Homework 1: Search in Pacman
DueAll those colored walls,
Mazes give Pacman the blues,
So teach him to search.
Introduction
In this project, your Pacman agent will find paths through his maze world, both to reach a particular location and to collect food efficiently. You will build general search algorithms and apply them to Pacman scenarios. The Pacman AI projects were developed at UC Berkeley by John DeNero and Dan Klein, and further developed by Brad Miller, Nick Hay, and Pieter Abbeel.
The code for this project consists of several Python files, some of which you will need to read and understand in order to complete the assignment, and some of which you can ignore. The entire project description can also be downloaded as a pdf from here.
Files you'll edit:  
search.py 

searchAgents.py 
Where all of your searchbased agents will reside. 
Files you might want to look at:  
pacman.py 
The main file that runs Pacman games. This file describes a Pacman GameState type, which you use in this project. 
game.py 
The logic behind how the Pacman world works. This file describes several supporting types like AgentState, Agent, Direction, and Grid. 
util.py 
Useful data structures for implementing search algorithms. 
Supporting files you can ignore:  
graphicsDisplay.py 
Graphics for Pacman 
graphicsUtils.py 
Support for Pacman graphics 
textDisplay.py 
ASCII graphics for Pacman 
ghostAgents.py 
Agents to control ghosts 
keyboardAgents.py 
Keyboard interfaces to control Pacman 
layout.py 
Code for reading layout files and storing their contents 
autograder.py 
Project autograder 
testParser.py 
Parses autograder test and solution files 
testClasses.py 
General autograding test classes 
test_cases/ 
Directory containing the test cases for each question 
searchTestClasses.py 
Project 1 specific autograding test classes 
Files to Edit and Submit: You will fill in portions of search.py
and searchAgents.py
during the assignment. You should submit these files with your code and comments. Please do not change the other files in this distribution or submit any of our original files other than these files.
Welcome to Pacman
After downloading the code, unzipping it, and changing to the directory, you should be able to play a game of Pacman by typing the following at the command line:
python pacman.py
Pacman lives in a shiny blue world of twisting corridors and tasty round treats. Navigating this world efficiently will be Pacman's first step in mastering his domain.
The simplest agent in searchAgents.py is called the GoWestAgent
, which always goes West (a trivial reflex agent). This agent can occasionally win:
python pacman.py layout testMaze pacman GoWestAgent
But, things get ugly for this agent when turning is required:
python pacman.py layout tinyMaze pacman GoWestAgent
If Pacman gets stuck, you can exit the game by typing CTRLc into your terminal.
Soon, your agent will solve not only tinyMaze
, but any maze you want.
Note that pacman.py
supports a number of options that can each be expressed in a long way (e.g., layout
) or a short way (e.g., l
). You can see the list of all options and their default values via:
python pacman.py h
Also, all of the commands that appear in this project also appear in commands.txt, for easy copying and pasting. In UNIX/Mac OS X, you can even run all these commands in order with bash commands.txt
.
Note: if you get error messages regarding Tkinter, see this page
Question 1 (3 points): Finding a Fixed Food Dot using Depth First Search
In searchAgents.py
, you'll find a fully implemented SearchAgent
, which plans out a path through Pacman's world and then executes that path stepbystep. The search algorithms for formulating a plan are not implemented  that's your job. As you work through the following questions, you might find it useful to refer to the object glossary (at bottom of page).
First, test that the SearchAgent
is working correctly by running:
python pacman.py l tinyMaze p SearchAgent a fn=tinyMazeSearch
The command above tells the SearchAgent
to use tinyMazeSearch
as its search algorithm, which is implemented in search.py
. Pacman should navigate the maze successfully.
Now it's time to write fullfledged generic search functions to help Pacman plan routes! Pseudocode for the search algorithms you'll write can be found in the lecture slides. Remember that a search node must contain not only a state but also the information necessary to reconstruct the path (plan) which gets to that state.
Important note: All of your search functions need to return a list of actions that will lead the agent from the start to the goal. These actions all have to be legal moves (valid directions, no moving through walls).
Important note: Make sure to use the Stack
, Queue
and PriorityQueue
data structures provided to you in util.py
! These data structure implementations have particular properties which are required for compatibility.
Hint: Each algorithm is very similar. Algorithms for DFS, BFS, UCS, and A* differ only in the details of how the fringe is managed. So, concentrate on getting DFS right and the rest should be relatively straightforward. Indeed, one possible implementation requires only a single generic search method which is configured with an algorithmspecific queuing strategy. (Your implementation need not be of this form to receive full credit).
Implement the depthfirst search (DFS) algorithm in the depthFirstSearch
function in search.py
. To make your algorithm complete, write the graph search version of DFS, which avoids expanding any already visited states.
Your code should quickly find a solution for:
python pacman.py l tinyMaze p SearchAgent
python pacman.py l mediumMaze p SearchAgent
python pacman.py l bigMaze z .5 p SearchAgent
The Pacman board will show an overlay of the states explored, and the order in which they were explored (brighter red means earlier exploration). Is the exploration order what you would have expected? Does Pacman actually go to all the explored squares on his way to the goal?
