the results tell you about the mrv heuristic? f. As you did in part d, experiment with a “random” MRV heuristic (break ties randomly). Comment on whether the variance in the number of tentative assignments differs from part c, and briefly (in a sentence or two) explain why or why not. g. In most applications of constraint satis…

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