This IPy notebook acts as supporting material for topics covered in **Chapter 6 Constraint Satisfaction Problems** of the book* Artificial Intelligence: A Modern Approach*. We make use of the implementations in **csp.py** module. Even though this notebook includes a brief summary of the main topics familiarity with the material present in the book is expected. We will look at some visualizations and solve some of the CSP problems described in the book. Let us import everything from the csp module to get started.

```
In [1]:
```from csp import *
# Needed to hide warnings in the matplotlib sections
import warnings
warnings.filterwarnings("ignore")

CSPs are a special kind of search problems. Here we don't treat the space as a black box but the state has a particular form and we use that to our advantage to tweak our algorithms to be more suited to the problems. A CSP State is defined by a set of variables which can take values from corresponding domains. These variables can take only certain values in their domains to satisfy the constraints. A set of assignments which satisfies all constraints passes the goal test. Let us start by exploring the CSP class which we will use to model our CSPs. You can keep the popup open and read the main page to get a better idea of the code.

```
In [ ]:
```%psource CSP

** _ init _ ** method parameters specify the CSP. Variable can be passed as a list of strings or integers. Domains are passed as dict where key specify the variables and value specify the domains. The variables are passed as an empty list. Variables are extracted from the keys of the domain dictionary. Neighbor is a dict of variables that essentially describes the constraint graph. Here each variable key has a list its value which are the variables that are constraint along with it. The constraint parameter should be a function

We use the graph coloring problem as our running example for demonstrating the different algorithms in the **csp module**. The idea of map coloring problem is that the adjacent nodes (those connected by edges) should not have the same color throughout the graph. The graph can be colored using a fixed number of colors. Here each node is a variable and the values are the colors that can be assigned to them. Given that the domain will be the same for all our nodes we use a custom dict defined by the **UniversalDict** class. The **UniversalDict** Class takes in a parameter which it returns as value for all the keys of the dict. It is very similar to **defaultdict** in Python except that it does not support item assignment.

```
In [2]:
```s = UniversalDict(['R','G','B'])
s[5]

```
Out[2]:
```

**f(A, a, B, b)**. In this what we need is that the neighbors must not have the same color. This is defined in the function **different_values_constraint** of the module.

```
In [ ]:
```%psource different_values_constraint

**parse_neighbors** which allows to take input in the form of strings and return a Dict of the form compatible with the **CSP Class**.

```
In [ ]:
```%pdoc parse_neighbors

**MapColoringCSP** function creates and returns a CSP with the above constraint function and states. The variables our the keys of the neighbors dict and the constraint is the one specified by the **different_values_constratint** function. **australia**, **usa** and **france** are three CSPs that have been created using **MapColoringCSP**. **australia** corresponds to ** Figure 6.1 ** in the book.

```
In [ ]:
```%psource MapColoringCSP

```
In [3]:
```australia, usa, france

```
Out[3]:
```

The N-queens puzzle is the problem of placing N chess queens on a N×N chessboard so that no two queens threaten each other. Here N is a natural number. Like the graph coloring, problem NQueens is also implemented in the csp module. The **NQueensCSP** class inherits from the **CSP** class. It makes some modifications in the methods to suit the particular problem. The queens are assumed to be placed one per column, from left to right. That means position (x, y) represents (var, val) in the CSP. The constraint that needs to be passed on the CSP is defined in the **queen_constraint** function. The constraint is satisfied (true) if A, B are really the same variable, or if they are not in the same row, down diagonal, or up diagonal.

```
In [ ]:
```%psource queen_constraint

**NQueensCSP** method implements methods that support solving the problem via **min_conflicts** which is one of the techniques for solving CSPs. Because **min_conflicts** hill climbs the number of conflicts to solve the CSP **assign** and **unassign** are modified to record conflicts. More details about the structures **rows**, **downs**, **ups** which help in recording conflicts are explained in the docstring.

```
In [ ]:
```%psource NQueensCSP

