What courses should you take in a given quarter? Answering this question requires balancing your interests, satisfying prerequisite chains, graduation requirements, availability of courses; this can be a complex tedious process. In this assignment, you will write a program that does automatic course scheduling for you based on your preferences and constraints. The program will cast the course scheduling problem (CSP) as a constraint satisfaction problem (CSP) and then use backtracking search to solve that CSP to give you your optimal course schedule.

You will first get yourself familiar with the basics of CSPs in Problem 0. In Problem 1, you will implement a heuristic you learned from lecture that will make CSP solving much faster. Lastly, in Problem 2, you will create the course scheduling CSP and solve it using the code from previous parts.

## Problem 0: CSP basics

1. Let's create a CSP. Suppose you have $n$ light bulbs, where each light bulb $i = 1, \dots, n$ is initially off. You also have $m$ buttons which control the lights. For each button $j = 1, \dots, m$, we know the subset $T_j \subseteq \{ 1, \dots, n \}$ of light bulbs that it controls. When button $j$ is pressed, it toggles the state of each light bulb in $T_j$ (for example, if $3 \in T_j$ and light bulb 3 is off, then after the button is pressed, light bulb 3 will be on, and vice versa). If multiple buttons—say $Z$ buttons—controlling the same light bulb are pressed, then that light bulb will be turned on if $Z$ is odd, or it will be turned off if $Z$ is even.

Your goal is to turn on all the light bulbs by pressing a subset of the buttons. Construct a CSP to solve this problem. Your CSP should have $m$ variables and $n$ constraints. For this problem only, you can use $m$-ary constraints: constraints that can be functions of up to $m$ variables. Describe your CSP precisely and concisely. You need to specify the variables with their domain, and the constraints with their scope and expression. Make sure to include $T_j$ in your answer.

Hint: If stuck, take a look at parts (b) and (c) of this problem to see how you could define the constraints using a boolean operator.

A clear description of your solution CSP including $m$ variables and their domains, and $n$ constraints.

2. Now, let's consider a simple CSP with 3 variables and 2 binary factors:

where $X_1,X_2,X_3 \in \{0,1\}$ and $t_1, t_2$ are XOR functions (that is $t_1(X) = x_1 \bigoplus x_2$ and $t_2(X) = x_2 \bigoplus x_3$).
1. What are the consistent assignments for this CSP?
2. Let's use backtracking search to solve the CSP without using any heuristics (MCV, LCV, forward checking, AC-3). The Backtrack() algorithm as defined in the lectures is a recursive algorithm, where new instances of Backtrack() are called within parent instances of Backtrack().

In this problem, we will ask you to produce the call stack for a specific call to Backtrack(). A call stack is just a diagram tracing out every recursive call. For our purposes, for each call to Backtrack() you should specify which variable is being assigned, the current domains, and which parent call to Backtrack() it's called within. For example, if the order in which we assign variables is $X_1$, $X_2$, $X_3$, the call stack would be as follows:
 {[01], [01], [01]} $\xrightarrow{X_1=0}$ {0, [01], [01]} $\xrightarrow{X_2=1}$ {0, 1, [01]} $\xrightarrow{X_3=0}$ {0, 1, 0} $\xrightarrow{X_1=1}$ {1, [01], [01]} $\xrightarrow{X_2=0}$ {1, 0, [01]} $\xrightarrow{X_3=1}$ {1, 0, 1}

The notation {1, [01], [01]} means that $X_1$ has been assigned value 1, while $X_2$ and $X_3$ are currently unassigned and each have domain $\{0,1\}$. Note that we omit the comma in the domain for easier reading. We also avoid the weight variable for simplicity; the only possible weights for this problem are 0 and 1. In this case, backtrack is called 7 times. Notice that Backtrack() is not called when there's an inconsistent partial assignment ($\delta=0$); for example, we don't call Backtrack() on $X_2 = 1$ when $X_1$ is already set to 1.

Using this call stack, we can produce the list of calls in the order they are explored. For this example where we assign variables in order $X_1$, $X_2$, $X_3$, the list would be $\{[01], [01], [01]\}, \{0, [01], [01]\}, \{0, 1, [01]\}, \{0, 1, 0\}, \{1, [01], [01]\}, \{1, 0, [01]\}, \{1, 0, 1\}$.

