Crossword

Write an AI to generate crossword puzzles.

$ python generate.py data/structure1.txt data/words1.txt output.png
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Background

How might you go about generating a crossword puzzle? Given the structure of a crossword puzzle (i.e., which squares of the grid are meant to be filled in with a letter), and a list of words to use, the problem becomes one of choosing which words should go in each vertical or horizontal sequence of squares. We can model this sort of problem as a constraint satisfaction problem. Each sequence of squares is one variable, for which we need to decide on its value (which word in the domain of possible words will fill in that sequence). Consider the following crossword puzzle structure.

In this structure, we have four variables, representing the four words we need to fill into this crossword puzzle (each indicated by a number in the above image). Each variable is defined by four values: the row it begins on (its i value), the column it begins on (its j value), the direction of the word (either down or across), and the length of the word. Variable 1, for example, would be a variable represented by a row of 1 (assuming 0 indexed counting from the top), a column of 1 (also assuming 0 indexed counting from the left), a direction of across, and a length of 4.

As with many constraint satisfaction problems, these variables have both unary and binary constraints. The unary constraint on a variable is given by its length. For Variable 1, for instance, the value BYTE would satisfy the unary constraint, but the value BIT would not (it has the wrong number of letters). Any values that don’t satisfy a variable’s unary constraints can therefore be removed from the variable’s domain immediately.

The binary constraints on a variable are given by its overlap with neighboring variables. Variable 1 has a single neighbor: Variable 2. Variable 2 has two neighbors: Variable 1 and Variable 3. For each pair of neighboring variables, those variables share an overlap: a single square that is common to them both. We can represent that overlap as the character index in each variable’s word that must be the same character. For example, the overlap between Variable 1 and Variable 2 might be represented as the pair (1, 0), meaning that Variable 1’s character at index 1 necessarily must be the same as Variable 2’s character at index 0 (assuming 0-indexing, again). The overlap between Variable 2 and Variable 3 would therefore be represented as the pair (3, 1): character 3 of Variable 2’s value must be the same as character 1 of Variable 3’s value.

For this problem, we’ll add the additional constraint that all words must be different: the same word should not be repeated multiple times in the puzzle.

The challenge ahead, then, is write a program to find a satisfying assignment: a different word (from a given vocabulary list) for each variable such that all of the unary and binary constraints are met.

Getting Started

• Download the distribution code from https://cdn.cs50.net/ai/2020/spring/projects/3/crossword.zip and unzip it.

Understanding

There are two Python files in this project: crossword.py and generate.py. The first has been entirely written for you, the second has some functions that are left for you to implement.

First, let’s take a look at crossword.py. This file defines two classes, Variable (to represent a variable in a crossword puzzle) and Crossword (to represent the puzzle itself).

Notice that to create a Variable, we must specify four values: its row i, its column j, its direction (either the constant Variable.ACROSS or the constant Variable.DOWN), and its length.

The Crossword class requires two values to create a new crossword puzzle: a structure_file that defines the structure of the puzzle (the _ is used to represent blank cells, any other character represents cells that won’t be filled in) and a words_file that defines a list of words (one on each line) to use for the vocabulary of the puzzle. Three examples of each of these files can be found in the data directory of the project, and you’re welcome to create your own as well.

Note in particular, that for any crossword object crossword, we store the following values:

• crossword.height is an integer representing the height of the crossword puzzle.
• crossword.width is an integer representing the width of the crossword puzzle.
• crossword.structure is a 2D list representing the structure of the puzzle. For any valid row i and column j, crossword.structure[i][j] will be True if the cell is blank (a character must be filled there) and will be False otherwise (no character is to be filled in that cell).
• crossword.words is a set of all of the words to draw from when constructing the crossword puzzle.
• crossword.variables is a set of all of the variables in the puzzle (each is a Variable object).
• crossword.overlaps is a dictionary mapping a pair of variables to their overlap. For any two distinct variables v1 and v2, crossword.overlaps[v1, v2] will be None if the two variables have no overlap, and will be a pair of integers (i, j) if the variables do overlap. The pair (i, j) should be interpreted to mean that the ith character of v1’s value must be the same as the jth character of v2’s value.

Crossword objects also support a method neighbors that returns all of the variables that overlap with a given variable. That is to say, crossword.neighbors(v1) will return a set of all of the variables that are neighbors to the variable v1.