** init** _ method takes only one parameter

```
In [4]:
```eight_queens = NQueensCSP(8)

We will now implement a few helper functions that will help us visualize the Coloring Problem. We will make some modifications to the existing Classes and Functions for additional book keeping. To begin we modify the **assign** and **unassign** methods in the **CSP** to add a copy of the assignment to the **assignment_history**. We call this new class **InstruCSP**. This will allow us to see how the assignment evolves over time.

```
In [5]:
```import copy
class InstruCSP(CSP):
def __init__(self, variables, domains, neighbors, constraints):
super().__init__(variables, domains, neighbors, constraints)
self.assignment_history = []
def assign(self, var, val, assignment):
super().assign(var,val, assignment)
self.assignment_history.append(copy.deepcopy(assignment))
def unassign(self, var, assignment):
super().unassign(var,assignment)
self.assignment_history.append(copy.deepcopy(assignment))

Next, we define **make_instru** which takes an instance of **CSP** and returns a **InstruCSP** instance.

```
In [6]:
```def make_instru(csp):
return InstruCSP(csp.variables, csp.domains, csp.neighbors, csp.constraints)

```
In [7]:
```neighbors = {
0: [6, 11, 15, 18, 4, 11, 6, 15, 18, 4],
1: [12, 12, 14, 14],
2: [17, 6, 11, 6, 11, 10, 17, 14, 10, 14],
3: [20, 8, 19, 12, 20, 19, 8, 12],
4: [11, 0, 18, 5, 18, 5, 11, 0],
5: [4, 4],
6: [8, 15, 0, 11, 2, 14, 8, 11, 15, 2, 0, 14],
7: [13, 16, 13, 16],
8: [19, 15, 6, 14, 12, 3, 6, 15, 19, 12, 3, 14],
9: [20, 15, 19, 16, 15, 19, 20, 16],
10: [17, 11, 2, 11, 17, 2],
11: [6, 0, 4, 10, 2, 6, 2, 0, 10, 4],
12: [8, 3, 8, 14, 1, 3, 1, 14],
13: [7, 15, 18, 15, 16, 7, 18, 16],
14: [8, 6, 2, 12, 1, 8, 6, 2, 1, 12],
15: [8, 6, 16, 13, 18, 0, 6, 8, 19, 9, 0, 19, 13, 18, 9, 16],
16: [7, 15, 13, 9, 7, 13, 15, 9],
17: [10, 2, 2, 10],
18: [15, 0, 13, 4, 0, 15, 13, 4],
19: [20, 8, 15, 9, 15, 8, 3, 20, 3, 9],
20: [3, 19, 9, 19, 3, 9]
}

**MapColoringProblem** class which inherits from the **CSP** Class. This means that our **make_instru** function will work perfectly for it.

```
In [8]:
```coloring_problem = MapColoringCSP('RGBY', neighbors)

```
In [9]:
```coloring_problem1 = make_instru(coloring_problem)

For solving a CSP the main issue with Naive search algorithms is that they can continue expanding obviously wrong paths. In backtracking search, we check constraints as we go. Backtracking is just the above idea combined with the fact that we are dealing with one variable at a time. Backtracking Search is implemented in the repository as the function **backtracking_search**. This is the same as **Figure 6.5** in the book. The function takes as input a CSP and few other optional parameters which can be used to further speed it up. The function returns the correct assignment if it satisfies the goal. We will discuss these later. Let us solve our **coloring_problem1** with **backtracking_search**.

```
In [10]:
```result = backtracking_search(coloring_problem1)

```
In [11]:
```result # A dictonary of assignments.

```
Out[11]:
```

Let us also check the number of assignments made.

```
In [12]:
```coloring_problem1.nassigns

```
Out[12]:
```

```
In [13]:
```len(coloring_problem1.assignment_history)

```
Out[13]:
```

Now let us explore the optional keyword arguments that the **backtracking_search** function takes. These optional arguments help speed up the assignment further. Along with these, we will also point out to methods in the CSP class that help make this work.