Hint: If you use a Stack
as your data structure, the solution found by your DFS algorithm for mediumMaze
should have a length of 130 (provided you push successors onto the fringe in the order provided by getSuccessors; you might get 246 if you push them in the reverse order). Is this a least cost solution? If not, think about what depthfirst search is doing wrong.
Question 2 (3 points): Breadth First Search
Implement the breadthfirst search (BFS) algorithm in the breadthFirstSearch
function in search.py
. Again, write a graph search algorithm that avoids expanding any already visited states. Test your code the same way you did for depthfirst search.
python pacman.py l mediumMaze p SearchAgent a fn=bfs
python pacman.py l bigMaze p SearchAgent a fn=bfs z .5
Does BFS find a least cost solution? If not, check your implementation.
Hint: If Pacman moves too slowly for you, try the option frameTime 0
.
Note: If you've written your search code generically, your code should work equally well for the eightpuzzle search problem without any changes.
python eightpuzzle.py
Question 3 (3 points): Varying the Cost Function
While BFS will find a fewestactions path to the goal, we might want to find paths that are "best" in other senses. Consider mediumDottedMaze
and mediumScaryMaze
.
By changing the cost function, we can encourage Pacman to find different paths. For example, we can charge more for dangerous steps in ghostridden areas or less for steps in foodrich areas, and a rational Pacman agent should adjust its behavior in response.
Implement the uniformcost graph search algorithm in the uniformCostSearch
function in search.py
. We encourage you to look through util.py
for some data structures that may be useful in your implementation. You should now observe successful behavior in all three of the following layouts, where the agents below are all UCS agents that differ only in the cost function they use (the agents and cost functions are written for you):
python pacman.py l mediumMaze p SearchAgent a fn=ucs
python pacman.py l mediumDottedMaze p StayEastSearchAgent
python pacman.py l mediumScaryMaze p StayWestSearchAgent
Note: You should get very low and very high path costs for the StayEastSearchAgent
and StayWestSearchAgent
respectively, due to their exponential cost functions (see searchAgents.py
for details).
Question 4 (3 points): A* search
Implement A* graph search in the empty function aStarSearch
in search.py
. A* takes a heuristic function as an argument. Heuristics take two arguments: a state in the search problem (the main argument), and the problem itself (for reference information). The nullHeuristic
heuristic function in search.py
is a trivial example.
You can test your A* implementation on the original problem of finding a path through a maze to a fixed position using the Manhattan distance heuristic (implemented already as manhattanHeuristic
in searchAgents.py
).
python pacman.py l bigMaze z .5 p SearchAgent a fn=astar,heuristic=manhattanHeuristic
You should see that A* finds the optimal solution slightly faster than uniform cost search (about 549 vs. 620 search nodes expanded in our implementation, but ties in priority may make your numbers differ slightly). What happens on openMaze
for the various search strategies?
Question 5 (3 points): Finding All the Corners
The real power of A* will only be apparent with a more challenging search problem. Now, it's time to formulate a new problem and design a heuristic for it.
In corner mazes, there are four dots, one in each corner. Our new search problem is to find the shortest path through the maze that touches all four corners (whether the maze actually has food there or not). Note that for some mazes like tinyCorners, the shortest path does not always go to the closest food first! Hint: the shortest path through tinyCorners
takes 28 steps.
Note: Make sure to complete Question 2 before working on Question 5, because Question 5 builds upon your answer for Question 2.
Implement the CornersProblem
search problem in searchAgents.py
. You will need to choose a state representation that encodes all the information necessary to detect whether all four corners have been reached. Now, your search agent should solve:
python pacman.py l tinyCorners p SearchAgent a fn=bfs,prob=CornersProblem
python pacman.py l mediumCorners p SearchAgent a fn=bfs,prob=CornersProblem
To receive full credit, you need to define an abstract state representation that does not encode irrelevant information (like the position of ghosts, where extra food is, etc.). In particular, do not use a Pacman GameState
as a search state. Your code will be very, very slow if you do (and also wrong).
Hint: The only parts of the game state you need to reference in your implementation are the starting Pacman position and the location of the four corners.
Our implementation of breadthFirstSearch
expands just under 2000 search nodes on mediumCorners. However, heuristics (used with A* search) can reduce the amount of searching required.
Question 6 (BONUS): Corners Problem: Heuristic
Note: Make sure to complete Question 4 before working on Question 6, because Question 6 builds upon your answer for Question 4.
Implement a nontrivial, consistent heuristic for the CornersProblem
in cornersHeuristic
.
python pacman.py l mediumCorners p AStarCornersAgent z 0.5
Note: AStarCornersAgent
is a shortcut for
p SearchAgent a fn=aStarSearch,prob=CornersProblem,heuristic=cornersHeuristic
.