Suppose we assign variables in the order $X_3$, $X_1$, $X_2$. Write the list of calls in the order they are explored and draw out the call-stack. How many calls do we make to Backtrack()? Why can this number change depending on the ordering?

3. We often add heuristics like AC-3 to speed up the backtracking search. How many calls to Backtrack() from your call stack in the previous question would we skip if we use AC-3? Briefly explain why we skip (or don't skip) calls in this search with AC-3.
3. For i., a list of all the consistent assignments (1 sentence). For ii., a list of calls in order of exploration, a drawing of the call stack, the number of times Backtrack() is called, and an explanation for why this number can change based on the order in which you assign variables (1-4 sentences). For this problem only you may hand-draw a call stack and paste a picture into the PDF, provided that the drawing is neat and everything is legible. For iii., the number of calls to Backtrack() that get skipped along with an explanation for why we skip these calls with AC-3 (1-2 sentences).
4. Now let's consider a general case: given a factor graph with $n$ variables $X_1,...,X_n$ and $n-1$ binary factors $t_1,...,t_{n-1}$ where $X_i \in \{0,1\}$ and $t_i(X) = x_i \bigoplus x_{i+1}$. Note that the CSP has a chain structure. Implement create_chain_csp() by creating a generic chain CSP with XOR as factors.

Note: We've provided you with a CSP implementation in util.py which supports unary and binary factors. For now, you don't need to understand the implementation, but please read the comments and get yourself familiar with the CSP interface. For this problem, you'll need to use CSP.add_variable() and CSP.add_binary_factor().

## Problem 1: CSP solving

We'll now pivot towards creating more complicated CSPs, and solving them faster using heuristics. Notice we are already able to solve the CSPs because in submission.py, a basic backtracking search is already implemented. For this problem, we will work with unweighted CSPs that can only have True/False factors; a factor outputs 1 if a constraint is satisfied and 0 otherwise. The backtracking search operates over partial assignments, and specifies whether or not the current assignment satisfies all relevant constraints. When we assign a value to a new variable $X_i$, we check that all constraints that depend only on $X_i$ and the previously assigned variables are satisfied. The function satisfies_constraints() returns whether or not these new factors are satisfied based on the unaryFactors and binaryFactors. When satisfies_constraints() returns False, any full assignment that extends the new partial assignment cannot satisfy all of the constraints, so there is no need to search further with that new partial assignment.

Take a look at BacktrackingSearch.reset_results() to see the other fields which are set as a result of solving the weighted CSP. You should read submission.BacktrackingSearch carefully to make sure that you understand how the backtracking search is working on the CSP.

1. Let's create an unweighted CSP to solve the n-queens problem: Given an $n\times n$ board, we'd like to place $n$ queens on this board such that no two queens are on the same row, column, or diagonal. Implement create_nqueens_csp() by adding $n$ variables and some number of binary factors. Note that the solver collects some basic statistics on the performance of the algorithm. You should take advantage of these statistics for debugging and analysis. You should get 92 (optimal) assignments for $n=8$ with exactly 2057 operations (number of calls to backtrack()).

Hint: If you get a larger number of operations or your code times out on the test cases, make sure your CSP is minimal. Try to define the variables such that the size of domain is O(n).

Note: Please implement the domain of variables as 'list' type in Python. You can refer to create_map_coloring_csp() and create_weighted_csp() in util.py as examples of CSP problem implementations. You can try these examples out by running:

python run_p1.py

2. You might notice that our search algorithm explores quite a large number of states even for the $8\times 8$ board. Let's see if we can do better. One heuristic we discussed in class is using most constrained variable (MCV): To choose an unassigned variable, pick the $X_j$ that has the fewest number of values $a$ which are consistent with the current partial assignment ($a$ for which satisfies_constraints() on $X_j=a$ returns True). Implement this heuristic in get_unassigned_variable() under the condition self.mcv = True. It should take you exactly 1361 operations to find all optimal assignments for 8 queens CSP — that's 30% fewer!

Some useful fields:

• In BacktrackingSearch, if var has been assigned a value, you can retrieve it using assignment[var]. Otherwise var is not in assignment.