The first of these is **select_unassigned_variable**. It takes in a function that helps in deciding the order in which variables will be selected for assignment. We use a heuristic called Most Restricted Variable which is implemented by the function **mrv**. The idea behind **mrv** is to choose the variable with the fewest legal values left in its domain. The intuition behind selecting the **mrv** or the most constrained variable is that it allows us to encounter failure quickly before going too deep into a tree if we have selected a wrong step before. The **mrv** implementation makes use of another function **num_legal_values** to sort out the variables by a number of legal values left in its domain. This function, in turn, calls the **nconflicts** method of the **CSP** to return such values.

```
In [ ]:
```%psource mrv

```
In [ ]:
```%psource num_legal_values

```
In [ ]:
```%psource CSP.nconflicts

**order_domain_values** governs the value ordering. Here we select the Least Constraining Value which is implemented by the function **lcv**. The idea is to select the value which rules out the fewest values in the remaining variables. The intuition behind selecting the **lcv** is that it leaves a lot of freedom to assign values later. The idea behind selecting the mrc and lcv makes sense because we need to do all variables but for values, we might better try the ones that are likely. So for vars, we face the hard ones first.

```
In [ ]:
```%psource lcv

**inference** can make use of one of the two techniques called Arc Consistency or Forward Checking. The details of these methods can be found in the **Section 6.3.2** of the book. In short the idea of inference is to detect the possible failure before it occurs and to look ahead to not make mistakes. **mac** and **forward_checking** implement these two techniques. The **CSP** methods **support_pruning**, **suppose**, **prune**, **choices**, **infer_assignment** and **restore** help in using these techniques. You can know more about these by looking up the source code.

**solve_simple** and **solve_parameters** and solve them using backtracking and compare the number of assignments.

```
In [14]:
```solve_simple = copy.deepcopy(usa)
solve_parameters = copy.deepcopy(usa)

```
In [16]:
```backtracking_search(solve_simple)
backtracking_search(solve_parameters, order_domain_values=lcv, select_unassigned_variable=mrv, inference=mac)

```
Out[16]:
```

```
In [17]:
```solve_simple.nassigns

```
Out[17]:
```

```
In [18]:
```solve_parameters.nassigns

```
Out[18]:
```

The `tree_csp_solver`

function (**Figure 6.11** in the book) can be used to solve problems whose constraint graph is a tree. Given a CSP, with `neighbors`

forming a tree, it returns an assignement that satisfies the given constraints. The algorithm works as follows:

First it finds the *topological sort* of the tree. This is an ordering of the tree where each variable/node comes after its parent in the tree. The function that accomplishes this is `topological_sort`

, which builds the topological sort using the recursive function `build_topological`

. That function is an augmented DFS, where each newly visited node of the tree is pushed on a stack. The stack in the end holds the variables topologically sorted.

Then the algorithm makes arcs between each parent and child consistent. *Arc-consistency* between two variables, *a* and *b*, occurs when for every possible value of *a* there is an assignment in *b* that satisfies the problem's constraints. If such an assignment cannot be found, then the problematic value is removed from *a*'s possible values. This is done with the use of the function `make_arc_consistent`

which takes as arguments a variable `Xj`

and its parent, and makes the arc between them consistent by removing any values from the parent which do not allow for a consistent assignment in `Xj`

.

If an arc cannot be made consistent, the solver fails. If every arc is made consistent, we move to assigning values.

First we assign a random value to the root from its domain and then we start assigning values to the rest of the variables. Since the graph is now arc-consistent, we can simply move from variable to variable picking any remaining consistent values. At the end we are left with a valid assignment. If at any point though we find a variable where no consistent value is left in its domain, the solver fails.

The implementation of the algorithm:

```
In [ ]:
```%psource tree_csp_solver

We will now use the above function to solve a problem. More specifically, we will solve the problem of coloring the map of Australia. At our disposal we have two colors: Red and Blue. As a reminder, this is the graph of Australia:

`"SA: WA NT Q NSW V; NT: WA Q; NSW: Q V; T: "`

Unfortunately as you can see the above is not a tree. If, though, we remove `SA`

, which has arcs to `WA`

, `NT`

, `Q`

, `NSW`

and `V`

, we are left with a tree (we also remove `T`

, since it has no in-or-out arcs). We can now solve this using our algorithm. Let's define the map coloring problem at hand:

```
In [19]:
```australia_small = MapColoringCSP(list('RB'),
'NT: WA Q; NSW: Q V')

We will input `australia_small`

to the `tree_csp_solver`

and we will print the given assignment.

```
In [20]:
```assignment = tree_csp_solver(australia_small)
print(assignment)

```
```

`WA`

, `Q`

and `V`

got painted with the same color and `NT`

and `NSW`

got painted with the other.