Admissibility vs. Consistency: Remember, heuristics are just functions that take search states and return numbers that estimate the cost to a nearest goal. More effective heuristics will return values closer to the actual goal costs. To be admissible, the heuristic values must be lower bounds on the actual shortest path cost to the nearest goal (and nonnegative). To be consistent, it must additionally hold that if an action has cost c, then taking that action can only cause a drop in heuristic of at most c.
Remember that admissibility isn't enough to guarantee correctness in graph search  you need the stronger condition of consistency. However, admissible heuristics are usually also consistent, especially if they are derived from problem relaxations. Therefore it is usually easiest to start out by brainstorming admissible heuristics. Once you have an admissible heuristic that works well, you can check whether it is indeed consistent, too. The only way to guarantee consistency is with a proof. However, inconsistency can often be detected by verifying that for each node you expand, its successor nodes are equal or higher in in fvalue. Moreover, if UCS and A* ever return paths of different lengths, your heuristic is inconsistent. This stuff is tricky!
NonTrivial Heuristics: The trivial heuristics are the ones that return zero everywhere (UCS) and the heuristic which computes the true completion cost. The former won't save you any time, while the latter will timeout.
Grading: Your heuristic must be a nontrivial nonnegative consistent heuristic to receive any points. Make sure that your heuristic returns 0 at every goal state and never returns a negative value. Depending on how few nodes your heuristic expands, you'll be graded:
Number of nodes expanded  Grade 

more than 2000  0 bonus points 
at most 2000  1 bonus point 
at most 1600  2 bonus points 
at most 1200  3 bonus points 
Remember: If your heuristic is inconsistent, you will receive no credit, so be careful!
Question 7 (BONUS): Eating All The Dots
Now we'll solve a hard search problem: eating all the Pacman food in as few steps as possible. For this, we'll need a new search problem definition which formalizes the foodclearing problem: FoodSearchProblem
in searchAgents.py
(implemented for you). A solution is defined to be a path that collects all of the food in the Pacman world. For the present project, solutions do not take into account any ghosts or power pellets; solutions only depend on the placement of walls, regular food and Pacman. (Of course ghosts can ruin the execution of a solution! We'll get to that in the next project.) If you have written your general search methods correctly, A*
with a null heuristic (equivalent to uniformcost search) should quickly find an optimal solution to testSearch with no code change on your part (total cost of 7).
python pacman.py l testSearch p AStarFoodSearchAgent
Note: AStarFoodSearchAgent
is a shortcut for p SearchAgent a fn=astar,prob=FoodSearchProblem,heuristic=foodHeuristic
.
You should find that UCS starts to slow down even for the seemingly simple tinySearch
. As a reference, our implementation takes 2.5 seconds to find a path of length 27 after expanding 5057 search nodes.
Note: Make sure to complete Question 4 before working on Question 7, because Question 7 builds upon your answer for Question 4.
Fill in foodHeuristic
in searchAgents.py
with a FoodSearchProblem
. Try your agent on the trickySearch
board:
python pacman.py l trickySearch p AStarFoodSearchAgent
Our UCS agent finds the optimal solution in about 13 seconds, exploring over 16,000 nodes.
Any nontrivial nonnegative consistent heuristic will receive 1 point. Make sure that your heuristic returns 0 at every goal state and never returns a negative value. Depending on how few nodes your heuristic expands, you'll get additional points:
Number of nodes expanded  Grade 

more than 15000  1 bonus point 
at most 15000  2 bonus points 
at most 12000  3 bonus points 
at most 9000  4 bonus points 
at most 7000  5 bonus points (hard!) 
Remember: If your heuristic is inconsistent, you will receive no credit, so be careful! Can you solve mediumSearch
in a short time? If so, we're either very, very impressed, or your heuristic is inconsistent.
Submission
See the course moodle page for submission instructions.
Object Glossary
Here's a glossary of the key objects in the code base related to search problems, for your reference:
SearchProblem (search.py)
 A SearchProblem is an abstract object that represents the state space, successor function, costs, and goal state of a problem. You will interact with any SearchProblem only through the methods defined at the top of
search.py
PositionSearchProblem (searchAgents.py)
 A specific type of SearchProblem that you will be working with  it corresponds to searching for a single pellet in a maze.
CornersProblem (searchAgents.py)
 A specific type of SearchProblem that you will define  it corresponds to searching for a path through all four corners of a maze.
FoodSearchProblem (searchAgents.py)
 A specific type of SearchProblem that you will be working with  it corresponds to searching for a way to eat all the pellets in a maze.
Search Function
 A search function is a function which takes an instance of SearchProblem as a parameter, runs some algorithm, and returns a sequence of actions that lead to a goal. Example of search functions are
depthFirstSearch
andbreadthFirstSearch
, which you have to write. You are providedtinyMazeSearch
which is a very bad search function that only works correctly ontinyMaze
SearchAgent
SearchAgent
is a class which implements an Agent (an object that interacts with the world) and does its planning through a search function. TheSearchAgent
first uses the search function provided to make a plan of actions to take to reach the goal state, and then executes the actions one at a time.