## Problem 2: Course Scheduling

In this problem, you will leverage our CSP solver for the problem of course scheduling. We have scraped a subset of courses that are offered from Stanford's Bulletin. For each course in this dataset, we have information on which quarters it is offered, the prerequisites (which may not be fully accurate due to ambiguity in the listing), and the range of units allowed. You can take a look at all the courses in courses.json. Please refer to util.Course and util.CourseBulletin for more information.

To specify a desired course plan, you would need to provide a profile which specifies your constraints and preferences for courses. A profile is specified in a text file (see profile*.txt for examples). The profile file has four sections:

• The first section specifies a fixed minimum and maximum (inclusive) number of units you need to take for each quarter. For example:

minUnits 0
maxUnits 3


• In the second section, you register for the quarters that you want to take your courses in. For example,

register Aut2019
register Win2020
register Spr2020


would sign you up for this academic year. The quarters need not be contiguous, but they must follow the exact format XxxYYYY where Xxx is one of Aut, Win, Spr, Sum and YYYY is the year.
• The third section specifies the list of courses that you've taken in the past and elsewhere using the taken keyword. For example, if you're in CS221, this is probably what you would put:

taken CS103
taken CS106B
taken CS107
taken CS109


• The last section is a list of courses that you would like to take during the registered quarters, specified using request. For example, two basic requests would look like this:

request CS224N
request CS229


Not every request must be fulfilled, and indeed, due to the additional constraints described below, it is possible that not all of them can actually be fulfilled.

Constrained requests. To allow for more flexibility in your preferences, we allow some freedom to customize the requests:

• You can request to take exclusively one of several courses by specifying:

request CS229 or CS229A or CS229T

Note that these courses do not necessarily have to be offered in the same quarter. The final schedule can have at most one of these three courses. Each course can only be requested at most once.

• If you want to take a course in one of a specified set of quarters, use the in modifier. For example, if you want to take one of CS221 or CS229 in either Aut2018 or Sum2019, do:

request CS221 or CS229 in Aut2018,Sum2019

If you do not specify any quarters, then the course can be taken in any quarter.

• Another operator you can apply is after, which specifies that a course must be taken after another one. For example, if you want to choose one of CS221 or CS229 and take it after both CS109 and CS161, add:
request CS221 or CS229 after CS109,CS161
Note that this implies that if you take CS221 or CS229, then you must take both CS109 and CS161. In this case, we say that CS109 and CS161 are prereqs of this request. (Note that there's no space after the comma.)

If you request course A and B (separately), and A is an official prerequisite of B based on the CourseBulletin, we will automatically add A as a prerequisite for B; that is, typing request B is equivalent to request B after A. Note that if B is a prerequisite of A, to request A, you must either request B or declare you've taken B before.

• Finally, the last operator you can add is weight, which adds non-negative weight to each request. To accommodate this, we will work with a standard CSP (as opposed to unweighted, like Problem 1), which associates a weight for each assignment $x$ based on the product of $m$ factor functions $f_1, \dots, f_m$: $$\text{Weight}(x) = \prod^m_{j=1}f_j(x)$$ where each factor $f_j(x)\geq 0$. Our goal is to find the assignment(s) $x$ with the highest weight. Notice that our backtracking search already works with normal CSPs; you should simply define factors that output real numbers. For CSP construction, you can refer to the CSP examples we have provided in util.py for guidance (create_map_coloring_csp() and create_weighted_csp()).

All requests have a default weight value of 1. Requests with higher weight should be preferred by your CSP solver. Note that you can combine all of the aforementioned operators into one as follows (again, no space after comma):

request CS221 or CS229 in Win2018,Win2019 after CS131 weight 5

Each request line in your profile is represented in code as an instance of the Request class (see util.py). For example, the request above would have the following fields:

• cids (course IDs that you're choosing one of) with value ['CS221', 'CS229']
• quarters (that you're allowed to take the courses) with value ['Win2018', 'Win2019']
• prereqs (course IDs that you must take before) with value ['CS131']
• weight (preference) with value 5.0

It's important to note that a request does not have to be fulfilled, but if it is, the constraints specified by the various operators after,in must also be satisfied.

You shall not worry about parsing the profiles because we have done all the parsing of the bulletin and profile for you, so all you need to work with is the collection of Request objects in Profile and CourseBulletin to know when courses are offered and the number of units of courses.