Next, take a look at generate.py. Here, we define a class CrosswordCreator that we’ll use to solve the crossword puzzle. When a CrosswordCreator object is created, it gets a crossword property that should be a Crossword object (and therefore has all of the properties described above). Each CrosswordCreator object also gets a domains property: a dictionary that maps variables to a set of possible words the variable might take on as a value. Initially, this set of words is all of the words in our vocabulary, but we’ll soon write functions to restrict these domains.

We’ve also defined some functions for you to help with testing your code: print will print to the terminal a representation of your crossword puzzle for a given assignment (every assignment, in this function and elsewhere, is a dictionary mapping variables to their corresponding words). save, meanwhile, will generate an image file corresponding to a given assignment (you’ll need to pip3 install Pillow if you haven’t already to use this function). letter_grid is a helper function used by both print and save that generates a 2D list of all characters in their appropriate positions for a given assignment: you likely won’t need to call this function yourself, but you’re welcome to if you’d like to.

Finally, notice the solve function. This function does three things: first, it calls enforce_node_consistency to enforce node consistency on the crossword puzzle, ensuring that every value in a variable’s domain satisfy the unary constraints. Next, the function calls ac3 to enforce arc consistency, ensuring that binary constraints are satisfied. Finally, the function calls backtrack on an initially empty assignment (the empty dictionary dict()) to try to calculate a solution to the problem.

The functions enforce_node_consistency, ac3, and backtrack, though, are not yet implemented (among other functions). That’s where you come in!

Specification

Complete the implementation of enforce_node_consistency, revise, ac3, assignment_complete, consistent, order_domain_values, selected_unassigned_variable, and backtrack in generate.py so that your AI generates complete crossword puzzles if it is possible to do so.

The enforce_node_consistency function should update self.domains such that each variable is node consistent.

• Recall that node consistency is achieved when, for every variable, each value in its domain is consistent with the variable’s unary constraints. In the case of a crossword puzzle, this means making sure that every value in a variable’s domain has the same number of letters as the variable’s length.
• To remove a value x from the domain of a variable v, since self.domains is a dictionary mapping variables to sets of values, you can call self.domains[v].remove(x).
• No return value is necessary for this function.

The revise function should make the variable x arc consistent with the variable y.

• x and y will both be Variable objects representing variables in the puzzle.
• Recall that x is arc consistent with y when every value in the domain of x has a possible value in the domain of y that does not cause a conflict. (A conflict in the context of the crossword puzzle is a square for which two variables disagree on what character value it should take on.)
• To make x arc consistent with y, you’ll want to remove any value from the domain of x that does not have a corresponding possible value in the domain of y.
• Recall that you can access self.crossword.overlaps to get the overlap, if any, between two variables.
• The domain of y should be left unmodified.
• The function should return True if a revision was made to the domain of x; it should return False if no revision was made.

The ac3 function should, using the AC3 algorithm, enforce arc consistency on the problem. Recall that arc consistency is achieved when all the values in each variable’s domain satisfy that variable’s binary constraints.

• Recall that the AC3 algorithm maintains a queue of arcs to process. This function takes an optional argument called arcs, representing an initial list of arcs to process. If arcs is None, your function should start with an initial queue of all of the arcs in the problem. Otherwise, your algorithm should begin with an initial queue of only the arcs that are in the list arcs (where each arc is a tuple (x, y) of a variable x and a different variable y).
• Recall that to implement AC3, you’ll revise each arc in the queue one at a time. Any time you make a change to a domain, though, you may need to add additional arcs to your queue to ensure that other arcs stay consistent.
• You may find it helpful to call on the revise function in your implementation of ac3.
• If, in the process of enforcing arc consistency, you remove all of the remaining values from a domain, return False (this means it’s impossible to solve the problem, since there are no more possible values for the variable). Otherwise, return True.

The assignment_complete function should (as the name suggests) check to see if a given assignment is complete.

• An assignment is a dictionary where the keys are Variable objects and the values are strings representing the words those variables will take on.
• An assignment is complete if every crossword variable is assigned to a value (regardless of what that value is).
• The function should return True if the assignment is complete and return False otherwise.

The consistent function should check to see if a given assignment is consistent.