Next, we define some functions to create the visualisation from the assignment_history of **coloring_problem1**. The reader need not concern himself with the code that immediately follows as it is the usage of Matplotib with IPython Widgets. If you are interested in reading more about these visit ipywidgets.readthedocs.io. We will be using the **networkx** library to generate graphs. These graphs can be treated as the graph that needs to be colored or as a constraint graph for this problem. If interested you can read a dead simple tutorial here. We start by importing the necessary libraries and initializing matplotlib inline.

```
In [21]:
```%matplotlib inline
import networkx as nx
import matplotlib.pyplot as plt
import matplotlib
import time

**make_update_step_function** which return such a function. It takes in as inputs the neighbors/graph along with an instance of the **InstruCSP**. This will be more clear with the example below. If this sounds confusing do not worry this is not the part of the core material and our only goal is to help you visualize how the process works.

```
In [22]:
```def make_update_step_function(graph, instru_csp):
def draw_graph(graph):
# create networkx graph
G=nx.Graph(graph)
# draw graph
pos = nx.spring_layout(G,k=0.15)
return (G, pos)
G, pos = draw_graph(graph)
def update_step(iteration):
# here iteration is the index of the assignment_history we want to visualize.
current = instru_csp.assignment_history[iteration]
# We convert the particular assignment to a default dict so that the color for nodes which
# have not been assigned defaults to black.
current = defaultdict(lambda: 'Black', current)
# Now we use colors in the list and default to black otherwise.
colors = [current[node] for node in G.node.keys()]
# Finally drawing the nodes.
nx.draw(G, pos, node_color=colors, node_size=500)
labels = {label:label for label in G.node}
# Labels shifted by offset so as to not overlap nodes.
label_pos = {key:[value[0], value[1]+0.03] for key, value in pos.items()}
nx.draw_networkx_labels(G, label_pos, labels, font_size=20)
# show graph
plt.show()
return update_step # <-- this is a function
def make_visualize(slider):
''' Takes an input a slider and returns
callback function for timer and animation
'''
def visualize_callback(Visualize, time_step):
if Visualize is True:
for i in range(slider.min, slider.max + 1):
slider.value = i
time.sleep(float(time_step))
return visualize_callback

Finally let us plot our problem. We first use the function above to obtain a step function.

```
In [23]:
```step_func = make_update_step_function(neighbors, coloring_problem1)

Next we set the canvas size.

```
In [24]:
```matplotlib.rcParams['figure.figsize'] = (18.0, 18.0)

**Visualize Button** will automatically animate the slider for you. The **Extra Delay Box** allows you to set time delay in seconds upto one second for each time step.

```
In [25]:
```import ipywidgets as widgets
from IPython.display import display
iteration_slider = widgets.IntSlider(min=0, max=len(coloring_problem1.assignment_history)-1, step=1, value=0)
w=widgets.interactive(step_func,iteration=iteration_slider)
display(w)
visualize_callback = make_visualize(iteration_slider)
visualize_button = widgets.ToggleButton(desctiption = "Visualize", value = False)
time_select = widgets.ToggleButtons(description='Extra Delay:',options=['0', '0.1', '0.2', '0.5', '0.7', '1.0'])
a = widgets.interactive(visualize_callback, Visualize = visualize_button, time_step=time_select)
display(a)

```
```