Well, that's a lot of information! Let's open a python shell and see them in action:
import util
bulletin = util.CourseBulletin('courses.json')
# retrieve information of CS221
cs221 = bulletin.courses['CS221']
print(cs221)
# look at various properties of the course
print(cs221.cid)
print(cs221.minUnits)
print(cs221.maxUnits)
print(cs221.prereqs)  # the prerequisites
print(cs221.is_offered_in('Aut2018'))
print(cs221.is_offered_in('Win2019'))

profile = util.Profile(bulletin, 'profile_example.txt')
profile.print_info()
# iterate over the requests and print out the properties
for request in profile.requests:
print(request.cids, request.quarters, request.prereqs, request.weight)


Solving the CSP. Your task is to take a profile and bulletin and construct a CSP. We have started you off with code in SchedulingCSPConstructor that constructs the core variables of the CSP as well as some basic constraints. The variables are all pairs of requests and registered quarters (request, quarter), and the value of such a variable is one of the course IDs in that Request or None, which indicates none of the courses should be taken in that quarter. We will add auxiliary variables later. We have also implemented some basic constraints: add_bulletin_constraints(), which enforces that a course can only be taken if it's offered in that quarter (according to the bulletin), and add_norepeating_constraints(), which constrains that no course can be taken more than once.

You should take a look at add_bulletin_constraints() and add_norepeating_constraints() to get a basic understanding how the CSP for scheduling is represented. Nevertheless, we'll highlight some important details to make it easier for you to implement:

• The existing variables are tuples of (request, quarter) where request is a Request object (like the one shown above) and quarter is a str representing a quarter (e.g. 'Aut2018'). For detail please look at SchedulingCSPConstructor.add_variables().
• The domain for quarter is all possible quarters (self.profile.quarters, e.g. ['Win2016', 'Win2017']).
• Given a course ID cid, you can get the corresponding Course object by self.bulletin.courses[cid].
1. Implement the function add_quarter_constraints() in submission.py. This is when your profile specifies which quarter(s) you want your requested courses to be taken in. This is not saying that one of the courses must be taken, but if it is, then it must be taken in any one of the specified quarters. Also note that this constraint will apply to all courses in that request.
2. Let's now add the unit constraints in add_unit_constraints().
1. In order for our solution extractor to obtain the number of units, for every course, you must add a variable (courseId, quarter) to the CSP taking on a value equal to the number of units being taken for that course during that quarter. When the course is not taken during that quarter, the unit should be 0.
2. You must take into account the appropriate binary factor between (request, quarter) and (courseId, quarter) variables.
3. You must ensure that the sum of units per quarter for your schedule are within the min and max threshold, inclusive. You should use the create_sum_variable() function we've implemented for you; pay careful attention to the arguments.

Hint: If your code times out, your maxSum passed to create_sum_variable() might be too large.

Note: Each grader test only tests the function you are asked to implement. To test your CSP with multiple constraints you can add to the profile text file whichever constraints that you want to add and run run_p2.py. Here is an example with profile2b.txt as input:
python run_p2.py profile2b.txt

Running this command will print information that may be helpful for debugging, such as profile information, the number of optimal assignments found (along with their weight and the number of times backtrack() is called while solving the CSP), one full optimal assignment, and the resulting course schedule.
3. Now try to use the course scheduler for any two quarters in the future (or more quarters if you wish, although this might lead to a slower search). Create your own profile.txt (take a look at some of the profile text files included in the assignment's main directory for inspiration) and then run the course scheduler:
python run_p2.py profile.txt

If the courses you wish to schedule are not listed in courses.json, feel free to add them in as you please! In addition, feel free to modify course details as well (e.g., you can change the list of quarters that a course is being offered in if it does not match the information on the current year's course calendar). You might want to turn on the appropriate heuristic flags to speed up the computation; in particular, self.ac3 = True applies the arc-consistency heuristic that we implement for you, and you can use your own MCV implementation. Does it produce a reasonable course schedule? Please include your profile.txt and the best schedule in your writeup (you can just paste it into the pdf that you submit); we're curious how it worked out for you! Please include your schedule and the profile in the PDF; otherwise you will not receive credit.

The profile.txt file (pasted into the pdf with the solutions), the corresponding outputted schedule, and a brief discussion (1-2 sentences) on whether or not it seems reasonable.