• An assignment is a dictionary where the keys are Variable objects and the values are strings representing the words those variables will take on. Note that the assignment may not be complete: not all variables will necessarily be present in the assignment.
• An assignment is consistent if it satisfies all of the constraints of the problem: that is to say, all values are distinct, every value is the correct length, and there are no conflicts between neighboring variables.
• The function should return True if the assignment is consistent and return False otherwise.

The order_domain_values function should return a list of all of the values in the domain of var, ordered according to the least-constraining values heuristic.

• var will be a Variable object, representing a variable in the puzzle.
• Recall that the least-constraining values heuristic is computed as the number of values ruled out for neighboring unassigned variables. That is to say, if assigning var to a particular value results in eliminating n possible choices for neighboring variables, you should order your results in ascending order of n.
• Note that any variable present in assignment already has a value, and therefore shouldn’t be counted when computing the number of values ruled out for neighboring unassigned variables.
• For domain values that eliminate the same number of possible choices for neighboring variables, any ordering is acceptable.
• Recall that you can access self.crossword.overlaps to get the overlap, if any, between two variables.
• It may be helpful to first implement this function by returning a list of values in any arbitrary order (which should still generate correct crossword puzzles). Once your algorithm is working, you can then go back and ensure that the values are returned in the correct order.
• You may find it helpful to sort a list according to a particular key: Python contains some helpful functions for achieving this.

The select_unassigned_variable function should return a single variable in the crossword puzzle that is not yet assigned by assignment, according to the minimum remaining value heuristic and then the degree heuristic.

• An assignment is a dictionary where the keys are Variable objects and the values are strings representing the words those variables will take on. You may assume that the assignment will not be complete: not all variables will be present in the assignment.
• Your function should return a Variable object. You should return the variable with the fewest number of remaining values in its domain. If there is a tie between variables, you should choose among whichever among those variables has the largest degree (has the most neighbors). If there is a tie in both cases, you may choose arbitrarily among tied variables.
• It may be helpful to first implement this function by returning any arbitrary unassigned variable (which should still generate correct crossword puzzles). Once your algorithm is working, you can then go back and ensure that you are returning a variable according to the heuristics.
• You may find it helpful to sort a list according to a particular key: Python contains some helpful functions for achieving this.

The backtrack function should accept a partial assignment assignment as input and, using backtracking search, return a complete satisfactory assignment of variables to values if it is possible to do so.

• An assignment is a dictionary where the keys are Variable objects and the values are strings representing the words those variables will take on. The input assignment may not be complete (not all variables will necessarily have values).
• If it is possible to generate a satisfactory crossword puzzle, your function should reutrn the complete assignment: a dictionary where each variable is a key and the value is the word that the variable should take on. If no satisfying assignment is possible, the function should return None.
• If you would like, you may find that your algorithm is more efficient if you interleave search with inference (as by maintaining arc consistency every time you make a new assignment). You are not required to do this, but you are permitted to, so long as your function still produces correct results. (It is for this reason that the ac3 function allows an arcs argument, in case you’d like to start with a different queue of arcs.)

You should not modify anything else in generate.py other than the functions the specification calls for you to implement, though you may write additional functions and/or import other Python standard library modules. You may also import numpy or pandas, if familiar with them, but you should not use any other third-party Python modules. You should not modify anything in crossword.py.

Hints

• For order_domain_values and select_unassigned_variable, it may be helpful to implement them first without worrying about the heuristics, and then add heuristics later. Your algorithm will still work: it just may end up exploring more assignments than it needs to before finding a solution.
• To run your program, you can run a command like python generate.py data/structure1.txt data/words1.txt, specifying a structure file and a words file. If an assignment is possible, you should see the resulting assignment printed. You may also add an additional command-line argument for an image file, as by running python generate.py data/structure1.txt data/words1.txt output.png, to generate an image representation of the resulting crossword puzzle as well.
• The Crossword class has a neighbors function you can use to access all of the neighbors (i.e., overlapping variables) of a particular variable. Feel free to use that any time you need to determine the neighbors of a particular variable!

How to Submit

If you don’t already have it installed, install submit50 by running pip3 install submit50. Then, execute the below, logging in with your GitHub username and password when prompted. For security, you’ll see asterisks (*) instead of the actual characters in your password.

submit50 ai50/problems/2020/spring/crossword