Just like the Graph Coloring Problem we will start with defining a few helper functions to help us visualize the assignments as they evolve over time. The **make_plot_board_step_function** behaves similar to the **make_update_step_function** introduced earlier. It initializes a chess board in the form of a 2D grid with alternating 0s and 1s. This is used by **plot_board_step** function which draws the board using matplotlib and adds queens to it. This function also calls the **label_queen_conflicts** which modifies the grid placing 3 in positions in a position where there is a conflict.

```
In [26]:
```def label_queen_conflicts(assignment,grid):
''' Mark grid with queens that are under conflict. '''
for col, row in assignment.items(): # check each queen for conflict
row_conflicts = {temp_col:temp_row for temp_col,temp_row in assignment.items()
if temp_row == row and temp_col != col}
up_conflicts = {temp_col:temp_row for temp_col,temp_row in assignment.items()
if temp_row+temp_col == row+col and temp_col != col}
down_conflicts = {temp_col:temp_row for temp_col,temp_row in assignment.items()
if temp_row-temp_col == row-col and temp_col != col}
# Now marking the grid.
for col, row in row_conflicts.items():
grid[col][row] = 3
for col, row in up_conflicts.items():
grid[col][row] = 3
for col, row in down_conflicts.items():
grid[col][row] = 3
return grid
def make_plot_board_step_function(instru_csp):
'''ipywidgets interactive function supports
single parameter as input. This function
creates and return such a function by taking
in input other parameters.
'''
n = len(instru_csp.variables)
def plot_board_step(iteration):
''' Add Queens to the Board.'''
data = instru_csp.assignment_history[iteration]
grid = [[(col+row+1)%2 for col in range(n)] for row in range(n)]
grid = label_queen_conflicts(data, grid) # Update grid with conflict labels.
# color map of fixed colors
cmap = matplotlib.colors.ListedColormap(['white','lightsteelblue','red'])
bounds=[0,1,2,3] # 0 for white 1 for black 2 onwards for conflict labels (red).
norm = matplotlib.colors.BoundaryNorm(bounds, cmap.N)
fig = plt.imshow(grid, interpolation='nearest', cmap = cmap,norm=norm)
plt.axis('off')
fig.axes.get_xaxis().set_visible(False)
fig.axes.get_yaxis().set_visible(False)
# Place the Queens Unicode Symbol
for col, row in data.items():
fig.axes.text(row, col, u"\u265B", va='center', ha='center', family='Dejavu Sans', fontsize=32)
plt.show()
return plot_board_step

**make_instru** function for keeping a history of steps.

```
In [27]:
```twelve_queens_csp = NQueensCSP(12)
backtracking_instru_queen = make_instru(twelve_queens_csp)
result = backtracking_search(backtracking_instru_queen)

```
In [28]:
```backtrack_queen_step = make_plot_board_step_function(backtracking_instru_queen) # Step Function for Widgets

**Visualize Button** will automatically animate the slider for you. The **Extra Delay Box** allows you to set time delay in seconds upto one second for each time step.

```
In [29]:
```matplotlib.rcParams['figure.figsize'] = (8.0, 8.0)
matplotlib.rcParams['font.family'].append(u'Dejavu Sans')
iteration_slider = widgets.IntSlider(min=0, max=len(backtracking_instru_queen.assignment_history)-1, step=0, value=0)
w=widgets.interactive(backtrack_queen_step,iteration=iteration_slider)
display(w)
visualize_callback = make_visualize(iteration_slider)
visualize_button = widgets.ToggleButton(desctiption = "Visualize", value = False)
time_select = widgets.ToggleButtons(description='Extra Delay:',options=['0', '0.1', '0.2', '0.5', '0.7', '1.0'])
a = widgets.interactive(visualize_callback, Visualize = visualize_button, time_step=time_select)
display(a)

```
```

Now let us finally repeat the above steps for **min_conflicts** solution.

```
In [30]:
```conflicts_instru_queen = make_instru(twelve_queens_csp)
result = min_conflicts(conflicts_instru_queen)

```
In [31]:
```conflicts_step = make_plot_board_step_function(conflicts_instru_queen)

```
In [32]:
```iteration_slider = widgets.IntSlider(min=0, max=len(conflicts_instru_queen.assignment_history)-1, step=0, value=0)
w=widgets.interactive(conflicts_step,iteration=iteration_slider)
display(w)
visualize_callback = make_visualize(iteration_slider)
visualize_button = widgets.ToggleButton(desctiption = "Visualize", value = False)
time_select = widgets.ToggleButtons(description='Extra Delay:',options=['0', '0.1', '0.2', '0.5', '0.7', '1.0'])
a = widgets.interactive(visualize_callback, Visualize = visualize_button, time_step=time_select)
display(a)

```
```

```
In [ ]:
```